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MONTESSORI UPPER ELEMENTARYMATH CURRICULUM ALIGNMENT

Based on the Maryland Voluntary State Curriculum

Montessori MathematicsGrades 4 to 6

June 2006

Prince George’s County Public Schools

PGIN 7690-3472

Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 1

BOARD OF EDUCATIONOF

PRINCE GEORGE’S COUNTY, MARYLAND

Beatrice P. Tignor, Ed.D., ChairHoward W Stone, Jr., Vice ChairJohn R. Bailer, MemberAbby L. W. Crowley, Ed.D., MemberCharlene M. Dukes, Ed.D., MemberRobert O. Duncan, MemberJose R. Morales, MemberJudy G. Mickens- Murray, MemberDean Sirjue, MemberLeslie Hall, Student Board MemberJohn E. Deasy, Ph. D., Chief Executive Officer

Shelley Jallow, Chief Academic OfficerPatricia Miller, Director of Curriculum and InstructionGladys Whitehead, Ph.D., Coordinating Supervisor, Academic ProgramsPamela Shetley, Ph.D., Director of the FOCUS Office

ACKNOWLEDGEMENTS

We wish to acknowledge the following teachers for their hard work and dedication to the creation of this outstanding document:

John Feeley Susan HolmesLaure Fleming Marion Lebensbaum

Janet Goodspeed Cynthia PeilGwendolyn C. Harris Kimberly Strayhorn


We also wish to thank all of the Curriculum Writing Production Center staff for their assistance.


Table of Contents

Introduction……………………………………………………………………………………………………………………………………………………………5

Montessori Math Materials List…………………………………………………………………………………………………………………………………….6

Voluntary State Curriculum Chart (VSC) ………………………………………………………………………………………………………………………..8

Quarterly Guidelines ……………………………………………………………………………………………………………………………………………….14

Montessori Alignment with VSC………………………………………………………………………………………………………………………………….15

Montessori Great Lessons ………………………………………………………………………………………………………………………………………..93

Montessori Sample Lessons ……………………………………………………………………………………………………………………………………104

Appendix ……………………………………………………………………………………………………………………………………………………………123History of Math Command CardsRounding and Estimating LessonsCalculator LessonsProblem Solving SequenceProblem Solving Activity CardsMeasurement Activity CardsMoney LessonsProbability Activity CardsStatistics Activity Cards

Introduction to Mathematics in the Montessori Elementary Classroom

The human mind is naturally mathematical. Mathematics is another language of communications. Mathematical inventions are a reflection of the culture. The child at the second plane of development (6-12 years old) is interested in exploring how and why mathematics was developed and its usefulness to humanity. To assist the child’s inner development, the elementary teacher offers experiences that lead towards mathematical abstraction. First, the Montessori Great Lessons are introduced. Next, concrete experiences are presented, using hands-on materials manipulative practice with individual concepts. Then the teacher gives math nomenclature as the first step leading to abstraction. Repetition through a variety of parallel, interrelated work is given to keep the child's interest. Also, interest is maintained through work on problem solving skills, as the child creates his/her own problems. After extensive work with the sensorial (manipulative) materials, the child naturally moves toward abstraction until only paper and pencil remain.


The child's work follows five steps:1. The teacher presents "The Story of Numbers,” giving a vision of the whole and inviting children to beginthe adventure of mastering mathematics.2. Children work individually with math materials, acquiring precision.3. The teacher presents mathematical vocabulary.4. Children are encouraged through questioning to begin to make generalizations.5. Children begin to work on paper (abstract level).

The mathematics program at the 6-12 level is designed to awaken the exploratory imagination of the reasoning mind. The child, from his work in the Montessori preschool, already has as a strong base for understanding math concepts. Correlation between quantity and symbol, base-ten place value to thousands, an introduction to the four operations using whole numbers, and memorization of math facts have been introduced and practiced. The major domains of study in the elementary class are:

1. Algebra, patterns, and functions2. Geometry3. Measurement4. Statistics5. Probability6. Number relationships and computation7. Processes of mathematics


Description of Montessori Math Materials

The 100 Board has 100 tiles numbered from 1-100, designed to be laid out in rows of ten on a 10 x 10 grid.

The Bead Bars are colored bead bars representing different quantities, used to explore patterns of numbers.

The Chain Cabinet holds ten cubes 1-10, the squares of the cubes, the chains of the cubes and the chains of the squares. Colors are the same as for the Bead Bars.

The Strip Boards are used to learn addition and subtraction facts and to lead children to discover numerical patterns. They are printed with a grid of squares, and use strips of wood cut in sizes from 1-9, and 1-18 to represent quantities.

The Multiplication Board is a wooden board with ten holes vertically and ten holes horizontally totaling 100 holes. Red beads are used to create the product.

The Division Board has nine holes vertically and nine holes horizontally totaling 81 holes. Green beads are used to create the quotient. Skittles represent the divisor.

The Finger Charts are series of charts with the basic math facts printed on them. They lead students to discover numerical patterns.

The Golden Bead Material is a three dimensional representation of place value. It uses single beads to represent units, ten beads on a wire to represent tens, 10 ten bars wired together in a square to represent hundreds, and 10 hundred squares stacked and held together in a cube to represent thousands.

The Stamp Game uses color-keyed "stamps" to represent the decimal system. Green stamps represent units and thousands, blue stamps represent tens, and red stamps represent hundreds. Place value is written on each stamp.

The Dot Game uses the same color coding system on a board divided into 5 columns. Each column has 25 rows of 10 small squares. The dot game reinforces the concept of exchanging tens in addition.

The Small and Large Bead Frames are color-coded abacuses.

The Racks and Tubes are used for division. They contain 700 color-coded beads sorted in tubes with ten each. The beads are used to represent the dividend. Color-coded boards with indentation to hold the beads are used to hold the quotient. The divisor is created with color coded skittles.

The Wooden Hierarchical Material is a large three-dimensional representation of place value to one million. It uses the same color coding system.

The Checkerboard is a rectangular board with four horizontal rows, each row containing color-coded nine squares.

The Bank Game is a set of cards to 9,000,000 in the hierarchical colors.


The Golden Bead Frame is an abacus is made of golden beads.

The Peg Board is a board with thirty holes vertically and thirty holes horizontally. Colored pegs use the same color coding system- green for units, blue for tens, red for 100. Color-coded skittles are also used with the pegboard.

The Fraction Insets, Divided Square Material, and Equivalency Insets are used in the study of fraction equivalence and operations on fractions.

The Decimal Material is laid out in columns to represent place values less than one unit.

The Constructive Triangle Boxes are a set of boxes holding various triangles color-coded to encourage exploration of geometric forms created by combining triangles.

The Geometry Cabinet has six drawers containing wooden figures to represent plane-closed figures.

The Geometry Sticks consist of sticks of several sizes and colors that can be fixed to a cork board to aid in the study of geometric figures such as lines and angles.

The Geometric Solids consist of nine different wooden forms painted blue.

The Yellow Area Material uses flat yellow quadrilaterals to encourage exploration of the concept of area.

The Volume Material is a series of containers and cubes used to explore volume.

The Binomial, Trinomial and Arithmetic Cubes contain colored cubes and prisms used in the preschool to refine the visual sense and discrimination of form. In the elementary a series of exercises leads students to discovery of the algebraic formulas.

The Numerical Decanomial helps in the memorization of multiplication tables, and is a preparation for squares, cubes and square and cube roots .


VSC and Montessori Upper ElementaryMath Curriculum Alignment

Table of Contents

Grade 4 Grade 5 Grade 6

VSC1.0

Algebra, Patterns, and Functions VSC1.0

Algebra, Patterns, and Functions VSC1.0

Algebra, Patterns, and Functions

1.A.1.a Page 15 1.A.1.a Page 15 1.A.1.a Page 151.A.1.b 15 1.A.1.b 15 1.A.1.b 151.A.1.c 16 1.A.1.c 16 1.A.1.c 161.A.1.d 16 1.A.1.d 16 1.A.1.d 161.A.2.a 171.A.2.b 171.A.2.c 181.B.1.a 18 1.B.1.a 18 1.B.1.a 181.B.1.b 18 1.B.1.b 18 1.B.1.b 18

1.B.1.c 19 1.B.1.c 191.B.1.d 19

1.B.2.a 19 1.B.2.a 19 1.B.2.a 191.B.2.b 21 1.B.2.b 21 1.B.2.b 21

1.B.2.c 211.B.2.d 211.B.2.e 22

1.C.1.a 23 1.C.1.a 23 1.C.1.a 231.C.1.b 24 1.C.1.b 24 1.C.1.b 24

1.C.1.c 261.C.2.a 251.C.2.b 25


VSC2.0

Geometry VSC2.0

Geometry VSC2.0

Geometry

2.A.1.a 26 2.A.1.a 26 2.A.1.a 262.A.1.b 26 2.A.1.b 26 2.A.1.b 26


2.A.1.c 27 2.A.1.c 27 2.A.1.c 272.A.2.a 28 2.A.2.a 282.A.2.b 28 2.A.2.b 28

2.A.2.c 282.A.2.d 29

2.B.1.a 29 2.B.1.a 292.B.1.b 30 2.B.1.b 302.B.2.a 31 2.B.2.a 312.C.1.a 2.C.1.a 32 2.C.1.a 32

2.C.1.b 332.C.1.c 33

2.D.1.a 34 2.D.1.a 34 2.D.1.a 342.E.1.a 35 2.E.1.a 35 2.E.1.a 35


VSC3.0

Measurement VSC3.0

Measurement VSC3.0

Measurement

3.A.1.a 36 3.A.1.a 363.A.1.b 37 3.A.1.b 373.A.1.c 373.B.1.a 38 3.B.1.a 38 3.B.1.a 383.B.2 39 3.B.2.a 39 3.B.2 393.C.1.a 40 3.C.1.a 40 3.C.1.a 403.C.1.b 41 3.C.1.b 41 3.C.1.b 413.C.1.c 41 3.C.1.c 41 3.C.1.c 41

3.C.1.d 42 3.C.1.d 423.C.1.e 42

3.C.2.a 42 3.C.2.a 423.C.2.b 43 3.C.2.b 433.C.2.c 44


VSC4.0

Statistics VSC4.0

Statistics VSC4.0

Statistics


4.A.1.a 45 4.A.1.a 45 4.A.1.a 454.A.1.b 45 4.A.1.b 45 4.A.1.b 45

4.A.1.c 46 4.A.1.c 464.A.1.d 464.A.1.e 474.A.1.f 48

4.B.1.a 48 4.B.1.a 48 4.B.1.a 484.B.1.b 49 4.B.1.b 49 4.B.1.b 49

4.B.1.c 50 4.B.1.c 504.B.1.d 504.B.1.e 51

4.B.2.a 52 4.B.2.a 52 4.B.2.a 524.B.2.b 52 4.B.2.b 52


VSC5.0

Probability VSC5.0

Probability VSC5.0

Probability

5.A.1.a 535.B.1.a 54 5.B.1.a 54 5.B.1.a 54

5.B.1.b 545.B.1.c 555.C.1.a 565.C.2 565.C.3 565.C.4 56


VSC6.0

Number Relationships and Computation

VSC6.0


VSC6.0


6.A.1.a 58 6.A.1.a 58 6.A.1.a 586.A.1.b 59 6.A.1.b 59 6.A.1.b 596.A.1.c 60 6.A.1.c 60 6.A.1.c 606.A.1.d 61 6.A.1.d 61 6.A.1.d 61

6.A.1.e 61 6.A.1.e 616.A.2.a 62


6.A.2.b 626.A.2.c 636.A.2.d 636.A.2.e 646.A.2.f 646.A.2.g 656.A.2.h 666.A.3.a 676.A.3.b 676.B.1.a 68 6.B.1.a 68 6.B.1.a 686.B.1.b 69 6.B.1.b 696.B.1.c 69 6.B.1.c 69

6.B.1.d 706.C.1.a 71 6.C.1.a 71 6.C.1.a 716.C.1.b 72 6.C.1.b 72 6.C.1.b 726.C.1.c 73 6.C.1.c 73 6.C.1.c 736.C.1.d 74 6.C.1.d 74 6.C.1.d 746.C.1.e 75 6.C.1.e 75 6.C.1.e 756.C.1.f 76 6.C.1.f 76 6.C.1.f 766.C.1.g 77 6.C.1.g 77

6.C.1.h 786.C.2.a 79 6.C.2.a 79 6.C.2.a 796.C.2.b 79 6.C.2.b 79

6.C.2.c 806.C.3.a 806.C.3.b 82


VSC7.0

Processes of Mathematics VSC7.0

Processes of Mathematics VSC7.0

Processes of Mathematics

7.A.1.a 82 7.A.1.a 82 7.A.1.a 827.A.1.b 82 7.A.1.b 82 7.A.1.b 82


7.A.1.c 82 7.A.1.c 82 7.A.1.c 827.A.1.d 83 7.A.1.d 83 7.A.1.d 837.A.1.e 83 7.A.1.e 83 7.A.1.e 837.A.1.f 83 7.A.1.f 83 7.A.1.f 837.A.1.g 84 7.A.1.g 84 7.A.1.g 847.A.1.h 84 7.A.1.h 84 7.A.1.h 847.B.1.a 85 7.B.1.a 85 7.B.1.a 857.B.1.b 85 7.B.1.b 85 7.B.1.b 857.B.1.c 85 7.B.1.c 85 7.B.1.c 857.B.1.d 86 7.B.1.d 86 7.B.1.d 867.C.1.a 87 7.C.1.a 87 7.C.1.a 877.C.1.b 87 7.C.1.b 87 7.C.1.b 877.C.1.c 87 7.C.1.c 87 7.C.1.c 877.C.1.d 88 7.C.1.d 88 7.C.1.d 887.C.1.e 89 7.C.1.e 89 7.C.1.e 897.C.1.f 89 7.C.1.f 89 7.C.1.f 897.C.1.g 89 7.C.1.g 89 7.C.1.g 897.C.1.h 89 7.C.1.h 89 7.C.1.h 897.D.1.a 90 7.D.1.a 90 7.D.1.a 907.D.1.b 90 7.D.1.b 90 7.D.1.b 907.D.1.c 91 7.D.1.c 91 7.D.1.c 917.D.1.d 92 7.D.1.d 92 7.D.1.d 92


Montessori Upper Elementary MathematicsQuarterly Overview for 4th, 5th, and 6th Grade Students

First Quarter Second Quarter Third Quarter Fourth Quarter

Statistics (All Indicators)

Algebra:1.) Patterns and Functions2.) Expressions / Order of

Operations3.) Coordinate Grids

Number Concepts:1.) Whole Number Place Value2.) Whole Number Operations3.) Integers and Exponents (6th

Grade)4.) Divisibility5.) Factors and Multiples

Processes: All Indicators

Algebra:Equations and Inequalities

Geometry:1.) Nomenclature (lines, angles,

and polygons)2.) Congruence and Similarity3.) Transformations4.) Analyzing Quadrilaterals,

Triangles, and CirclesMeasurement:

1.) Length (Standard and Metric)2.) Degrees of an Angle

Numbers:1.) Analyze Fractions2.) Equivalent Forms3.) Fraction Operations (+,-,x)


Probability (All Indicators)

Measurement:1.) Weight and Capacity

(Standard and Metric)2.) Applications of Formulas

(Area, Perimeter, Volume)3.) Composite Figures4.) Time

Number Concepts:1.) Decimal Place Value2.) Decimal Operations3.) Equivalent Forms4.) Money5.) Percent


Accelerated Curriculum with Extended Assessment Limits:

Number Concepts:1.) Whole Number Operations2.) Fraction Operations3.) Decimal Operations4.) Integers Operations5.) Ratios and Scale Models

Geometry:1.) Analyze Circles

Measurement:1.) Use Measurement Tools2.) Apply Measurement

Formulas (Surface Area, Volume)


Montessori Math and the Maryland Voluntary State CurriculumThis sequence has been kept as closely aligned to the Montessori math curriculum as possible, but adjustments have been made based on the indicators tested quarterly in Prince George’s County Public Schools. If teachers systematically incorporate these indicators into their presentations and discussions, students will be comfortable with quarterly benchmark tests and the Maryland School Assessment (MSA). The Montessori Math Lessons are designed to be presented individually or in small groups. Each presentation isolates one learning objective. Where students begin in the sequence, and how quickly they progress, depends on the developmental needs of each individual student. The teacher observes and responds to each individual child’s learning needs. Grade level expectations are intended to be used only as an aid to planning. Teachers will use their Montessori curriculum albums as their primary guides, and each student will progress at his or her own unique and appropriate pace.


Montessori Upper Elementary Math Curriculum Alignment

VSC-Mathematics-Standard 1.0 Knowledge of Algebra, Patterns, and Functions: Students will algebraically represent, model, analyze, or solve mathematical or real-world problems involving patterns or functional relationships.


Topic 1.A. Patterns and Functions Topic 1.A. Patterns and Functions Topic 1.A. Patterns and Functions1.A.1. Identify, describe,extend, and create numeric patterns and functions

Quarter1

1.A.1. Identify, describe,extend, and create numeric patterns and functions

Quarter1

1.A.1 Identify, describe, extend, and create numeric patterns and functions

Quarter1

a. Represent and analyze numeric patterns using skip countingAssessment limit: Use patterns of 3, 4, 6, 7, 8, or 9 starting with any whole number (0 – 100)

Montessori Lessons:Short and Long Bead Chains, Bead Bars, Flash Cards, Multiplication Charts

Albanesi Cards:9.A -K10.A –G

POW (06-07): Week 1

SFAW 2-9; TE 1DIS: M-35; p. 69-70, 127SFAW 2-10 / 2-11; TE 1DIS: M-15; p. 29-30, 107DIS: J-18; p. 35-36, 100

MSA Finish Line:Pages 8-11

a. Interpret and write a rule for a one-operation (+, -, x, ÷ with no remainders) function table

Assessment limit: Use whole numbers or decimals with no more than 2 decimal places (0 – 1000)

Montessori Materials:Command Cards

SFAW: 2-14; TE 1DIS: J-8; p. 15-16, 90


a. Identify and describe sequences represented by a physical model or in a function table

Montessori Lessons:Short and Long Bead Chains, Bead Bars; Student Created Models

Glencoe: 7-6, 7-6a

b. Create a one-operation(+ or -) function table to solve a real world problem

Montessori Lessons:Word Problem Cards

b. Create a one-operation(x, ÷ with no remainders) function table to solve a real world problem


SFAW: 3-15; TE 1

b. Interpret and write a rule for one-operation (+, -, x, ÷ ) function tableAssessment limit: Use whole numbers or decimals with no more than two decimal places (0 – 10,000)


GLencoe: 9-6a, 9-6


Grade 4 Grade 5 Grade 6c. Complete a function table using a one operation (+, -, ×, ÷ with no remainders) rule

Assessment Limit: Use operational symbols (+, -, x) and whole numbers (0-200)


SFAW 3-13; TE 1DIS: J-13; p. 25-26, 95

c. Complete a one-operation function tableAssessment limit: Use whole numbers with +, -, x, ÷ (with no remainders) or use decimals with no more than two decimal places with +, - (0 – 200)


POW (06-07): Week 3

SFAW: 2-14; TE 1DIS: J-8; p. 15-16, 90

c. Complete a function table with a given two-operation rule

Assessment limit: Use the operations of (+, -, x), numbers no more than 10 in the rule, and whole numbers (0 - 50)



d. Describe the relationship that generates a one-operation rule


d. Apply a given two operation rule for a pattern

Assessment limit: Use two operations (+, -, x) and whole numbers (0 – 100)


POW (06-07): Week 4

SFAW: 3-15; TE 1


1.A. 2. Identify, describe, extend, analyze, and create a non-numeric growing or repeating pattern

Quarter1

a. Generate a rule for the next level of the growing patternAssessment limit: Use at least 3 levels but no more than 5 levels

Montessori Materials:Command Cards; Student Created Models

POW (06-07): Week 2


Grade 4 Grade 5 Grade 6b. Generate a rule for a repeating pattern

Montessori Materials:Command Cards;


Assessment limit: Use no more than 4 objects in the core of the pattern

StudentCreated Models


c. Create a non-numeric growing or repeating pattern

Montessori Materials:Command Cards; Student Created Models


Grade 4 Grade 5 Grade 6Topic 1.B. Expressions, Equations, and Inequalities Topic 1. B. Expressions, Equations, and Inequalities Topic 1. B. Expressions, Equations, and Inequalities

1.B.1 Write and identify expressions

Quarter1

1.B.1 Write and identify expressions

Quarter1

1.B.1 Write and evaluate expressions

Quarter1

a. Represent numeric quantities using operational symbols (+, -, ×, ÷ with no remainders)

Assessment limit: Use whole numbers (0 – 100)

Montessori Lessons:Word Problem Cards; Math Vocabulary Card Set; Chart for four basic operations;Problem Solving Sequence

SFAW 2-12; TE 1DIS: J-18; p. 35-36, 100

SFAW 3-13 / 3-14; TE 1DIS: J-20; p. 39-40, 102DIS: J-21; p. 41-42, 103

a. Represent unknown quantities with one unknown and one operation (+, -, ×, ÷ with no remainders)

Assessment limit: Use whole numbers (0 – 100) or money ($0 - $100)

Montessori Lessons:Word Problem Cards; Math Vocabulary Card Set; Chart for four basic operations; Problem Solving Sequence

SFAW: 2-12; TE 1

a. Write an algebraic expression to represent unknown quantities

Assessment limit: Use one unknown and one operation (+, -) with whole numbers, fractions with denominators as factors of 24, or decimals with no more than two decimal places (0-200)

Montessori Lessons:Word Problem Cards; Math Vocabulary Card Set; Chart for four basic operations; Problem Solving SequenceGlencoe: 1-1 / 1-6

PGCPS 6th CFPG: Supplemental Lesson 1

b. Determine equivalent expressions


Montessori Materials:Math Fact Families; Bead Bars

SFAW 2-13; TE 1DIS: J-19; 37-38, 101


b. Determine the value of algebraic expressions with one unknown and one operation

Assessment limit: Use +, - with whole numbers (0-1000) or ×, ÷ (with no remainders) with whole numbers (0-100) and the number for the unknown is no more than 9


POW (06-07): Week 22

SFAW: 2-13; TE 1


b. Evaluate an algebraic expression

Assessment limit: Use one unknown and one operation (+, -) with whole numbers (0 – 200), fractions with denominators as factors of 24 (0 – 50), or decimals with no more than two decimal places (0 – 50)


Albanesi Cards:72.A-G

Glencoe: 1-6, 1-7

MSA Finish Line:Pages12-15



Grade 4 Grade 5 Grade 6c. Use parenthesis to evaluate a numeric expression

Fourth QuarterAccelerated CurriculumMontessori Materials:Command CardsPOW (06-07) Week 35SFAW: 3-13 and 3-16; TE 1DIS: J-23; p. 45-46,105

c. Evaluate numeric expressions using the order of operations

Assessment limit: Use no more than 4 operations(+, -, x, ÷ with no remainders) with or without 1 set of parentheses or a division bar and whole numbers (0-100)

Montessori Lessons:Integers

POW (06-07) Week 2

Glencoe: 1-5, 9-1


d. Represent algebraic expressions using physical models, manipulatives, and drawings

Montessori Materials:Command Cards, Student Created Models

1.B.2. Identify, write, solve, and apply equations and inequalities

Quarter2


Quarter2


Quarter2

Grade 4 Grade 5 Grade 6a. Represent relationships using relational symbols(>, <, =) and operational symbols (+, -, ×, ÷) on either side

Assessment limit: Use operational symbols (+, -, ×) and whole numbers (0 – 200)

Montessori Lessons:Word Problem Cards; Math Vocabulary Card Set / Chart for four basic operations;Problem Solving Sequence

POW (06-07): Week 31

SFAW 12-1; TE 4DIS: J-10; p. 19-20, 92SFAW 12-2; TE 4

a. Represent relationships by using the appropriate relational symbols (>, <, =) and one operational symbol (+, -, ×, ÷ with no remainders) on either side



SFAW: 3-15; TE 1Reasoning andProblem SolvingDIS: J-13; p. 25-26, 95SFAW: 3-15; TE 1

a. Identify and write equations and inequalities to represent relationships

Assessment limit: Use a variable, the appropriate relational symbols (>, <, =), and one operational symbol (+, -, ×, ÷) on either side and use fractions with denominators as factors


Glencoe: 9-1


DIS: J-11; p. 21-22, 93SFAW 12-3; TE 4DIS: J-12; p. 23-24, 94

Extend lesson toreach limitSFAW: 12-2; TE 4(Teacher will need toinclude inequalities.)

of 24 (0 – 50) or decimals with no more than two decimal places (0 – 200)


Grade 4Grade 5 Grade 6

b. Find the unknown in an equation with one operation

Assessment limit: Use multiplication (×) and whole numbers (0-81)

Montessori Lessons:Euclids’s Laws/Balancing Equations; Word Problem Cards

SFAW 3-15; TE 1


b. Find the unknown in an equation use one operation (+, -, ×, ÷ with no remainders)



SFAW: 2-12; TE 1DIS: J-13; p. 25-26, 95SFAW: 2-15, TE 1DIS: J-21; p. 41-42, 1


Fourth QuarterAccelerated Curriculum:POW (06-07): Week 36SFAW: 12-2; TE 4DIS J-25; pg. 49-50,107SFAW: 12-3; TE 4DIS J-26; pg 51-52, 108(Include decimal examples.)

b. Determine the unknown in a linear equation

Assessment limit: Use one operation (+, -, ×, ÷ with no remainders) and use positive whole number coefficients using decimals with no more than two decimal places (0 – 100)


Albanesi Cards:41.A-D

Glencoe: 9-2, 9-2a, 9-3, 9-4, 9-4b


Fourth Quarter Accelerated Curriculum:Montessori Binomial and Trinomial Squares and CubesAlbanesi Cards: 41.E; 44.A-H; 45.A-E; 47.A-BSolving Two-Step Equations:POW (06-07): Week 34Glencoe: 9-5, 9-5b

c. Solve for the unknown in a one-step inequality


d. Identify or graph Montessori Lessons:


solutions of a one-step inequality on a number line.

Euclids’s Laws/Balancing Equations; Word Problem Cards

e. Apply given formulas to a problem solving situation




Grade 4 Grade 5 Grade 6Topic1.C. Numeric and Graphic Representations of Relationships

Topic 1.C. Numeric and Graphic Representations of Relationships

Topic 1.C. Numeric and Graphic Representations of Relationships

1.C.1 Locate points on a number line and in a coordinate grid

Quarter 1 – 1.C.1.b and c

Quarter 2- 1.C.1.a

1.C.1. Locate points on a number line and in a coordinate grid

Quarter 1 – 1.C.1.b

Quarter 2- 1.C.1.a

1.C1. Locate points on a number line and in a coordinate plane

Quarter1

a. Represent mixed numbers and proper fractions on a number line

Assessment limit: Use proper fractions with a denominators of 6, 8, or 10

Montessori Materials:History of Numbers; Classification of Numbers

POW (06-07): Week 17

SFAW 10-4; TE 4DIS: H-29; p. 57-58, 115

a. Represent decimals and mixed numbers on a number line

Assessment limit: Use decimals with no more than two decimal places (0 – 100) or mixed numbers with denominators of 2, 3, 4, 5, 6, 8, or 10 (0 - 10)

Montessori Materials:History of Numbers; Classification of Numbers

SFAW: 7-5; TE 3DIS: H-23 p. 45-46, 109SFAW: 7-14; TE 3

Fourth Quarter Accelerated Curriculum:Use integers -20 to 20.POW(06-07): Week 37, 38

a. Represent rational numbers on a number line

Assessment limit: Use integers (-20 to 20)

Montessori Materials:History of Numbers; Classification of Numbers; Negative Snake Game; Integers Lessons; Social Studies Cross-Curricular Lesson: BC/CE Timeline

Albanesi Cards:38.A – K; 39.A – C; 42.A- H; 46.A – B

Glencoe: 8-1



Grade 4 Grade 5 Grade 6b. Identify positions in a coordinate plane

Assessment limit: Use the first quadrant and ordered pairs of whole numbers (0 - 20)

Montessori Lessons:Integrated Lessons in Science and Geography (latitude and longitude)

Albanesi Cards:101.D – I

SFAW 4-9; TE 2DIS: L-4; p. 7-8, 66


b. Create a graph in a coordinate plane

Assessment limit: Use the first quadrant and ordered pairs of whole numbers (0 – 50)

Montessori Lessons:Integrated Lessons in Science and Geography (latitude and longitude)Albanesi Cards:25.J – O

POW (06-07): Week 2

SFAW: 3-14


Fourth Quarter Accelerated Curriculum: Introduce coordinate grid with four quadrants.SFAW: 12-9; TE 4

b. Graph ordered pairs in a coordinate plane.

Assessment limit: Use no more than 3 ordered pairs of integers (-20 to 20) or no more than 3 ordered pairs of fractions/mixed numbers with denominators of 2 (-10 to 10)

Montessori Lessons:Integrated Lessons in Science and Geography (latitude and longitude)

Glencoe: 8-6


Fourth Quarter Accelerated Curriculum:Glencoe: 9-5, 9-7

c. Represent decimals on a number line

Fourth QuarterAccelerated CurriculumMontessori Lessons: Command CardsSFAW 7-14, TE 3

c. Graph linear data from a function table

Fourth QuarterAccelerated Curriculum:Montessori Lessons: Story Problems; Command CardsAlbanesi Cards: 41.A-DGlencoe: 9-5, 9-7POW: Week 26

1.C.2. Analyze linear relationships

Quarter1


VSC-Mathematics-Standard 1.0 Knowledge of Algebra, Patterns, and Functions: Students will algebraically represent, model, analyze, or solve mathematical or real-world problems involving patterns or functional relationships.Grade 4 Grade 5 Grade 6

a. Identify and describe the change represented in a graph

Assessment limit: Identify increase, decrease, or no change

Montessori Lessons:Student Surveys, Student Created Graphs, Integrated Lessons in Science

Glencoe: 2-8


b. Translate the graph of a linear relationship onto a table of values that illustrates the type of change

Montessori Lessons:Student Surveys, Student Created Graphs, Integrated Lessons in Science

QUARTER 2

REVIEWMSA Finish Line of Algebra, Patterns, and Functions:Pages 7, 28-30

QUARTER 2


QUARTER 2



VSC-Mathematics-Standard 2.0 Knowledge of Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of objectsGrade 4 Grade 5 Grade 6Topic 2.A. Plane Geometric Figures Topic2. A. Plane Geometric Figures Topic 2.A. Plane Geometric Figures2.A.1. Analyze theproperties of planegeometric figures

Quarter2

2.A.1. Analyze theproperties of planegeometric figures

Quarter2

2.A.1. Analyze theproperties of planegeometric figures

Quarter2

a. Identify properties ofangles usingmanipulativeand pictures

Montessori Lessons:Geometry NomenclatureCards: Three PeriodLessons; Angle Search in

the Classroom / AngleSearch in Yarn Webs;Geometric Stick Box

SFAW 8-3; TE 3DIS: K-46; p. 91-92, 160

a. Identify and describerelationships of linesand line segments ingeometric figures orpictures

Assessment limit: Use parallel or perpendicular lines and line segments

Montessori Lessons:Geometry Nomenclature Cards: Three Period Lessons; Line Treasure Hunt in Classroom / Yarn Webs; Geometric Stick Box

POW (06-07): Week 8

SFAW 6-1; TE 2DIS: K-46; p. 91-92, 160

a. Identify, describe, andlabel points, lines, rays,line segments, vertices,angles, and planesusing correct symbolicnotation

Montessori Lessons:Geometry Nomenclature Cards; Geometric Stick Box; Drawing and labeling points, lines, rays, vertices, and angles using rulers and protractors.

