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GRADE 5 MATHEMATICS
CURRICULUM GUIDE
Loudoun County Public Schools 2016-2017
Overview, Scope and Sequence, Unit Summaries, The First 20 Days Classroom Routines, Curriculum Framework, Learning Progressions
(additional attachments: Intervention Ideas, NCSM Great Tasks SOL alignment, Math Literature Connections)
Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the
Elementary Math Resources VISION site: http://loudounvision.net/. Search: Math—Elementary Resources; Enrollment key: MATH (all caps)
INTRODUCTION TO LOUDOUN COUNTY’S MATHEMATICS CURRICULUM GUIDE
This CURRICULUM GUIDE is a merger of the Virginia Standards of Learning (SOL) and the Mathematics Achievement Standards for Loudoun County Public Schools. The CURRICULUM GUIDE includes
excerpts from documents published by the Virginia Department of Education. Other statements, such as suggestions on the incorporation of technology and essential questions, represent the professional
consensus of Loudoun’s teachers concerning the implementation of these standards. This CURRICULUM GUIDE is the lead document for planning, assessment, and curriculum work.
NAVIGATING THE LCPS MATHEMATICS CURRICULUM GUIDE
The Curriculum Guide is created to link different components of the guide to related information from the
Virginia Department of Education, resources created by Loudoun County Public Schools, as well as vetted outside resources. To navigate the curriculum guide, click on the hyperlink (if in MSWord, hold the [ctrl] button and left click with the mouse on the
hyperlink). It will direct you to either another resource within the curriculum guide, or to a website resource. If you’re directed to a resource within the curriculum guide, to “go back,” hold the [alt] key and press the left arrow button.
Mathematics Internet Safety Procedures 1. Teachers should review all Internet sites and links prior to using it in the classroom. During this review, teachers need to ensure the appropriateness of the content on the site, checking for broken links, and paying attention to any inappropriate pop-ups or solicitation of information. 2. Teachers should circulate throughout the classroom while students are on the internet checking to make sure the students are on the appropriate site and are not minimizing other inappropriate sites. 3. Teachers should periodically check and update any web addresses that they have on their LCPS web pages. 4. Teachers should assure that the use of websites correlates with the objectives of the lesson and provide students with the appropriate challenge.
LCPS Grade 5 Mathematics Curriculum Guide 2016-2017
2009 Virginia SOL Testing Blueprint: Test Items by Strand Dates of LCPS Quarters
SOL Reporting Category 5th Grade SOL Number of Test Items (CAT)
NEW
Number of Items
Traditional
Number and Number Sense 5.1, 5.2a-b*, 5.3a-b 5 7
Computation and Estimation 5.4*, 5.5a-b*, 5.6*, 5.7* 9 13
Measurement and Geometry 5.8a-e, 5.9, 5.10, 5.11,
5.12a-b, 5.13a-b 8 12
Probability, Stats, Patterns,
Functions, & Algebra
5.14, 5.15, 5.16b-d, 5.17,
5.18a-d, 5.19 13 18
*Items measuring these SOL will be completed without the use of a calculator.
2016 – 2017 School Calendar
Starts Ends
First Quarter August 29 November 4
Second Quarter November 9 January 26
Third Quarter January 30 April 6
Fourth Quarter April 17 June 9
Quarter 1 Quarter 2 Quarter 3 Quarter 4
P = Teacher Workday/Planning Day H = Holiday/ No School F = First Day of School TI = Teacher Institute for new professionals NH = New Hire Workday SD = In School Staff Development days CS = County Wide Staff Development Days
LCPS Grade 5 Mathematics Curriculum Guide 2016-2017
Grade 5 Nine Weeks Overview
1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
Unit 1-Classroom Routines: “The First 20 Days Classroom Routines” NUMBER TALKS 5.4 Problem Solving (whole numbers) Unit 2-Whole Number Operations & Applications 5.4 Single- and Multi-step Practical Problems Using Whole Number Operations 5.3 Prime/Composite, Odd/Even 5.18 Variables, Expressions, and Equations 5.8a Perimeter and Area (whole numbers) Unit 4-Comparing & Applying Rational Number Concepts 5.10 Elapsed Time Unit 3-Patterns & Properties 5.19 Distributive Property 5.7 Order of Operations 5.17 Relationships in Numerical and Geometric Patterns
Unit 1-Classroom Routines: NUMBER TALKS 5.4 Problem Solving 5.10 Elapsed Time 5.15 Graphs Unit 4-Comparing & Applying Rational Number Concepts 5.1 Decimal Place Value and Rounding 5.2 Fraction/Decimal Equivalents, Comparing, Ordering 5.14 Probability & Sample Space 5.17 Relationships in Numerical Patterns (see unit summary) Unit 5-Rational Number Operations & Measurement Applications 5.6 Adding and Subtracting Fractions in Single- and Multi-Step Problem Solving, Simplest Form
Unit 1-Classroom Routines: NUMBER TALKS 5.4 Problem Solving 5.10 Elapsed Time 5.15 Graphs Unit 5 (cont’d)-Rational Number Operations & Measurement Applications 5.5 Decimal Operations 5.8 Metric & Customary Measurement, Perimeter/Area/Volume with Fractions and Decimals 5.17 Relationships in Numerical Patterns (conversions--see unit summary) Unit 6-Classifying & Subdividing Plane Geometric Figures 5.11 Angles 5.12 Angles and Triangles 5.13 Plane Figures, Combining and Subdividing 5.9 Circles Unit 7-Data & Statistics 5.16 Mean, Median, Mode, Range