Glencoe: 13-1, 13-2

b. Identify, compare,classify, and describe

angles in relationshipto another angle

Assessment limit: Use acute, right, or obtuse angles

Montessori Lessons:Geometry Nomenclature Cards; Geometric Stick Box

POW (06-07): Week 10

SFAW 8-4; TE 3DIS: K-41, K-42; p. 81-84, 155-156


b. Identify polygons withina composite figureAssessment limit: Use polygons with no more than 8 sides as part of a composite figure comprised of triangles or quadrilaterals

Montessori Lessons:Geometry Nomenclature Cards; Constructive Triangles; Geometric Cabinet

SFAW 6-6; TE 2DIS: K-5; p.101-102, 165


b. Identify and describeline segmentsAssessment limit: Use diagonal line segments

Montessori Lessons:Geometry Nomenclature Cards

Glencoe: 13-3a



VSC-Mathematics-Standard 2.0 Knowledge of Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of objects

Grade 4 Grade 5 Grade 6c. Identify parallel and

intersecting linesegments

Montessori Lessons:Geometry Nomenclature Cards; Geometric Stick Box

SFAW 8-3; TE 3

c. Identify and describethe radius and diameterof a circle

Montessori Lessons:Geometry Nomenclature Cards; Circle Search in the Classroom; Linear Parts of a Circle; Stick Box

c. Identify and describe theparts of a circle

Assessment limit: Use radius, diameter, or

Montessori Lessons:Geometry Nomenclature Cards; Parts of a Circle Chart; Concentric Circles; Linear Parts of a Circle: Red Fraction Inset Material

Fourth Quarter Accelerated Curriculum:Parts of a circle.Montessori Lessons: Geometry Nomenclature Cards; Circle Search in the Classroom; Linear Parts of a CirclePOW (06-07): Week 38SFAW 8-5


circumference Glencoe: 4-6

PGCPS 6th CFPG:Supplemental Lesson 2

2.A.2. Analyze geometricrelationships

Quarter2

2.A.2. Analyze geometricRelationships

Quarter2


VSC-Mathematics-Standard 2.0 Knowledge of Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of objectsGrade 4 Grade 5 Grade 6

a. Compare and classifyquadrilaterals by lengthof sides and types ofangles (Include theangle symbol <ABC)Assessment limit: Use squares, rectangles, rhombi, parallelograms, and trapezoids

Montessori Lessons:Stick Box, Quadrilateral Who Am I? Game, GEO Boards, Geometric Cabinet, Nomenclature Cards

POW (06-07): Week 9

SFAW 6-9; TE 2DIS: K-48; p. 95-96, 162


a. Compare and classifytriangles by sidesAssessment limit: Use scalene, equilateral, or isosceles

Montessori Lessons:Constructive Triangles, Stick Box, Triangle Logical Adjective Game, GEO Boards, Geometric Cabinet, Geometry Nomenclature Cards

Glencoe: 13-4

b. Compare triangles bysides

Montessori Lessons:Constructive Triangles, Stick Box, Triangle Logical Adjective Game, GEO Boards, Geometric Cabinet, Geometry Nomenclature Cards

b. Compare and classifytriangles by anglemeasureAssessment limit: Use equiangular, obtuse, acute, or right

Montessori Lessons:Constructive Triangles, Stick Box, Triangle Logical Adjective Game, GEO Boards, Geometric Cabinet, Geometry Nomenclature CardsMSA Finish Line:Pages 44-47

c. Determine a third anglemeasure of a triangle given two angle measures

Montessori Lesson:Sum of the angles of a triangle

Assessment limit: Use the concept of the sum of angles in any triangle is 180° without using a diagram

POW (06-07): Week 22

Glencoe: 13-4b



VSC-Mathematics-Standard 2.0 Knowledge of Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of objectsGrade 4 Grade 5 Grade 6

d. Identify and comparethe relationshipbetween parts of acircle

Assessment limit: Use radius, diameter and circumference ( =3.14)

Montessori Lessons:Circle as a “Regular Polygon,” Sensorial Exploration of Circumference; Geometry Nomenclature Cards

Glencoe: 4-6


Topic 2.B. Solid Geometric Figures

Quarter2


Quarter2


2.B.1. Analyze the properties of solid geometric figures

2.B.1. Analyze the properties of solid geometric figures

a. Identify cones,cylinders, prisms, andpyramidsAssessment limit: Use cones or cylinders

Montessori Lessons:Sensorial Classification of Geometric Solids, Geometric Nomenclature Cards, Solid Geometry Classroom Scavenger Hunt

SFAW: 8-1DIS: K-38, p. 75-76, 152

a. Identify and classifypyramids and prisms bythe number of edges,faces, or verticesAssessment limit: Use triangular pyramids, rectangular pyramids, triangular prisms, or rectangular prisms

Montessori Lessons:Sensorial Classification Geometric Solids, Geometric Nomenclature Cards, Solid Geometry Classroom Scavenger Hunt

SFAW 10-1; TE 4DIS: H-28; p. 55-56, 114



Grade 4 Grade 5 Grade 6b. Describe solidgeometric figures bythe number of edges,faces, or verticesAssessment limit: Use triangular pyramids, rectangular pyramids, triangular prisms, or rectangular prisms

Montessori Lessons:Sensorial Classification of Geometric Solids, Geometric Who Am I? Game, Geometric Solids Scavenger Hunt

SFAW 8-1; TE 3DIS: K-38; p. 75, 75, 152

b. Identify and classifypyramids and prisms bythe baseAssessment limit: Use triangular prisms and pyramids or rectangular prisms and pyramids


POW (06-07): Week 17

SFAW 10-2; TE 4DIS: H-29; p. 57-58, 113



Grade 4 Grade 5 Grade 62.B.2. Analyze the relationship between plane geometric figures and surfaces of solid geometric figures

Quarter2

2.B.2. Analyze the relationship between plane geometric figures and faces of solid geometric figures

Quarter2

a. Compare a plane figureto surfaces of solidgeometric figureAssessment limit: Analyze or identify the number or arrangement of squares needed to make a cube and triangles/rectangles needed to make a triangular pyramid or rectangular pyramid.


POW (06-07): Week 9

SFAW: 8-1 cont’d, 8-2DIS: K-45, p. 89-90, 159


a. Compare a plane figureto faces of solidgeometric figureAssessment limit: Analyze and identify the number or arrangement of rectangles needed to make a rectangular prism, number of triangles/rectangles needed to make a triangular prism, and the number of circles/rectangles needed to make a cylinder.

Montessori Lessons:Sensorial Classification of Geometric Solids, Geometric Who Am I? Game, Geometric Solids Scavenger Hunt, Creating Nets for Geometric Solids



VSC-Mathematics-Standard 2.0 Knowledge of Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of objectsGrade 4 Grade 5 Grade 6Topic 2.C. Representation of Geometric Figures Topic 2.C. Representation of Geometric Figures Topic 2.C. Representation of Geometric Figures2.C.1. Represent planegeometric figures

Quarter2

2.C.1. Represent planegeometric figures

Quarter2

2.C.1. Represent planegeometric figures

Quarter2

a. Sketch acute, right,obtuse angles, andparallel andintersecting linesegments

Montessori Lessons:Geometry Nomenclature Cards: Students create booklet; Geometry command cards

a. Identify, describe, anddraw angles, parallelline segments, andperpendicular linesegmentsAssessment limit: Provide their dimensions as whole numbers (0 - 20) or angle measurements (0° - 179°)

Montessori Lessons:Sensorial Exploration of Angles with Montessori Protractor and Red Circle Fractions, Stick Box, Traditional Protractor; Geometry Nomenclature cards: Student created booklets; Geometry command cards

SFAW: 6-2DIS: K-49, p. 97-98, 163


a. Draw geometric figures using a variety of tools

Assessment limit: Draw triangles given the measures of 2 sides and one angle or 2 angles and 1 side using whole numbers (0-20) and angle measures (0°-179°)

Montessori Lessons:Traditional Protractor; Standard and Metric Rulers; Geometry Nomenclature Cards: Student created booklets; Geometry Command Cards

Glencoe: 13-4, 13-4b

b. Identify, describe, ordraw a polygonAssessment limit: Use the first quadrant given no more than six coordinates

Montessori Lessons:Traditional Protractor; Standard and Metric Rulers; Geometry Nomenclature Cards: Student created booklets; Geometry command cards




Grade 4 Grade 5 Grade 6b. Identify or describeangle relationships

Assessment limit: Use perpendicular bisectors or angle bisectors

Montessori Lessons:Geometry Nomenclature: Supplementary and Complementary Angles; Bisecting Angles with a Compass

POW (06-07): Week 21

Glencoe: 13-1, 13-3



VSC-Mathematics-Standard 2.0 Knowledge of Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of objectsGrade 4 Grade 5 Grade 6Topic 2.D. Congruence Topic 2.D. Congruence Topic 2.D. Congruence2.D.1. Analyze geometric figures

Quarter2

2.D.1. Analyze similar figures to

Quarter2

2.D.1. Analyze congruent figures

Quarter2

a. Identify and describegeometric figures ascongruent

Assessment limit: Identify the result in a transformation as being congruent to the original figure

Montessori Lessons:Constructive Triangles, Stick Box, Geometric Equivalence Materials

SFAW 8-6, TE 3DIS: K-43, K-48, p. 85-86, 95-96, 157, 162

a. Identify or describegeometric figures as similarAssessment limit: Use same shape and different size


SFAW: 6-9DIS: K-48, p. 95-96, 162


a. Identify and describecongruent polygons andtheir correspondingparts


POW (06-07): Week 36


VSC-Mathematics-Standard 2.0 Knowledge of Geometry: Students will apply the properties of one-, two-, or three-dimensional geometric figures to describe, reason, or solve problems about shape, size, position, or motion of objectsGrade 4 Grade 5 Grade 6Topic 2.E. Transformations Topic 2.E. Transformations Topic 2.E. Transformations1. Analyze a transformation

Quarter2

1. Analyze a transformation

Quarter2

1. Analyze a transformation on a coordinate plane

Quarter2

a. Identify and describethe results oftranslations, reflections,and rotationsAssessment limit: Use a horizontal line translation, reflection over a vertical line, or rotation of 90° clockwise around a given point of a geometric figure or picture

Montessori Lessons:Blue Constructive Triangles, GEO Boards

POW (06-07): Week 11

SFAW 8-7,TE 3DIS: K-44 p. 87-88, 158


a. Identify and describethe results oftranslations, reflections,and rotations ofgeometric figuresAssessment limit: Use translation along a vertical line, reflection over a horizontal line, or rotation 90° or 180° around a given point

Montessori Lessons:Blue Constructive Triangles, GEO Boards

SFAW: 6-10DIS: K-52, p. 103-104, 166


a. Plot the result of onetransformation(translation, reflection,

rotation) on a coordinate plane

Montessori Lessons:Grid Paper, Command Cards

POW (06-07): Week 23


QUARTER 2

REVIEW or GeometryMSA Finish LinePages 31, 44-46

QUARTER 2


QUARTER 2



VSC Mathematics Standard 3.0 Knowledge of Measurement: Students will identify attributes, units, or systems of measurements or apply a variety of techniques, formulas, tools or technology for determining measurements

Grade 4 Grade 5 Grade 6Topic 3.A. Measurement Units Topic 3.A. Measurement Units Topic 3. A. Measurement Units3.A.1. Read customary and metric measurement units

Quarter 2- 3.A.1.aQuarter 3- 3.A.1.b and c

3.A.1 Read customary and metric measurement units

Quarter3

3.A.1. Read customary and metric measurement units

Quarter3

a. Estimate and determine length and heightAssessment limit: Use the nearest millimeter or ¼ inch

Montessori Lessons:Measuring Object Boxes

Albanesi Cards:(See Appendices – Measurement Activities)

SFAW 10-8, TE 4DIS: K-1, p. 1-2, 115

SFAW 10-7, TE 4DIS: K-2, p. 3-4, 116

Quarter 4Accelerated CurriculumSFAW 10-8, TE 4DIS: K-3, p. 5-6, 117

SFAW 10-10, TE 4DIS: K-8, p. 15-16, 122

SFAW 10-11, TE 4DIS: K-4, p. 7-8, 116

a. Estimate and determine weight or mass

Assessment limit: Use the nearest ounce for weight and the nearest gram for mass

Montessori Lessons:Scales: Balance, Spring


SFAW 9-8, TE 3DIS: K-28, p. 55-56, 142




Grade 4 Grade 5 Grade 6b. Estimate and determine weight or mass

Montessori Lessons:Scales: Balance, Spring


POW(06-07): Week 36


Revisit SFAW11-2, TE 4DIS: I-5, p. 9-10, 97

b. Estimate and determine capacity

Assessment limit: Use the nearest ounce

Montessori Lessons:Sensorial Exploration using household containers, graduated Cylinder material, Scales: Balance, Spring


SFAW 10-6, TE 4SFAW 10-7, TE 5

c. Estimate and determine capacity

Montessori Lessons:Graduated Cylinders in Multiple Sizes


SFAW 10-9, TE 4



Grade 4 Grade 5 Grade 6Topic 3.B. Measurement Tools Topic 3.B. Measurement Tools Topic 3.B. Measurement Tools3.B.1. Measure in customary and metric units

Quarter2

3.B.1. Measure in customary and metric units

Quarter2

3.B.1. Measure in customary and metric units

Quarter2

a. Select and use appropriate tools and units

Assessment limit: Use the nearest millimeter or ¼ inch with a ruler

Montessori Lessons:The Story of Standard Measurement


POW(06-07): Week 18 and 19

SFAW 11-9, TE 4DIS: K-5, p. 9-10, 119SFAW 11-12, TE 4(metric)DIS: K-5, p. 9-10, 119MSA Finish Line:Pages 48-51

Quarter 4Accelerated CurriculumSFAW 10-9, TE 4


Assessment limit: Measure length to 1/8 inch with a ruler

Montessori Lessons:The Story of Standard Measurement; Intro to the Metric System, Sensorial Exploration using yardsticks, meter sticks, and dual sided rulers.


POW(06-07): Week 14

SFAW 9-2, TE 3DIS: K-3 p.5-6, 117


Assessment limit: Measure length to the nearest 1/16 inch with a ruler

Montessori Lessons:The Story of Standard Measurement; Intro to Metric System


Glencoe: 12-2, 12-3, 12-3b, 12-4, 12-6

MSA Finish Line:

Pages 60-63


DIS: K-6, p. 11-12, 120Measure Angles:SFWA 8-3, TE 3EnrichmentActivity, p. 443DIS: K-49, p. 97-98, 163


Grade 4 Grade 5 Grade 63.B.2. Compare right angles to a corner

Quarter2

3.B.2. Measure angles Quarter2

3.B.2. Measure angles in polygons

Quarter2

Montessori Lessons:Constructive Triangles

Quarter 4Accelerated CurriculumRefer to SFAW 8-3, TE 3

a. Measure a single angle and angles in regular polygonsAssessment limit: Measure an angle between 0 and 180 to the nearest degree

Montessori Lessons:Constructive Triangles

SFAW 6-4, TE 2DIS: K-45 p. 89-90, 159See Sample Lesson PlanInvestigation TE 2 p.326

MSA Finish Line:

Pages 60-63

Montessori Lessons:Equivalence Material, Yellow area material, Constructive Triangles


Quarter 4Accelerated Curriculum:Montessori Lessons: Bisecting angles, Addition and Subtraction of angles

Topic 3.C. Applications in Measurement Topic 3.C. Applications in Measurement Topic 3.C. Applications in Measurement3.C.1. Apply measurement concepts

Quarter3

3.C.1. Estimate and apply measurement formulas

Quarter2

3.C.1. Estimate and apply measurement formulas

Quarter2



a. Determine perimeter

Assessment limit: Use polygons with no more than 6 sides given the length of the sides in whole numbers (0 – 100)

Montessori Lessons:Sensorial Exploration of Perimeter, Constructive Triangles, Equivalence Materials, GEO Boards, Real World Problems, Measurement Command Cards

Albanesi Cards:(See Appendices, Measurement Cards: 69.Q)

SFAW 8-10, TE 3DIS: K-26, p. 51-52, 140


Quarter 4Accelerated CurriculumRevisitSFAW 8-10, TE 3DIS: K-31, p. 61-62, 145

a. Determine perimeter

Assessment limit: Use polygons with no more than 8 sides and whole numbers (0 –500)

Montessori Lessons:Sensorial Exploration of Perimeter, Constructive Triangles, Equivalence Materials, GEO Boards, Real World Problems, Measurement Command Cards

Albanesi Cards:(See Appendices, Measurement Cards: 69.Q)

POW(06-07): Week15

SFAW 9-5, TE 3DIS: K-26 p.51-52, 140

a. Estimate and determine the area of a polygon

Assessment limit: Use triangles and whole number dimensions (0 – 200)

Montessori Lessons:Yellow area material, Equivalence material

Albanesi Cards:(See Appendices)

POW(06-07): Week 18

Glencoe: 1-8, 12-1a, 14-1, 14-2, 14-2a, 14-2b


Quarter 4Accelerated Curriculum:Montessori Lessons: Area for Rhombus, and regular polygon, area of a circle material, use of equivalence Material, Pythagorean Theorem lesson with sensorial Pythagoras material

Grade 4 Grade 5 Grade 6b. Determine area Montessori Lessons: b. Determine area Montessori Lessons: b. Estimate and Montessori Lessons:


Assessment limit: Use rectangles with the length of the sides in whole numbers (0 – 100)

Egyptian Rope Stretchers, Yellow Area Material, Equivalence Material, Geo Boards

Albanesi Cards:(See Appendices, Measurement Cards: 36.T)

POW(06-07): Week 12

SFAW 8-11, InvestigationTE 3 (432J)DIS: K-25, p. 49-50, 140



Assessment limit: Use rectangles and whole numbers (0 – 200)

Egyptian Rope Stretchers, Yellow Area Material, Equivalence Material, Geo Boards

Albanesi Cards:(See Appendices, Measurement Cards: 36.T)

SFAW 9-7, TE 3(Re-teaching Lesson: TEp.548B)DIS: K-25 p.4-50, 139


determine the volume of a rectangular prism

Assessment limit: Use rectangular prisms and whole number dimensions (0 – 1000)

Yellow Volume Material, Volume Cube BoxStory of Archimedes

POW(06-07): Week 19

Glencoe: 14-5, 14-6a


Quarter 4Accelerated Curriculum:Montessori Lessons: Volume and surface area of hexagonal, trapezoidal , and rhomboid prisms, pyramids, cylinders, and cones, Materials: Whole and divided rhomboid, hexagonal prisms, hollow volume material

POW(06-07): Week 39 and 40

Glencoe: 14-6VSC Mathematics Standard 3.0 Knowledge of Measurement: Students will identify attributes, units, or systems of measurements or apply a variety of techniques, formulas, tools or technology for determining measurements

Grade 4 Grade 5 Grade 6c. Determine start time, elapsed time, and end time

Assessment limit: Use

Montessori Lessons:Flash Cards, Clocks

Albanesi Cards:107.A-D, (See Appendices,

c. Find the area and the perimeter of any closed figure on a gridAssessment limit: Use whole and partial units (0-50)

Montessori Lessons:Geo Board Activities, Grid Paper Lessons,

See AttachedLesson PlanSFAW 9-7

c. Estimate and determine the area of a composite figure

Montessori Lesson:Constructive Triangles, Yellow area Material

PGCPS 6th CFPGSupplemental Lesson 3


hour and half hour intervals

Measurement: 69.N.9)

POW(06-07): Week 5

SFAW 4-2, TE 2DIS: K-14, p. 27-28, 128

SFAW 4-3, TE 2DIS: K-16, p. 31-32, 130


Assessment limit: Use composite figures with no more than four polygons (triangles or rectangles) and whole number dimensions (0 – 500)


d. Estimate and determine volume by counting

Montessori Lessons:Story of ArchimedesVolume Cubes

SFAW 10-5, TE 4

d. Determine missing dimension of a quadrilateral given the perimeter lengthAssessment limit: Find length in a quadrilateral given the perimeter with whole number dimensions (0 – 200)

Montessori Lessons:See Appendices, Story Problems

POW(06-07): Week 17



Grade 4 Grade 5 Grade 6e. Determine the missing dimension of rectanglesAssessment limit: Find length in a square or rectangle given the area and whole number

Montessori Lessons:See Appendices, Story Problems



dimensions (0 – 200)

3.C.2. Calculate equivalent measurements

Quarter 2 – 3.C.2.aQuarter 3 – 3.C.2.b,c

3.C.2. Calculate equivalent measurements

Quarter3

a. Determine equivalent units of lengthAssessment limit: Use 36 inches = 1 yard and whole numbers (0-100)

Montessori Lessons:Sensorial Exploration

SFAW 10-11, TE 4(customary)only lengthDIS: K-19, p. 37-38,133


a. Determine start, elapsed, and end timeAssessment limit: Use the nearest minute

Montessori Lessons:Flash Cards, Clocks

Albanesi Cards:(See Appendices, Measurement: 69.N.9)

POW(06-07): Week 26

SFAW 9-13, TE 3DIS: K-16,17 p.31-34,130-131



Grade 4 Grade 5 Grade 6b. Determine equivalent units of time

Montessori Lessons:Sensorial Exploration Sundial Observation, Memorization Games

b. Determine equivalent units of measurement

Assessment limit: Use seconds, minutes, and hours or pints, quarts, and gallons

Montessori Lessons:Sensorial Exploration using Household Containers

POW(06-07): Week16

SFAW 9-12, TE 3


DIS: K-14 p.27-28, 128SFAW 10-6, TE 3DIS: K-6 p.11-12, 120


c. Determine equivalent units of capacity and weight within the same system

Albanesi Cards:(See Appendices - Measurement)

REVIEWMSA FINISH LINEPages 47, 68-70




VSC – Mathematics Standard 4.0 Knowledge of Statistics: Students will collect, organize, display, analyze, or interpret data to make decisions or predictions.

Grade 4 Grade 5 Grade 6Topic 4.A. Data Displays Topic 4.A. Data Displays Topic 4.A. Data Displays4.A.1. Collect, organize, and display data

Quarter1

4.A.1. Collect, organize, and display data

Quarter1

4.A.1. Organize and display data

Quarter1

a. Collect data by conducting surveys to answer a question

Montessori Lessons:Student Generated Surveys; Student Created Graphs; Statistics Command Cards (See Appendix)

Albanesi Cards:101.A-C25.A

SFAW 4-4, TE 2DIS: M-19, p. 37-38, 111SFAW 4-6; TE 2DIS: L-2, p. 3-4, 64

a. Collect data by conducting surveys to answer a question

Montessori Lessons:Student Generated Surveys; Student Created Graphs; Statistics Command Cards (See Appendix)

SFAW: 5-1, TE 2DIS: H-37, p. 73-74, 123

a. Organize and display data to make frequency tablesAssessment limit: Use no more than 5 categories or ranges of numbers and total frequencies of no more than 25

Montessori Lessons:Student Generated Surveys; Student Created Graphs; Statistics Command Cards (See Appendix); Cross-Curricular Lessons: Science Fair

Albanesi Cards:62.A-B

Glencoe: 2-1, 2-2a

b. Organize and display data in line plots and frequency tables using a variety of categories and sets of data

Assessment limit: Use line plots with no more than 20 pieces of unorganized data and a range of no more than 10 and whole numbers (0 – 100)

Montessori Lessons:Student Generated Surveys; Student Created Graphs; Statistics Command Cards; Cross-Curricular Lessons: Science Fair

POW(06-07): Week 3

SFAW 4-13, TE 2SFAW 4-14, TE 2

b. Organize and display data in stem-and-leaf plots

Assessment limit: Use no more than 20 data points and whole numbers (0 – 100)


SFAW 5-4, TE 2DIS L-28, pg. 55-56, 90

b. Organize and display data to make stem-and-leaf plots

Assessment limit: Use no more than 20 data points and whole numbers (0–999)


POW(06-07): Week 4

Glencoe: 2-5




Grade 4 Grade 5 Grade 6c. Organize and display data in line plotsAssessment limit: Use no more than 20 pieces of data with a range of no more than 20 and whole numbers (0 – 200)


SFAW: 5-5, TE 2

c. Organize and display data using a back-to-back stem-and-leaf plot


Quarter 4Accelerated CurriculumBox and Whisker plot

POW(06-07): Week 30

Glencoe: 2-7bd. Organize and display data in double bar graphsAssessment limit: Use no more than 4 categories and intervals of 1, 2, 5, or 10 and whole numbers (0 – 100)

Montessori Lessons:Student Generated Surveys; Student Created Graphs; Statistics Command Cards

Albanesi Cards:25.B-H

SFAW: 5-2, TE 2DIS: L-3, p. 5-6, 65



Grade 4 Grade 5 Grade 6e. Organize and display data in line graphsAssessment limit: Use y-axis with intervals of 1, 2, 4, 5, or 10 and x-axis with no more than 10 time intervals and whole numbers (0 – 100)


Albanesi Cards:25.I

SFAW 5-3, TE 2DIS: L-6, pg. 11-12, 68

f. Determine the appropriate type of graph to effectively display data


SFAW 5-8, TE 2



Grade 4 Grade 5 Grade 6Topic 4.B. Data Analysis Topic 4.B. Data Analysis Topic 4.B. Data Analysis4.B.1. Analyze data Quarter

14.B.1. Analyze data Quarter

14.B.1. Analyze data Quarter

1a. Interpret line plotsAssessment limit: Use no more than 20 pieces of data with a range no more than 10 and whole numbers (0 – 100)

Montessori Lessons:Integrated Lessons in Science and Social Studies;Statistics Command Cards; Analyze data displays encountered in classroom periodicals such as “Weekly Reader,” “Scholastic News,” “Time for Kids,” etc.

SFAW 4-7, TE 2DIS: L-25, 49-50, 87


a. Interpret and compare data in stem & leaf plotAssessment limit: Use no more than 20 data points and whole numbers (0 – 100)


POW(06-07): Week 6

SFAW 5-4, TE 2


a. Interpret frequency tablesAssessment limit: Use no more than 5 categories or ranges of numbers and frequencies of no more than 25


POW(06-07): Week 3 and 15

Glencoe: 2-1



VSA – Mathematics Standard 4.0 Knowledge of Statistics: Students will collect, organize, display, analyze, or interpret data to make decisions or predictions.

Grade 4 Grade 5 Grade 6b. Interpret line graphsAssessment limit: Use the x-axis representing no more than 6 time intervals, the y-axis consisting of no more than 10 intervals with scales as factors of 100 using whole numbers (0 – 100)


SFAW 4-10; TE 2DIS: L-5, p. 9-10, 67SFAW 4-11, TE 2DIS: M-31, p. 61-62, 123


b. Interpret and compare data in line plotsAssessment limit: Use no more than 20 pieces of data with a range of no more than 20 and whole numbers (0 – 100)


POW(06-07): Week 5MSA Finish Line:Pages 84-87

b. Read and analyze circle graphsAssessment limit: Use no more than 5 categories using data in whole numbers or percents (0 – 1000)


POW(06-07): Week 3

Glencoe: 2-3MSA Finish Line:Pages 92-95

QUARTER 4Accelerated Curriculum;Making Circle Graphs Glencoe: 14-3b



Grade 4 Grade 5 Grade 6c. Interpret and compare data in double bar graphsAssessment limit: Use no more than 4 categories and intervals of 1, 2, 5, or 10 and whole numbers (0 – 1000)



c. Interpret data from a stem-and-leaf plot


d. Interpret and compare data in double line graphs

Assessment limit: Use y-axis with intervals of 1, 2, 5, or 10 and x- axis with no more than 10 time intervals and whole numbers (0-100)

Montessori Lessons:Integrated Lessons in Science Social Studies; Statistics Command Cards; and Analyze data displays encountered in classroom periodicals such as “Weekly Reader,” “Scholastic News,” “Time for Kids,” etc.

SFAW: 5-8, TE 2(Extension on page 291)SFAW Chapter 5 Review(Page 294)MSA Finish Line:Pages 92-95

Quarter 4Accelerated Curriculum:Bar Graphs and Line Graphs:Glencoe: 2-2Making Bar Graphs and Line Graphs using a SpreadsheetGlencoe: 2-2bMaking Predictions:Glencoe: 2-4Analyzing Data:Glencoe: 2-8



Grade 4 Grade 5 Grade 6e. Read circle graphsAssessment limit: Use no more than 4 categories and data in whole numbers or percents which are multiples of 5 and whole numbers (0 – 100)


SFAW 5-7, TE 2




Grade 4 Grade 5 Grade 64.B.2. Describe a set of data

Quarter1

4.B.2. Describe a set of data (mean, median, mode)

Quarter1

4.B.2. Describe a set of data

Quarter1

a. Determine median, mode, and rangeAssessment limit: Use no more than 8 pieces of data and whole numbers (0 – 100)

Montessori Lessons:Statistics Command Cards; Cross-Curricular Lessons: Science Fair

POW(06-07): Week 4

SFAW 4-12; TE 2DIS: L-26, p. 51-52, 88

a. Determine the mean of a given data set or data display

Assessment limit: Use no more than 8 pieces of data and whole numbers without remainders (0 – 1000)


SFAW 5-6, TE 2


a. Apply measures of central tendency (mean, median, mode)


Albanesi Cards:62.C-G


Glencoe: 2-6, 2-7

QUARTER 4Accelerated Curriculum:Spreadsheets and Mean, Glencoe 2-6b

b. Model the mean of a set of data

Montessori Lessons:Statistics Command Cards

SFAW 7-12, TE 3

b. Apply the range and measures of central tendency to solve a problem or answer a question

Montessori Lessons:Statistics Command Cards; Story Problem Cards

QUARTER 1

Review of StatisticsMSA Finish Line

Pages 71, 84-86

QUARTER 1

Review of StatisticsMSA Finish LinePages 79, 104-106

QUARTER 1

Review of StatisticsMSA Finish LinePages 83, 96-98

VSC –Mathematics-Standard 5.0 Knowledge of Probability: Students will use experimental methods or theoretical reasoning to determine probabilities to make predictions or solve problems about events whose outcomes involve random variation.


Grade 4 Grade 5 Grade 6Topic 5. A. Sample Space Topic 5.A. Sample Space Topic 5.A. Sample Space

5. A.1. Identify possible outcomes

Quarter3

a. Determine possible outcomes of independent eventsAssessment limit: Use two independent events with no more than 4 outcomes each and an organized list or tree diagram

Montessori Lessons:Probability Problem Solving Cards (See Appendix); Small Group Experiments

POW (06-07): Week 7

SFAW 5-10, TE 2DIS: L-16, pg. 31-32, 78

SFAW 5-11, TE 2

MSA Finish LinePages 108-111



Grade 4 Grade 5 Grade 6Topic 5.B. Theoretical Probability Topic 5.B. Theoretical Probability Topic B. Theoretical Probability5.B.1. Determine the probability of one simple event comprised of equally likely outcomes

Quarter3

5.B.1. Determine the probability of one simple event comprised of equally likely outcomes

Quarter3

5.B.1. Determine the probability of one simple event comprised of equally likely outcomes

Quarter3

a. Express the probability as a fractionAssessment limit: Use a sample space of no more than 6 outcomes

Montessori Lessons:Probability Problem Solving Cards (See Appendix)

POW (06-07) Week: 26,30

SFAW 12-7, TE 4DIS: l-12, pg. 23-24, 75


a. Make predictions and express the probability as a fractionAssessment limit: Use a sample space of no more than 20 outcomes


SFAW 5-12, TE 2Investigation from TE Page 258 J


a. Express the probability of an event as a fraction.


Glencoe: 11-1, 11-2

b. Express the probability of an event as a decimalAssessment limit: Use a sample space of 10, 20, 25, or 50 outcomes

Montessori Lessons:Probability Problem Solving Cards (SeeAppendix); Small Group Experiments

Glencoe: 11-1, 11-2





c. Express the probability of an event as a percent

Montessori Lessons:Probability Problem Solving Cards (See Montessori Math Supplement); Small GroupExperiments

POW (06-07): Week 14

Glencoe: 11-1, 11-2

Accelerated Curriculum:Probability and Area:Glencoe: 11-4Probability of Independent Events: Glencoe: 11-Problem Solving with Probability: Glencoe 11-5a




Topic 5.C. Experimental Probability Topic 5.C. Experimental Probability Topic 5.C. Experimental Probability1. Analyze the results of a probability experiment

Quarter3

a. Make predictions and express the experimental probability as a fraction, a decimal, or a percentAssessment limit: Use no more than 30 results in the sample space

Montessori Lessons:Probability Problem Solving Cards (See Montessori Math Supplement); Small Group Experiments

POW (06-07): Week 13

Glencoe: 11-1b, 11-3


2. Conduct a probability experiment

Montessori Lessons:Probability Problem Solving Cards (See Montessori Math Supplement); Small Group Experiments

3. Compare outcomes of theoretical probability with the results of experimental probability

Glencoe: 11-1b

4. Describe the difference between theoretical and experimental probability

Glencoe: 11-1b



QUARTER 3

REVIEWMSA Finish LinePages 87, 92-94

QUARTER 3


QUARTER 3



VSC – Mathematics-Standard 6.0 Knowledge of Number Relationships and Computation/Arithmetic: Students will describe, represent, or apply numbers or their relationships or will estimate or compute using mental strategies, paper/pencil or technology.

Grade 4 Grade 5 Grade 6Topic 6.A. Knowledge of Number and Place Value Topic 6.A. Knowledge of Number and Place Value Topic 6.A. Knowledge of Number and Place Value6.A.1. Apply knowledge of whole numbers and place value

Quarter1

6.A.1. Apply knowledge of fractions, decimals, and place value

Quarter 2 – 6.A.1.a,c,d

Quarter 3 – 6.A.1.b,e

6.A.1. Apply knowledge of rational numbers and place value

Quarter 1 – 6.A.1.a, b, e

Quarters 2 and 3– 6.A.1.c,da. Read, write, and represent whole numbers using symbols, words, and modelsAssessment limit: Use whole numbers (0 - 1,000,000)

Montessori Lessons:Golden Beads, Wooden Hierarchical Material, Long Bead Frame, Infinity Street, Golden Mat with Bead Bars

Albanesi Cards:92.A-M6.A-E

SFAW 1-1, TE 1DIS: F-7, pg. 13-14, 85

a. Read, write, or represent fractions or mixed numbers using symbols, models, and wordsAssessment limit: Use denominators that are factors of 24 and numbers (0 – 200)

Montessori Lessons:Red Fraction Materials, Story Problem Cards, Student Created Models

Albanesi Cards:104.A-F1.A-B

SFAW 7-3, TE 3DIS: H-15, pg. 29-30, 101


a. Read, write, and represent whole numbersAssessment limit: Use exponential form with powers of 10 (0 - 100,000)

Montessori Lessons:Powers of Two, Wooden Squaring Material, Scientific Notation ; Exponential Notation, Paper and Pencil Abstract Lessons

Albanesi Cards:92.A-P; 6.A-1; 7.A

Glencoe: 4-1


Fourth Quarter Accelerated Curriculum:Exponential FormAlbanesi Cards:POW (06-07): Week 29Glencoe: 1-4



Grade 4 Grade 5 Grade 6b. Express whole numbers in expanded formAssessment limit: Use whole numbers (0 - 1,000,000)


Albanesi Cards:92.P-S6.H-I

b. Read, write, or represent decimals using symbols, words, or modelsAssessment limit: Use no more than 3 decimal places (0 – 100)

Montessori Lessons:Decimal Board with Cubes

Albanesi Cards:110.A-B5.A-C, E-H22.A

SFAW 1-3, TE 1


Fourth Quarter Accelerated Curriculum:Read write and represent integers (-100 to 100). Compare and order integers.

Montessori Lessons:Positive and Negative Numbers Board, Integers, Cross-Curricular Connection: BC/CE Timeline

SFAW 12-5, TE 4DIS: J-1, PG. 1-2, 83

b. Read, write, and represent integersAssessment limit: Use integers from (-100 to 100)

Montessori Lessons:Positive and Negative Numbers Board, Integers, Cross-Curricular Connection: BC/CE Timeline

Albanesi Cards:38.A; 42.A

Glencoe: 8-1




Grade 4 Grade 5 Grade 6c. Identify the place value of a digit in a numberAssessment limit: Use whole numbers (0 - 1,000,000)


Albanesi Cards:92.N-O6.F-G

POW (06-07): Week 6

SFAW 1-2, TE 1DIS: F-10, pg. 19-20, 88


c. Identify and determine equivalent forms of proper fractions

Assessment limit: Use denominators that are factors of 100, decimals, or percents (0 – 200)

Montessori Lessons:Red Fraction Materials, Student Created Models, LCM / LCD with the Pegboard

Albanesi Cards:93.H-I3.G-S20.A-H

SFAW 7-8, TE 3DIS: H-24, pg. 47-48, 110

SFAW 7-10, TE 3DIS: H-17, pg. 33-34, 103


Fourth Quarter Accelerated CurriculumInclude ratios and decimalsPOW (06-07): Week 25SFAW 11-8, TE 4DIS: I-135, pg.. 69-70, 127

c. Identify and determine equivalent forms of fractions as decimals, as percents, and as ratiosAssessment limit: Use proper fractions with denominators as factors of 100, decimals, percents, or ratios (0 – 1000)

Montessori Lessons:Red Fraction Insets with Centesimal Square / Percentage Wheel, paper and pencil lessons in abstract problem solving

Albanesi Cards:118.F-H; 3.S-W; 16.A-L;20.A-H; 22.A-H; 33-A-G;36.A-F; 50.A-H; 56.A-H;57.A-D; 64.A-F

POW (06-07): Week 27

Glencoe: 5-6 and 5-7Glencoe: 10-1,10-2, 10-4




Grade 4 Grade 5 Grade 6d. Compare, order, and describe whole numbersAssessment limit: Use no more than 4 whole numbers with or without using the symbols (<, > , =) and whole numbers (0 - 1,000,000)


SFAW 1-5, TE 1DIS: F-12, pg. 23-24, 90


d. Compare or order fractions with or without using the symbols (<, >, or =)Assessment limit: Use no more than 4 fractions or mixed numbers with denominators that are factors of 100 and numbers (0 – 100)

Montessori Lessons:Red Fraction Materials, Student Created Models, Finding Common Denominators

Albanesi Cards:93.A-G; 109.J-L; 1.O-Q;31.A-E

POW (06-07): Week 11

SFAW 7-12, TE 3DIS: H-20, pg. 39-40, 106SFAW 7-13, TE 3SFAW 7-14, TE 3Include inequalities: <, >, =

d. Compare and order fractions, decimals alone or mixed together, with and without relational symbols (<, >, =)Assessment limit: Include no more than 4 fractions with denominators with factors of 100 or decimals with up to 2 decimal places (0 – 100)

Montessori Lessons:Paper and Pencil Lessons in Abstract Problem Solving

Albanesi Cards:93.D-G; 104.G; 109.J-L;110.C; 118.A; 1-O-Q;5.D; 31.A-E, 51.H

POW (06-07): Week 5,8,9

Glencoe: 3-1 and 3-2Glencoe: 5-4, 5-5


e. Compare, order, and describe decimals with or without using the symbols (<, >, or =)Assessment limit: Use no more than 4 decimals with no more than 3 decimal places and numbers (0 – 100)


Albanesi Cards:104.G; 110.C; 118.A; 5.D; 51.HPOW (06-07): Week 18SFAW 1-4, TE 1MSA Finish Line:Pages 132-135

e. Compare and order integers

Montessori Lessons:Integers Lessons, Positive and Negative Numbers Board

Albanesi Cards:42.A



Grade 4 Grade 5 Grade 66.A.2. Apply knowledge of fractions and decimals

Quarter 2 – 6.A.2.a,b,d,gQuarter 3- 6.A.2.e,f,h

a. Read, write, and represent proper fractions of a single region using symbols, words, and modelsAssessment limit: Use denominators 6, 8, and 10

Montessori Lessons:Red Fraction Materials

Albanesi Cards:1.A,B,D,F

POW (06-07): Week 13

SFAW 9-1, TE 3DIS H-12, pg. 23-24, 99

b. Read, write, or represent proper fractions of a set which has the same number of items as the denominator using symbols, words, and modelsAssessment limit: Use denominators of 6, 8, and 10 with sets of 6, 8, and 10, respectively

Montessori Lessons:Story Problem Cards, Student Created Classroom Models

Albanesi Cards93.A-C

POW (06-07): Week 13

SFAW 9-2, TE 3DIS H-14, pg. 27-28, 101




Grade 4 Grade 5 Grade 6c. Find equivalent fractions

Fourth Quarter Accelerated Curriculum:


Albanesi Cards:100.A-C3.G-SPOW (06-07): Week 34SFAW 9-6, TE 3DIS: H-20, pg. 39-40, 104SFAW 9-7, TE 3DIS: H-17, pg. 33-34, 103

d. Read, write, and represent mixed numbers using symbols, words, and models


Albanesi Cards:16.A-J

POW (06-07): Week 15

SFAW 9-10, TE 3DIS: H-15, pg. 29-30, 101



Grade 4 Grade 5 Grade 6e. Read, write, and represent decimals using symbols, words and modelsAssessment limit: Use no more than 2 decimal places and numbers (0-100)


Albanesi Cards:104.A-G5.A-C, and 5.E-H

SFAW 11-1, TE 4DIS: I-4, pg. 7-8, 96

Fourth Quarter Accelerated CurriculumExpand to 3 decimal placesDIS I-5, pg. 9-10, 97

f. Express decimals in expanded formAssessment limit: Use no more than 2 decimal places and numbers (0-100)


POW (06-07): Week 23

SFAW 11-2, TE 4DIS: I-4, pg. 7-8, 96




Grade 4 Grade 5 Grade 6g. Compare and order fractions and mixed numbers with or without using the symbols (<, >, or =)Assessment limit: Use like denominators and no more than 3 numbers (0-20)


Albanesi Cards:93.D-E1.O-Q

SFAW 9-8, TE 3DIS: H-19, pg. 37-38, 105

SFAW, 9-11, TE 3DIS: H-20, pg. 39-40, 106


Fourth Quarter Accelerated Curriculum:Expand to include unlike denominators.POW (06-07): Week 35SFAW 9-9, TE 3DIS: H-20, pg. 39-40, 104



Grade 4 Grade 5 Grade 6h. Compare, order, and describe decimals with or without using the symbols (<, >, or =)Assessment limit: Use no more than 3 decimals with no more than 2 decimal places and numbers (0 – 100)


Albanesi Cards:104.G, 110.C, 118.A, 5.D,51.H

POW (06-07): Week 23

SFAW 11-3, TE 4DIS: I-7, pg. 13-14, 99


Fourth Quarter Accelerated Curriculum:Expand to 3 decimal places and 4 decimals.POW (06-07): Week 37DIS I-8, pg 15-16, 100

6.A.3. Apply knowledge of money

Quarter3



Grade 4 Grade 5 Grade 6a. Compare the value of sets of mixed currencyAssessment limit: Use 2 sets of mixed currency and money ($0 - $100)

Montessori Lessons:Currency Flash Cards, Cash Box

Albanesi Cards:21.A-E

SFAW 1-10, TE 1DIS: F-1, pg. 1-2, 79


b. Determine the change from $100

Montessori Lessons:Currency Flash Cards, Cash Box, Practical Life: Shopping from menus or catalogs, Story Problem Cards

POW (06-07): Week 22

SFAW 1-11, TE 1DIS: F-2, pg. 3-4, 80



Grade 4 Grade 5 Grade 6Topic 6.B. Number Theory Topic 6.B. Number Theory Topic 6.B. Number Theory6.B.1. Apply number relationships

Quarter1

6.B.1. Apply number relationships

Quarter1

6.B.1. Apply number relationships

Quarter1

a. Identify and use divisibility rulesAssessment limit: Use the rules for 2, 5, or 10 with whole numbers (0 – 1000)

Montessori Lessons:Factoring on the Pegboard, Flash Cards, Multiplication Chart, Exploration with the Unit Division Board

Albanesi Cards:94.A-B, and G-H2.A,B,F

POW (06-07): Week 8

SFAW 7-11, TE 3DIS: H-1, pg. 1-2, 87

a. Identify or describe numbers as prime or compositeAssessment limit: Use whole numbers (0 – 100)

Montessori Lessons:Pegboard, Paper and Pencil Lessons in Abstract Problem Solving

Albanesi Cards:98.A-1; 14.A-H

SFAW 3-11, TE 1DIS: H-3, pg. 5-6, 89

a. Determine prime factorizations for whole numbers and express them using exponential form

Montessori Lessons:Pegboard, Paper and Pencil Lessons in Abstract Problem Solving

Albanesi Cards:See fifth grade list, and63.A-E; 79, A-D

POW (06-07): Week 6

Glencoe: 1-2 and 1-3



Grade 4 Grade 5 Grade 6b. Identify factorsAssessment limit: Use whole numbers (0 – 24)

Montessori Lessons:Factoring on the Pegboard, Flash Cards, Multiplication Chart

Albanesi Cards:97.A-F; 15.A-D

POW (06-07): Week 14

SFAW 3-1, TE 1DIS: G-14, pg. 27-28, 150

b. Identify and use rules of divisibilityAssessment limit: Use rules for 2, 3, 5, 9, or 10 and whole numbers (0 - 10,000)

Montessori Lessons:Factoring on the Pegboard, Flash Cards, Multiplication Chart, Exploration with the Unit Division Board

Albanesi Cards:97.F; 2.A-C and E-H

POW (06-07): Week 1SFAW 3-10, TE 1DIS: H-2, PG. 3-4, 88MSA Finish Line:Pages 136-139

c. Identify multiplesAssessment limit: Use the first 5 multiples of any single digit whole number

Montessori Lessons:Long and Short Bead Chains, Flash Cards, Multiplication Chart

Albanesi Cards:96.A-F9.A-F

POW (06-07): Week 14

SFAW 3-2, TE 1DIS G-9, pg. 17-18, 145


c. Identify the greatest common factorAssessment limit: Use 2 numbers whose GCF is no more than 10 and whole numbers (0 – 100)

Montessori Lessons:Pegboard, Prime Factor Trees, Paper and Pencil Lessons in Abstract Problem Solving

Albanesi Cards:99.A-F; 15.A-D

POW (06-07): Week 11

SFAW 7-9, TE 3DIS: H-4, pg. 7-8, 90



Grade 4 Grade 5 Grade 6d. Identify a common multiple and the least common multipleAssessment limit: Use no more than 4 single digit whole numbers

Montessori Lessons:Pegboard, Prime Factor Trees, Paper and Pencil Lessons in Abstract Problem Solving

Albanesi Cards:96.A-F; 9.H-R

POW (06-07): Week 12

SFAW 8-1, TE 3DIS: H-29, pg. 57-58, 115


REVIEW: RelationshipsMSA Finish Line:Pages 95, 128-130





Grade 4 Grade 5 Grade 6Topic 6.C. Number Computation Topic 6.C. Number Computation Topic 6.C. Number Computation6.C.1. Analyze number relations and compute

Quarter 1 - 6.C.1.a,b,c,dQuarter 2 - 6.C.1.eQuarter 3 – 6.C.1.f,g

6.C.1. Analyze number relations and compute

Quarter 1 - 6.C.1.a,b,cQuarter 2 – 6.C.1.dQuarter 3 – 6.C.1.e,f,g,h

6.C.1. Analyze number relations and compute

Quarter 1 – 6.C.1.fQuarter 2 – 6.C.1.a,bQuarter 3 – 6.C.1.c,d,e

a. Add whole numbersAssessment limit:Use up to 3 addends with no more than 4 digits in each addend and whole numbers (0 - 10,000)

Montessori Lessons:Stamp Game, Golden Mat, Long Bead Frame

Albanesi Cards:95.A-D; 4.A-G;120.A-B; 35.A-B

SFAW 2-5, TE 1DIS: F-36, 71-72, 114

SFAW 2-6, TE 1

a. Multiply whole numbersAssessment limit:Use a 3-digit factor by another factor with no more than 2-digits and whole numbers (0 - 10,000)

Montessori Lessons: Checkerboard, Golden Bead Frame

Albanesi Cards:102.A-E; 106.A-G;113.A-F; 116.A-C;119.A; 120.C;7.F-H; 11.A-C;35.C

SFAW 2-4, TE 1DIS: G-59, pg. 117-118, 195

SFAW 2-8, TE 1DIS: I-119, pg. 37-38, 111


a. Add and subtract fractions and mixed numbers and express answers in simplest formAssessment limit: Use proper fractions and denominators as factors of 60 (0–20)

Montessori Lessons:Paper and pencil lessons in abstract problem solving, Review of Red Fraction Box

Albanesi Cards:See fifth grade and,100.D-E, J-M109.C-G,M-N49.A-G; 54.A-J; 58.A-C;65.G-K; 77.A-F

POW (06-07): Week 10,11

Glencoe: 5-2, 5-3 (Review of Simplest Form & LCM)

Glencoe: 6-3, 6-4, 6-5, 6-6




Grade 4 Grade 5 Grade 6b. Subtract whole numbersAssessment limit: Use a minuend and subtrahend with no more than 4 digits in each and whole numbers (0 – 9999)

Montessori Lessons:Stamp Game, Golden Mat, Long Bead Frame

Albanesi Cards:95.C; 120.A-B;

POW (06-07): Week 21

SFAW 2-7, TEDIS: F-37, pg. 73-74, 115


b. Divide whole numbersAssessment limit: Use a dividend with no more than a 4-digits by a 2-digit divisor and whole numbers (0 - 9,999)

Montessori Lessons: Stamp Game, Division with Racks and Tubes

Albanesi Cards:103.A-E; 105.A-F;111.A-G; 114.A-E;120.E-G; 17.A-G;35.D-G

SFAW 3-7 and 3-8SFAW 4-1, 4-2, and 4-5

b. Multiply fractions and mixed numbers and express in simplest formAssessment limit: Use denominators as factors of 24 not including 24 (0 – 20)

Montessori Lessons:Paper and pencil lessons in abstract problem solving, Review of Red Fraction Box

Albanesi Cards:100.F-I; 109.A-B; 3.A-F;34.A-J; 40.A-G; 58.D-J; 77.G-R

POW (06-07): Week 12, 20

Glencoe: 7-2, 7-3, 7-4




Grade 4 Grade 5 Grade 6c. Multiply whole numbersAssessment limit: Use a one 1-digit factor by up to a 3-digit factor using whole numbers (0 – 1000)

Montessori Lessons:Stamp Game, Golden Mat, Long Bead Frame, Checkerboard

Albanesi Cards:102.A-E; 7.A-E; 11.A-BPOW (06-07): Week 6SFAW 5-4, TE 2DIS G-39, pg. 77-78,175SFAW 5-5, TE 2SFAW 5-6, TE 2DIS G-42, pg. 83-84,178


Fourth Quarter Accelerated CurriculumExpand to three-digit by two-digit whole numbers.POW (06-07): Week 32SFAW 6-1, TE 2DIS: G-55, pg. 109-110, 191SFAW 6-5, TE 2DIS: G-58, pg. 115-116, 194SFAW 6-6, TE 2DIS: G-59, pg. 117-118, 195

c. Interpret quotients and remainders mathematically and in the context of a problemAssessment limit: Use dividend with no more than a 3-digits by a 1 or 2 digit divisor and whole numbers (0 – 999)

Montessori Lessons:Story Problem Cards

Albanesi Cards:97.A-E; 15.A-D

SFAW 3-12


c. Multiply decimalsAssessment limit: Use a decimal with no more than 3 digits multiplied by a 2-digit decimal (0 – 1000)

Montessori Lessons:Decimal Checkerboard, Decimal Board, Paper and pencil lessons in abstract problem solving

Albanesi Cards:110.E-G; 117.D; 118.C-E;21.I-J; 43.A-D;43.A-D; 5.F-J; 26.A-F; 51.A-G

Glencoe: 4-1 and 4-2




Grade 4 Grade 5 Grade 6d. Divide whole numbersAssessment limit: Use up to a 3-digit dividend by a 1-digit divisor and whole numbers with no remainders (0 - 999)

Montessori Lessons:Stamp Game, Division with Racks and Tubes

Albanesi Cards:91.A-K; 111.A-G; 112.A-B;120.E; 10.A-G

POW (06-07): Week 7SFAW 7-5, TE 3DIS G-50, pg. 99-100,186SFAW 7-6, TE 3DIS M-23, pg. 45-46, 115SFAW 7-7, TE 3DIS G-51, pg. 101-102, 187


Fourth Quarter Accelerated Curriculum:Expand to include 4-digit dividend by 2-digit divisor, and interpret quotients and remainders in Story Problems.SFAW 7-14, TE 3DIS: G-62, pg. 123-124, 198POW (06-07): Week 33

d. Add and subtract proper fractions and mixed numbers with answers in simplest formAssessment limit: Use denominators as factors of 24 and numbers (0 – 20)

Montessori Lessons:Red Fraction Materials, Paper and pencil lessons in abstract problem solving

Albanesi Cards:93.J-M; 100.D-E; 109.M-N;115.B-E; 1.I-9; 24.A-J;28.A-P; 30.A-DPOW (06-07): Week 13SFAW 8-3, TE 3DIS: H-30, pg. 59-60,116SFAW 8-4, TE 3DIS: H-31, pg. 61-62, 117SFAW 8-5, TE 3DIS: H-32, pg. 63-64, 118MSA Finish Line:Pages 156-159

Fourth Quarter Accelerated Curriculum:Multiply fractions and mixed-numbers and express in simplest form.POW (06-07): Week 32SFAW 8-11, TE 3DIS: H-38, pg. 75-76, 124SFAW 8-12, TE 3DIS, H-39, pg. 77-78, 125SFAW 8-13, TE 3DIS: H-40, pg. 79-80, 126

d. Divide decimalsAssessment limit: Use a decimal with no more than 5 digits divided by a whole number with no more than 2 digits without annexing zeros (0 – 1000)

Montessori Lessons:Decimal Cubes and Skittles, Paper and pencil lessons in abstract problem solving

Albanesi Cards:5.K-N; 21.K-M; 22.A-L;55.A-F; 59.A-E; 61.A-H

Glencoe: 4-2




Grade 4 Grade 5 Grade 6e. Add and subtract proper fractions and mixed numbers

Assessment limit: Use 2 proper fractions with a single digit like denominators, 2 mixed numbers with single digit like denominators, or a whole number and a proper fraction with a single digit denominator and numbers (0 – 20)

Montessori Lessons:Red Fraction Materials, Student Created Models

Albanesi Cards:93.J-M100.D-E

POW (06-07): Week 16

SFAW 10-2, TE 4DIS: H-29, 57-58, 115


Fourth Quarter Accelerated Curriculum:Extend to Unlike DenominatorsSFAW 10-3, TE 4DIS H-31, pg. 61-62, 117SFAW 10-5, TE 4DIS H-31, 61-62, 117

Montessori Lessons:Paper and pencil lessons in abstract problem solving

Albanesi Cards:109.H-I; 36.G-I; 57.E-H;58.K-M

POW (06-07): Week 28

Glencoe: 10-7




Grade 4 Grade 5 Grade 6f. Add 2 decimalsAssessment limit: Use the same number of decimal places but no more than 2 decimal places and no more than 4 digits including monetary notation and numbers (0 – 100)


Albanesi Cards:104.H

POW (06-07): Week 24

SFAW 11-6, TE 4DIS I-11, pg. 21-22, 103SFAW 11-7, TE 4

SFAW 2-5, TE 1 (Money)

f. Subtract decimals including moneyAssessment limit: Use a minuend and subtrahend with no more than 3 decimal places and numbers (0 – 1000)

Montessori Lessons:Decimal Board with Cubes, Cash Box, Story Problem Cards

Albanesi Cards:110.D; 117.B-C;118.B; 18.C-D;21.F-H

POW (06-07): Week 19

SFAW 1-13, TE 1DIS: I-117, pg. 33-34, 109


f. Simplify numeric expressions using the properties of addition and multiplicationAssessment limit: Use the distributive property to simplify numeric expressions and whole numbers (0 – 1000)

Montessori Lessons:Paper and Pencil Lessons in Abstract Problem Solving

POW (06-07): Week 24

Glencoe: 9-1




Grade 4 Grade 5 Grade 6g. Subtract decimalsAssessment limit: Use the same number of decimal places but no more than 2 decimal places and no more than 4 digits including monetary notation and numbers (0 – 100)


Albanesi Cards:104.I; 117.B-C

POW (06-07): Week 24

SFAW 11-6, TE 4DIS: I-11, pg. 21-22, 103SFAW 11-7, TE 4

SFAW 2-7, TE 1 (Money)


g. Multiply decimalsAssessment limit: Use a decimal in monetary notation by a single digit whole number and numbers (0 – 100)

Montessori Lessons:Decimal Checkerboard, Decimal Board with Cubes, Cash Box, Story Problem Cards

Albanesi Cards:110.E-G; 117.D;5.F-J; 21.I-J; 43.A-D

SFAW 2-9, TE 1


Fourth Quarter Accelerated Curriculum:Multiply decimals by decimal.POW (06-07): Week 33SFAW 2-10, 2-11, TE 1DIS: I-121, pg. 43-46DIS: I-123, pg. 114-115

SEVENTH GRADE INDICATOR: 6.C.1.aAdd subtract, multiply and divide integers. Use one operation. (-100 to 100)

Fourth Quarter Accelerated CurriculumMontessori Lessons:Negative Snake Game, Integers lessons

POW (06-07): Week 32, 33

Glencoe: 8-2, 8-3, 8-4, 8-5



Grade 4 Grade 5 Grade 6h. Divide decimals by whole numbers

Fourth Quarter Accelerated Curriculum:(See 6th Grade 6.C.1.d for assessment limit)Montessori Lessons:Decimal Board with Cubes, Cash Box, Story Problem CardsAlbanesi Cards:110.H-J; 117.E-F;5.K-N; 21.K-LPOW (06-07): Week 34SFAW 4-11, TE 2DIS: I-126, pg. 51-52, 118Fourth Quarter Accelerated Curriculum:SEVENTH GRADE INDICATOR: 6.C.1.aAdd integers (-100-100).SFAW 12-6SFAW 12-9


Grade 4 Grade 5 Grade 66.C.2. Estimation Quarter 1 6.C.2. Estimation Quarter 1 – 6.C.2.b

Quarter 3 6.C.2.a,c6.C.2. Estimation Quarter 3

a. Determine the approximate sum and difference of 2 numbersAssessment limit: Use no more than 2 decimal places in each and numbers (0 – 100)

Montessori Lessons:Rounding with the Golden Beads and Number Cards, Rounding on the Golden Mat, Abstract Addition and Subtraction

POW (06-07): Week 24

SFAW 11-5, TE 4DIS: I-12, pg. 23-24, 104


a. Determine the approximate sum and difference of decimalsAssessment limit: Use no more than 3 addends with no more than 3 decimal places in each addend or the difference of a minuend and subtrahend with no more than 3 decimal places and numbers (0 – 1000)

Montessori Lessons:Rounding on the Decimal Board, Abstract Addition and Subtraction

SFAW 1-9, TE 1DIS: I-112, pg. 23-24, 104

a. Determine the approximate products and quotients of decimalsAssessment limit: Use a decimal with no more than a 3 digits multiplied by a 2-digit whole number, or the quotient of a decimal with no more than 4 digits in the dividend divided by a 2-digit whole number (0 – 1000)

Montessori Lessons:Rounding on the Decimal Board, Abstract Multiplication and Division

Glencoe: 3-3


b. Determine the approximate product or quotient of 2 numbersAssessment limit: Use a 1-digit factor with the other factor having no more than 2-digits or a 1-digit divisor and no more than a 2-digit dividend and whole numbers (0 – 1000)

Montessori Lessons:Fact Families, Rounding on the Golden Mat, Abstract Multiplication and Division

POW (06-07): Week 25

SFAW 5-2, TE 2DIS: G-37, pg 73-74, 173

SFAW 7-2, TE 3DIS: G-38, pg. 75-76, 174


b. Determine approximate product and quotient of whole numbersAssessment limit: Use a 1-digit factor with the other factor having no more than 3 digits or a dividend having no more than 3 digits and a 1-digit divisor and whole numbers (0 – 5000)

Montessori Lessons:Fact Families, Rounding on the Golden Mat, Abstract Multiplication and Division

POW (06-07): Week 20POW (06-07): Week 23, 24

SFAW 2-2, TE 1DIS: G-56, pg. 111-112, 192

SFAW 3-3, TE 1



Grade 4 Grade 5 Grade 6c. Determine the approximate product of decimalsAssessment limit: Use a decimal in monetary notation and a single digit whole number (0 – 100)

Montessori Lessons:Fact Families, Rounding on the Golden Mat, Cash Box Lessons on Estimating Cost

POW (06-07): Week 21

SFAW 2-8, TE 1


6.C.3. Analyze ratios, proportions, and percents

Quarter3

a. Represent ratios in a variety of forms

Montessori Materials:Ratios on the Pegboard, Paper and Pencil Lessons in Abstract Problem Solving, Story Problems

Glencoe: 10-5, 10-6



Grade 4 Grade 5 Grade 6b. Use ratios and unit rates to solve problems

Fourth Quarter Accelerated Curriculum:Montessori Materials:Ratios on the Pegboard, Paper and Pencil Lessons in Abstract Problem Solving, Story Problems, Student Created Scale ModelsPOW (06-07): Week 35, 37, and 38Glencoe 10-3

QUARTER

REVIEW of ComputationMSA Finish Line:Pages 131, 160-162

QUARTER


QUARTER



VSC - Mathematics Standard 7.0 Processes of Mathematics: Students demonstrate the processes of mathematics by making connections and applying reasoning to solve problems and to communicate their findings.