Unit 1-Classroom Routines: NUMBER TALKS 5.4 Problem Solving 5.10 Elapsed Time 5.15 Graphs Unit 7-Data & Statistics 5.15 Stem and Leaf, Line Graphs 5.16 Mean, Median, Mode, Range Review for SOL Assessment & Post SOL Topics
48 days 45 days 48 days 39 days
LCPS Grade 5 Mathematics Curriculum Guide 2016-2017
Grade 5 Scope & Sequence Quarter 1: 48 days
Days Unit Standard Content Strand Topic All year Unit 1-Classroom
Routines
“The First 20 Days Classroom Routines” and NUMBER TALKS, Problem Solving, Elapsed Time, Graphs
24 Unit 2-Whole Number Operations & Applications
5.4 Computation and Estimation
Whole Number Problem Solving
5.3 Number and Number Sense
Prime/Composite/Even/Odd
5.18 Patterns, Functions, and Algebra
Variables, Expressions, and Equations
5.8a Measurement Perimeter and Area with Whole Numbers
15 Unit 3-Patterns & Properties
5.19 Patterns, Functions, and Algebra
Distributive Property
5.7 Computation and Estimation
Order of Operations
5.17 Patterns, Functions, and Algebra
Relationships in Numerical and Geometric Patterns
5 Unit 4-Comparing & Applying Rational Number Concepts
5.10
Measurement Elapsed Time
4 Assessment, Review, and Intervention
LCPS Grade 5 Mathematics Curriculum Guide 2016-2017
Quarter 2: 45 days
Days UNIT Standard Content Strand Topic All
year Unit 1-Classroom Routines
NUMBER TALKS, Problem Solving, Elapsed Time, Graphs
27
Unit 4-Comparing & Applying Rational Number
Concepts
5.1 Number and Number Sense
Decimal Place Value and Rounding
5.2 Number and Number Sense
Fraction/Decimal Equivalents, Comparing, Ordering
5.14 Probability and Statistics Probability and Sample Space
5.17 Patterns, Functions, and Algebra
Relationships in Numerical Patterns
14 Unit 5-Rational Number Operations &
Measurement Applications
5.6 Computation and Estimation
Adding and Subtracting Fractions in Single- and Multistep Problem Solving, Simplest Form
4 Assessment, Review, and Intervention
LCPS Grade 5 Mathematics Curriculum Guide 2016-2017
Quarter 3: 48 days
Days UNIT Standard Content Strand Topic All
year
Unit 1-Classroom
Routines
NUMBER TALKS, Problem Solving, Elapsed Time, Graphs
25 Unit 5 (cont’d)-Rational Number
Operations & Measurement Applications
5.5 Computation and Estimation
Decimal Operations
5.8 Measurement Metric and Customary Systems, Perimeter/Area/Volume with Fractions and Decimals
5.17 Patterns, Functions, and Algebra
Relationships in Numerical Patterns (conversions—see unit summary)
15
Unit 6-Classifying & Subdividing
Plane Geometric Figures
5.11 Measurement Angles
5.12 Geometry Angles and Triangles
5.13 Geometry Plane Figures, Combining and Subdividing
5.9 Measurement Circles
5 Unit 7-Data & Statistics
5.16 Probability and Statistics Mean, Median, Mode, and Range
3 Assessment, Review, and Intervention
LCPS Grade 5 Mathematics Curriculum Guide 2016-2017
Quarter 4: 39 days
Days UNIT Standard Content Strand Topic All
year
Unit 1-Classroom
Routines
NUMBER TALKS, Problem Solving, Elapsed Time, Graphs
11 Unit 7-Data & Statistics
5.16 Probability and Statistics Mean, Median, Mode, and Range (continued)
5.15 Probability and Statistics Stem and Leaf Plot, Line Graphs
28 Assessment, Review, and Intervention
SOL Tests
LCPS MATH Unit Summary Grade 5 2016-2017
Unit: 1 Quarters 1-4
Classroom Routines
VDOE Standards of Learning:
1st quarter: The First 20 Days Classroom Routines 5.4 The student will create and solve single-step and multistep practical problem involving addition, subtraction, multiplication, and division with and without remainders of whole numbers.
5.10 The student will determine the amount of elapsed time in hours and minutes within a 24-hour period.
VDOE Process Goals: Problem Solving, Communication, Connections, Reasoning, Representations
Learning Targets:
I can create, estimate, and solve addition, subtraction, multiplication, and division problems that have two or more steps involved in order to find the answer.
I can determine the elapsed time between two events within a 24-hour period.
Big Ideas Essential Questions
Classroom routines (about 10 minutes each day)
can be used to introduce, support, and extend
topics throughout the yearly curriculum in order to
provide students with regular practice in important
mathematical ideas.
Whole number operations in the context of
practical problems (integrated in a variety of
mathematical topics throughout the year).
Apply knowledge of number and number sense to
investigate and solve practical problems
Elapsed time within a 24-hour period (possibly
spanning from a.m. to p.m.) with the goal of
becoming a daily routine.
Can you describe the structure of the story problem? (for example: a part plus a part equals a whole in addition; a whole minus a part equals a part in subtraction; multiplication is repeated addition; division is repeated subtraction or fair shares)
What are some strategies for solving multistep story problems?
What are two strategies for determining elapsed time?