Grade 4 Grade 5 Grade 6Topic 7.A. Problem Solving Topic 7.A. Problem Solving Topic 7.A. Problem Solving7.A.1. Apply a variety of concepts, processes, and skills to solve problems

Quarters1,2,3, and 4

7.A.1. Apply a variety of concepts, processes, and skills to solve problems


7.A.1. Apply a variety of concepts, processes, and skills to solve problems


a. Identify the question in the problem

Montessori Lessons:Understanding the Language of the Four Basic Operations; Key Terms Card Set and Chart; Story Problem Cards

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

b. Decide if enough information is present to solve the problem

Montessori Lessons:Modeling the Problem; Story Problem Cards

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

c. Make a plan to solve a problem


POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



Grade 4 Grade 5 Grade 6Topic 7.A. Problem Solving Topic 7.A. Problem Solving Topic 7.A. Problem Solving

d. Apply a strategy, i.e., draw a picture, guess and check, finding a pattern, writing an equation

Montessori Lessons:Modeling the Problem; Symbolizing the Problem and Calculating; Story Problem Cards

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

e. Select a strategy, i.e., draw a picture, guess and check, finding a pattern, writing an equation

Montessori Lessons:Modeling the Problem; Symbolizing the Problem and Calculating; Independence in Solving Story Problems Lesson (See Montessori Math Problem Solving Sequence)

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

f. Identify alternative ways to solve a problem

Montessori Lessons:Independence in Solving Story Problems Lesson (See Montessori Math Problem Solving Sequence); Small Group Discussions

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

g. Show that a problem Montessori Lessons: g. Show that a problem Montessori Lessons: g. Show that a problem Montessori Lessons:


might have multiple solutions or no solution

Independence in Solving Story Problems Lesson (See Montessori Math Problem Solving Sequence); Small Group Discussions

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40


Grade 4 Grade 5 Grade 6Topic 7.A. Problem Solving Topic 7.A. Problem Solving Topic 7.A. Problem Solving

h. Extend the solution of a problem to a new problem situation








Grade 4 Grade 5 Grade 6Topic 7.B. Reasoning Topic 7.B. Reasoning Topic 7.B. Reasoning

7.B.1. Justify ideas or solutions with mathematical concepts or proofs






a. Use inductive or deductive reasoning

Montessori Lessons:Sensorial Explorations of Key Mathematical Concepts followed by Teacher Presentation of Vocabulary, Formulas, and Problem Solving Steps





b. Make or test generalizations






c. Support or refute mathematical statements or solutions

Montessori Lessons:Use of Montessori Manipulative Materials to Provide Control of Error; Use of Rubrics







Grade 4 Grade 5 Grade 6d. Use methods of proof, ie. direct, indirect, paragraph, or contradiction

Montessori Lessons:Use of Montessori Manipulative Materials to Provide Control of Error; writing in Math Journals, Small Group Discussions, Student Created Models; Use of Rubrics

d. Use methods of proof, i.e., direct, indirect, paragraph, or contradiction


d. Use methods of proof, i.e., direct, indirect, paragraph, or contradiction




Grade 4 Grade 5 Grade 6Topic C. Communications Topic C. Communications Topic C. Communications7.C.1. Present mathematical ideas using words, symbols, visual displays, or technology


7.C.1. Present mathematical ideas using words, symbols, visual displays, or technology


7.C.1. Present mathematical ideas using words, symbols, visual displays, or technology


a. Use multiple representations to express concepts or solutions

Montessori Lessons:Independence in Solving Story Problems Lesson (See Montessori Math Problem Solving Sequence); Small Group Discussions; Story Problem Cards

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

b. Express mathematical ideas orally

Montessori Lessons:Small Group Discussions; Mentoring in the Classroom; Math Night Presentations in which students give lessons to parents





c. Explain mathematically ideas in written form

Montessori Lessons:Modeling the Problem; Symbolizing the Problem; Story Problem Cards

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



Grade 4 Grade 5 Grade 6Topic C. Communications Topic C. Communications Topic C. Communications

d. Express solutions using concrete materials


POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



Grade 4 Grade 5 Grade 6Topic C. Communications Topic C. Communications Topic C. Communications

e. Express solutions using pictorial, tabular, graphical, or algebraic methods


POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

f. Explain solutions in written form

Montessori Lessons:Independence in Solving Story Problems Lesson; Writing in Problem Solving (See Montessori Math Problem Solving Sequence); Math Journals; Story Problem Cards

POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40



POW (06-07): Weeks 1-40

g. Ask questions about mathematical ideas or problems

Montessori Lessons:Individual and Small Group Lessons across the Mathematics Curriculum; Writing in Math Journals





h. Give or use feedback to revise mathematical thinking

Montessori Lessons:Individual and Small Group Lessons across the Mathematics Curriculum; Writing in






Math Journals Math Journals Math Journals



Grade 4 Grade 5 Grade 6Topic 7.D. Connections Topic 7.D. Connections Topic 7.D. Connections7.D.1. Relate or apply mathematics within the discipline, to other disciplines, and to life


7.D.1. Relate or apply mathematics within the discipline, to other disciplines, and to life


7.D.1. Relate or apply mathematics within the discipline, to other disciplines, and to life


a. Identify mathematical concepts in relationship to other mathematical concepts

Montessori Lessons:Activate Prior Knowledge, Individual and Small Group Lessons across the Mathematics Curriculum; Writing in Math Journals





b. Identify mathematical concepts in relationship to other disciplines

Montessori Lessons: Cross-curricular lessons in Science (data collection, data representation, data analysis); Social Studies (BC/CE Timeline, Latitude and Longitude), Language Arts (Understanding Graphs in Non-fiction Text), Art (Shape, Line, and Pattern); Music (Rhythm) , etc.







Grade 4 Grade 5 Grade 6c. Identify mathematical concepts in relationship to life

Montessori Lessons:Solving Real-world Problems; Creating Surveys and Displaying Data; Science Fair Projects; Cross-Curricular Lessons; Student Created Story Problems

c. Identify mathematical concepts in relationship to life


c. Identify mathematical concepts in relationship to life




Grade 4 Grade 5 Grade 6Topic 7.D. Connections Topic 7.D. Connections Topic 7.D. Connectionsd. Use the relationship among mathematical concepts to learn other mathematical concepts

Montessori Lessons:Activate Prior Knowledge, Individual and Small Group Lessons across the Mathematics Curriculum

d. Use the relationship among mathematical concepts to learn other mathematical concepts


d. Use the relationship among mathematical concepts to learn other mathematical concepts



Montessori Great Lessons for Math

The History of NumbersWhat Is a Number?From the earliest of times people have asked the question, “How many?” Long ago, this question was answered with a simple comparison such as, “as many as those trees on the hill”or“ as many as these stones in the bag.” The most convenient and ever-accessible reference was one’s fingers. “This many” (with two fingers raised) was a readily available way to communicate precisely. Over time, words came to be associated with raised fingers meant to convey how many. The word for one is remarkably similar in many languages: one (English), un (French), uno (Spanish), ein (German), unus, (Latin), monos (Greek). All spring from an ancient common root and share the sound of “n.” These words for telling how many are called numbers.More Names for Larger QuantitiesAs the need to count larger quantities arose, new names were invented. The word eleven came from an old English word meaning “one left.” If you counted the first ten on your fingers, there was still one left. Twelve, similarly, came from an old word meaning “two left.” Thirteen was literally, “three and ten.” This pattern continued until twenty, which originated in an old English word meaning “two tens.” The rest of the tens were counted in this way. When ten tens were reached, so was the need to think of a new name to call them. Hundred came from a very old English word, which is no longer used. The last of the ancient number names in our English language is thousand, since there was seldom a need to discuss numbers larger than this.Recording Numbers – The Invention of NumeralsWhen the need arose to record numbers, people used inventions like knots on a piece of rope or slashes made by a knife on a branch. In ancient civilizations all over the world, the number one came to look like a line, much like one raised finger. Before there were ever alphabets, people recorded numbers. In ancient Egypt, while collecting revenue for the Pharaoh, the tax collectors recorded a mark for each bushel of grain brought by a farmer. They made these easier to read by spacing them in different ways:

|= 1 ||| = 3 ||| ||||| = 5 |||||| = 9When they reached 10, they used a symbol like an inverted U. For one hundred, they wrote a scroll ~.They put these symbols together to make larger numbers. 567 would have been written~~~~~ ∩∩∩∩∩∩ |||||||

Every time the Egyptians came to ten of one symbol, they would make up another symbol. These symbols were strung together, so that, even though they might look very long, any number could be written.

Around this same time the Sumerian civilization came up with a number system based on 60. The Babylonians were renowned ancient astronomers. They studied the stars and based their calculations on 360. Our measurements of time to this day are related to their discoveries and their numbers. The ancient Mayans in South America and the Chinese in Asia also developed great civilizations, languages, and number systems. The problem with most ancient number systems was in having to memorize so many new symbols.Letters As Numbers


The Phoenicians were skillful traders who lived on the coast of the Mediterranean Sea. Their need to keep careful records motivated them to invent the first alphabet. The Greeks adopted their alphabet and soon became the most educated civilization of their time. To avoid the problem of memorizing more symbols to mean numbers, the Greeks and also the Hebrews, used their alphabet letters to mean numbers, too. If we used our alphabet this way, it would look like this:A = 1, B = 2, C = 3, D = 4, E = 5, F = 6, G = 7, H = 8, I = 9J = 10, K = 20, L = 30, M = 40, N = 50, O = 60, P = 70, Q = 80, R = 90S = 100, T = 200, U = 300, V = 400, W = 500, X = 600, Y = 700, Z = 800

We would have to make up a new symbol for 900, but otherwise, we could write any number we wanted with our alphabet. 134 would be written SLD.

When writing thousands, the Greeks put a line over the letter to mean thousand (e.g., 4,692 would read DXRB. This was certainly very ingenious for recording, but it must have taken real skill to add and subtract!

The Romans decided on another system for counting and recording numbers. Like the Greeks, they used their alphabet letters for numbers, but they only used a few of their letters. To avoid the Egyptian problem of very long numbers, the Romans changed the letter every time they get to five instead of ten. These are the Roman numerals:I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000

A line over any letter multiplied its value by 1,000. To compare Roman numerals to the Egyptian system, 567 would be written DLXVII, 6 digits instead of 18. In later years, the Romans developed a technique to make their numerals even more concise, but they were certainly a challenge when it came to dividing!


The Invention of Place Value

It was the ancient Hindus who invented the number system which we still use today. They designed nine symbols which have evolved into the numerals we know so well: 1, 2, 3, 4, 5, 6, 7,8, 9.

The way they looked at it, the symbol 7 meant 7 of something, whether it was 7 units, 7 hundreds, or 7 millions. So they designed a system of places to put the 7 where it would mean different things. In the first place, it would mean 7 units – 7. In the second place, it would mean 7 tens – 73. In the third place, it would means 7 hundreds – 755, and so forth, increasing the value of each place to the left by ten. This worked very well for a long time, but gradually, a problem arose. In large numbers, how would you tell what place the 7 was in? 7 3 Did this mean seventy-three? Or seven hundred and three? Or seven thousand and three? The Hindus eventually added a dot to show the empty places 7..3, for seven thousand three. They called the dot “sunya.”

Around the year 800 AD, the Hindu number system spread north to the lands then occupied by people who spoke Arabic. Their civilization at that time covered all of North Africa and spread east to northwest India and north to include Spain. An Arabic mathematician named Al-Khwarizmi (al-kwar-iz-mee) wrote a book explaining this new number system in 820 A D. In 967 AD, while Europe was in the period called “the Dark Ages,” a Frenchman named Gerbert traveled to Spain and learned of this new system. He brought back his knowledge and tried to convince European mathematicians of the value of these “Arabic Numerals.” Even when he became Pope Sylvester II in 999 AD, European traders preferred to stick with the traditional Roman numerals. After the Dark Ages had passed and Europe entered a much more prosperous era, Leonardo Fibonacci, a mathematician from Pisa, Italy, visited North Africa. Here he learned of their number system and published a book about it in the year 1202 AD. Italian merchants embraced the simplicity of this system for calculation and called it “sunya” by the Arabic name for nothing - “sifr.” They soon changed its pronunciation to “zepiro.” You can easily see how this evolved into our “zero.” The dots used in the Hindu system for sunya grew into circles and eventually became the same size as the other numerals. By the time Columbus discovered America, all of Europe was using the “Arabic” numerals. Today they are in use all over the world.

The information for this write-up was taken from Isaac Asimov’s book, How Did We Find Out About Numbers? It is part of the Asimov Science Library, copyright 1973, Walker and Company, New York. Other interesting information about numbers and place value can be found in the World Book Encyclopedia.


Introduction to Geometry

Geometry has its roots in the ancient world. We have several stories to illustrate the origins of this mathematical discipline. Below, we have included four stories: two Greek, one Egyptian, and the story of Pythagoras who traveled between the two ancient countries.

1.) Euclid, the Father of Geometry

One of the most influential mathematicians of ancient Greece, Euclid, was born around 365 BC. Very little is known about his life. It is believed that he was educated at Plato's academy in Athens, Greece. His 13 books, the Elements, are some of the most famous books in the world. He wrote them at about 300 B.C. Euclid started a mathematical school in Alexandria, Egypt. (One of his students was Archimedes.)

Euclid created the geometry called Euclidean Geometry. He collected all the geometric facts known at his time, added his own, and included proofs for each fact. He arranged this information in 13 books, called the Elements. This 13 volume series eventually became the most influential geometry textbook.

Quite possibly the most important book of the Elements is the first book. It has the definitions of: points, lines, planes, angles, circles, triangle, quadrilaterals, and parallel lines. Some of the facts in this book are:

Two points determine a line. A line segment extended infinitely in both directions produces a straight line. A circle is determined by a center and a distance. All right angles are equal to one another. If a straight line falling on two straight lines forms interior angles on the same side less than 180°, the two straight lines, if produced indefinitely, will meet

on that side.

For his work in the field of geometry he is known as The Father of Geometry.

2.) The Origin of the Constructive TrianglesLife in ancient Egypt centered on the Nile River and the mineral rich soil which provided fertile ground for agriculture. During the spring season each year, the valley of Egypt was subject to periods of great rainfall which lead to flooding. When the Nile overflowed its banks each year, there was a covering of rich black earth left behind on the land when the waters receded. This earth covered over the landmarks that showed the borders of people’s property. Each year, neighbors had to remark their property. The Egyptians had to develop a fair way to re-measure and ensure that each family had their assigned land to cultivate and farm. They used geometry.


Ancient Egyptian surveyors used ropes to measure and mark the boundaries. These surveyors were called “harpedonapta.” Harpedonapta means “rope-stretcher.” Surveys were made by groups of three slaves holding a rope with knots in a ratio of 3 to 4 to 5. Each slave stood by a knot, pulled, and held it taut so that a right-angled triangle was formed. The right-angle of the triangle was then flipped while the other angles were held in place. This formed a second adjacent triangle and allowed the surveyors to mark off a perfectly rectangular field. Longer ropes created more and more accurate angles and polygons. This same method of rope stretching was used to mark the foundation for the base of buildings and even pyramids. The ancient Egyptians had discovered the strength of the mighty triangle.

3.) Story of PythagorasThere once lived a very wise man whose name was Pythagoras. Pythagoras was born on the island of Samos in Greece, around 569 BC. Not much is known of his early years. He traveled through Egypt learning, among other things, mathematics. It seems he got caught up in a war there and was kept as a slave in Babylon for a time. While there, he learned secrets from the Babylonian priests about the stars and how they moved in the sky.

Pythagoras formed a group, the Brotherhood of Pythagoreans, a secret religious society devoted to the study of mathematics. He was also mystic – that’s like a magician. People said Pythagoras had extraordinary powers of the mind. He believed that numbers ruled the universe, and the Pythagoreans gave numerical values to many objects and ideas. He found out about the number relationships in right-angled triangles.

In Egypt, he learned about the “harpenodapta.” He was fascinated by their secret of the three numbers making a right angled triangle. So later, with his brotherhood, he studied the right-angled triangle. Pythagoras’ most famous discovery was about triangles. He found the theorem which today is still called the “Pythagorean Theorem.” It reveals the number relationships of the sides of a right-angled triangle: The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse.

If we take an isosceles right triangle with legs of measure 1, the hypotenuse will measure square root 2. But this number cannot be expressed as a length that can be measured with a ruler and that deeply disturbed the Pythagoreans, who believed that "All is number." They called these numbers "alogon," which means "unutterable."

4.) Calculating Volume - The Story of ArchimedesTo find the volume of a sphere is a problem. Not only do the sides not rise up perpendicularly, but there is no base, either. Many thousands of years ago, a man named Archimedes was commissioned with the same problem. You may remember Euclid, the mathematician we call the Father of Geometry. Archimedes was his student.

Our friend, Archimedes (third century BC), experimented with spheres and with irregular solids. One day, the king commissioned his help with a problem: the king had a goldsmith make a crown with a lump of gold the king had supplied. However, the king was not sure whether the goldsmith had cheated him and not used all the gold for the crown; he thought he might have kept some for himself.

Archimedes weighed the crown. It was the same weight as the gold. But, if the goldsmith had replaced some of the gold with a lighter metal, it would weigh the same, but would be larger. How could the volume of the crown be measured?

Archimedes was really plagued by this problem. He was so exhausted; he decided to take a bath. When he climbed into the tub, he found the answer he’d been looking for. He saw the level of the water in the tub rise as his body sank into the water. He knew how to measure the volume of the crown!


He was so excited that he jumped out of the tub, ran straight to the king (naked as he was, through the streets) yelling “Eureka!” which means “I found it!” Well, this is how much a man can get excited about measuring volume. His idea was to dip the solid crown into water and measure how much water would be displaced. We can do this too, to measure the volume of our sphere.


Money

Introduction to the Study of Money

The study of money provides a wonderful opportunity to teach the Montessori curriculum in the true cosmic sense. It can be integrated with the study of History, Geography, Language and much more. For example, children can investigate the metals and alloys coins are made of, or they can research the origins of paper money. Older children might explore the architectural styles of the buildings or the significance of the people on paper money. Also, the study of money can be used to introduce new vocabulary words such as “barter” and “self-sufficient.” There are many possibilities. The study of money should start with a story the history of money, thus opening up many related cultural activities to the exploration of the child. Following those activities, others such as:

• Introducing nomenclature through matching coins with names• Nomenclature cards• Naming monetary values ( for quantities under $ 1.00) can be introduced

For further work, the child should be familiar with linear counting, skip counting, fractions and decimals.Word problems should be an integral part of the study of money.

The Story of Money

We have heard about the Phoenicians and their trading and selling throughout the ancient Mediterranean in our story “Communication in Signs.” We have heard about the way in which the Babylonians recorded business transactions using cuneiform writing. Today, I would like to focus on how another great tool was developed to make trade between people throughout the world possible. This great tool is called Money.

We know that the first people on earth lived in very small, nomadic tribes. Everyone needed to find the things which were needed to survive. They traveled about their homeland looking for food and avoiding danger. Life was very hard and people spent all their time meeting their most basic needs (finding food and shelter, and later clothing and means of defense.) If there was no food in one place, people had to leave there and go to somewhere their needs could be met. If a larger group of people wanted to hunt in their homeland, a small tribe would have to move on.

As generations passed, some peoples began to establish seasonal hunting routes. Every spring they might travel to the ocean or a bay and every winter they might return to their caves in the mountain foothills. On their annual circuits they traveled near the homelands of other groups. People began to see that some tribes were expert in crafts or types of hunting unfamiliar to their own tribe. They began to tradewith these new peoples. That means that people in one tribe would give one thing (such as dried fish) for another (such as a storage pot). Over the years, people began to look for certain tribes to get their special crafts and goods. In some places various tribes began to meet where rivers flowed together to trade goods.

More time passed and people’s lives began to change. They did less hunting and gathering and they did more farming and raised more livestock. People began to gather in settlements and traveled less. They traded more. As more trading was done, more complex trading problems occurred. If one person had a large water jug and another had wheat, how much wheat should be given for the jug? How many pigs should be given for a cow? How stone or iron tools should be given for a horse? People began to have trouble agreeing on fair exchanges.


In order to make sure that everyone had a thing that was considered valuable enough for trading, people began to carry salt along for trading days. If someone had an object you wanted but they didn’t need the trade item you made, grew, or caught, you could offer them salt instead. Everyone wanted salt. They used it to preserve food, and it added taste to bland foods. Barley was also an item that people used for trading. This was true in ancient Sumeria. You can imagine the problems people had with these two trade items. If you got caught in a rain storm on your way to a trading meeting, your salt would be dissolved. If your barley got wet, it would spoil. The Sumerians thought about this problem and one day, one of them thought of a new way of trading. We don’t know his or her name but we do know that person was wise, and I wouldn’t be surprised if this inventor was a merchant also.

The wise merchant thought, "I need something that everyone wants, that is easy to carry, and which won't be destroyed. Barley just won't do!" The merchant looked around and saw many people wearing jewelry in the market on Trading Day. Some of the jewelry was made of a shiny metal which we now call silver. Not many people wore it, even though it was greatly admired because it was rare. This merchant decided that he would begin to offer people silver for things that he wanted and gradually people in his town became used to the idea. They wanted silver nuggets to make into jewelry. They wore silver necklaces, drank from silver cups, and used silver mirrors. And above all its uses, everyone agreed silver was beautiful. It glittered in the light, and even in a shady place it glowed like the moon.

The people began to agree that certain amounts of silver were worth goods and services. They also began to use silver and another rare and beautiful metal, gold, for special uses, such as tribute to a king or warlord, to give as a bride's dowry, or blood money (to compensate a family for the death of a member through accident or in battle). Precious metals were a part of business.

Of course all silver and gold pieces were not the same. Some were full of holes and some were solid lumps. In order to make sure the same amount of silver and gold were paid for similar goods and services people began to weigh gold and silver right in the market to make sure they were being treated fairly and not being given a lot of hollow lumps.

Using silver and gold in business was a great advance for people. If you had a herd of cattle and you needed to go on a long trip, you could sell your cattle for gold. Then you could take a bag of gold instead of your herd with you on the trip and still have the value of a large herd to buy goods and services. With all of the people buying and selling goods in the marketplace, weighing gold and silver before each transaction was inconvenient. Also, some merchants tampered with their scales in order to cheat their customers. A very wealthy king solved this problem for us. His name was Croesus. He ruled in the kingdom of Lydia, which is now part of Turkey in the sixth century BC. When he came to power the people of Lydia used a naturally occurring mixture of gold and silver for trading. This mixture was called electrum. Croesus decided that these lumps of metal were not good enough tools for trade. People had to weigh each lump in order to insure that they were getting the right amount of electrum for their goods. Some people complained of being cheated by merchants who used incorrectly calibrated scales.

Croesus decided that the government would take over the job of weighing gold and silver, and to show its weight and value his officials would stamp the metal with the king's special design (a lion attacking an ox). The stamp guaranteed that the coin was weighed accurately and could be used for payment without weighing. These new stamped, weighed coins were called “staters.” King Croesus not only weighed the new money, but he also made it. He had complete control of this money. He never forgot to tell his officials that while they were minting new coins they should make some more for him. He was fabulously wealthy. That is why today you might hear someone say, "He's as rich as Croesus."

For over a thousand years people used money that was gold or silver or copper. Over time paper money was developed. In western countries this paper money could be redeemed for gold or silver, like a certificate. But in China, in the 600's AD, the Chinese emperor issued paper money of which he guaranteed the value. Eventually as gold and silver became more and more rare western countries began to accept the Chinese idea that the government makes, controls and guarantees the value of its money.


Today, our money in this country is like the money of 7th century China in that it is made, controlled and guaranteed by our government. We have paper money and coins, and though they are still made of metal like the stater, they are not made of silver and gold but of durable alloys which will not bend or lose their imprint in use. Because of the growing tolerance and acceptance of other people in Prehistoric times, and the recognition of the great skills that other people have, trading began between peoples. Because of the discovery of the wise merchant who thought of trading for precious metals, the use of coins eventually came to be. Because of King Croesus, we have standardized money which inspires confidence in the people who use it. Because of the understanding of the Chinese Emperor, we have modern money which symbolizes the prosperity of our country. Whenever we use money, we should think with gratitude of all these people.


History of Measurement

Measurement is fundamental and we can’t date when it began. We can probably speculate that ancient people measured things with a stick. Measurement became essential in trade and barter. Some ancient civilizations developed measurements based on units that were the length of certain parts of the body. About 3,000 BC in Egypt, the cubit (the length of a man’s elbow to the tip of his middle finger when his arm was outstretched) was the measurement. In Latin, elbow is “cubitim.” This measurement varied too much, so people went by the royal cubit (the pharaoh’s cubit.) This wasn’t without problems either, because the pharaoh changed. Lengths to be measured varied, too, so there was another unit called “palm,” the distance across the hand. A horse is measured in hands. About 7 palms were equal to a cubit. A finger was even a smaller measurement.

The Hebrews adopted the Egyptian system when they were in Egypt and they took it back with them to their land and added some things. A span was the distance from the outstretched thumb to the little finger. A pace was 2 cubits (which is about 1 yard in English.) A royal foot was 18 fingers and was about two-thirds of the royal cubit. These measurements passed around the world .The Greeks adapted this and added some measurements that were multiples of the fingers. The Romans adopted many of these measurements and added the “uncia” (width of the thumb.) The English word “inch” is derived from this. Twelve unicia are equal to 1 foot. Three feet are equal to 1 yard (the distance from a person’s nose to the top of the outstretched hand.)

The Roman measurements were adopted in Europe during the Middle Ages. In 1855, the yard was standardized in Europe. The yard became the standard measurement. The inch is 1/36th of a yard. A foot is 1/3 of a yard. A rod is 5 ½ yards. A furlong is 220 yards. A mile is 1,760 yards. A rod comes from when tax collectors measured land with a rod, so the instrument used to measure the land became the name of the measurement. The furlong became the standard length to plow a furrow.

British tradition said that one inch was equal to 3 rounds, dry barley corns. The foot is the length of the Emperor Charlemagne’s foot. The stick that was 12 inches long and represented Charlemagne’s foot became the “ruler.” The Romans used a length that was 5,000 feet, called a “milliare.” It was 1,000 paces of the Roman army at a forced march. This distance evolved into what we call the “mile.” The distance of this mile was law. A law can also be called a statute. So, the standard mile was called a statute mile. There is also a nautical mile which is 6,076.1 feet.

Since the 1600’s, people talked of standardizing measurement. In 1790, French scientists developed a system that was never to change. In 1795, the system was in place but its use wasn’t required until 1840. This system used the unit of measure called a meter, which was based on a measurement of the earth. The meter is a fraction of the earth’s surface –one ten-millionth of the distance from the North Pole to the equator along the longitude that happened to fall near Dunkerque, France. The name “meter” was chosen from the Greek word “metron” which means measure. Once the meter was standardized other measurements were derived from it. A decimeter is the meter divided into 10 equal parts. The centimeter is the meter divided into 100 parts (“centi” means 100.) The millimeter is the meter divided into 1,000 parts. Everything else was derived from the meter, including weight and mass. The liter is defined as the volume of a cubic decimeter A gram was the mass of a cubic centimeter of water at a temperature when it weighs the most (about 40 degrees Centigrade.) These standards have changed and are now very precise. A meter is now defined by a wavelength of light, undisturbed by temperature, gravity, or pressure. The hope of the French was to unify the world by using this metric system of measurement. The United States is the only major country not using the metric system of measurement. (We continue to use the Standard system.)


Montessori Sample Lesson Plan

Lesson Title: Addition and Subtraction of Whole Numbers (VSC Indicator: 4.6.C.1.a and b) Age: 7-9 Years

Montessori Materials: Long Bead Frame, Long Bead Frame Paper, Colored Pencils

ßStatement of Objective: What should students know and do as a result of the lesson?

ßKey Vocabulary: What critical vocabulary will be part of this lesson?

ßWarm-Up: How will you engage students in learning? How will you connect the lesson to their prior knowledge?

Access Prior Knowledge(Engagement)

Students will use Montessori manipulative material in order to add or subtract whole numbers.

AddendSumSubtrahendMinuendDifference

The teacher will question students on their previous work with the small/ large bead frame. The teacher will review the hierarchy of numbers from units to millions. If necessary, have students count and exchange beads. Remind students that adding means “putting together” and subtracting means “taking away.” Tell students which operation you will be working on today.

Teacher Directed Activities: How will you aid students in constructing meaning of new concepts? How will you introduce/model new skills or procedures?

Teacher-Monitored Activities: What will students do together to use new concepts or skills? How will you assist students in this process?

FirstPeriod (Explanation)

Introductory and/or Developmental Activities

The teacher will model writing an addition or subtraction problem on the bead frame paper, and invite students to write the problem on their paper. The teacher will guide the students to compose the first addend or the subtrahend on the bead frame. Next, guide student in adding the second addend or taking away the minuend, exchange (re-group) beads when necessary. The teacher will model noting the sum on the bead frame paper and show regrouping marks with colored pencils. Students count the beads to check the sum or difference written on their paper.

Second Period(Exploration)

Guided Practice

The students will repeat the process of adding or subtracting on the bead frame and continue recording on paper whenever they exchange beads. The students will calculate sums and differences and check their work by counting the beads. The teacher will observe and assist, sending students off for independent, Third Period work as soon as each one is ready.


Extension, Refinement, and Practice Activities: What opportunities will students have to use the new skills and concepts in a meaningful way? How will students expand and solidify their understanding of the concept and apply it to a real-world situation? How will students demonstrate their mastery of the essential learning outcomes?

ThirdPeriod

(Elaboration)

Independent Activities and/or Meaningful-Use Tasks

Students will complete a series of addition or subtraction problems using the bead frame and record the work in his/her math journal.Students will work with a partner to create and solve problems using the bead frame.Students will complete story problem cards using the bead frame to compute addition and subtraction.

ßOngoing Assessment: How will you monitor student progress throughout the lesson?

Check for Mastery(Teacher Evaluation)

Assessment

The teacher will provide opportunities for students to demonstrate their knowledge during the second period of the lesson, and will provide additional support for below level students prior to them engaging in the Third Period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning?The teacher will ask the child to demonstrate solving an addition or subtraction problem on the long bead frame and recording correctly on paper. The teacher may model an appropriate BCR or SR during this time using the language addend and sum or minuend, subtrahend, and difference in the written portion of the problem. The teacher may then assign students to complete another BCR in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?

Follow-Up Work(Student Evaluation)

During the Montessori un-interrupted work period, individuals and small groups of students will select addition and subtraction problems sets from a variety of resources to complete using the bead frame and the paper and pencil recording method taught. Students will write in their math journals to explain the process of addition and subtraction using key vocabulary from the lesson.

ßExtension Activities: How will you differentiate follow-up lessons for students who do not master the initial concept? What follow up lessons will you offer to students once they have mastered the concepts taught?

Extending the Lesson

The teacher will observe and re-teach students who have not mastered the concepts and draw attention to points of interest which address an individual’s weakness in understanding. Once students have mastered the concept, the teacher may extend the lesson by showing properties of addition, increasing the number of addends, asking the students to complete the problems in the abstract, and/or encouraging students to generate their own addition and subtraction problems.



Lesson Title: Multiplication of Whole Numbers – VSC Indicator 4.6.C.1.c Age: 8-9 Years

Montessori Materials: Montessori Checkerboard, Gray and White Tiles, and Bead Bars





Students will use Montessori manipulative material in order to multiply whole numbers

MultiplicandMultiplierProduct

Teacher will set out and explain the checkerboard material to the students by directing student’s attention to the numbers on the vertical and horizontal edges of the board. Remind the student of their previous work with the large/small bead frame. The teacher will point to the numbers and direct the students to read the place values listed (vertical edge: units/ones, tens, hundreds, thousand; horizontal edge: units/ones, tens, hundreds, thousands, ten thousands, hundred thousand, millions). Teacher will review student’s knowledge of place value by removing a bead bar from the box of beads and placing it on a colored square within the center of the board. Teacher will direct the student to state the value of the bead when placed in a colored box. (Ex. 4 bar placed in the units green square-student reads as four units, 4 bar placed on the blue tens square-student reads as forty, 5 bar place on red hundred square-student reads as five hundred, etc.) Tell students that we will be using the checkerboard for multiplication. Elicit from the students that multiplication means putting together the same quantity over and over again.





Teacher will write out the multiplication problem to be solved on the checkerboard (ex. 734 x 4) and ask the students to copy the problem into their math journals. The gray multiplier tile with the number 4 is Second Period

Guided Practice

The students will go through the process of multiplying on the checkerboard and continue recording on paper whenever they exchange beads. The students will


placed on the ones category on the gray vertical edge and the white multiplicand tiles (7-3-4) are placed on the horizontal edge in the respective categories 7-on the hundreds, 3 on the tens, 4 on the ones. Teacher will model laying out bead bars in the correct squares on the checkerboard to represent the quantity 734 made 4 times. Teacher will discuss and guide the student to remove, place, count, and exchange the appropriate number of bead bars in order to solve the multiplication problem. The teacher will model the process of regrouping and exchanging as necessary, reminding students to record the exchange on their papers. After the multiplication is complete, the teacher will guide the students in reading the value of the product using the bottom hierarchies as a guide. The teacher will repeat the process asking for increased student participation.

(Exploration) record products and check their work by counting the beads. The teacher will observe and assist, sending students off for independent, Third Period work as soon as each one is ready.


ThirdPeriod

(Elaboration)


Students will complete a series of multiplication problem using the checkerboard and record the work in his/her math journal.Students will work with a partner to create and solve problems using the checkerboard.Students will complete story problem cards using the checkerboard to compute multiplication.



AssessmentThe teacher will provide opportunities for students to demonstrate their knowledge during the second period of the lesson, and will provide additional support for below level students prior to them engaging in the Third Period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How


will you assess their learning?The teacher will ask the child to demonstrate solving a multiplication problem on the checkerboard. The teacher may model an appropriate BCR or SR using the language multiplicand, multiplier, and product in the written portion of the problem. The teacher may then assign students to complete another SR or BCR in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?


During the Montessori un-interrupted work period, individuals and small groups of students will select multiplication problem sets from a variety of resources to complete using method taught. Students will write in their math journals to explain the process of multiplication using key vocabulary from the lesson.


Extending the LessonThe teacher will observe and re-teach students who have not mastered the concepts and will draw attention to points of interest which address an individual’s weakness in understanding. Once students have mastered the concept, the teacher may extend the lesson by showing properties of multiplication, asking the students to complete the problems in the abstract, and/or encouraging students to generate their own multiplication problems.



Lesson Title: Dividing Whole Numbers -VSC Indicator 4.6.C.1.dAge: 6-10Montessori Materials: Racks and Tubes





Students will use Montessori manipulative material in order to divide whole numbers and to interpret quotients and remainders in the context of a problem.

Dividend, divisor, quotient,

Explain the manipulative material to the students. Review the concept of place value with the students. Review with students the hierarchy of numbers. Remind students of their previous work with the golden bead material.





Explain the concept of dividing as a sharing. Write a three digit division problem with a one digit divisor (ex. 963 ÷ 3). Guide the students in determining which set of division boards to use and the amount of beads to place in the corresponding cups (3-units/ones, 6 tens, 9 hundreds). Identify the divisor and remove the corresponding number of skittles. Explain that in division, we begin sharing with the highest category. Guide


Guided Practice

Teacher will repeat with a variety of problems. Remain with student and guide student in completing problems. Ask a variety of questions to gage students’ understanding of the process and correct use of manipulative.


the students in distributing the beads from the hundreds cup evenly across the board. Ask the student, “How many beads did each skittle receive?” Record student’s correct response on the paper. Repeat process with the beads in the tens container and units/ones container. Exchange/ regroup beads when needed.


ThirdPeriod

(Elaboration)


Students will complete a series of problems using the racks and tubes material. Students will work with a partner to create and solve additional problems.



Assessment

Teachers will provide opportunities for students to demonstrate their knowledge during the second period lesson and will provide additional support for below level students prior to them engaging in the third period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning?Teacher will ask the child to demonstrate how to complete a division problem. The teacher may model an appropriate BCR or SR. Teacher may assign student to complete another BCR or SR independently in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?


During the Montessori uninterrupted work period individuals and small groups of students will select Albanesi cards and continue using racks and tubes. Student will write in his/her math journal using key vocabulary from the lesson.



Extending the LessonThe teacher will observe and re-teach students who have not mastered the concepts and will draw attention to points of interest which address an individual’s weakness and understanding. Student will create division word problems and solve using racks and tubes.



Lesson Title: Addition and Subtraction of Fractions and Mixed-Numbers with Like Denominators– VSC Indicator 4.6.C.1.eAge: 9-10 yearsMontessori Materials: Fraction Inset Materials, Box of Red Plastic Fractions





Add and subtract proper fractions and mixed numbers with like denominators.

FractionMixed Number NumeratorDenominatorAdditionSubtraction

The teacher will gather a group of students and let them know they will be continuing their work with fractions. The teacher will review the concept that a fraction is a part of a unit and is smaller than one whole and review other vocabulary. Students take turns setting up fraction materials from the box and labeling them with correct numerators and denominators.





The teacher will let the students know that we can perform the same mathematical operations with fractions that we perform with whole numbers. Write an addition problem for fractions such as 1/5 + 3/5. The students will copy the problem into their math journals. The teacher will model “putting together” the two addends to create a new fraction: 4/5. The students will record the answer in their math journals.


Guided Practice

The students will repeat the process of adding or subtracting with the fraction materials and recording the problems and answers. The teacher will observe and assist, drawing attention to how the numerator changes but the denominator stays they same. The teacher will send students off for independent, Third Period work as soon as each one is ready.



ThirdPeriod

(Elaboration)


Students will complete a series of addition or subtraction problems using the fraction materials and will record the work in their math journals.Students will work with a partner to create and solve problems using the fraction materials.Students will complete story problem cards using the fraction materials to compute addition and subtraction.



Assessment

The teacher will provide opportunities for students to demonstrate their knowledge during the second period of the lesson, and will provide additional support for below level students prior to them engaging in the Third Period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning?The teacher will ask the child to demonstrate solving an addition or subtraction problem with fractions and recording correctly on paper. The teacher may model an appropriate BCR or SR during this time using the language addend and sum or minuend, subtrahend, and difference in the written portion of the problem. The teacher may then assign students to complete another BCR in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?


During the Montessori un-interrupted work period, individuals and small groups of students will select fractions addition and subtraction problem sets from a variety of resources. Students will write in their math journals to explain the process of addition and subtraction with fractions using key vocabulary such as numerator and denominator.



The teacher will observe and re-teach students who have not mastered the concepts and draw attention to points of interest which address an individual’s weakness in understanding. Once students have mastered the concept, the teacher may extend the lesson by showing properties of increasing the number of addends, asking the students to complete the problems in the abstract, and/or encouraging students to generate their own addition and subtraction problems for fractions. The next lessons in this series involve finding equivalent fractions, the least common denominator, and adding and subtracting fractions with unlike denominators.



Lesson Title: Fractions and Percents – VSC Indicator 6.6.A.1.c Age: 11-12

Montessori Materials: Montessori Centesimal Protractor (aka Black Percentage Square), Red Plastic Fractions





Identify and determine equivalent forms of fractions and percents

FractionNumeratorDenominatorEquivalentHundredthPercent

Review the red fraction materials with the students and have them practice making some equivalent fractions such as 1/4 = 2/8 and 1/2 = 5/10 and record the equivalencies in their math journals. Elicit form the students that even though the fractions look different they are “equivalent.” Explain that today we will look at “percent,” another way to make equivalent fractions. Have students break apart the word “percent” to discover it has something to do with hundredths.





Take a red plastic tenth and ask the students to name it. Elicit from the students that a tenth can be divided into smaller parts such as twentieths by breaking it in half. Using scissors, cut a tenth of a tenth. Elicit from the students that it is a hundredth. Place it into the black percentage square to show that 1/100 = 1%. Show the students that a whole is equal to 100%. Percent is another way of talking about a fraction with a denominator of 100.


Guided Practice

Have the students take turns choosing fractions such as 3/4 and 2/10, placing them into the percentage square, and reading the measurements: 3/4 = 75% and 2/10 = 20%. The teacher will show students how to find equivalent fractions and percents in the abstract by using common denominators when the original fraction has a denominator that is a factor of 100 (1/4 = 25/100 = 25%).



(SEE APPENDIX)ThirdPeriod

(Elaboration)


Students will complete a table in their math journals recording the equivalent forms of fractions and percents through 10/10.

Fraction # / 100 Percent1/2 50/100 50%2/2 100/100 100%1/3 33.3 / 100 33.3%2/3 66.6 / 100 66%3/3 100/100 100%

10/10 100 / 100 100%



Assessment

The teacher will provide opportunities for students to demonstrate their knowledge during the second period of the lesson, and will provide additional support for below level students prior to them engaging in the Third Period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning?The teacher will ask the child to name the percent equivalent to a given numeric fraction. The teacher may model an appropriate BCR or SR during this time. The teacher may then assign students to complete another BCR in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?


During the Montessori un-interrupted work period, individuals and small groups of students will select fraction and percent problems sets from a variety of resources. Students will write in their math journals to explain the process of changing fractions to percents.



The teacher will observe and re-teach students who have not mastered the concepts and draw attention to points of interest which address an individual’s weakness in understanding. Once students have mastered the concept, the teacher may extend the lesson by showing how to use division to find equivalent fractions and percents when the denominator is not a factor of 100. Students can use a calculator and compare results between the calculator and the centesimal protractor.



Lesson Title: Identifying Angles-VSC Indicator 4.2.A.1.a and b Age: 6-7

Montessori Materials: Box of Sticks





Students will use Montessori manipulative material in order to identify, compare, and describe angles in relationship to another angle.

Whole angleRight angleObtuse angleAcute angleStraight angleVerticesRay

Introduce the students to stick box. Show the students the different colored sticks and explain that each stick represents a different length. Model for students how to use the push pins to secure the sticks on the board. Ask students to share the types of lessons they have had using the stick box. State that today they will learn how to make angles using the stick box. Give the child two sticks and a push pin. One stick should be slightly longer than the other. Secure both ends and one end of the top stick to the board so that the top stick will act as a sweep hand. (Refer to diagram in your Montessori album.)





Insert a pencil through the sweep hand and begin to trace a complete 360 degree angle while the child observes. Note that the line traced has nothing to do with the angle. Explain to the child that the space between the two sticks is called an angle. Identify the whole angle is when it goes all the way around. Repeat rotating this time stopping when the sticks form a straight line, explain this is a straight angle. Repeat the exercise as above, this time stopping when the sticks form a right angle. Use the measuring angle from the box and place it in the center of the angle. Explain that because it measures 90 degrees it is called a right angle. Repeat


Guided Practice

Teacher will write out a label for each angle: whole angle, straight angle, right angle, obtuse angle, acute angle. Teacher will guide the students in making the angles with the sticks and labeling them. After all the angles have been made and labeled, teacher will remove the labels and mix them up. Teacher will make angles and have the students label them. Ask the child to make a certain angle and then place the proper label with it. As the child makes each angle, ask the child how he/she knows what the angle is.


the exercise, stopping when the sticks form an acute angle. Use the measuring angle to show that it is less than 90 degrees and explain that it is called an acute angle. Repeat the exercise, this time stopping when the sticks form an obtuse angle. Use the measuring stick to show it is greater than 90 degrees.


ThirdPeriod

(Elaboration)


Students will complete a series of problems using the stick box. Students will work with a partner to create and label angles, students will create booklets showing a variety of acute angles, obtuse angles, and right angles with vertices rotated in different positions.



Assessment

Teachers will provide opportunities for students to demonstrate their knowledge during the second period lessons and will provide additional support for below level students prior to them engaging in the third period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning?Teacher will ask the child to demonstrate how to make angles. The teacher may model an appropriate BCR or SR. Teacher may assign student to complete another BCR or SR in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?


During the Montessori uninterrupted work period individuals and small groups of students will select geometry command cards and continue using the stick box to make angles. Student will write in his/her math journal using key vocabulary from the lesson.



Extending the LessonThe teacher will observe and re-teach students who have not mastered the concepts and will draw attention to points of interest which address an individual’s weakness and understanding. Teacher will gather a group of students to create a yarn web and identify and label angles.


Montessori Lesson Plan

Lesson Title: Identify Polygons in a Composite Figure 5.2.A.1.bAge: 9-12Montessori Materials: Constructive Triangles





Students will use Montessori constructive triangle materials in order to identify polygons in a composite figure.

Similar, congruent, equilateral, scalene, isosceles, obtuse, acute, right , polygons

Teacher will remove the triangles from the triangular box. Direct students to review nomenclature by placing labels next to triangles, such as equilateral triangle, scalene triangle, right angle triangle. Students will identify similar triangles (large gray equilateral and small red equilateral triangle).





Teacher will model for students how to create congruent figures by putting the same colored triangles together to form large equilateral triangles. Teacher will explain to the students that these are called constructive triangles because we can use them to build other polygons. Teacher will model using different combinations of constructive triangles to create new polygons. For example: the large gray equilateral triangle surrounded by three medium obtuse angle isosceles triangles form a large hexagon, two green right angled scalene triangles can come together to form a rectangle or a parallelogram depending on how you place the edges together.


Guided Practice

Teacher will write labels of new vocabulary terms used during the lesson. Teacher will ask the students to place the label on the appropriate item. Once all items are labeled properly, teacher will remove the labels and mix-up the triangles. Teacher will direct the students in re-labeling the items. Teacher will ask students a variety of questions about the triangle in order to gage students’ understanding of information presented. This is repeated until the children are familiar with the terms and their meanings.



ThirdPeriod

(Elaboration)


Student will use other triangle boxes in order to create and identify polygons in a composite figure.



Assessment

Teachers will provide opportunities for students to demonstrate their knowledge during the second period lessons and will provide additional support for below level students prior to them engaging in the third period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning?Teacher will ask the child to identify polygons in a composite figure. the teacher may model an appropriate BCR or SR. Teacher may assign student to complete another BCR or SR in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?


During the Montessori uninterrupted work period individuals and small groups of students will select the constructive triangles and create booklets, charts, and nomenclature cards of polygons in composite figures. Students will write in his/her math journal using key vocabulary from the lesson.


Extending the Lesson The teacher will observe and re-teach students who have not mastered the concepts and will draw attention to points of interest which address an individual’s weakness and understanding.



Lesson Title: Determining Area of a Triangle VSC Indicator 6.3.C.1.a Age: 11-12

Montessori Materials: Yellow Area Material





Students will use Montessori Area Material in order to determine the area of a triangle.

Figure, base, height,congruent

Review and remind students of previous lessons and activities using the yellow area material (area of a rectangle and area of a parallelogram). Elicit from students that a parallelogram can be transformed into a rectangle. Review with students area of a rectangle = b x h, area of a parallelogram is b x h. Preview any new vocabulary words





Remove and identify the isosceles acute triangle. Have students label the base and the height of the triangle. Question students on how to find the area (by making a rectangle). Bring out the divided triangle and show that the two triangles are congruent. Slide the two sections of the divided triangle to form a rectangle with the first triangle. (teacher may review album for pictures) Ask the students to count the units across the base and to count the units across the height to determine the area of the newly formed rectangle. Remove the two sections of the divided triangle and point out that the area of the triangle is half of the area of the rectangle.


Guided Practice

Make labels to place on the triangle. Ask a variety of questions to help the child to verbalize the formula. Move the labels to make the formula (A = ½ bh). Repeat the process using the right angled scalene triangle, and the obtuse angle triangle. Student may transform the material into a rectangle by doubling and count the squares if needed. Substitute the numbers in the appropriate places of the formula and solve. Emphasize that we have to divide (take half of) the rectangle or parallelogram in order to find the area of the triangle.



ThirdPeriod

(Elaboration)


Students will use the geometric cabinet to trace triangles, double them, and find half the area of the rectangle or parallelogram formed. Students will perform operations in the abstract using the formula A=1/2 bh.



Assessment

Teachers will provide opportunities for students to demonstrate their knowledge during the second period lesson and will provide additional support for below level students prior to them engaging in the third period work.ßCulminating Assessment: How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning?Teacher will ask the child to demonstrate concrete and abstract use of formula/ finding area of triangle. The teacher may model an appropriate BCR or SR. Teacher may assign students to complete another BCR or SR in order to demonstrate their mastery.ßFollow-Up Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for the next lesson? What other work will be assigned to help students practice, prepare, or elaborate on a concept or skill taught?


During the Montessori uninterrupted work period individuals and small groups of students will select measurement command cards. Students will use Albanesi cards to complete additional activities. Student will write in his/her math journal using key vocabulary from the lesson.


Extending the LessonThe teacher will observe and re-teach students who have not mastered the concepts and will draw attention to points of interest which address an individua’ls weakness and understanding. Students will create booklets, posters and other presentations of determining area of triangles.


Appendix


History of Mathem

atics More Nam

es for Larger Quantities

Research the names for the periods of num

bers beyond thousand in the World

Book Encyclopedia.W

here did these names com

e from?

What do they m

ean?Does every num

ber have a name?

What is the m

ost recently named num

ber?Has there ever been a child to nam

e a number?

Write a num

ber that begins with decillions. Label its periods.W

rite a number that begins with googol. Label its periods where possible.

Make up your own nam

es for those periods without official names.

Write a num

ber that begins with centillion. Label the periods of numbers.

For those periods which do not have official names, leave zeros or m

ake upyour own nam

es.

History of Mathem

atics - What is a Num

ber?

Design as many m

odels of the answer to this question as you can in the classroom,

“How many years have you lived?”

Answer the following questions using the phrase “as many as” and the objects or

people in the classroom as com

parisons. “How many tim

es did you walk in aline today?” (e.g., “As m

any times as there are girls at that table”)

1. “How many buttons are on your shirt or dress?”

2. “How many barrettes are in your friend’s hair?”

3. “How many days did you com

e to school this week?”4. “How m

any things on your workplan did you do yesterday?”


History of Mathem

atics Recording Numbers – The Invention of Num

erals B

Research the Mayan num

ber system. W

hat was it based on? Why do you think they used

this number as their base?

When President Lincoln began his Gettysburg Address, “Four score and seven years ago...,”

how many years was he referring to?

How many are in the English num

ber “one dozen?” Look up the meaning of the num

ber“gross.” W

hatlanguage does this word com

e from? W

hy do you think a dozen became such a convenient

number to use? List all the factors of one dozen. W

hat is the origin of the number “naught?”

Does it remind you of any other words in our language?

History of Mathem

atics - Recording Numbers – The Invention of Num

erals A

Research the Sumerian civilization. W

hat types of numerals did they have? W

hy was 60an im

portant number to them

?

Research the Babylonian astronomers. W

hy did they use the number 360? W

hat othercontributions did they m

ake to our knowledge today?

Write the following num

bers in Egyptian notation:264568197

Do you think the Egyptian system was a good one? Explain one advantage and one disadvantage.

Research the Chinese number system

. Share your knowledge in an oral presentation to the class.


Rounding and Estimating

Introduction to Rounding Whole Numbers - The Decimal SystemRounding is a technique used in mathematics which allows us to simplify large numbers and manipulate them mentally with greater ease. The rules for rounding are simple:

1. To round any number to a selected place, examine the number in the previous place.2. If the number in the previous place is half of ten (5), or more, add one to the selected place.3. Drop the remaining value of the number in all places less than the rounded place.

Presentation One - What Is About a Whole?Tell the students they are going to learn about rounding. To understand rounding, they need to learn how the word "about" is used mathematically. Using familiar objects, for example a measuring cup of water or a sandwich or an apple, demonstrate by removing one small part at a time. Each time a part is removed, ask the question, "Is this about a cup? (or about an apple or about a sandwich?) Establish with each example, that when there is a half or more we say that we have about a whole.

Direct Aim: Students will understand the use of the word "about" in rounding, and the importance of one half.

Presentation Two - What Is Half of Ten?Prerequisites: Understanding the decimal system, including place value, is foundational to the concepts of estimating whole numbers. Students should have had initial fraction presentations. The concept of one-half must be thoroughly understood and applied to whatever unit of measure is to be estimated.

Materials: Golden Bead material and Detachable numeral cards

Using Golden Bead material, revisit the places of the decimal system: units, tens, hundreds, and thousands, and their relationships to each other. Review the basic meaning of the decimal system: that each place signifies a quantity ten times greater than the place before it. Review the process of exchanging when a quantity of ten is reached in any place. Have the child get 10 units from the golden beads. Ask her to make these ten units into two equal groups. Identify each group of five units as HALF of ten. Five is half of ten. Label each group of 5 units, half of ten.

Move one unit from the first half to the second half. Now the child sees two groups, 4 units and 6 units. Which one is more than half? Yes, 6. Which one is less than half? Yes, 4. Six is more than half of ten.

Label the group of 6 units, 6 > half of ten. Four is less than half of ten. Label the group of 4 units, 4 < half of ten. Move another bead from the smaller group into the larger group. Now the child sees two groups, 3 units and 7 units. Which one is more than half? Yes, 7. Which one is less than half? Yes, 3. Seven is more than half of ten. Label (7>half of ten). Three is less than half of ten. Label (3<half of ten).

Move another bead from the smaller group into the larger group. The child sees two groups, 2 units and 8 units. Is 2 more than half or less than half of ten? Yes, 2 is less than half of ten. Label (2<half of ten). Is 8 more than half or less than half of ten? Yes, 8 is more than half of ten. Label (8>half of ten).

Move one more bead from the smaller group to the larger group. The child sees two groups, 1 unit and 9 units. Which is less than half of ten? Which is more than half of ten? Yes, 1 is less than half of ten. Label (1<half of ten). Nine is more than half of ten. Label (9>half of ten).


Using the detachable numeral cards for units, ask the child to select the numeral which represents half of ten. (5) Place the 5 card on the mat. Label (5=half of ten). Ask the child to select the other numerals which represent quantities less than ten. (4,3,2,1) Place these numerals beneath the 5. Label (< half of ten). Ask the child to select the numerals representing quantities greater than half of ten. (6,7,8,9) Place these numerals above the 5. Label (>half of ten).

Repeat this exercise with 10 ten bars and detachable numeral cards for tens to establish that 5 tens equal one-half of 10 tens, or one hundred. Repeat this exercise with 10 hundred squares and detachable numeral cards for hundreds to establish that 5 hundreds equal half of 10 hundreds, or one thousand. Via the child's imagination and logical reasoning, extend this concept throughout the decimal system, so that she understands that 5 in any place represents one-half of the next larger place.

Direct Aim: Student will gain experience with concrete materials and corresponding symbols of the concept of half of ten.

Presentation Three - RoundingLay out 10 units, 10 tens, and 10 hundreds in their appropriate places on the mat. Ask the child if she remembers the process for exchanging in the decimal system. Have her verbalize that every time we find 10 of any order (units, tens, hundreds, etc.), we exchange - the search for ten. Allow the child to complete the exchanging process with the golden beads you placed on the mat.

Explain that today we will learn a technique called rounding. The purpose of rounding is to make it easier for us to work with large numbers mentally. The rules of rounding are much like the rules of exchanging, with some variations.

Ask the child to lay out 4,681 on the mat with the golden beads. Explain that we will now experience what rounding means by following some simple rules. The basic rule for rounding states:

When rounding to a specific place, examine the quantity in the previous place. If this quantity is one-half of ten or more, we exchange it for one more in the rounded place, returning any quantities smaller than that place to the bank. We call this rounding up.

If the quantity in the previous place is less than one-half of ten, we leave the value of the rounded place as it is, returning any quantities smaller than that place to the bank. We call this rounding down.

We will now round 4,681 to the tens place. What is the place before the tens? (units) What is one-half of ten? (5) Which values are one-half of ten or more? (5, 6, 7, 8, 9) Which values are less than one-half of ten? (4, 3, 2, 1) What is the value in the units place? (1) Is this five or more? (no) Round down!


Put the unit back in the bank. What is our number now? (4,680) We have just rounded it to the tens place. We can say (and write) that 4,681 is about 4,680. We rounded down!

Compose another number (352) to round to the hundreds place. Ask the child, What is the place before the hundreds place? (tens) Is the number in the tens place 5 or more? Yes! Round the hundreds place up to 400 and return the tens and units to the bank. We can now write and say that 352 rounded to the hundreds place is 400. This is called rounding up.

Front-end rounding is the name given to rounding the largest place of a numeral.

Ask the older children to compare and contrast the process of rounding with that of exchanging in the decimal system in their math journals, using correct nomenclature for each.Extensions - Estimating:When estimating length, height, perimeter, or area, standard units of English and metric linear and square measurement must be previously experienced and understood.

Digital and analog clocks and the basic units of time measurement of must be understood to estimate time.

Units for measuring weight and liquid must be explored and their relationships established in order for estimation in these areas to be meaningful.

Degrees Celsius and Fahrenheit and their correlation with temperature must be previously known before estimating temperature.

Fractions and their decimal equivalents are the key domains preceding the estimation of mixed numbers and money.

Direct Aim: Student will understand the process of rounding and how it compares to exchanging.

Presentation Four - Rounding Decimal NumbersWhen decimal concepts and numerals are understood by the students, rounding work may begin. Use the decimal board or the decimal golden mat and the decimal numeral cards to create quantity and symbol representing decimals. Apply the same rules for whole numbers to round decimals to various places.

CalculatorsMaterials: Calculator diagram, white board and markers, calculators (enough for a small group)

Presentation:Place calculator diagram on lesson mat. Calculators are wonderful tools. We can use them to quickly complete equations. When do you think it would be useful to use a calculator? On white board write addition equation: 5 + 2 = 7. Using the diagram, identify the keys used to compute the equation. Distribute calculators. Invite children to compute the equation. Continue lesson with equations for all previously introduced operations: +, , x, ˜

Direct Aim: To familiarize children with the calculator.

Extensions:Represent decimals to hundredths on a calculator.Use the calculator to do multiplication with more than two digits in the multiplier.Use the calculator to do or verify division and interpret results.Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 128

Montessori Problem Solving Sequence

Reading Skills in Problem Solving

Story problems should be part of the daily reading program of the elementary child. Familiarity with mathematical nomenclature and thinking, modeled on a daily basis by the teacher and classmates, will allow each child to move naturally into practical application of mathematical knowledge. There are many children’s books available from the library, which enhance mathematical thinking skills while making reading about math a joy.

Prerequisite SkillsThe domain of problem solving concerns the application of mathematical knowledge to real life situations.There are five sub skills required for children to experience success in this area:

1. Language: understanding the language used in identifying the four basic operations (addition, subtraction, multiplication, and division);

2. Modeling: the ability to picture in concrete ways the interactions and relationships described in the situation;

3. Calculating: calculating skills, including use of calculators, for performing operations with whole numbers, fractions, and integers;

4. Symbolizing: understanding the symbols and meanings of number sentences (equations and inequalities expressing relationships between quantities using symbols);

5. Writing Skills: explaining the process of solving the problem.

Skill One: Language of the Four Basic OperationsMaterialsWrite the words listed below on cards with the symbol for the operation on the back. Have the children sort the words into columns with the operational symbol at the top of each column. Make a chart for control of error.

Addition/Multiplication KeyWords

SubtractionKey Words

MultiplicationKey Words

DivisionKey Words

combinein all

differencegreater than

sameeach

sharingdividing


altogethertotaladdedbuyaddcollectplusbringput togethergatherhow manyhow muchhow oftenincreaseincludemore

more thanless thanfewer thanolder thanyears agoyounger thanremainingleftgive awaysellwent awaytake awaymissingleft overextraenough

bundle oftimesgroups ofteams ofsets ofgroups in allgroupsaltogethersquaredcubedin each herdon each teamin each boxlaps in eachcrayons perracebox

taking apartseparatingequal groupsequal sharesequal partsin eachhow many groupshow many perequal sectionsshared equallyequal amountssplitting aparthow many ineach getlaps per raceeach groupdividedeach grouphow manyon eachteamsboxesteam


Skill Two: Modeling the Problem

In order for the child to manipulate the information in a story problem, she must first understand the situation. This can be facilitated in a variety of ways:

1. Choose children to act out the parts of the story. Encourage them to use objects in the environment to make the situation concrete.

2. Encourage the child to use the golden beads and other Montessori materials in the classroom in a variety of ways to illustrate the dynamics of the situation. For example, if the story describes sharing a pizza, use the fraction insets for the dividing process.

3. Model for the children how to draw the characters and dynamics of the situation in a variety of ways.

4. Create the quantities described in the situation with Montessori materials or other objects in the environment. Label these quantities, telling what they represent.

5. Create a label for the information you want to know.

6. Use the real children and the real things in the environment as the basis of the story problems. Make up one story every day for the whole class to solve, using the names and interests of the children in the classroom.

Skill Three: Calculating

Children should use a wide repertoire of calculating tools in the classroom environment to solve story problems. The Golden Bead material is applicable to all situations. The stamp game, bead frames, the checkerboard, charts of simple equations, and all math materials can and should be used in reference to real life situations, both in initial presentations and in daily practice. The use of calculators is allowed in many testing situations. Using the calculator is an opportunity for the child to focus on the concept of number sentences.

Skill Four: Symbolizing the ProblemOverview:A number sentence is an expression, written in mathematical symbols, of relationships between numerical or spatial quantities. Number sentences can be equations or inequalities. The most frequently used symbols in number sentences are as follows:

+ add < is less than- subtract > is greater thanx multiply is congruent to≅Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 131

÷ divide ≈ is approximately equal to= is equal to % percent≠ is not equal toIn a number sentence, quantities and symbols are combined to show relationships. Number sentences represent the procedure used to solve the problem. A number sentence is also the expression of the active process of solving problems to establish these relationships.

Four birds sat on a tree branch. Two flew away. How many are left?

The number sentence reflects the thinking process:

4 birds (at first) − 2 birds (flew away) = 2 (left)

Sequence is an important factor in writing number sentences. (Commutative and associative properties of addition and multiplication are relevant here.) Using a calculator is a good way to focus on creating number sentences. The number sentence is exactly the sequence of buttons you push on the calculator to find the correct answer to the equation. Conversely, any number sentence may be solved with a calculator by entering its symbols correctly in sequence.

Materials:1. paper slips with symbols: >, <, =2. small items from environment: pencils, crayons, erasers, etc.

Presentation:Invite children to organize into two groups. Ask children to compare groups. Summarize the children's findings. Explain that in mathematics, if one group has more members we say that it is greater than the other group. The group with fewer members is said to be less than the other group.Organize children into two groups of the same number. Explain that if there are the same number of members in both groups, the groups are called equal.

Place a group of 7 pencils and a group of 4 crayons on the lesson mat.

Place symbol slips on mat. Place > symbol between the groups. Review the symbols for greater than and less than.

1. Explain that this symbol means greater than. The group of pencils is greater than the group of crayons.

2. Reverse groups on mat. Place < symbol between the groups. This symbol means less than. The group of crayons is less than the group of pencils.

3. Organize both groups into equal amounts. Place = symbol between groups. This symbol means equal to. The group of crayons is equal to the roup of pencils.