Prerequisite Skills Vocabulary
VDOE Vertical Alignment document 3-6
4.4 d) solve single‐step and multistep add/sub/mult problems with whole numbers 4.9 determine elapsed time in hours/min within 12‐hour period
VDOE Vocabulary Word Wall Cards
addend subtrahend minuend divisor dividend quotient remainder difference product sum part whole elapsed time hour, minute, second analog/digital clock
LCPS MATH Unit Summary Grade 5 2016-2017
Achievement Criteria How to Assess Achievement
“The content of the mathematics standards is intended to support the following five process goals for students:
• becoming mathematical problem solvers • communicating mathematically • reasoning mathematically • making mathematical connections and • using mathematical representations to model and
interpret practical situations.” -2009 Mathematics Standards of Learning
Click here for a brief audio PowerPoint slide with more information about the Process Goals
Pre and Post Unit Assessments for each standard
can be found in PowerSchool Assessment (formerly Interactive Achievement)
Sample Math Tasks are available in VISION:
Search: LCPS-Math: K-12 Math Tasks Enrollment key: MATH
NCSM Great Tasks
(available in all LCPS Elementary Schools—click link)
Classroom Routines
1st quarter: The First 20 Days Classroom Routines NUMBER TALKS: Example: http://www.mathsolutions.com/videopage/videos/Final/Classroom_NumberTalk_Gr3.swf Number Talks sample flipcharts available on the Elementary Math Resources VISION site INVESTIGATIONS: Measurement Benchmarks Estimation and Number Sense Guess My Number PROBLEM SOLVING: Math Playground’s Thinking Blocks (problem solving) ELAPSED TIME:
Randomly, throughout the school day, announce, “Start time!” and have students record the time. Later, announce, “End time!” and have students compute the amount of time elapsed. Do this activity daily for several weeks, and periodically throughout the weeks following elapsed time instruction.
Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the Elementary Math Resources VISION site: http://loudounvision.net/ Search: Math—Elementary Resources Enrollment key: MATH (all caps)
LCPS MATH Unit Summary Grade 5 2016-2017
Unit: 2 Quarter 1
Whole Number Operations & Applications
VDOE Standards of Learning:
5.4 The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers.
5.3 The student will a) identify and describe the characteristics of prime and composite numbers; and b) identify and describe the characteristics of even and odd numbers.
5.18 The student will Investigate and describe the concept of variable; Write an open sentence to represent a given mathematical relationship, using a variable; model one-step linear equations in one variable, using addition and subtraction; and create a problem situation based on a given open sentence, using a single variable. 5.8a The student will a) Find area and perimeter in standard units of measure for grid areas only (Volume will be introduced in 3rd quarter).
VDOE Process Goals: Problem Solving, Communication, Connections, Reasoning, Representations
Learning Targets:
I can create, estimate, and solve addition, subtraction, multiplication, and division problems that have two or more steps involved in order to find the answer.
I can demonstrate examples of prime, composite, even, and odd numbers using models, numbers, and words.
I can describe and write an open sentence (including a variable) to represent a mathematical relationship. I can then model solving an open sentence (limited to addition or subtraction) and create a problem situation based on a given open sentence.
The student will, given a problem situation, decide if the problem requires perimeter, area, and/or volume and estimate and then measure, using appropriate units, to solve the problem.
Big Ideas Essential Questions
Whole number operations in context of prob solving
Model prime/composite and odd/even numbers
Area and perimeter with whole numbers
Modeling Equations (solving equations is a 6th gr SOL)
Interpreting mathematical situations and models by using symbols and variables to interpret patterns
Division can be represented by sharing (partitive model) or grouping (quotitive model). For example:
Sharing (partitive)—Mrs. Smith has 225 pencils and wants to share them with her class. There are 25 students in her class. How many pencils will each student get?
Grouping (quotitive)—Mrs. Gomez has 288 pencils that come in packs (groups) of 12. How many packs of pencils does Mrs. Gomez have?
Partial products and partial quotients are strategies that allow for students to engage in number sense while using computation and the distributive property:(8 x 57 = 8 x (50 + 7) = (8 x 50) + (8 x 7) =400 + 56= 456)
How can you demonstrate, explain, and justify at least two ways to show a number is even or odd?
What is a prime number? a composite number?
How do prime and composite numbers compare and contrast?
What are strategies for estimating the sum, difference, product, or quotient of two numbers?
What is a variable? a variable expression?
What is an open sentence and what are its parts?
How do you express a word problem using an open sentence?
How are perimeter and area different?
Why is area expressed in square units?
Where would you use area and perimeter in your everyday life?
LCPS MATH Unit Summary Grade 5 2016-2017
Prerequisite Skills Vocabulary
VDOE Vertical Alignment document 3-6
4.4 d) solve single‐step and multistep add/sub/mult problems with whole numbers 4.5 a) determine common multiples/ factors 4.16 a) recognize/demonstrate meaning of equality in equation 4.3 a) read/write/represent/ID decimals through thousandths; b) round to whole, tenth, hundredth; c) compare/order; d) write decimal and fraction equiv from a model 4.6 a) estimate/measure weight/mass, describe results in U.S. Cust/metric units; b) ID equiv measurements between units within U.S. Cust system and between units within metric system
VDOE Vocabulary Word Wall Cards
prime composite even odd term expression algebraic expression whole number more than balance equation less than factor inverse operation open sentence symbol twice area variable perimeter rounding standard form word form square units expanded form addend subtrahend minuend dividend quotient remainder divisor product sum difference partitive (sharing) quotitive (grouping)
Achievement Criteria How to Assess Achievement
“The content of the mathematics standards is intended to support the following five process goals for students:
• becoming mathematical problem solvers • communicating mathematically • reasoning mathematically • making mathematical connections and • using mathematical representations to model and
interpret practical situations.” -2009 Mathematics Standards of Learning
Click here for a brief audio PowerPoint slide
with more information about the Process Goals
Pre and Post Unit Assessments for each standard
can be found in PowerSchool Assessment (formerly Interactive Achievement)
Sample Math Tasks are available in VISION:
Search: LCPS-Math: K-12 Math Tasks Enrollment key: MATH
NCSM Great Tasks
(available in all LCPS Elementary Schools—click link)
LCPS MATH Unit Summary Grade 5 2016-2017
Differentiation Resources
Have students enter large numbers in a calculator or a voice-supported word processing program that translates the number into words.