4. Remove items from mat. Continue three-period lesson. Show me the greater than symbol. What is this symbol?

5. Invite the children to create their own groupings and place symbols.

Extensions:Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 132

Create number sentences requiring placement of >, <, and =.Create similar games for the other symbols as needed.

Direct Aim:Student will identify >, <, and = symbols and apply concept to number sentences.

Control Of Error:Teacher


Writing Skills in Problem Solving

Initially, first year elementary students may be asked to draw representations, give oral explanations, and write number sentences for story problems. As their writing skills increase, they should label the parts of their drawing and write simple sentences. Their math journals will be a vehicle for developing the ability to analyze and sequence the steps of their mathematical explorations. As they move toward mastery, they will be guided to explain in detail the how and why of applying their mathematical knowledge to real life situations.

Putting it All Together: Mathematical Thinking Skills

The process of mathematical thinking and the language that facilitates it can be effortlessly imparted on a daily basis in any Montessori classroom. In the beginning, highlight a particular mathematical concept, exploring the language needed to understand it and giving concrete presentations to illustrate its process.

1. Language: Create a story about one of the children in the class, and ask the children what mathematical questions arise in their minds. Ask them to predict what they think they will find out. For example, will their answer by more or less? Show how to investigate the language of the story,like a detective, to find the clues that will reveal the operation to be followed.

2. Modeling: Ask helpers to act out the story or show how to draw a representation of the story or ask a child to use concrete materials to show the problem.

3. Calculating: Have a child calculate the answer with materials. Ask the children if this answer makes sense.

4. Symbolizing: Ask the children to explain what happened in the story and how the problem was solved. Create a number sentence to show what has been done. Model writing the steps, in sequence, as the children describe them, then use the calculator to verify the number sentence.

5. Writing about Math: Ask the students to explain what they wanted to find out, and what they did to find it. Beginning students may simply make drawings and label them. Third year students should be able to write an analysis of a multi-step problem solving process.

After being guided through the steps of this process of mathematical thinking, the children will begin to internalize it. On subsequent days, they will anticipate the questions, the steps, the thinking, and develop fluency with the mathematical language used to explain these. As each new concept arises to be explored, this same format will be used to extend and integrate the children’s ability to apply mathematical knowledge to real life situations.

The children following this format will very soon begin to formulate their own story problems from their home and classroom experiences. They will see mathematical questions and their answers everywhere they look. They will model for each other and extend each other’s thinking in their day to day work with numbers.Independence in Solving Story ProblemsPrerequisite:Students should have worked with the activity: “Language of the Four Basic Operations” and the teacher should have modeled the five skills of problem solving many times with the whole class.

Materials:• “Language of the Four Basic Operations” chart• labels: Find Important InformationPicture the SituationWrite a Number SentenceCalculate and CheckMontessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 134

Explain

Presentation:Review the “Language of the Four Basic Operations” chart. Say: “Today we are going to discuss how to solve story problems. Within every story problem is a math equation for you to solve. Solving a story problem is like solving a puzzle. It can be fun if you follow five easy steps.”

Place the following problem on the mat and ask children to read it aloud:

Place the first label “Find Important Information” on the mat. Say: “The first step in solving a story problem is to find the important information. What question do we see here?” When children identify the question, underline it. Ask: What numbers do you see in the story?” When the children find them, circle the numbers 12 and three. Ask: “Do you see any of the language of the four basic operation in this story?” Children should be able to identify “left” as a key word for subtraction. Circle the word “left.” Finally, ask children to predict if the answer will be larger than 12 or smaller.

Place the second label “Picture the Situation” on the mat. Say: “The second step in solving a story problem is to picture the situation. What are some ways that we can solve this story problem? Invite children to choose their own method (For example: the children might use skittles from the stamp game to represent the 12 children playing. A ruler could represent the slide. A book could be the jungle gym. A paper could represent the swings. Each item should be labeled. )

Place the third label “Calculate and Check” on the mat. Say: “The third step in solving a story problem is to calculate the answer.” Have the children enact the situation with the model they have created. (In the above example, three skittles will move to the slide, leaving nine at the swings.) Then ask what is the answer to the question, and does the answer make sense? Twelve children were playing and three went away. Should the answer be larger than twelve or smaller? Invite discussion. Emphasize checking the reasonableness of the answer.

Place the fourth label “Write a Number Sentence” on the mat. Say: “The fourth step in solving a story problem is to write a number sentence. How many children were on the swings? How many children left to go play on the slide? What mathematical operation did we determine was needed?” Write the number sentence (12 children – 3 children = 9 children.) Have one of the children check the answer with one of the Montessori materials or a calculator.

Place fifth label “Explain” on the mat. Say: “The fifth step in solving a story problem is to explain how you got your answer.” Invite children to write about their approach to this story problem.

Extensions:Children create their own story problems.

Direct Aim:Student will identify the five steps to successful problem solving.

Control of Error:Materials, Teacher

Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools

There were 12 children playing on the swings. Three of the children went to play on the slide. The jungle gym was next to the swings. How many children are left playing on the swings?

135

Key Terms

+ combine in all how often

altogether total added More

buy add collect increase

plus bring put together Include

gather how many how much


-Difference take away greater than sell

more thanless than

extra fewer than missing

older than Left over years ago younger than enough

remaining Give away Leftwentaway

tookaway

x same each squared on each team


bundle of times groups of in each boxlaps in eachrace

teams of sets of groups in allcrayons perbox on each team

groupsal-together in each herd

cubedmiles one way

÷ sharing laps per race dividing

taking apart separating on eachequal groups


equal shares equal parts each groupeach get

how many groups How many perteams

splitting apart how many in allequal sections

Problem Solving Strategies (Ages 6-12)

Introduction to Problem Solving with Addition

First Period: (Engagement)Read the story problem together. “Ms. Orbos has 8 red crayons and 6 blue crayons for the art project. How many crayons does she have inall?”

Instruct the students to get 8 red crayons and 6 blue crayons. Say: “What are we going to do with these crayons to answer the question?What is the question?” (How many crayons does she have in all?) Record student responses on board. (add them, put them together, count them, combine, etc.)

What words in the question told you to add? (and / in all) “And” and “in all” are key words for addition. What other words did you think of that mean addition? (put them together, combine, gather, total). There are many key words that mean addition. Show card set with addition key words.

“Let’s practice using the words for addition with our crayons.” Have the children demonstrate, “Combine the 8 red and the 6 blue crayons. What is the total ?” “Collect 8 red crayons and 6 blue crayons. How many do you have altogether?” “Do these words tell us to do the same thing as before?”

Have students pick a card from the addition key word card set and make up a story problem using addition and tell it to a partner.

Second Period: (Guided Exploration)Review the addition key word card set. Read the story problem together. Joey collected 20 stamps from Italy and 30 stamps from Canada. How many stamps did he collect altogether? What are the key words in the story problem? (collect, and, altogether) Use the golden beads to show the numbers in the story. (2 tens and 3 tens) What is the answer? (5 tens or 50) Let’s write our work in a number sentence. A number sentence uses numbers and math symbols to tell what we did. 20 + 30 = 50. The key words tell


us which symbol to use. Have students pick a card from the addition key word card set and make up a story problem using addition with golden beads and write it as a number sentence. Share with a partner.

Second Period: (Independent Exploration)Get the addition key word card set and a stamp game from the shelf to help you solve the following problems.Follow the 4 steps below:1. Write down the key words that tell you to add.2. Use the stamps to show the numbers.3. Find the answer.4. Write a number sentence to show your work.

Mrs. Brown collected 10 children from one class and 4 more children from another class to go to the library. What is the total number of children going to the library?

Mark brought 20 chocolate and 10 oatmeal cookies for the party. How many cookies did he bring in all?The class bought 10 bananas and 20 pears for the picnic. How many pieces of fruit did they bring altogether?

Second Period: (Explanation)Read the story problem together: Tim put together 20 blue marbles and 8 red marbles in a box. How many marbles are in the box?

Follow the 4 steps below:1. Write down the key words that tell you to add.2. Use the stamps to show the numbers.3. Find the answer.4. Write a number sentence to show your work.

Now we will explain how we know our answer is correct. What did you do first to find the answer? What did you do second? Third?”Summarize students’ responses in a sentence or sentences on the board. (Example: We got 20 stamps and added 30 more stamps to make 50.) Have students copy the sentence.

Third Period: (Evaluation)Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 140

Read the following story problem:I ate 3 peanuts. I ate 2 more peanuts. How many peanuts did I eat altogether?

Which of the following key words tells you to add?Howmanypeanutsaltogether

2. Use golden beads to do the problem, then choose the picture that shows what you did.

OOO + OOO = OOOOOO OOO + OO = OOOOO O + OOO = OOOO

3. How many peanuts did I eat altogether?9

1356

4. Choose the correct number sentence to show the work you did in the problem.3 + 3 = 63 + 2 = 55 + 5 = 101 + 3 = 4

Third Period – (Evaluation)Students will read the story problem and complete the 5 steps independently. Rachel found three frogs in her backyard. After the rain, she found 5 more. How many frogs did Rachel find altogether?

Key words:

Make a model with materials or by drawing:

Find the answer:


Write the number sentence:

Explain how you know your answer is correct using words and numbers:

Introduction to Problem Solving with SubtractionFirst Period: (Engagement)Read the story problem together.Lena caught 10 crickets in the field. She put them in a shoebox. Four of the crickets got away. How many are left?

Instruct the students to get 10 paper clips or chips or marbles to represent the 10 crickets. Say: “What are we going to do with these paper clips to answer the question? What is the question?” (How many crickets were left?) Record student responses on board. (subtract 4, take away 4, make 4 jump away, let 4 escape)

What words in the question told you to add? (got away/left) “Got away” and “left” are key words for subtraction. What other words did you think of that mean subtraction? (take away, jump away, escape). There are many key words that mean subtraction. Show card set with subtraction key words.

“Let’s practice using the words for subtraction with our paper clips.” Have the children demonstrate, “I have 10 paper clips and give away 4. How many remain?” “I have ten paper clips. You have 4. What is the difference?” “I have 100 paper clips and you have 4. How many more do I have than you?”“Do these words also tell us to subtract?” Have students pick a card from the subtraction key word card set and make up a story problem using subtraction.

Second Period: (Guided Exploration)Review the subtraction key word card set. Read the story problem together. Andrea is eight years old. Her sister is four. How much older than her sister is Andrea?What are the key words in the story problem? (older than) Use the subtraction strip board to show the numbers in the story. (8 and 4) What is the answer? (4)

Let’s write our work in a number sentence. A number sentence uses numbers and math symbols to tell what we did. 8 – 4 = 4. The key words tell us which symbol to use. Have students pick a card from the subtraction key word card set and make up a story problem using the subtraction strip board. Write it as a number sentence. Share with a partner.

Second Period: (Independent Exploration)Get the subtraction key word card set and the golden beads from the shelf to help you solve the following problems.

Follow the 4 steps below:

1. Write down the key words that tell you to subtract.2. Make a model using materials from the classroom or by drawing a diagram.3. Find the answer.4. Write a number sentence to show your work.

Mrs. Brown invited 14 of her students to a lesson. At 10:00 three of them had to go to the library for TAG. How many students finished the lesson with Mrs. Brown?

Mark brought 20 chocolate cookies for the party. After the party, there were 2 cookies left on the plate to take home. How many cookies were eaten?

The class brought 20 bananas and 10 pears for the picnic. How many more bananas than pears did they bring?Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 142

Second Period: (Explanation)Read the story problem together:Tim had 20 blue marbles in a box. After losing six of them playing marbles, how many marbles remained in the box?Follow the 4 steps below:1. Write down the key words that tell you to add.2. Make a model to show the problem, using materials or by drawing.3. Find the answer.4. Write a number sentence to show your work.

“Now we will explain how we know our answer is correct. What did you do first to find the answer? What did you do second? Third?” Summarize students’ responses in a sentence or sentences on the board. (Example: I drew 20 marbles and crossed out 6. 20 – 6 = 14. 14 marbles remained in the box.) Students may copy the explanation from the board.

Third Period: (Evaluation)

Jane bought 15 bugs for her tarantula to eat in two weeks. If it ate 7 bugs the first week, how many wereleft to eat the second week?1. Which of the following key words tells you to subtract?

howmanyleftbugs

2. Use golden beads to do the problem, then choose the picture that shows what you did.OOOOOOOOOO OOOOO -OOOOOOO = OOOOOOOO

OOOOOOOO + OOO =OOOOOOOOOOO

OOOOOOOOOOOOOOO -OOO =OOOOOOOOOOOO

3. How many bugs were left to eat the second week?81572

4. Choose the correct number sentence to show the work you did in the problem.7 – 2 = 515 – 2 = 1315 – 7 = 815 – 15 = 0

Third Period – (Evaluation)


Students will read the story problem and complete the 5 steps independently.

Rachel counted 23 worms in her backyard after the rain. The next day, when it was hot and sunny, she only counted 11. How many more worms did Rachel count on the rainy day than on the sunny day?

Key words:Make a model with materials or by drawing:Find the answer:Write the number sentence:Explain how you know your answer is correct using words and numbers:ExtensionTake the white card sets for addition and subtraction key words and mix them together. On a mat, lay out an addition and subtraction symbol. Sort the cards into two columns by deciding whether the key word indicates addition or subtraction. Check your work with the control of error on the back of the cards. Copy the lists into your notebook.

Introduction to Problem Solving with MultiplicationFirst Period: (Engagement)Read the story problem together.Brian wants to bring cookies for the class to celebrate his birthday. He wants to give each of his friends 2cookies. There are 20 students in Brian’s class. How many cookies does he need to bring in all?

Instruct the students to use bead bars to make a model of the problem. Say: “What are we going to do with these bead bars to answer the question?What is the question?” (How many cookies were does he need in all?) Record student responses on board. (get a 2 for each student, get a 2 twenty times, count twenty twos, add 2 twenty times, put together twenty two bars, multiply 20 times 2)

What words in the question told you to make groups of the same number? (each) What words in the question told you to add? (in all) “Each” and “in all” are key words for multiplication. Multiplication is adding the same number a certain number of time. What other words did you think of that mean multiplication? (times) There are many key words that mean multiplication. Show card set with multiplication key words.

“Let’s practice using the words for multiplication with our bead bars.” Have the children demonstrate, “I have 3 boxes with 8 crayons in each. How many crayons altogether?”

“I read two chapters of my book each day for 5 days. How many total chapters did I read?”

“Six basketball teams have six players each. How many players on all six teams?”

“There are four herds of buffalo with 50 in each herd. How many buffalos altogether?

“Do these words also tell us to multiply?”Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 144

Have students pick a card from the multiplication key word card set and make up a story problem usingmultiplication with bead bars. Share with a partner or the group.

Second Period: (Guided Exploration)Review the multiplication key word card set. Read the story problem together.

Sharyse wants to make a necklace for herself and her two best friends. She needs eight beads for each necklace. How many beads does Sharyse need altogether?

What are the key words in the story problem? (and/each/altogether)Draw a diagram to show the numbers in the story. (3 friends, 8 beads)What is the answer? (24)

Let’s write our work in a number sentence. A number sentence uses numbers and math symbols to tell what we did. 3 x 8 = 24. The key words tell us which symbol to use. Have students pick a card from the multiplication key word card set and make up a story problem using drawing to model the problem. Write it as a number sentence. Share with a partner.

Second Period: (Independent Exploration)Get the multiplication key word card set and the golden beads from the shelf to help you solve the following problems.

Follow the 4 steps below:1. Write down the key words that tell you to multiply.2. Make a model using materials from the classroom or by drawing a diagram.3. Find the answer.4. Write a number sentence to show your work.

Devin plays basketball after school on a neighborhood team. His team competes against other teams in his league. There are 6 players on the teams and 5 teams in the league. What is the total number of basketball players in the league?Bianca helped her teacher sort crayons onto 7 trays for a class art project. She placed 6 different colors on each tray. How many crayons were needed for the art project in all?

Karessa set up tables for a class party. She placed 4 chairs at each of 6 round tables. How many chairs did Karessa place altogether?

Second Period: (Explanation)Read the story problem together: Kandace likes to run races. She ran 3 laps around the track every day after school for one week. Howmany laps did Kandace run after school that week?


Follow the 4 steps below:1. Write down the key words that tell you to multiply.2. Make a model to show the problem, using materials or by drawing.3. Find the answer.4. Write a number sentence to show your work.

“Now we will explain how we know our answer is correct.What did you do first to find the answer? What did you do second? Third?”Summarize students’ responses in a sentence or sentences on the board. (Example: I read the problem. I made a diagram with the school days. 5 x 3 = 15, so she ran 15 laps that week.) Students may copy the explanation from the board.

Third Period: (Evaluation)Read the following story problem: Ronald loves to go with his dad to race his Ferrari at the racetrack. Ronald’s dad races his Ferrari aroundthe track one time in 3 minutes. How many minutes will it take his dad to race around the track 7 times?

1. Which of the following key words tells you to multiply?howmany

timesFerrari

2. Use bead bars to do the problem, then choose the picture that shows what you did.

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

OOOOOOOOOOOOOOO

OOO OOO OOOOOO OOO OOO OOO

3. How many minutes will it take Ronalds’s dad to race around the track 7 times?732125

4. Choose the correct number sentence to show the work you did in the problem.7 x 3 = 213 x 5 = 157 + 3 = 103 x 8 = 24


Third Period – (Evaluation)Students will read the story problem and complete the 5 steps independently.

Rebecca’s guinea pig had babies for the third time. In each litter there were four baby guinea pigs. Howmany babies has Rebecca’s guinea pig had so far?

Key words:

Make a model with materials or by drawing:

Find the answer:

Write the number sentence:

Explain how you know your answer is correct using words and numbers:

Use the rubric to score your work.

ExtensionTake the white card sets for addition, subtraction and multiplication key words and mix them together. On a mat, lay out the symbols for addition, subtraction and multiplication. Sort the cards into three columns by deciding whether the key word indicates addition,subtraction or multiplication. Check your work with the control of error on the back of the cards. Copy the lists into your notebook.

Introduction to Problem Solving with Division

First Period: (Engagement)Read the story problem together.There are 24 chairs and 6 tables in the classroom. If each table has the same number of chairs, how many chairs will we place at each table?

Instruct the students to use the Stamp Game to make a model of the problem. Say: “What are we going to do with these stamps to answer the question?What is the question?” (How many chairs will we place at each table?) Record student responses on board. (e.g., use skittles for tables, get out 24 stamps for chairs, pass out the stamps to the skittles, divide 24 stamps around 6 skittles, make 6 groups of chairs)

What words in the question told you to make equal groups? (same)What words in the question told you to divide? (place at each)“Place at each” and “same” are key words for division. Division is making equal smaller groups from a larger number.What other words did you think of that mean division? (pass out, make groups of) There are many key words that mean division. Show card set with division key words.


“Let’s practice using the words for division with our stamps.” Have the children demonstrate, “I have eighteen crayons to put into 3 boxes. How many crayons will be in each box, if each box has an equal number of crayons?”“Donna and her sister want to share the gummy bears equally. If they share 12 gummy bears, how many will they each get?”“I have fifteen fish to put into five fish tanks. If each tank has the same number of fish, how many fish will be in each tank?”“Twenty-four kids want to play basketball on the neighborhood teams. If each team needs six players, how many teams can be formed?”“Do these words also tell us to divide?”

Have students pick a card from the division key word card set and make up a story problem using division with stamps. Share with a partner or the group.

Second Period: (Guided Exploration)

Review the division key word card set.Read the story problem together.Sharyse wants to make a necklace for herself and her two best friends. She has 24 beads. If she puts an equal number of beads on each necklace, how many beads will be on each necklace?

What are the key words in the story problem? (and/equal/each)Draw a diagram to show the numbers in the story. (3 neckalces, 24 beads)What is the answer? (8)

Let’s write our work in a number sentence. A number sentence uses numbers and math symbols to tell what we did. 24 ÷ 3 = 8. The key words tell us which symbol to use.

Have students pick a card from the division key word card set and make up a story problem drawing a diagram to model the problem. Write it as a number sentence. Share with a partner.

Second Period: (Independent Exploration)Get the division key word card set and the golden beads from the shelf to help you solve the following problems.

Follow the 4 steps below:1. Write down the key words that tell you to divide.2. Make a model using materials from the classroom or by drawing a diagram.3.Find the answer.4. Write a number sentence to show your work.5.Write an explanation using words, numbers and key words to explain how you know your answer is correct.

Bianca is helping her teacher sort crayons onto trays for a class art project. She has 35 crayons to divide evenly among 5 trays. How many crayons will Bianca put on each tray?

Carmen’s mom drives 50 miles in the carpool each week. Since there are 5 school days in a week, how many miles does Carmen’s mom drive in a day?

If two brothers share 36 baseball cards evenly, how many will they each get?

Second Period: (Explanation)Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 148

Read the story problem together:Kandace is a track star. This weekend she ran 14 laps around the track in 7 different races. How many laps were in each of Kandace’s races?

Follow the 5 steps below:1. Write down the key words that tell you to divide.2. Make a model to show the problem, using materials or by drawing.3. Find the answer.4. Write a number sentence to show your work.

“Now we will explain how we know our answer is correct.What did you do first to find the answer? What did you do second? Third?”Summarize students’ responses in a sentence or sentences on the board. (Example: I read the problem. I made a diagram of 7 races and 14 laps. 14 ÷ 7 = 2, so she ran 2 laps in each race.) Students may copy the explanation from the board.

Third Period: (Evaluation)Read the following story problem:Missy arranged 60 cookies on 5 plates. If each plate had an equal number of cookies, how many cookies did Missy put on each plate?

1. Which of the following key words tells you to divide?howmanyon eachcookie

2. Use golden beads to do the problem, then choose the picture that shows what you did.

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

OOOOO OOOOOOOOOO OOOOOOOOOO OOOOOOOOOO OOOOOOOOOO OOOOO

OOO OOO OOOOOO OOO OOO OOOOOO OOOOOO OOO OOO

3. How many cookies were on each plate?7 12

5 604. Choose the correct number sentence to show the work you did in the problem.

5 x 10 = 5060 ÷ 5 = 1260 ÷ 10 = 6


60 ÷ 6 = 10Third Period – (Evaluation)Students will read the story problem and complete the 5 steps independently. Rebecca’s guinea pig had 12 baby guinea pigs in three different litters. Each litter had the same number of babies. How many baby guinea pigs were in each litter?Key words:Make a model with materials or by drawing:Find the answer:Write the number sentence:Explain how you know your answer is correct using words and numbers.


Problem Solving Activity Cards (Ages 6-12)

Problem Solving Practice - Addition

1. Find the key words and numbers.

2. Show the numbers using beads.

3. Calculate the answer. 4. W

rite a number sentence.

5. Use what you know about addition to explain your answer, using words and num

bers.

1. Jim collected 10 green and 6 blue pencils. How m

any pencils did he collect altogether? 2. Don collected 20 red and 8 blue pencils. How m

any pencils did he collect altogether? 3. Sam

collected 40 green and 7 blue pencils. How many pencils did he collect altogether?

4. Tom collected 50 red and 5 blue pencils. How m

any pencils did he collect altogether? 5. Ben collected 30 brown and 9 red pencils. How m

any pencils did he collect altogether?

Problem Solving Practice - Addition

• Find the key words and numbers.

• Show the numbers using ten bars.

• Calculate the answer. • W


• Use what you know about addition to explain your answer with words and numbers.

1. There are 40 jellybeans in the jar. Joe added 10 more. How m

any jellybeans are in the jar? 2. There are 10 jellybeans in the jar. Ann added 50 m

ore. How many jellybeans are in the jar?

3. There are 30 jellybeans in the jar. Bob added 40 more. W

hat is the total number of jellybeans?

4. There are 20 jellybeans in the jar. Sue added 50 more. W

hat is the total number of jellybeans?

5. There are 60 jellybeans in the jar. Tim added 30 m

ore. What is the total num

ber of jellybeans?


For example:

added, how many, 20

jellybeans, 40 more

20 + 40 = 60I know sixty is the answer because the words 'added'and 'how m

any' told m

e to add, and if you add 20jellybeans plus 40 jellybeans the answer is sixtyjellybeans.

For example:

collected, altogether, 30 pencils, 3 pencils30 + 3 = 33I know 33 is the right answer because the words'collected' and 'altogether' tell you to add, and 30 pencils plus 3 pencils equals 33 pencils.

For example:

added, how many, 20 jellybeans, 40 m

ore20 + 40 = 60I know sixty is the answer because the words 'added'and 'how m

any' told me to add, and if you

add 20jellybeans plus 40 jellybeans the answer is sixtyjellybeans.

151

Problem Solving Practice - Addition U.S. M

oney


• Show the numbers using coins.



• Explain your answer, using words and numbers.

1. There are four dimes and two quarters in m

y pocket. How much do I have in m

y pocket? 2. There are three nickels and one dim

e in my pocket. How m

uch do I have in my pocket?

3. There are two quarters and one nickel in my pocket. How m

uch do I have in my pocket?

4. I have 10 pennies in my pocket. I added 6 pennies m

ore. How much do I have in m

y pocket altogether? 5. I have two dim

es in my pocket. I added five pennies m

ore. How much do I have in m

y pocket altogether?

Problem

Solving Practice - Addition • Find the key words and num

bers. • Show the num

bers using a ruler. • Calculate the answer. • W


• Explain your answer, using words and num

bers.

1. A ladybug crawled 10 cm east. It continued crawling 8 cm

more. How far did it crawl in all?










For example:

in all, more,12 cm

, 4 cm12+ 4 = 16I figured out the answer is 16 cm

, because I saw thewords 'in all' and 'm

ore' in the problem,

so I knew I hadto add. 12 centim

eters plus 4 centim

eters equals 16centim

eters.

For example:

in all, more,12 cm





eters plus 4 centimeters

equals 16 centimeters.

For example:

how much, 3 dim

es, 1 nickel30¢ + 5¢ = 35¢The right answer is 35¢. I know this is right, becausethe words 'how m

uch' in the story tell you to add, and 3dim

es equals 30¢ and 5 pennies equals 5¢. 30+5 = 35cents.

152

Problem Solving Practice - Addition U.S. M

oney


• Show the numbers using coins.



• Explain your answer, using words and numbers.

1. Anne collected one half dollar and four nickels from Bob. How m

uch did Anne collect in all? 2. Anne collected three quarters and five nickels from

Ben. How much did Anne collect in all?

3. Anne collected one half dollar and six dimes from

Sam. How m

uch did Anne collect in all? 4. Anne collected two quarters and 3 pennies from

Jane. How much did Anne collect in all?

5. Anne collected seven dimes and nine pennies from

Joe. How much did Anne collect in all?

Problem

Solving Practice – Addition, Fractions • Find the key words and num

bers. • Show the num

bers using a ruler. • Calculate the answer. • W


• Explain your answer, using words and num

bers.

1. Jane ate 1/4 of the pie. I ate 2/4 of the pie. How many portions of the pie did we eat altogether?

2. Jane ate 1/3 of the pie. I ate 1/3 of the pie. How many portions of the pie did we eat altogether?

3. Jane ate 1/5 of the pie. Jim ate 2/5 of the pie. I ate 1/5 of the pie. How m

any portions of the pie did we eat altogether? 4. Jane ate 4/7 of the pie. I ate 2/7 of the pie. How m

any portions of the pie did we eat altogether? 5. Jane ate 2/8 of the pie. Jim

ate 1/8 of the pie. I ate 3/8 of the pie. How many portions of

the pie did we eat altogether?


For example:

in all, more,12 cm





eters plus 4 centim

eters equals 16centim

eters.

For example:

Collected, 1 half dollar, 2 dimes

50¢ + 20¢ = 70¢M

y answer is 70 cents, because if you see 'collected' in a problem

, it means

add, so I added 50¢ from the half dollar

and 20¢ from the two dim

es. 50¢+20¢ equals seventy cents.

153

Problem Solving Practice - Subtraction

• Find the key words and the numbers (the m

inuend and the subtrahend) in the story problem.

• Show the numbers with m

aterials or by drawing. • Calculate the answer. • W


• Use what you know about subtraction to explain your answer, using words and numbers.

1. There were twenty-four trading cards on the table. Later there were only four cards o the table. How m

any are missing?

2. There were thirty-three toy men in the box. Later there were sixteen toy m

en. How many

are missing?

3. There were eighteen cookies in the cookie jar. Later there were three cookies in the jar. How m

any are missing?

4. There were twelve sheep in the pen. Later there were only eight. How many are m

issing? 5. There were thirty-five birds in the cage. Later there were only twelve. How m

any are m

issing?








1. There were cookies and Popsicles for snack. Fifteen cookies were eaten. Seven Popsicles were eaten. How m

any more cookies than Popsicles were eaten?

2. There were twenty bananas and six oranges eaten for snack. How many m

ore bananas than oranges were eaten? 3. There were Eskim

o pies and fudgesicles for snack. Eleven Eskimo pies were eaten and

nine fudgesicles were eaten. How many m

ore Eskimo pies than fudgesicles were eaten?

4. There were cheese crackers for snack. Tamm

y ate twenty-six and Reggie ate thirteen. How m

ore did Tamm

y eat than Reggie?









1. Maria wanted to buy a vase. It cost $1.00 but she did not have enough m

oney. She only had $.75. How m

uch more did she need?

2. Dimone wanted an action figure. It cost eighty-seven cents. He only had seventy-eight

cents. How much m

ore does he need?

3. Helen wanted a book. It cost four dollars and ninety-five cents. She had three dollars and fifty cents. Did she have enough? How m

uch more did she need?

4. Marcy and Patty wanted to share a pizza. Together they had five dollars. The pizza

was eight dollars and forty-seven cents. How much m

ore did they need?

5. Matt wanted to ride the Ferris wheel. It cost $3.50. He only had $2.50. How m

uch m

ore did he need?








1. There were eight boys and seven girls and one boy left. Then eight boys came and

three girls left. Then six girls and nine boys came and four m

ore girls came and five

boys left. How many children were there then?

2. There were seven gulls, three pelicans and eight sandpipers on the beach. Four gulls flew away. Then six m

ore pelicans, nine more gulls and eight m

ore sandpipers cam

e. Then two pelicans left. Six more gulls and nine m

ore sandpipers came.

Seven sandpipers left. How many birds were left on the beach?

3. There were eight deer, three raccoons, five rabbits and two squirrels in the field. The eight deer ran away. Then six m

ore deer and nine more rabbits cam

e into the field. Four deer left and nine m

ore squirrels came into the field. Eight squirrels left.

Six more raccoons, two m

ore rabbits and six more deer cam

e into the field. How m

any animals were there in the field then?


Problem Solving Practice – Dynam

ic Subtraction







1. It is 2,438 miles to M

exico City. Jane has traveled 989 miles. How m

uch farthe m

ust she travel to get to Mexico City?

2. There were 2,487 milliliters in the tub. Bob took 1,031 m

illiliters out. How much

was left in the tub?

3. There were 649 gallons of water in the fishpond. Bill used a bucket to take 38 gallons out. How m

uch was left in the pond?

4. The elephant weighs 756 pounds. The hippo weighs 345 pounds. How much

more does the elephant weigh than the hippo?








1. Shawn had sixty minutes to stay at the pool on Friday. Thirty-five m

inutes have passed. How m

uch time does he have rem

aining?

2. Thomas had one hundred and twenty m

inutes to go on rides at King’s Dominion.

Fifty minutes have passed. How m

uch time does he have rem

aining?

3. Theresa had fifty-four minutes to shop with her M

om. Thirty-two m

inutes have passed. How m

uch time does she have rem

aining?

4. Betty has one hundred and eighty minutes to swim

at the pool. Sixty-two minutes

have passed. How much tim

e does she have remaining?

5. David must wait for his dad for thirty m

inutes. Twenty minutes have passed.

How much tim

e does he have remaining?


Problem Solving Practice - Memorization of Multiplication Facts

• Find the key words and the numbers (multiplier and multiplicand) in the problem.• Lay out the problems using geometric multiplication on graph paper.• Calculate the answer.• Write a number sentence.• Use what you know about multiplication to explain your answer, using words and numbers.

1. There are ten swim lesson groups at the pool. There are four children in each group. How manychildren are in swimming lessons in all?

2. There are two soccer teams. There are twelve kids on each team. How many children are soccerplayers?

3. There are four swim teams. There are eight children on each group. How many children are on swimteams?

4. There are nine scout patrols. There are five boys in each patrol. How many boys are there in patrolsin all?



1. There are eight children in a group. There is the same number in all nine groups. How many childrenare in groups in all?

2. There are three children in a group. There is the same number in all ten groups. How many childrenare in groups in all?

3. There are four children in a group. There is the same number in all eight groups. How many childrenare in groups in all?

4. There are nine children in a group. There is the same number in all three groups. How many childrenare in groups in all?

5. There are five children in a group. There is the same number in all six groups. How many children arein groups in all?


• Find the key words and the numbers (multiplier and multiplicand) in the problem.• Lay out the problems using geometric multiplication on graph paper.• Calculate the answer.• Write a number sentence.• Use what you know about multiplication to explain your answer, using words and


• Find the key words and the numbers (multiplier and multiplicand) in the problem.• Lay out the problems using geometric multiplication on graph paper.• Calculate the answer.• Write a number sentence.• Use what you know about addition to explain your answer, using words and


numbers.

1. There are eight cards in a set. James has fifteen sets. How many cards does he have in all?2. The bike race is twenty miles long. Hope was in the race three times. How many miles did she bike?3. There are two miles in each lap of the car race. The race cars do twenty laps in a race. How long isthe race?4. There are 367 cattle in a herd. There are five herds in the valley. How many cattle are in the valley?

numbers.

1. Pete brought boxes of paint to the art area three times. Each box contained eight bottles of paint. Howmany bottles of paint did Pete bring to the table altogether?

2. Maria brought boxes of binders to the classroom five times. Each box contained twelve binders. Howmany binders did Maria bring altogether?

3. Jane brought boxes of balls to the classroom four times. Each box contained five balls. How manyballs did Jane bring altogether?

4. Adrienne brought bags of potatoes to the kitchen three times. Each bag contained nine potatoes. Howmany potatoes did Adrienne bring altogether?



1. You have a seven square. How many beads are there in the square? What is seven squared?2. You have a nine square. How many beads are there in the square? What is nine squared?3. You have a cube of five. How many beads are there in the cube? What is five cubed?

Problem Solving Practice - Subtraction of Fractions

• Find the key words and the numbers (the minuend and the subtrahend) in the story problem.• Lay out the problems with the fraction insets.• Calculate the answer.• Write a number sentence.• Use what you know about subtraction and fractions to explain your answer, using words and numbers.

1. The Browns had pizza last night. The pizzas were divided into tenths. They had 8/10 left last night. They ate 4/10 today. How many tenths are left today?

2. The Kings had pie for supper last night. They 6/4 of the pies left last night. They ate 2/4 more today. How many fourths are left now?

3. The Blacks had pie last night. There were 12/4 of the pies left last night. They ate 6/4 today. How many fourths are left now?


4. You have a cube of ten. How many beads are there in the cube? What is ten cubed?5. You have a square of eight. How many beads are there in the square? What is eight squared?

4. The Greens had pie last night. There were 14/5 of the pies left last night. They ate 6/5 today. How many fifths are left now?

5. The Powell family had pizza last night. There 6/9 of a pizza left last night. They ate 6/9 today. How many ninths are left now?

Problem Solving Practice - Addition and Subtraction of Fractions

• Find the key words and the numbers (the minuend and the subtrahend) in the story problem.• Lay out the problems with the fraction insets.• Calculate the answer.• Write a number sentence.• Use what you know about math to explain your answer, using words and numbers.

1. The Browns had pizza last night. The pizzas were divided into tenths. They had 8/10 left last night. They ate 4/10 today. They bought another pizza with 10/10. How many tenths are left today?

2. The Kings had pie for supper last night. They 6/4 of the pies left last night. They ate 2/4 more today. They bought 2 more pies with 8/4. How many fourths are left now?

3. The Blacks had pie last night. There were 12/4 of the pies left last night. They ate 6/4 today. Then they bought another pie with 4/4. How many fourths are there now?

4. The Greens had pie last night. There were 14/5 of the pies left last night. They ate 6/5 today. They bought three more pies with 15/5. How many fifths are there now?

5. The Powell family had pizza last night. There was 6/9 of a pizza left last night. They ate 6/9 today. Then they bought two more pizzas with 18/9. How many ninths

Problem Solving Practice - Subtraction with US Money

• Find the key words and the numbers (the minuend and the subtrahend) in the story problem.• Lay out the problems with the money materials.• Calculate the answer.• Write a number sentence.• Use what you know about subtraction of money to explain your answer, using words and numbers.

1. Tim had $.75. He spent $.32. How much money does he have left?2. Jill had $4.36. She spent $1.25. How much money does she have left?3. Kate had $6.30. She spent $2.40. How much money does she have left?4. Jonece had $3.60. She spent $1.25. How much money does she have left?5. Casey had $5.17. He spent $2.34. How much money does he have left?


are there now?

Problem Solving Practice - Dynamic Subtraction with US Money

• Find the key words and the numbers (the minuend and the subtrahend) in the story problem.• Lay out the problems with the money materials.• Calculate the answer.• Write a number sentence.• Use what you know about subtraction of money to explain your answer, using words and numbers.

1. Tim had $15.82. He spent $3. How much money does he have left?2. Jill had $38.19. She spent $7.45. How much money does she have left?3. Kate had $11.86. She spent $2.49. How much money does she have left?4. Jonece had $43.05 She spent $12.78. How much money does she have left?5. Casey had $15.35. He spent $.75. How much money does he have left?

Problem Solving Practice - Subtraction with US Money

• Find the key words and the numbers (the minuend and the subtrahend) in the story problem.• Show the numbers with materials or by drawing.• Calculate the answer.• Write a number sentence.• Use what you know about subtraction of money to explain your answer, using words and numbers.

1. There were 3, 402 people on the beach at Ocean City. At lunchtime, 1,365 people left the beach. How many are left on the beach?

2. There were 3,081 cars parked in the garage at the airport. By nighttime, 2,436 cars had driven away. How many cars are left in the garage?

3. There are 6,843 people living in our beach town during the summer. During the colder months, 1,395 people go back to their houses in town. How many people stay in our beach town during the colder months?

4. Pete had $4,382 in the bank. He spent $1,649. How much does he have now?

5. There are 600 chocolate peanuts in the candy box. Marcy and Kevin ate 274. How many are left in the candy box.

Problem Solving Practice – Subtraction of Fractions

• Find the key words and the numbers (the minuend and the subtrahend) in the story problem.• Lay out the problems with the fraction insets.

Problem Solving Practice -Division

• Find the key words and the numbers (divisor and dividend) in the problem.• Lay out the problems using the golden beads and skittles.• Calculate the answer.


• Calculate the answer.• Write a number sentence.• Use what you know about subtraction of fractions to explain your answer, using words and numbers.

1. The Browns had pizza last night. The pizzas were divided into tenths. They had 7/10 left last night. They ate 2/10 today. How many tenths are left today?

2. The Kings had pie for supper last night. They 3/8 of the pies left last night. Someone ate another 1/8 today. How many eighths are left?

3. The Blacks had pie last night. There were 5/6 of the pies left last night. They ate 2/6 today. How many sixths are left now?

4. The Greens had pie last night. There were 5/9 of the pies left last night. They ate 4/9 today. How many ninths are left now?

5. The Powell family had pizza last night. There 7/7 of a pizza left last night. They ate 4/7 today. How many sevenths are left now?

• Write a number sentence.• Use what you know about division to explain your answer, using words and numbers.

1. There are 320 children. If there are 8 buses, how many equal groups of children will each bus have?2. There are 106 children. If there are 4 buses, how many equal groups of children will each bus have?3. There are 180 children. If there are 9 buses, how many equal groups of children will each bus have?4. There are 280 children. If there sre7 buses, how many equal groups of children will each bus have?5. There are 87 children. If there are 3 buses, how many equal groups of children will each bus have?

Problem Solving Practice- Division

• Find the key words and the numbers (divisor and dividend) in the problem.• Lay out the problems using the golden beads and skittles.• Calculate the answer.• Write a number sentence.• Use what you know about division to explain your answer, using words and numbers.

1. There are 42 apples. Seven apples are in each box. How many boxes are there?2. There are 64 apples. Eight apples are in each box. How many boxes are there?3. There are 63 apples. Nine apples are in each box. How many boxes are there?4. There are 18 apples. Six apples are in each box. How many boxes are there?5. There are 30 apples. Five apples are in each box. How many boxes are there?

Problem Solving Practice - Division


1. Mom shared 48 pencils among 6 children. How many pencils did each child get?2. Mom shared 54 pencils among 9 children. How many pencils did each child get?3. Mom shared 28 pencils among 4 children. How many pencils did each child get?4. Mom shared 50 pencils among 10 children. How many pencils did each child get?


5. Mom shared 35 pencils among 7 children. How many pencils did each child get?



1. There are 1,240 beads shared equally. If there are 4 jars, how many beads will each jar get?2. There are 3, 235 beads shared equally. If there are 5 jars, how many beads will each jar get?3. There are 865 beads shared equally. If there are 3 jars, how many beads will each jar get?4. There are 2,592 beads shared equally. If there are 8 jars, how many beads will each jar get?5. There are 966 beads shared equally. If there are 7 jars, how many beads will each jar get?



1. Kenny had 240 tickets for the raffle. He got everyone to buy a book of ten tickets. To how many people did he sell tickets?

2. Maria had 96 binders. She divided them into twelve equal groups. How many binders were in each group?

3. Jane brought 1,478 tiny balls to the room. She divided them into 43 equal groups. How many balls were in each group?

4. Adrienne had 869 potatoes. She separated them into 16 equal groups. How many potatoes were in each group?

5. Arthur had 2,227 apples. He separated them into 26 crates. How many apples were in each crate?



• Find the key words and the numbers (divisor and dividend) in the problem.• Lay out the problems using the large bead frame.• Calculate the answer.• Write a number sentence.• Use what you know about division to explain your answer, using words and numbers.

1. Twelve people pooled their money to buy a lottery ticket. They won the jackpot. How much money does each one have?

2. The college theater has 8,625 seats. There are an equal number of seats in each row. How many seats are in each row?

3. There are 7,686 addresses in Baxterville. They are divided evenly into zip codes. How many addresses are there in each zip code?

4. There are 6,936 people in Suntown. They are divided into equal voting precincts. How many people are in each voting precinct?


• Find the key words and the numbers (divisor and dividend) in the problem.• Lay out the problems using the stamp game.• Calculate the answer.• Write a number sentence.• Use what you know about division to explain your answer, using words and numbers.

1. Pete made twenty-six bundles of straws for the art supply box. There were 2,782 straws altogether. How many straws were in each bundle?

2. Alice made fifty-three bundles of sticks. There were 4,247 straws altogether. How many sticks were in each bundle?

3. Horace made eighteen stacks of cards. There were 7,392 cards altogether. How many cards were in each stack?

4. Linda made forty-seven groups of tulip bulbs. There were 9,783 bulbs in all. How many tulip bulbs were in each group?

5. Josh made twenty-four groups of pebbles. There were 6,013 pebbles in all. How many pebbles were in each group?

Measurement Activity Cards

Measurement - Length in Inches

* Find the key words (units of measurement) and the numbers in the directions.* Use a ruler to measure the object.* Record the measurement.* Report your results in a graph or a narrative when all your measurement is finished.

Measurement - Length in Nonstandard Units

* Find the key words (units of measurement) and the numbers in the directions.* Use a paper clip to measure the object.* Record the measurement.* Report your results in a graph or a narrative when all your measurement is finished.


1. Measure the length of 10 floor tiles in inches.2. Measure the width of 5 file cabinets in inches.3. Measure the height of 10 light switch plates in inches.4. Measure the length of 10 metal inset frames in inches.5. Measure the height of 3 chairs in inches.

1. Measure the length of 10 floor tiles in your feet. (Put your feet heel to toe.)2. Measure the width of 5 file cabinets in your feet.3, Measure the height of 3 light switch plates using the width of your hand.4. Measure the height of 4 chairs using the width of your hand.

Measurement - Word Problems

* Find the key words (units of measurement) and the numbers in the problem.* Use a ruler or a clock to calculate the measurement.* Record the measurement.* Explain your answer.

1. Mary looked at the clock. It said 4 o’clock. Later she looked up and saw that the clock said 4:25. How much time has gone by?

2. Peter needed to measure the couch. He laid down a foot long ruler 6 times end to end on the couch. What is the length of the table?

3. Andrew looked at the clock. It said 7 o’clock. Later he looked up and saw that the clock said 7:10. How much time has gone by?

4. Paul needed to measure the car. He laid down a foot long ruler 9 times end to end beside the car. What is the length of the car?

5. Ted looked at the clock. It said 8 o’clock. Later he looked up and saw that the clock said 8:15. How much time has gone by?

Measurement - Length in Inches and Feet

*Find the key words (unit of measure).*Use a ruler to measure the object.*Record the measurement.*Report your results in a graph or a narrative when all your measurement is finished.

1. Measure the length of the short-chain of 6 in inches.2. Measure the length of the short-chain of 8 in inches.3. Measure the length of the short-chain of 7 in inches.4. Measure the length of the long-chain of 5 in feet.5. Measure the length of the long chain of 9 in feet.


Measurement - Weight in Ounces

*Find the key words (unit of measure).*Use a balance scale to measure the object.*Record the measurement.

Report your results in a graph paper or a narrative when all your measurement is finished.For example: (Measure the weight of a box of crayons in ounces)box of crayons = 20 ouncesI measured the weight of a box of crayons using the balance scale and it weighed 20 ounces.

1. Measure the weight of 30 blue pegs in ounces.2. Measure the weight of 2 grammar boxes in ounces.3. Measure the weight of the 8-cube material in ounces.4. Measure the weight of the 9-cube material in ounces.5. Measure the weight of the 10-cube material in ounces.

Measurement - Weight in Pounds

*Find the key words (unit of measure).*Use a balance scale to measure the object.*Record the measurement.*Report your results in a graph paper or a narrative when all your measurement is finished.

1. Measure the weight of 4 grammar boxes in pounds.2. Measure the weight of 60 green pegs in pounds.3. Measure the weight of 2 books in pounds.4. Measure the weight of 3 pencil boxes in pounds.5. Measure the weight of 10 erasers in pounds.

Extension: Three children may go to the school nurse to get their weights. Record each person's weight on a chart. When the class finished recording their weights, make a graph showing the weight of each child.


Measurement - Measurement in Cups and Pints

*Find the key words (unit of measure).*Use the measuring cup to measure the object. Fill the measuring cup with water. Find how many cups ofwater would fill the object. Take note that 2 cups = 1 pint*Record the measurement.*Report your results in a graph paper or a narrative when all your measurement is finished.

1. Measure the capacity of a pitcher in pints.2. Measure the capacity of a water bottle in pints.3. Measure the capacity of a flower vase in pints.4. Measure the capacity of a milk carton in pints.5. Measure the capacity of a can of juice in pints.

Measurement - Estimating Perimeter and Area

*Find the key words(unit of measure).*Use an inch ruler to measure the object.*Record the measurement.*Report your results in a graph paper or a narrative when all your measurement is finished.

Note: To measure the perimeter, add the length of all the sides of the object.To measure the area, count the number of squares in the object.

1. Measure the perimeter of a clipboard.2. Measure the perimeter of the pegboard.3. Measure the perimeter of the checkerboard.4. Measure the area of a shape with 40 black and 25 white squares.5. Measure the area of a bathroom floor with 130 white and 260 blue square tiles.

Measurement - Word Problems

*Find the key words.*Show the numbers with the unit of measure.*Record the measurement.*Report your results in a graph paper or a narrative when all your measurement is

Measurement - Standard Units Inches (to the Half Inch)

* Find the key words (units of measurement) and the numbers in the directions.* Use a ruler to measure the object.* Record the measurement.* Report your results in a graph or a narrative when all your measurement is


finished.

1. Mom sewed a curtain 64 inches long. She added ruffles measuring 4 inches long. What is the length of the whole curtain in inches?

2. Tim will make a vegetable garden. Two sides of the garden will be 3 ft. long and two sides will be 6 ft. long. What will be the perimeter of Tim's garden?

3. A recipe needs 3 cups of milk. If I have a pint of milk, do I have enough milk for the recipe? Why or why not?

4. Last month, Donna's weight was 120 pounds. This month, her weight increased by 6 pounds. How much does she weigh at present?

5. Mary has a rug with a design of 40 green squares, 20 red squares and 80 white squares. What is the area of the rug in square units?

finished.

1. Measure the length of 10 metal inset frames in inches.2. Measure the length of 5 books in inches.3. Measure the length of 10 new pencils in inches.4. Measure the length of 10 markers in inches.5. Measure the length of 10 bead bars in inches.

Problem Solving Practice -Probability 1

Imagine the circumstances in the problem.Consider what you know about this situation.Draw a model of the problem.Write a prediction using the terms likely or unlikely in a sentence.

For example: The class is choosing someone for the special privilege of presenting the principal with a plaque for goodservice. Everyone is putting his or her name in the hat. There are 26 students and one teacher in theclassroom. Is it likely or unlikely that the teacher’s name will be chosen?

It is unlikely that the teacher’s name will be chosen, as there are so many more students than tteachers. There is only one chance in 27 that the teacher’s name will be chosen.

1. There are 500 daisies and10 jack-in-the-pulpits growing in a field. Is it likely or unlikely that a little girl will make a bouquet with more jack-in-the-pulpits than daisies?

2. There are 864 black horses and 435 white horses in the corral. Is it likely or unlikely that a black horses will be the first horse out of the gate?

3. There is a box of coins on the table. It contains 437 quarters and 129 pennies. Is it likely or unlikely that Tom will pick a quarter if he chooses from the box without looking?


4. Jane is bird watching. There are 754 finches and 75 orioles in the flyway. Is it likely or unlikely that Jane will see a finch?

5. There is a box of chips on the table. It contains 85 blue chips and 25 red chips. Is it likely or unlikely that you will pick a blue chip if you choose from the box without looking?

Problem Solving Practice -Probability 2

Imagine the circumstances in the problem.Consider what you know about this situation.Draw a model of the problem.Write a prediction using the terms equally likely, more likely, and less likely.

For example:The class is choosing a student for the special privilege of presenting the principal with a plaque for good service. Everyone is putting his or her name in the hat. There are 26 students including thirteen boys and thirteen girls. Is it equally likely, more likely, and less likely that a boy’s name will be chosen?

BBBBBBBBBBBBBBGGGGGGGGGGGGG

It is equally likely that a boy or a girl’s name might be chosen from the hat. This is because there is an equal number of boys and girls

1. There are 200 violets and 3 daffodils growing in a field. Is it equally likely, more likely, or less likely that a little girl will make a bouquet with more daffodils than violets?

2. There are 33 black goats and 4 brown goats in the barn. Is it equally likely, more likely, or less likely that a black goats will be the first goat out of the door?

3. There is a box of marbles on the table. It contains 247 red marbles and 17 yellow marbles. Is it equally likely, more likely, or less likely that Tim will pick a red marble if he chooses from the box without looking?

4. Harry is bird watching. There are 35 robins and 129 sparrows in the flyway. Is it equally likely, more likely, or less likely that Harry will see a robin?

5. There is a box of chips on the table. It contains 37 green chips and 92 white chips. Is it equally likely, more likely, or less likely that you will pick a green chip if you choose from the box without looking?


Problem Solving Practice - Probability 3

Imagine the circumstances in the problem.Consider what you know about this situation.Draw a model of the problem.Write a prediction using the terms certain, impossible, equally likely, more likely, and less likely.

For example:The class is choosing a student for the special privilege of presenting the principal with a plaque for good service. Every child is putting his or her name in the hat. There are 26 students including thirteen boys and thirteen girls. Is it certain or impossible that the teacher will be chosen?BBBBBBBBBBBBBGGGGGGGGGGGGIt is impossible that the teacher’s name will be chosen from the hat. This is because there are only students’ names in the hat.

1. There are 500 thistles, 900 dandelions, and 10 jack-in-the-pulpits growing in a field. Is it equally likely, more likely, or less likely that a little girl will make a bouquet with more jack-in-the-pulpits than thistles?

2. There are 864 black horses and 435 chestnut horses in the corral. Is it equally likely, more likely, or less likely that a black horses will be the first horse out of the gate?

3. There is a box of dimes and pennies on the table. It contains 23 dimes and 23 pennies. Is it certain, impossible, equally likely, more likely or less likely that Bill will pick a dime if he chooses from the box without looking?

4. Julie is bird watching. There are 754 thrushes and 75 bluebirds in the flyway. Is it equally likely, more likely, or less likely that Julie will see a thrush?

5. There is a box of blue and red pencils on the table. It contains 85 blue pencils and 85 red pencils. Is it certain, impossible, equally likely, more likely, or less likely that you will pick a blue or a red pencil from this box?

Problem Solving Practice – Probability 4 (Fairness and Equal Likelihood)

Imagine the circumstances in the problem.Consider what you know about this situation.Draw a model of the problem.Write a prediction using the terms certain, impossible, equally likely, more likely, and less likely.


For example:

Dr. Feeley is making a new board game with a spinner. There are three colors on the spinner: red, blue, and green. Each color is a third of the spinner. The colors determine on which space you place your skittle to move around the board. Will you have a fair chance of landing on red on your fist spin of the spinner?

1. Since each color on the spinner takes up an equal portion of the card, I will have a fair chance of landing on red.

2. There is a prize for the person who is the first to choose a gold card out of the hat. There are ten of each; gold, blue, and pink cards. Will Nathaniel have a fair chance at picking a gold card if all cards are put back in the hat after each child’s turn?

3. Mr. Feeley is making a new board game with a spinner. There are three colors on the spinner: yellow, blue, and green. Yellow takes up half of the spinner while blue and green each take up a fourth of the spinner. The colors determine on which space you place your skittle to move around the board. Will you have a fair chance of landing on green on your fist spin of the spinner?

4. There is a target at the class picnic for the beanbag toss. The hole for one hundred points is two inches wide. The hole for fifty points is four inches wide. The holes for twenty-five points are eight inches wide. Will you have a fair and equal chance of scoring one hundred points using this target?

5. Julie bought a lottery ticket. Fifty tickets were sold. Every ticket has a different number on it and there is only one winning number. No one could buy more than one ticket. Does Julie have a fair and equal chance of winning the lottery?


Measurement Activities Grades 4-6Measuring Length using Standard MeasuresMaryland 4th - 6th Grade Math Standards: Students will estimate or determine length to the nearest centimeter or 1/8

inch and measure length to the nearest millimeter or 1/16 inch using a ruler.

DEFINITIONS: Inch - An inch is a unit used for measuring length. An inch is this longHalf inch - A half inch is half the distance of an inch. There are two half inch measures in an inch. A half inch is this

longAbbreviations for the word inch are in. (l in.) or these marks "(1 ").

OBSERVATIONS: A ruler is divided into numbered segments called inches.

PRACTICE:1. There are half-inch segments in one inch.

2. Draw a ruler like the one on this card. Be sure to label inches with numbers and draw lines only for the half inches.

3. Use a classroom ruler to measure the line segments below.

a. b. c.d. e. f.

Measuring Length using Standard MeasuresMaryland 4th - 6th Grade Math Standards: Students will estimate or determine length to the nearest centimeter or 1/8 inch and measure length to the nearest millimeter or 1/16 inch using a ruler.

DEFINITIONS: Quarter Inch. A quarter inch is one fourth of an inch. There are four quarter inches in an inch. There are two quarter inches in a half inch. Two quarters is the same as one half inch.1/4 inch is this long2/4 inch is this long3/4 inch is this long4/4 inch is this long

PRACTICE:1. There are _____ quarter inches in an inch, there are ______ quarter inches in a half inch.

2. Draw a ruler like the one on this card. Be sure to label inches with numbers and draw lines for half inch and quarter inch segments.

3. Use a classroom ruler to measure or to draw the segments referenced below.a. b. c. d.e. 1 1/4 in f. 3/4 in g. 1 1/2 in h. 2 1/4 in.

4. Two other ways to say one inch are with the abbreviations _______ and _______.

5. The line that shows 2 quarter inches is the same as the line on the ruler for _____ inch.Measuring Length using Standard MeasuresMaryland 4th - 6th Grade Math Standards: Students will estimate or determine length to the nearest centimeter or 1/8 inch and measure length to the nearest millimeter or 1/16 inch using a ruler.

DEFINITIONS: Eighth Inch. An eighth inch is one eighth of an inch. There are 8-eighth inch spaces in an inch. There are 4-eighth inch spaces in a half inch. There are two eighth inch spaces in a quarter inch.


1 in. 2 in. 3 in. 4 in. 5 in.1/2 in 1/2 in

171

PRACTICE:1. There are _______ eighth inch spaces in an inch, _____ eighth inch spaces in a half inch.2. Draw a ruler like the one shown on this card. Be sure to label inches with numbers, and draw lines to show each

eighth inch segment.3. Use a classroom ruler to measure or to draw the line segments referenced below.a. b. c. d. e.f. 1 3/8 in. g. 5/8 in. h. 2 1/8 in. i. 4/8 in. j. 3 2/8 in4. The line on the ruler that shows two eighth inch segments is the same as the line for _____.5. The line on the ruler that shows four eight in. segments is the same as the line for _______.6. Six eights of an inch is the same distance as _____quarters of an inches.7.Eight eights equals _______ inch.


DEFINITIONS: Sixteenth Inch. A sixteenth inch is one sixteenth of an inch. There aresixteen sixteenth inch spaces in an inch. 8 sixteenths is a half inch.

PRACTICE1. Copy and complete the equivalents chart.2. Name each point marked with an arrow. Name as sixteenths. Give an equivalents fraction where appropriate.



OBSERVATIONSYou have learned how to use a ruler to measure inches and fractions of an inch. Sometimes you may want to measure an object when you do not have a ruler. Finding a part of your hand that is approximately equal to one inch, approximately equal to 1/2 inch, and approximately equal to 3 inches will help you estimate lengths up to 6 inches when you do not have a ruler.

PRACTICE EXAMPLE1. Trace your hand. Measure differentparts of your hand. On your drawing,label a part of your hand that is:a) approximately 1/2 inch longb) approximately 1 inch longc) approximately 3 inches long

2. Trace your hand again, label themeasurements of your hand that areshown in the brackets in the example.


OBSERVATIONS: A standard ruler can be used to record the exact length of an object in inches or parts of an inch. Follow these steps to get an accurate measure.1. Align the ruler next to the object you want to measure.2. Match the beginning of the ruler with the beginning of the object.3. Record the measurement you see at the end of the object as the object's length.

EXAMPLE: The nail 2 1/2 inches long.

PRACTICE:1. Measure the objects in the object box. Record the name of each object and it's length.2. Find some objects to measure in the classroom. Record each object's name and length.(examples of objects to measure : unit bead, large paper clip, pencil, sheet of paper, etc)


OBSERVATIONS: Sometimes you need to measure an object that is longer than a few inches.When you need to measure longer objects, standard units of length called foot and yard are used.

DEFINITIONS:Foot: A foot is a standard (customary) unit of length equal to 12 inches. Many rulers are 1 foot.Yard: A yard is a standard (customary) unit of length equal to 3 feet or 36 inches.One big step is about a yard, or from your nose to the fingertips of an outstretched arm.Length & Width The longest side of an object is its length, the shortest side is its width.Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools Le

ngth

Width

173

Abbreviations: Foot can be written as ft. or 1'. Yard can be written as yd.

PRACTICE: Copy and complete this chart.Predict (estimate), the length of the items below. Then measure to find the exact length.

ITEM PREDICTION (Estimate) ACTUAL MEASURETable LengthBook widthPencil lengthDoorway widthFile cabinet widthOne big stepYour nose to fingertipYour height


OBSERVATION: Standard or customary lengths longer than a yard are measured using a mile.Mile: A mile(mi.) is a customary unit of length equal to 5,280 ft. or 1,760 yds.Equivalents Table12 inches = 1 foot3 feet = 1 yd.1 yard = 36 inchesPRACTICE: Use the equivalents table and what you know about a mile to answer the questions. Be sure to copy and complete each statement. (See your teacher if you needhelp with this)

1. Copy the equivalents table. CHALLENGE QUESTIONS2. 2 feet = _____ inches 6. 1 1/2 feet = _____ inches3. 2 yards = ____ inches or ____ feet 7.48 inches = _____ feet4. 2 miles = ____ feet or ____ yards 8. 1 mile = ____ inches5. 1 mile is about how many "big steps?" _____ 9. 75 inches = ____ feet and ___ inches


OBSERVATION:* Find the key words, units of measurement and the numbers in the problem.* Show each step of your work.* Record the answer with the proper label.* Explain how you got your answer.

PRACTICE:

1. A large kangaroo can cover a distance of 25 feet in a single leap. How many yards and feet would that be?

2. Mom measured the windows and determined she would need 28 feet of fabric for the new curtains. How many yards and feet will she have to buy?

3. Joe counted 880 ‘big steps' to reach the end of the playground. What total distance in miles did he walk going to the end of the playground and coming back home.

4. A standard 36-inch yardstick was broken into two pieces. One piece was 12 inch long. How long was the other piece?


Lessons with Money

Naming Monetary Values

Prerequisites:This lesson is for the child who has had experience using the Stamp Game, where a quantity such as ten is represented symbolically by a single square.Materials:• a box containing real money, including a penny, a nickel, a dime, a quarter, a half-dollar, and a dollar bill• labels on which are written the corresponding values---1¢, 5¢, 10¢, 25¢, 50¢, 100¢Presentation:Using the three - period lesson format, introduce each coin and place the corresponding label next to it:1. “The penny equals one cent. The nickel equals five cents,” etc.2. Then, “Show me five cents. Show me ten cents” etc.3. Finally, scramble the labels and coins and ask the child to match them, first showing only heads, then only tails, then mixed.Extensions:Make booklets using the money stamp. Stamp the coins and write the monetary amount, copying from the labels, showing both heads and tails.Direct Aim:The student will learn the monetary value of our American currency.Indirect Aim:The student will be prepared for counting money.The student will be prepared for writing money notation.Control of Error:TeacherChart with each coin and its appropriate monetary value

Showing Equivalent Monetary Amounts Using Different Coins/Bills

Prerequisite:The child should have experience in linear counting, skip-counting, finding equivalent fractions and decimals.