Have students use visual cues and/or mnemonic devices to help them remember place value order.
Have students write large numbers in a sand tray.
Have students color-code the different periods of the number line.
Have students work in groups to participate in a mock bank, using large-denomination play money.
Have students work in pairs to check each other when writing and reading large numbers.
Have students solve story problems using pictures, numbers and words.
Intervention Ideas (available in all LCPS Elementary Schools—click link)
ELL Model Performance Indicators
(click to link)
Investigations: Building on Numbers You Know (whole # operations) Mathematical Thinking at Grade 5: Investigation 1: Exploring Numbers and Number Relationships Measurement Benchmarks Investigation 1: Measures of Length and Distance Investigation 2: Measures of Weight and Liquid Volume Containers and Cubes Investigation 1: The Packaging Factory Investigation 2: Packing Problems Investigation 3: Measuring the Space in Our Classroom ESS Lessons: 5.4—Take a Trip 5.3—Sieve of Eratosthenes: An Ancient Algorithm 5.3—Partners and Leftovers 5.18—Variables and Open Sentences 5.8—Rolling Rectangles Learn Zillion: Multi-step word problems Find all the factor pairs of a number using area models Find all factor pairs using a rainbow factor line Find all factor pairs of a number using a t-chart Determine if a number is prime or composite using area models Algebraic expressions Brain Pop Prime Numbers Other: Illuminations: Factor Game Algebra Balance scales (NLVM) Math Literature Connections (click link) Hands-on Equations Book 1 (and Hands-on Equations Whiteboard software) Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the Elementary Math Resources VISION site: http://loudounvision.net/ Search: Math—Elementary Resources Enrollment key: MATH (all caps)
LCPS MATH Unit Summary Grade 5 2016-2017
Unit: 3 Quarter 1
Patterns & Properties
VDOE Standards of Learning:
5.19 The student will investigate and recognize the distributive property of multiplication over addition.
5.7 The student will evaluate whole number numerical expressions, using the order of operations limited to parenthesis, addition, subtraction, multiplication, and division.
5.17 The student will describe the relationship found in a number pattern and express the relationship.
VDOE Process Goals: Problem Solving, Communication, Connections, Reasoning, Representations
Learning Targets:
I can recognize the distributive property of multiplication over addition, identify examples of the property, and model the property using pictures, numbers, and words.
I can simplify expressions with more than two operations using the order of operations and explain each step.
I can describe the mathematical relationships found in patterns using symbols.
Big Ideas Essential Questions
Modeling and identifying the distributive property Partial products and partial quotients are strategies that allow for students to engage in number sense while using computation and reinforce the distributive property: (ie: 8 x 57 = 8 x (50 + 7) = (8 x 50) + (8 x 7) = 400 + 56 = 456) Order of operations (without exponents) **note that after parenthesis, multiplication/division is solved left to right, then addition/subtraction is also solved left to right. Input/output (function) table Using patterning as a problem solving tool Patterns represented and modeled in a variety of ways including numeric, geometric, and algebraic formats.
How does the distributive property relate to multiplication and addition?
Why would you use the distributive property?
How could the distributive property be used in computation strategies for whole numbers?
What is the order of operations?
Why is there a specific order of operations?
When would you use the order of operations?
How does the order of operations simplify an expression?
How can you use algebraic symbols to represent change in a pattern?
What are the connections between a pattern or function and the words, table, and symbols that represent it? How does an input/output table work?
Prerequisite Skills Vocabulary
VDOE Vertical Alignment document 3-6 4.16 b) investigate/ describe associative property for addition/multiplication 4.15 recognize/create/extend numerical/geometric patterns
VDOE Vocabulary Word Wall Cards distribute equality distributive property pattern function parentheses multiplication division addition subtraction function table evaluate input/output table relationship expression algebraic symbol equation sequence operation order of operations
LCPS MATH Unit Summary Grade 5 2016-2017
Achievement Criteria How to Assess Achievement
“The content of the mathematics standards is intended to support the following five process goals for students:
• becoming mathematical problem solvers • communicating mathematically • reasoning mathematically • making mathematical connections and • using mathematical representations to model and
interpret practical situations.” -2009 Mathematics Standards of Learning
Click here for a brief audio PowerPoint slide with more information about the Process Goals
Pre and Post Unit Assessments for each standard
can be found in PowerSchool Assessment (formerly Interactive Achievement)
Sample Math Tasks are available in VISION:
Search: LCPS-Math: K-12 Math Tasks Enrollment key: MATH
NCSM Great Tasks
(available in all LCPS Elementary Schools—click link)
Differentiation Resources
Have students follow the order of operations when solving multistep word problems.
Guess My Rule: Have students take turns giving the teacher a number to “input” on an input-output T-table. Using a predetermined rule, write the appropriate “output” number in the table. Next, provide an input number and ask a student who thinks he/she knows the rule being followed to determine the output—but do not allow the student to say the rule aloud. The student will prove that he/she knows the rule by being able to give the correct output. When most of the students seem to know the rule, have the students state the rule to the rest of the class.
Create a Venn diagram to explain the difference between a repeating pattern and a growing pattern.
How does the distributive property work with the problem 3 × 43? Use pictures and symbols.
Show how the distributive property works with variable expressions. Ask students how to write “3 times n + 1.” Lead them to see that this is written 3 × (n + 1). Ask students to represent this with their base-10 blocks, using a rod for n and a unit for one. Have them also represent this with symbols to help them see the connection between concrete and symbolic representations.
Intervention Ideas (available in all LCPS Elementary Schools—click link)
ELL Model Performance Indicators
(click to link)
Investigations: Patterns of Change: Investigation 1: Number patterns in changing shapes ESS Lessons: 5.19—Exploring the Distributive Property 5.7—Order Out of Chaos 5.17—Pick Your Pattern Learn Zillion: Find the rule for a function machine using a vertical table (and 9 other lessons) Input Output Brain Pop Intro to Krypto Illuminations Krypto Game Distributive Property Order of Operations Math Literature Connections (click link) Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the Elementary Math Resources VISION site: http://loudounvision.net/ Search: Math—Elementary Resources Enrollment key: MATH (all caps)
LCPS MATH Unit Summary Grade 5 2016-2017
Unit: 4 Quarter 2
Comparing & Applying Rational Number Concepts
VDOE Standards of Learning:
5.10 The student will determine an amount of elapsed time in hours and minutes within a 24- hour period.
5.1 The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.
5.2 The student will a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least
5.14 The student will make predictions and determine the probability of an outcome by constructing a sample space.
5.17 The student will describe the relationship found in a number pattern and express the relationship. *in the context of equivalent fraction tables and skip counting.
VDOE Process Goals: Problem Solving, Communication, Connections, Reasoning, Representations
Learning Targets:
I can determine the elapsed time between two events within a 24-hour period.
I can round decimal numbers expressed through thousandths to the nearest whole number, tenth, or hundredth, and represent my thinking using symbols, pictures, numbers, and words.
I can recognize and name fractions in their equivalent decimal form and compare and order fractions and decimals using models and multiple strategies.
I can make predictions and determine the probability of an outcome by using tools (spinners, number cubes, etc.) and models (sample space, number line, tree diagram, chart, etc.).
Big Ideas Essential Questions
Elapsed time in the context of fractions
Decimal place value, rounding decimals using a variety of strategies including number lines
Fraction/Decimal equivalents, Comparing, Ordering
Representing probability with fractions and sample spaces, including tree diagrams
Investigate and develop an understanding of number sense by modeling numbers, using different representations such as physical materials, diagrams, mathematical symbols, and word names
Apply knowledge of number and number sense to investigate and solve practical problems
Fractions and decimals both represent parts of wholes and students need to be flexible in how they view these numbers. Area models, set models, and linear models should all be used in addition to symbols and numerals in developing a conceptual understanding of fractions and decimals.
Patterns can be utilized as a strategy for finding equivalent fractions (equivalent fraction table):
3 = 3 I 6 I 9 I 12 I 15 (skip count by 3’s) 4 4 I 8 I 12 I 16 I 20 (skip count by 4’s)
What are two strategies for determining elapsed time?
Can you explain and justify various strategies for rounding decimal numbers?
How can you compare fractions that have the same denominator? Same numerator?
How can you compare fractions using a landmark number such as ½ or 1?
Why would you use estimation with rational numbers?
How can the same quantity be represented as both a fraction and a decimal?
How do you create and solve a practical problem using estimation of decimals and/or fractions?
How do you know that a fraction is in simplest form?
What are some strategies for adding and subtracting fractions with unlike denominators?
What are strategies for adding and subtracting mixed numbers and improper fractions?
What strategies are used to determine the sample space for an experiment?
LCPS MATH Unit Summary Grade 5 2016-2017
Prerequisite Skills Vocabulary
VDOE Vertical Alignment document 3-6 4.9 determine elapsed time in hours/min within 12‐hour period 4.1 a) ID orally/in writing place value for each digit in a whole number through millions; b) compare two whole numbers through millions w/symbols; c) round whole numbers through millions to nearest 10/100/1,000/10,000/100,000 4.2 a) compare and order fractions/mixed numbers; b) represent equivalent fractions; c)ID division statement that represents a fraction 4.13 a) predict the likelihood of simple event; b) represent probability as a number between 0 and 1
VDOE Vocabulary Word Wall Cards elapsed time probability hour, minute, second tree diagram analog/digital clock equally likely place value likely fraction unlikely numerator certain denominator uncertain simplest form/simplify sample space multiple outcome lowest terms decimal reduce decimal point greatest common factor base ten unit least common denominator tenth hundredth thousandth rational numbers improper fraction mixed number factor least greatest
Achievement Criteria How to Assess Achievement
“The content of the mathematics standards is intended to support the following five process goals for students:
• becoming mathematical problem solvers • communicating mathematically • reasoning mathematically • making mathematical connections and • using mathematical representations to model and
interpret practical situations.” -2009 Mathematics Standards of Learning
Click here for a brief audio PowerPoint slide with more information about the Process Goals
Pre and Post Unit Assessments for each standard
can be found in PowerSchool Assessment (formerly Interactive Achievement)
Sample Math Tasks are available in VISION:
Search: LCPS-Math: K-12 Math Tasks Enrollment key: MATH
NCSM Great Tasks
(available in all LCPS Elementary Schools—click link)
Differentiation Resources
Have students create elapsed time situations using television guides from newspapers and magazines. Have students trade problems to solve.