.Materials:• classroom money kit containing at least one hundred pennies, twenty nickels, ten dimes, four quarters, two half dollars, and a dollar bill• separate, multiple labels showing 1¢, 5¢, 10¢, 25¢, 50¢, 100¢, equal signs (=)• money pocket chart (optional)

.Presentation:1. Review monetary amounts. Place the coins and matching labels, inserting = between them.Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 175

2. “We can show the same amount of money using different coins.” Invite the child to count with you as you place five pennies on the mat. Then place = 5¢ next to the row of pennies. Similarly, place count ten pennies and show = 10¢. Do the same with two nickels, five tens, ten tens, etc.

3. “Let’s find groups of coins that equal the same amount.” Then form the groups in a row, such as five pennies = one nickel =5¢, or ten nickels = five dimes = a half-dollar = 50¢, etc.

4. “Let’s find out how many different ways we can show the same amount of money.” This time a group of coins may be mixed to show an equivalence, such as two dimes, one nickel = one quarter = 25¢.

5.“Let’s find out how many of each kind of coin we need to equal one dollar.” Using the money pocket chart,place one dollar in the top row. On each subsequent row, use only one kind of coin to count to $1.00—e.g., twenty nickels, ten dimes, etc.

Extensions:6. Using the Decimal Board, show the parallel quantities.7. Point out that in money, we also use groups of five.8. Using the money stamps, record the work that was done on the mat.

Direct Aim:1. The student will demonstrate money equivalencies.2. The student will demonstrate the value of each coin in relation to one dollar.

Indirect Aim:1. The student will be prepared for counting random amounts of money2. The student will be prepared for “regrouping” coins to the least number of coins needed.Control of Error: Teacher

Naming Monetary Values Using the “$” SymbolPrerequisite:The child should have experience in linear counting, skip-counting, finding equivalent fractions and decimals.

Materials:• a box containing real money, including a penny, a nickel, a dime, a quarter, a half dollar, and a dollar bill• labels on which are written the corresponding values---$0.01, $0.05, $0.10, $0.25, $0.50, $1.00

Presentation:Using the three - period lesson format, introduce each coin and place the corresponding label next to it:1. “The penny equals one cent. The nickel equals five cents.”2. Then, “Show me five cents. Show me ten cents.”3. Finally, scramble the labels and coins and ask the child to match them, first showing only heads, then only tails, then mixed.

Extensions:Make Booklets/charts using the money stamp, stamp the coins and write the monetary amount, copying from the labels, showing both heads and tails.

Direct Aim: The student will learn the monetary value of our American currency

Indirect Aim:1. The student will be prepared for counting money;2. The student will be prepared for writing money notation.

Control of Error:TeacherMontessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 176

Chart with each coin and its appropriate monetary value

Counting Money and Naming the Value up to $5.00

Materials:• money kit containing at least 100 pennies, 20 nickels, ten dimes, four quarters, two half dollars, and a dollar bill• coin matching puzzle cards• money bingo• money dominoes

Presentation:1. Count each group of coins separately. “How much money do we have in pennies?”2. As the child counts from one to one hundred, guide him/her in placing the pennies into groups of ten on the mat. Or, simply count groups of ten pennies instead of going from one to one hundred. Then, finish by counting the groups of ten to find the total.3. Next, “How much money do we have in nickels?” The child skip counts by fives as he/she places the nickels on the mat and names the total.4. Continue, “How much money do we have in dimes?” The child counts by tens as he/she places the dimes on the mat and names the total.5. The teacher places a small mix of coins on the mat. “Let’s begin by counting the coins that are the greatest amount.” Guide the child in counting several such mixes of coins.

Direct Aim:The student will learn how to count money accurately.

Indirect Aim:The student will be prepared for writing money notation.

Control of Error: Teacher


Writing Money Notation

Materials:• a box containing separate, moveable pieces showing a dollar sign, a decimal point, several of each numeral from zero through nine• labels with monetary amounts written in words---e.g., “two dollars and sixteen cents”• a box containing a moveable cents sign; labels with monetary amounts under one dollar written inwords--e.g., “thirty-two cents”

Presentation:1. “Today we are going to focus on how to record money accurately.” Take a word label and read it: ”two dollars and sixteen cents.”

2. Using the manipulative, demonstrate that the dollar sign is placed first, then the number showing how many dollars. Place the decimal point to mean “and,” then show the cents to the right of the decimal point. Proceed in this manner with another label, until the child is ready to take over.

3.“When we have amounts that are less than one dollar, we sometimes show iit using a cents sign only.”Proceed as above using word labels and showing the correct amount with the manipulative. Point out theabsence of the dollar sign or the decimal point. Emphasize that the cents sign is placed to the right ofthe numerals.

Extensions:1. Introduce games such as the Coin Matching Puzzle Cards, Money Bingo, and Money Dominoes that require the student to count money accurately and match it to the amount in notation on a card or a domino.2. A box containing different objects labeled with different prices under $5.00, and money kit. Child can choose an object and find the corresponding price in coins.3. Prepared practice sheets on which are written the words and the child must write the corresponding notation.

Direct Aim:The student will write money notation using both the cents and the dollar sign.


Finding the Least Amount of Coins Needed

Materials:• a classroom money kit• labels with different amounts of money written on each and with the corresponding coins pictured on the reverse side

Presentation:


1. Review counting money. Place a mix of coins on the mat, such as three dimes, two nickels, and six pennies. Ask the child to count it aloud and name the amount: “forty-six cents!” Then ask, “How many coins did we use to show $ .46?” Eleven coins!

2. Say, “Let’s see if we can show $ .46 using fewer coins.” Based on previous lessons in showing equivalent amounts, suggest that we exchange two dimes and one nickel for one quarter. Let the child take over the process until finally there are one quarter, two dimes and a penny on the mat. “Now how many coins do we have to show $. 46?” Only four coins! Repeat using different amounts of coins.

3. Place a label with an amount indicated--e.g., $ .73. “Can you find coins to equal seventy-three cents?” Allow the child to place whatever combination of coins equals that amount--e.g., seven dimes and three pennies. “Yes! Seven dimes, three pennies equal seventy-three cents.” Count the coins out loud. Then, “You have used ten coins to show seventy-three cents. Now see if you can exchange / regroup some of them to use the least possible number of coins.” If the child hesitates, demonstrate that five dimes equal one half dollar. One half-dollar, two dimes, and three pennies equal seventy-three cents. “We only need six coins to show $ .73.”

4. When the child is ready to take over, place the other labels on the mat and challenge him/her to show each amount using the least number of coins.

Extensions:5. Child can write an amount, find the least number of coins to represent it, and write it as a sentence e.g., $.85 = one half dollar, a quarter, and a dime.6. Use the Money Stamps to record and label each amount.

Direct Aim: The student will show monetary amounts using the least number of coins needed.

Indirect Aim: The student will review the concepts of regrouping and equivalency using money.

Control of Error: TeacherThe reverse side of the money labels, the appropriate group of coins is pictured

Making Change: Purchases up to $10.00

Materials:• handmade rectangular making change chart with several rows: across the top row are the headings Cost, Amount Given, Penny, Nickel, Dime, Quarter, Half Dollar, Dollar Bill, Total Change• cards on which are written various monetary amounts (both rounded to exact dollar and mixed dollars and cents)• blank papers to place under total change• classroom money kit

Presentation:1. “Sometimes we don’t have the exact money to pay for things, so we have to give more than the itemcosts, and then we get change back. Today we are going to practice making change.”

2. Show the Making Change Chart. “Suppose the cost is $3.25.” Place the card showing $3.25 under theCost heading. “I don’t have that exact amount in my wallet, but I have four dollar bills.” Place the cardshowing $4.00 under the Amount Given heading. “The question is, how much money/change should Iget back? To find out, we can count from $3.25 to $4.00. We count from the cost and stop at the amountgiven.”

3. Demonstrate how this is done by counting aloud and placing appropriate coins under the variousheadings on the mat: “$3.25, $3.50.” Place a quarter under the quarter heading. “$3.50, $3.75,” asanother quarter is placed in that space say “$3.75, $4.00,” placing a third quarter in the quarter space.Montessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 179

“Now I have counted from $3.25 to $4.00. Let’s see what the total change is.” Slide the three quartershorizontally to the right so they are all in the space under the Total Change heading.

4. “Now let’s count the money in this space to find the total change: $.25, $.50, $.75.... Seventy—five centsis my change!” Record this among on a blank paper and place in under the Total Change heading,returning the coins to the Classroom Money Kit. Or, instead of sliding the coins to the right, leave them inplace on the mat. Count them and record the total. Continue as above until the child is confident intaking over the procedure.

Direct Aim: The student will learn how to count from the cost to the amount given in order to make change.


Fractions of a Dollar

Prerequisites:1. The child has had experience showing equivalent amounts of money and is familiar with the number of each coin needed to show one dollar.2. The child has had introductory lessons on fractions and knows the terms numerator and denominator.

Materials:• classroom money kit• blank labels on which to write 1/100, 1/20, 1/10, 1/4, 1/2• a calculator

Presentation:1. “Let’s review how many of each coin we need to equal one dollar.” Place a dollar bill at the top of themat. Then guide the child in counting ten rows of ten pennies--”100 pennies!” Below the pennies, countout the nickels up to $1.00, and so on with the dimes, quarters and half-dollars.

2. Then say, “Today we are going to think of each coin as a fraction of a dollar. How many pennies are in adollar?’’ “One hundred.” “Yes! So one hundred is the denominator because it shows the total amount ofpennies in a dollar.”

3. Use a blank label, draw a line in the middle, and write 100 under the line. Isolate one penny: “This pennyis one out of one hundred. It is one hundredth of a dollar. We put one in the numerator to show that it isone of a hundred.” Write the numeral 1 above the line on the label so the fraction is complete, showing1/100.

4. Continue in this fashion, demonstrating that a nickel is 1/20, a dime is 1/10, a quarter is 1/4, a half dollaris 1/2.

5. “Now let’s write fractions for more than one of a coin.” Isolate three pennies. “What is our denominator?”If the child hesitates, remind him/her that the denominator is the total number. ”The denominator is still100.” Use a blank label, draw a line in the middle, and write 100 under the line. Point to the threepennies. “What is our numerator?” “Three!” “Yes, we are showing three out of one hundred pennies.”Write 3 in the numerator and read the fraction: “three hundredths.” Continue in this fashion usingnickels, dimes, quarters, half dollars.


Direct Aim: The student will identify each coin as a fraction of a dollar.


Rounding Money to the Nearest Dollar

Prerequisite:1. The child needs to understand that $ .50 is half of a dollar and to be able to recognize amounts that are greater than or less than $ .50.2. It is necessary that the child have prior or parallel experience in rounding.

Materials:• money number line or several money number lines in different segments, such as from $5.00 to $6.00, with intervals of $ .10 ($5.00, $5.10, $5.20, $5.30, etc.)• classroom money kit with labels showing $1.00, $2.00, $3.00, etc. up to $10.00• cards showing several money amounts with dollars and cents up to $9.99--e.g., $1.36, $5.69

Presentation:1. “I have some money in my pocket.” Take it out and place it on the mat. “ABOUT how much money do Ihave?” Some children may count and state the exact amount- - e.g., “$5.80!”

2. “Yes, I have EXACTLY five dollars and eighty cents, but ABOUT how much money do I have? Is itABOUT five dollars or ABOUT six dollars?”

3. Place the money number line on the mat. “Find $5.80. Is it closer to $5.00 or to $6.00?” The children cansee that it is closer to $6.00. “We have rounded $5.80 to the nearest dollar. When we round to thenearest dollar, we drop the cents, but we must decide whether to round up or to round down. Let’s dosome other examples and see if you can figure out when we round up to the next dollar or down.”Present other examples, using the money number line.

4. Then make sure the rule is stated clearly: “When there is $ .50 (half of a dollar) or more, round up to thenext dollar. When there is less than $ .50, half of a dollar, round down.”

5. Place the labels which are rounded to dollars in a row across the mat: $1.00, $2.00, $3.00, $4.00, etc. upto $10.00. Take a card showing dollars and cents--e.g., $1.28. “If we round one dollar and twenty-eightcents to the nearest dollar, will that be $1.00 or $2.00?” Guide the child in figuring this, then place thecard showing $1.28 under the label $1.00. Continue using other cards showing dollars and cents,rounding them to the nearest dollar and placing them under the appropriate dollar amount.

Direct Aim: The student will round money to the nearest dollar, up to $10.00.

Control of Error: TeacherCorrect rounded amounts on the backs of the cards showing dollars and cents.

Adding Amounts of MoneyMontessori Curriculum Alignment- Mathematics- Grades 4, 5, 6Prince George’s County Public Schools 181

Prerequisite:The child should have experience with adding decimals with Decimal Board.

Materials:classroom money kitcards on which are written different money addition and subtraction problems blank slips of paper on which to record the sums

Presentation:1.) “Today we are going to put together different amounts of money to find the sum.” Select a card such as$ .56 + $ .32= . Invite the child to show those amounts using dimes and pennies from the Money Kit. Asin using the Golden Bead Material or the Stamp Game, demonstrate how to combine the units (pennies)first and then the tens (dimes):“$ .88!” Record the sum to complete the number sentence:$ .56 + $ .32 = $ .88

2.) Select a card which will require regrouping/exchanging, such as $ .27 + $ .65 = . Invite the child to show those amounts. Combine the pennies--twelve pennies. Demonstrate how to exchange ten pennies for one dime. Then combine and count the dimes: $ .92! Record the sum and complete the number sentence. Continue with several examples requiring regrouping into the one-dollar and ten-dollar places.

Extension:Show the child how to record the work vertically, lining up the decimal points and writing the $ in the sum.

Direct Aim:The student will add money and record the work.

Control of Error:TeacherCalculator

Subtracting Amounts of Money

Prerequisites:Subtraction using Decimal Board

Materials:Classroom money kitMaking change chart and labels

Presentation:1.) Review the steps for making change, counting up from the cost to the amount given. Then say, “Thechange is the difference between the cost and the amount given. We can also find this difference bysubtracting the cost from the amount given.”


2.) Begin with an example that does not involve regrouping/exchanging: “Suppose the cost is $ .53. I don’thave the exact coins, so I pay with three quarters, $ .75. I can figure the change, the difference, bysubtracting $ .53 from $ .75.”

3.) Invite the child to display $ .75. “Let’s exchange the three quarters for seven dimes and five pennies tocorrespond to the decimal places.” Proceed as with the Stamp Game to take away the units (pennies)first, then the tens (dimes). “Two dimes and two pennies! We have twenty-two cents left!”

4.) Use the Making Change Chart to demonstrate that twenty-two cents is also the Total Change whencounting up from the cost to the amount given.

5.) Record the number sentence: $ .75 - $ .53 = $ .22.

6.) Demonstrate subtraction as above, now using as example which requires exchanging/regrouping.

7.) Proceed as in doing subtraction with the Golden Bead Material or the Stamp Game. Record the number sentence.

Extension:Demonstrate how to write the subtraction work vertically, lining up the decimal points and writing the dollar sign in the difference.

Direct Aim:The student will subtract money accurately and record the work.

Control of Error:Counting up on the Making Change ChartCalculator


Probability - Let’s Get Popping With Math!!!

Activity 1 - Math Can Answer Your Questions!Look at the container. You will see a group of colorful pop beads. What mathematical questions come to your mind? Write as many mathematical questions as you can about the collection of beads.

Math can be used to help answer any of the questions you have in mind.

Activity 2 - Use Math To Estimate and Predict!Did you wonder how many pop beads are in one bag? Examine the bag without opening it. When you tell about how many there are, you are estimating. Write your estimate in a complete sentence.

Did you wonder which color had the most pop beads? When you guess which color has the most, you are making a prediction. Write your prediction in a complete sentence.

Now open the bag. Count the pop beads. How does the actual number of pop beads in the bag compare with your estimate?There are ____ pop beads in the bag. My estimate was _____ (for example, too high by 3, too low by 10, exactly correctly)

Activity 3 - Tallying and Graphing Can Help You See Results!Sort your pop beads by color. Tally your results on the chart below. When you tally, you make a mark for every one you count. First, write in the colors of the beads you found in your bag in the column on the left.

Colors of Pop Beads

Colors Tally


Was your prediction correct?


A pictograph uses pictures to show a number. Fill in your data on the pictograph below. Draw one pop bead in each row for each pop bead you have of that color. Be sure to label your pictograph and give it a title.

_______________________

________________________

Activity 4 – Interpreting Your Data Can Be Fun!

Look at your pictograph. You can find many interesting relationships within your data. Which color had the most? Which color had the least? Do any colors have the same number of pop beads? What is the difference between the highest and the lowest number on your pictograph? Do any two colors equal half of the total number of pop beads in the bag?

Look carefully at your pictograph. Write down as many observations about the relationships in your data as you can.

Activity 5 – Math Can Help You Compare Your Results!Did you wonder how many yellow pop beads were in each bag? Math can help you compare your results with those of your classmates. Choose two classmates to help you compare results. How many yellow beads were in each of the three bags?

Bag 1 ______ Bag 2 ______ Bag 3 ______

When you found out about how many yellow beads were in each bag, you are finding an average.Add together the total sum of the yellow pop beads in all three bags.Divide this sum by the number of bags (3).If your remainder is two or more, round up. If your remainder is one, round down.Show your work.


Compare your average with other groups of students in the class.

Choose another question about averages. You may decide to find the average of another color of pop beads. You may even choose to find the average number of pop beads in three bags. Formulate your question:

About how many _______________________ are in each bag?

Bag 1 ______ Bag 2 ______ Bag 3 ______

Remember to find the total sum and divide by the number of bags. Round up when your remainder is more than half the number of bags. Show your work.Compare your results with your classmates’.Write your findings in a complete sentence.

Activity 4: ReviewLook back at the activities that you read about and performed. Write the definition of these math skills:Estimating PredictionTallying PictographAveraging

Probability - Let’s Take a Chance

Activity 1: Certainty and Impossibility

When we say something is probable, we mean that it is likely to happen. It is probable that we will come to school on Monday morning. Write about another event that is probable.

Look in your bag of marbles. There are 11 marbles in your bag: 1 is purple, 3 are white, 3 are red, and 4 are yellow. Sort your marbles into groups of the same color. Draw a picture of your sorted marbles.

What is the probability that, without looking, you would pick a marble of each color. The probability of picking a purple marble is 1 out of 10. There is only 1 purple marble in a bag containing 11 marbles.

The probability that you’d pick a white marble is _____ out of _____

The probability that you’d pick a red marble is _____ out of _____

The probability that you’d pick a yellow marble is _____ out of _____

When something will definitely occur we say it is certain. For example, it is certain that you will pick a purple, white, red, or yellow marble from your bag. When something will definitely not occur we say it is impossible. For example, it is impossible for you to pick a green marble from your bag.

Explain why it is certain that you’ll pick a purple, white, red, or yellow marble but impossible that you’ll pick a green marble.


Activity 2: More Likely, Less Likely, Equally Likely

When you picked a marble from your bag without looking, you did not know what your result, or outcome, would be. You didn’t know if you would pick a purple, or white, or red, or yellow marble. By closing your eyes before picking, you made sure your pick was at random. By picking at random you made sure that each marble had the same chance to be chosen.

Look at the marbles in your bag. Do you think one color is more likely to be picked? Explain why.

Do you think one color is less likely to be picked? Explain why.

Do you think any two color are equally likely to be picked? Explain why.

Activity 3: Dependent and Independent EventsWhen you closed your eyes and picked a marble from the bag, each marble had the same probability of being picked. However, the probability changes as the number of possible outcomes changes.

Remove one marble from your bag and put it on the table. How many marbles are now in your bag? _____Now close your eyes and pick another marble out of the bag. What was the probability of picking this marble? _____ out of _____.Place this marble on the table. How many marbles are now in your bag? _____Now close your eyes and pick another marble out of the bag. What was the probability of picking this marble? _____ out of _____.Explain why the probability changes after you remove marbles from the bag.

Activity 4: ReviewLook back at the activities you completed. Write the definition for each of the words below.Probability CertainImpossible OutcomeRandom


Probability - How Do I Love Math? Let Me Count the Ways!

Activity 1 - Math Can Answer Your Questions!

Look at the boxes of conversation hearts. What mathematical questions come to your mind? Write one or two of the questions you would like to answer.


Activity 2 - Use Math To Estimate and Predict!

Did you wonder how many hearts are in one box? Take a minute now and examine a box of conversation hearts. When you guess how many hearts are in the box, you are estimating. Write your estimate in a complete sentence.

Did you wonder which color in the box had the most hearts? When you guess which color has the most, you are making a prediction. Write your prediction in a complete sentence. I predict that __________________________

Now open the box. Count the hearts. How does the actual number of hearts in the box compare with your estimate?There are ____ hearts in the box. My estimate was ______

Turn the page to the following activity to verify your prediction about the colors of the hearts.

Activity 3 - Tallying and Graphing Can Help You See Results!Sort your conversation hearts into colors. Tally your results on the chart below.

Colors of Hearts

PurpleYellowWhiteGreenOrangePink


Was your prediction correct?A pictograph is a way to display data that uses a picture to represent a number. Fill in your data on the pictograph below. Draw one in each row for each conversation heart of that color. Be sure to label your pictograph and give it a title.

Activity 4 – Interpreting Your Data Can Be Fun!

Look at your pictograph. You can find many interesting relationships within your data. Which color had the most? Which color had the least? Do any colors have the same number of hearts? What is the difference between the highest and the lowest number on your pictograph? Do any two colors equal half of the total number of hearts in the box?

Look carefully at your pictograph. Write down as many observations about the relationships in your data as you can.

Activity 5 – Math Can Help You Compare Your Results!Did you wonder how many yellow hearts were in each box? Math can help you compare your results with those of your classmates. Choose two classmates to help you compare results. How many yellow hearts were in each of the three boxes?Box 1____ Box 2____ Box 3____

When you find out about how many were in each box, you are finding an average. Add together the total sum of the yellow hearts in all three boxes. Divide this sum by the number of boxes (3). If your remainder is two or more, round up. If your remainder is one, round down.

The average number of yellow hearts in our three boxes was ______. Compare your average with other groups of students in the class.

Choose another question about averages. You may decide to find the average of another color of hearts. You may even choose to find the average number of hearts in three boxes.

Formulate your question: About how many _________________are in each box?

Compare your results with your classmates’. Remember to find the total sum and divide by the number of boxes. Round up when your remainder is more than half the number of boxes.Write your findings in a complete sentence. The average number of _____________________________ in ___boxes was_____________.

Write what you remember about these math skills in your math journal:


Estimating Predicting TallyingPictograph Averaging

Probability - Math Can Be Egg-citing!

Activity 1 - Math Can Answer Your Questions!Look at the collection of eggs. These eggs contain a variety of colorful jelly beans. What mathematical questions come to your mind? Write as many mathematical questions as you can about the eggs and jelly beans.


Activity 2 - Use Math To Estimate and Predict!Did you wonder how many jelly beans are in one egg? Select an egg of your choice. Examine it without opening it. When you guess how many jelly beans are in the egg, you are estimating. Write your estimate in a complete sentence.

Did you wonder which color had the most jelly beans? When you guess which color has the most, you are making a prediction. Write your prediction in a complete sentence.

Now open the egg. Count the jelly beans. How does the actual number of jelly beans in the egg compare with your estimate?There are ____ jelly beans in the egg. My estimate was _____.

Activity 3 - Tallying and Graphing Can Help You See Results!Sort your jelly beans by color. Tally your results on the chart below.Colors of Jelly BeansPurpleYellowWhiteGreenOrangePinkRed

Was your prediction correct?


VSC Standard 4.0 Knowledge of Statistics:A.1.a Collect data by conducting a survey to answer a question (Grade 4, 5 indicators)A.1.b Organize and display data in line plots and frequency tables using a variety of categories and sets of data. (Grade 4) (Grade 6 - A.1.a)B.1.a Interpret frequency tables (Grade 6 indicator)Directions:Study the sample below. Make a frequency table to display and organize data you collect.Show the teacher your survey question and your blank table before you ask students the question.

Favorite Type of Books (Genre) Frequency TableSurvey Question:What is your favorite type of book?

Interpretive and Summary SentencesStudents preferred fantasy over mystery and historical fiction. An equal number of students preferred mystery and historic fiction.

Steps to make a frequency table1. Write a survey question.

2. Draw a blank frequency table to record the data.3. Ask classmates the survey question and record each answer as a tally mark on the table.

4. Count the tallies and record the number in the frequency column.5. Write two sentences that interpret and summarize your results.

Vocabulary:Data – pieces of collected information Survey – collecting information by asking questions

VSC Standard 4.0 Knowledge of StatisticsA.1.f Determine the appropriate type of graph to effectively display data (Grade 5 indicator)

Directions:Read and copy the definitions and respond to the tasks.

Definitions of GraphsPictograph: uses pictures or symbols to show data

Line Graph: shows how data changes over timeBar Graph: uses bars to show data that can be countedLine Plot: compares data by showing clusters of information ona number lineCircle Graph: shows how parts of data relate to the total

Tasks: Tell which graph listed above would best be used to:

1. Display the temperature in degrees over a two week period of time.

2. Display the size in acres of six different parks in Prince George’sCounty.

3. Display the percent of a person’s budget used for various expenses.

4. Display each score from 1-20 received by students on a science test.

5. Display the number of people of 300 surveyed, who prefer various vacation destinations.

VSC Standard 4.0 Knowledge of StatisticsA.1.b Organize and display data in stem and leaf plots. (Grade 5 and 6)B.1.a Interpret and compare data in stem and leaf plots(Grade 5, Grade 6- B.1.c)Directions:Read the “stem and leaf plot” section in a math text to review steps to make a

VSC Standard 4.0 Knowledge of StatisticsA.1.b Interpret and compare data in line plots (Grade 4; Grade 5 A.1.c)B.1.a Interpret and compare data in line plots (Grade 4;Grade 5 - B.1.b)

Directions: Read about line plots in a math resource. Study the line plot below and respond to the tasks. Make a line plot for data set A and for data set


Item Tally FrequencyMystery //// 4Fantasy //// /// 8Historical Fiction //// 4None of Above // 2

stem and leaf plot. Respond to the tasks below.

Visitors over 10 days Tasks1. List in order all the two and three digit

numbers displayed at left.2. Which stem has the most leaves?3. Explain what the zero represents in the

leaf column.4. What was the largest number of

visitors recorded?5. Interpret the information with a summary statement.

Make a stem and leaf plot for each set of data below.Data Set A:Test scores: 77, 98, 65, 79, 82, 75, 93, 72, 72, 91, 68, 79, 100, 88, 72, 94, 96, 87

Data Set B:Height of plants in inches: 36, 27, 33, 42, 29, 30, 45, 35, 47, 20,103, 38, 26, 37, and 27

VocabularyStem:The tens digit in each number is a stem. Stems are arranged in order vertically

from least to greatest.Leaf:The ones digit in each number is a leaf. Leaves are arranged horizontally from left to right in the same row as the appropriate stem.

B.

Heights of Plantsx

x x xx x x xI I I I I 14 15 16 17 18Tasks

1. How many plants are 15 inches tall?2. What is the height of the smallest plant?3. What is the height of the tallest plant?4. Why are there no x’s over the 16?5. How would you summarize the data?

Make a line plot using each of these data sets.

Data Set A:Points scored by each player: 11, 14, 8, 10, 11, 12, 9, 8, 10, 11

Data Set B:Library books read last month: 5, 2, 0, 3, 3, 2, 4, 1, 6, 3, 2

VSC 4.0 Knowledge of StatisticsA.1.d Organize and display data in double bar graphs (Grade 5 indicator)B.1.c Interpret and compare data in double bar graphs (Grade 5 indicator)

1. Read information on bar graphs in a classroom resource.2. Find a double bar graph in a science or social studies text, newspaper, or

magazine.3. Interpret the data on the graph. Write summary sentences. Make a

prediction.4. Draw the graph.5. Label these items on your bar graph:a) Title

VSC 4.0 Knowledge of StatisticsA.1.e Organize and display data in line graphs (Grade 5 indicator)B.1.d Interpret and compare data in double line graphs (Grade 5 indicator)

1. Read information on line graphs in a classroom resource.2. Find a double line graph in a science or social studies text, newspaper, or

magazine.3. Interpret the data by identifying trends in the data. Make a prediction based

on the data.Remember, if the part of a line between two points is rising from left to right, the numbers are increasing. If the part of the line between two points is falling from left to right, the


Stem Leaf6 4,6,6,98 09 1,5,712 8

b) Keyc) Horizontal Axis Labeld) Vertical Axis Labele) Scale (includes numbers that span from least to greatest numbers on axis)f) Interval (difference between numbers on the axis)

data numbers are decreasing.4. Draw the graph.5. Label these items on your line graph:a) Titleb) Vertical Axis Labelc) Horizontal Axis Labeld) Interval (difference between numbers on an axis)e) Scale (includes numbers that span from least to greatest numbers on axis)

6. Write three questions that can be answered using the graph. Answer the questions.

Double Line Graphs


VSC Standard 4.0 Knowledge of Statistics4B. 1. d Interpret and compare data

in double line graphs. (Grade 5 indicator)

194

Definition Steps to Make a Double Line Graph A double line graph uses two different-colored lines

to compare changes over time or two different numeric values.

1. Choose two similar sets of data that you wish to compare.2. Choose an interval for each scale. Draw the graph. Label the

axes.3. Write a key for the two lines.4. Graph the data by making a point for each intersection of the

data for both items in your key/legend.5. Draw a line from each point to the next one in order for both

items on your key/legend.6. Title your graph.

Interpretive and Summary Sentences Examples of comparisons to use in a double line graph: In the double line graph above we compared the

amount of money spent on food and gas over a six month period.

The largest amounts of monies spent were in March for both food and gas.

During half of the months more money was spent on food, and the other half of the months it was the opposite.

Compare your absences from school with a friend’s over the course of a quarter.

Compare your television time with your reading time over the period of a week.

Compare your servings of fruits and vegetables with those of a classmate over the course of a week.

Compare the time you spent on reading and math with the time a friend spent on the same subjects, during the school week.


Double Bar Graphs

Definition Steps to Make a Double Bar GraphA double bar graph uses two different-colored or shaded bars to compare two similar sets of data that can be counted.

1. Choose two similar sets of data that you wish to compare.2. Decide on a scale and its intervals. Draw the graph. Label the

axes.3. Write a key for the two bars.4. Graph the data by drawing bars of the correct length or

height.5. Title your graph.

Interpretive and Summary Sentences Examples of comparisons to use in a double bar graph: In the double bar graph above, we compared the

points scored by the Robert Goddard boys’ and girls’ basketball teams over a three week period.

In each of the three weeks, the boys scored more points than the girls.

Week one shows the largest difference in point scores.

you and a friend's test scores for three separate weeks favorite basketball team's scores for last three games with scores

of your next favorite team for at least 3 games number of hours you and your friend watched television over a

period of at least 3 days number of pages you and a classmate read during the course of a

week


VSC Standard 4.0 Knowledge of StatisticsA. 1. d Organize and display data in

double bar graphs. (Grade 5 indicator)

B. 1. c Interpret and compare data in double bar graphs. (Grade 5 indicator)

Weeks

Points

Robert Goddard Playoff Scores

196

Circle Graphs

38%

Definition


VSC Standard 4.0 Knowledge of StatisticsB 1.e Read circle graphs (Grade 5

indicator)

ExtensionsA circle graph represents all (100%) of a set o

Read and analyze circle graphs (Grade 6 indicator)

197

mount vernon field trip, t - pgcps · web viewmontessori upper elementary math curriculum alignment...

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