Allow students to continue to use the demonstration clocks as they use the timeline strategies.
Focus on computing elapsed time only in minutes. Then, focus on computing elapsed time in hours. Incorporate both hours and minutes once students are proficient in both separately.
Tape a decimal or fraction to the back of each student. Then have them ask the other students yes or no questions to try to figure out their number.
Investigations: Name that Portion Between Never and Always Measurement Benchmarks Investigation 3: It’s About Time ESS Lessons: 5.10—What Time is it? 5.1—Decimal Round-Up/Round-Down 5.2—Order Up! 5.14—It’s in the Bag
LCPS MATH Unit Summary Grade 5 2016-2017
Use a variety of models (area, set, linear) to represent fractions and decimals.
Have students look at recipes and convert the fractions used into decimals.
Students can make connections to science by using both decimals and fractions to measure things.
Have students create their own lesson to share with small groups of third or fourth graders. Have students develop a lesson plan as well as an activity sheet with a probability situation. Use a rubric to grade students on their performance.
Draw a number line on the board that starts with 0 and goes to 1. Label a few points on the line. Above the 0, write “impossible.” Above the 1, write “certain.” Give students strips of paper with situations on them. (You will see a dinosaur as you walk home today. It will get dark tonight. It will rain tomorrow.) Have them read their paper to the class and then place it where they feel it should go on the number line. Include situations that will fall at different locations on the line.
Intervention Ideas
(available in all LCPS Elementary Schools—click link)
ELL Model Performance Indicators (click to link)
Learn Zillion: 5.10 Identifying the start time, change of time, and end time in real-world elapsed time problems (and 5 subsequent lessons) 5.1 Round decimals to the nearest whole number using a number line (and 3 subsequent lessons) 5.2 Convert decimals to fractions to the tenths place using number lines (and three more lessons) Convert fractions into decimals to the tenths place (and 2 more lessons) Compare Decimals Using Fractions 5.14 Find the probability of a compound event by creating a tree diagram Brain Pop: Fractions to decimals Math Literature Connections (click link) Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the Elementary Math Resources VISION site: http://loudounvision.net/ Search: Math—Elementary Resources Enrollment key: MATH (all caps)
LCPS MATH Unit Summary Grade 5 2016-2017
Unit: 5 Quarters 2-3
Rational Number Operations & Measurement Applications
VDOE Standards of Learning:
5.6 The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form.
5.5 The student will a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit) b) create and solve practical problems involving decimals.
5.8 The student will a) find perimeter, area, and volume in standard units of measure c) identify equivalent measurements within the metric system d) estimate and then measure to solve problems, using U.S. Customary and metric units. e) choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units.
5.17 The student will describe the relationship found in a number pattern and express the relationship. *see Big Ideas for connection to measurement
VDOE Process Goals: Problem Solving, Communication, Connections, Reasoning, Representations
Learning Targets:
I can develop strategies and use them to compute the sum or difference of fractions and mixed numbers in practical single and multi-step problems.
I can investigate, create, and solve single and multi-step problems involving decimal operations (addition, subtraction, multiplication, and division; decimals through thousandths and divisors with only one nonzero digit).
The student will, given a problem situation, decide if the problem requires perimeter, area, and/or volume and estimate and then measure, using appropriate units, to solve the problem.
I can describe the mathematical relationships found in patterns using symbols.
Big Ideas Essential Questions
Number and operation sense and application to practical problems
Adding and subtracting fractions in the context of practical problems
Adding, subtracting, multiplying and dividing decimals
Measuring in the metric and customary systems, finding equivalent measures within the metric system
Perimeter /Area (rectangles, squares, right triangles)/Volume
Using fractions and decimals in the context of measurement situations
Conversions within systems of measurement can be modeled using a double number line, for example:
What strategies are used to estimate and compute addition and subtraction of fractions?
How are least common multiple and least common denominator important when adding and subtracting fractions?
How do you know if a fraction is in simplest form?
How can you use estimation strategies for the sum, difference, product, or quotient of two numbers?
What is the meaning of mathematical operations and how do these operations relate to one another when creating and solving single-step and multistep word problems?
How is multiplication and division of whole numbers and decimals alike and different?
LCPS MATH Unit Summary Grade 5 2016-2017
How can the distributive property help with computation involving decimals?
How might the distributive property help find perimeter? * 2 ( L + W ) *
Why are equivalent measures within the metric system useful? When is it appropriate to convert from one unit to measure to another?
What are three examples of objects measured by each unit of measure in both the U.S. Customary and metric systems?
What are the connections between a pattern or function and the words, table, and symbols that represent it?
How can numerical or algebraic symbols to represent change in a pattern?
Prerequisite Skills Vocabulary
VDOE Vertical Alignment document 3-6 4.5 d) solve single‐/multistep practical problems involving add/sub fractions and decimals 4.6 a) estimate/measure weight/mass, describe results in U.S. Cust/metric units; b) ID equiv measurements between units Within U.S. Cust system and between units Within metric system 4.15 recognize/create/extend numerical/geometric patterns
VDOE Vocabulary Word Wall Cards estimate measure weight mass customary measurement metric system length width volume feet inches yards meter liter gram gallon pint quart cup miles ounce pound
Achievement Criteria How to Assess Achievement
“The content of the mathematics standards is intended to support the following five process goals for students:
• becoming mathematical problem solvers • communicating mathematically • reasoning mathematically • making mathematical connections and • using mathematical representations to model and
interpret practical situations.” -2009 Mathematics Standards of Learning
Click here for a brief audio PowerPoint slide with more information about the Process Goals
Pre and Post Unit Assessments for each standard can be found in PowerSchool Assessment
(formerly Interactive Achievement)
Sample Math Tasks are available in VISION: Search: LCPS-Math: K-12 Math Tasks
Enrollment key: MATH
NCSM Great Tasks (available in all LCPS Elementary Schools—click link)
LCPS MATH Unit Summary Grade 5 2016-2017
Differentiation Resources
Bring in catalogs, sale flyers, and menus. Have the students use calculators to figure out the cost of dinner for their family. Have them choose gifts for their friends and family, given a budget of $75.00.
Have students complete all activities with a partner, taking turns being the recorder and measurer.
Focus on just one system at a time: first measure only in metric units, next, measure only in U.S. Customary units.
Intervention Ideas (available in all LCPS Elementary Schools—click link)
ELL Model Performance Indicators (click to link)
Investigations: Measurement Benchmarks Investigation 1: Measures of Length and Distance Investigation 2: Measures of Weight and Liquid Volume Name that Portion Fractions Investigation 2: 1-9 Decimals Investigation 3: 1-8 ESS Lessons: 5.5—Party Time 5.6—Enough Room? 5.8—Rolling Rectangles 5.8—Measurement Mania 5.17—Pick Your Pattern Learn Zillion: Improper and Mixed Numbers Add and Subtract Unlike Denominators Multiplying and Dividing Decimals Adding and Subtracting Decimals with Number Line Other: Math Playground’s Thinking Blocks (problem solving) Math Literature Connections (click link) Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the Elementary Math Resources VISION site: http://loudounvision.net/ Search: Math—Elementary Resources Enrollment key: MATH (all caps)
LCPS MATH Unit Summary Grade 5 2016-2017
Unit: 6 Quarter 3
Classifying & Subdividing Plane Geometric Figures
VDOE Standards of Learning:
5.11 The student will measure right, acute, obtuse, and straight angles.
5.12 The student will classify a) angles as right, acute, obtuse, or straight; and b) triangles as right, acute, obtuse, equilateral, scalene, or isosceles.
5.13 The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will a) develop definitions of these plane figures; and b) investigate and describe the results of combining and subdividing plane figures.
5.9 The student will identify and describe the diameter, radius, chord, and circumference of a circle.
VDOE Process Goals: Problem Solving, Communication, Connections, Reasoning, Representations
Learning Targets:
I can use a protractor or angle ruler to measure an angle in degrees and solve for a missing angle using addition or subtraction.
I can sort angles by their measures (acute, right, obtuse, or straight) and classify triangles by their characteristics (acute, right, obtuse, equilateral, scalene, or isosceles.
I can identify plane figures based on their definitions and describe how shapes change when they are combined with other shapes or divided into smaller parts.
I can identify and describe the relationship between the measures of the parts of a circle (radius, diameter, chord, and circumference).
Big Ideas Essential Questions
Measuring and classifying angles
Classifying triangles
Properties of plane figures
Combining and subdividing plane figures
Parts of circles
Developing knowledge about how geometric figures relate to each other and beginning to use mathematical reasoning to analyze and justify properties and relationships among figures.
Discovering geometric relationships by constructing, drawing, measuring, comparing, and classifying geometric figures.
Visualizing, drawing, and comparing figures help to develop an understanding of the relationships and to develop spatial sense.
What are the measures (in degrees) of acute, right, obtuse and straight angles?
What tools are used to measure and draw acute, right, obtuse and straight angles?
What processes are used to measure and draw angles?
How are triangles classified by the size of their largest angle?
How are triangles classified by length of side?
What is a plane figure? What are the characteristics of a plane figure?
What are the definitions of plane figures?
What happens when plane figures are subdivided or combined?
What is the chord, diameter, and radius of a circle?
What is the relationship between the radius of a circle and its diameter?
What is the relationship between the radius of a circle and its circumference?
LCPS MATH Unit Summary Grade 5 2016-2017
Prerequisite Skills Vocabulary
VDOE Vertical Alignment document 3-6 4.10 a) ID/describe representations of points/lines/line segments/rays/angles; b) ID representations of lines illustrating parallelism/perpendicularity 4.12 a) define polygon; b) ID polygons with 10 or fewer sides 4.11 a) investigate congruence of plane figures after transformations; b) recognize images of figures from transformations
VDOE Vocabulary Word Wall Cards point congruent vertex similar plane figures classify line line segment right triangle ray acute triangle polygon obtuse triangle quadrilateral equilateral triangle parallelogram isosceles triangle rectangle scalene triangle rhombus angle square circle trapezoid acute angle obtuse right angle subdivide obtuse angle combine straight angle straight edge diameter radius chord circumference protractor parallel perimeter area straight edge angle ruler
Achievement Criteria How to Assess Achievement
“The content of the mathematics standards is intended to support the following five process goals for students:
• becoming mathematical problem solvers • communicating mathematically • reasoning mathematically • making mathematical connections and • using mathematical representations to model and
interpret practical situations.” -2009 Mathematics Standards of Learning
Click here for a brief audio PowerPoint slide with more information about the Process Goals
Pre and Post Unit Assessments for each standard
can be found in PowerSchool Assessment (formerly Interactive Achievement)
Sample Math Tasks are available in VISION:
Search: LCPS-Math: K-12 Math Tasks Enrollment key: MATH
NCSM Great Tasks
(available in all LCPS Elementary Schools—click link)
LCPS MATH Unit Summary Grade 5 2016-2017
Differentiation Resources
Have students work individually or in groups on a project to construct scale models of their homes, while others may choose to compare the angles of triangles in different sized structures. Have all of the students present their projects to the rest of the class.
Use manipulatives including sets of fractals, pattern blocks, pinwheel tiling, platonic solids and more. Using such resources encourage your students to come to the front of the classroom and visually and physically engage with the stimuli on the interactive whiteboard.
Use string and/or geoboards to create the circle and parts of a circle.
Using the hands of a clock and a protractor, have students find the angle of the following times: 1:00, 2:00, 3:00, 4:00, 5:00, and 6:00. Is there a pattern?
When introducing types of triangles, focus on one parameter at a time. Have students identify triangles based on only angles first, then only on sides. Once students are comfortable with each, lead students to categorize triangles by both sides and angles.
Intervention Ideas (available in all LCPS Elementary Schools—click link)
ELL Model Performance Indicators
(click to link)
Investigations: Picturing Polygons
ESS Lessons: 5.11—Angles Are Everywhere 5.12—Triangle Sort 5.13—All Cracked Up 5.9—Human Circles Learn Zillion: Circumference Perimeter Area and Perimeter Brain Pop: Angles Similar Figures Circles Math Literature Connections (click link) Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the Elementary Math Resources VISION site: http://loudounvision.net/ Search: Math—Elementary Resources Enrollment key: MATH (all caps)
LCPS MATH Unit Summary Grade 5 2014-15
Unit: 7 Quarter 3 - 4
Data & Statistics
VDOE Standards of Learning:
5.15 The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots and line graphs.
5.16 The student will a) describe mean, median, and mode as measures of center b) describe mean as fair share c) find the mean, median, mode, and range of a set of data d) describe the range of a set of data as a measure of variation
VDOE Process Goals: Problem Solving, Communication, Connections, Reasoning, Representations
Learning Targets:
I can interpret data organized in line graphs and stem-and-leaf plots and answer experimental questions quantitatively.
I can calculate measures of center (mean, median, mode) and range and describe them as measures of center and variation.
Big Ideas Essential Questions
Further develop and investigate data collection strategies.
Collecting and organizing data into meaningful representations based on issues related to practical experiences.
Identifying and justifying comparisons in data representations and communicating the interpretation of data.
Recognizing data analysis methods as powerful means for decision making.
Demonstrating mean as a fair share visually with manipulatives.
Using mean, median, mode, and range in the context of other characteristics of the data in order to best describe the results.
How does data collected help determine the type of representation used?
What are the measures of center and what do they measure in the data set?
How does range measure variation in data?
How can you model mean as a “fair share”?
Can you determine the mean of a set a data represented in a stem-and-leaf plot? Median? Mode? Range?
Prerequisite Skills Vocabulary
VDOE Vertical Alignment document 3-6 4.14 collect/organize/display/interpret data from a variety of graphs
VDOE Vocabulary Word Wall Cards stem-and-leaf plot measures of center line graph mean as a fair share line plot mode bar graph median variation range data collect organize display interpret representation
LCPS MATH Unit Summary Grade 5 2014-15
Achievement Criteria How to Assess Achievement
“The content of the mathematics standards is intended to support the following five process goals for students:
• becoming mathematical problem solvers • communicating mathematically • reasoning mathematically • making mathematical connections and • using mathematical representations to model and
interpret practical situations.” -2009 Mathematics Standards of Learning
Click here for a brief audio PowerPoint slide with more information about the Process Goals
Pre and Post Unit Assessments for each standard
can be found in PowerSchool Assessment (formerly Interactive Achievement)
Sample Math Tasks are available in VISION:
Search: LCPS-Math: K-12 Math Tasks Enrollment key: MATH
NCSM Great Tasks
(available in all LCPS Elementary Schools—click link)
Differentiation Resources
Have students work individually or in groups of two or three to collect data on topics of their choice (e.g., favorite sandwich choice, favorite fast food restaurant, favorite ice cream flavor, favorite after school activity). Have them first organize the data they collect by using tally tables or charts. Then, have them create appropriate bar graphs to display the organized data. Instruct them to write two or more questions that can be answered by reading the graphs.
Challenge students to find different types of graphs (e.g., bar, line, circle or pie) in magazines, newspapers and various other publications. Have them create a poster or display of the graphs they find and write an explanation of the use of each one under it.
Provide pairs of students with graphs from newspapers, magazines, and textbooks for them to use in discussing the various ways such information is displayed and the advantages of each.
Allow students to draw pictures instead of writing in their journals to summarize this activity.
Use large sized graph paper.
Intervention Ideas (available in all LCPS Elementary Schools—click link)
ELL Model Performance Indicators (click to link)
Investigations: Patterns of Change Investigation 2: Motion Stories, Graphs, and Tables Investigation 3: Computer Trips on Two Tracks Data, Kids, Cats, and Ads Investigation 1: Balancing Act Investigation 2: Examining Cats ESS Lessons: 5.15—Mystery Data 5.16—What Does It Mean? Learn Zillion: Mean Range Math Literature Connections (click link) Additional resources (including FLIPCHARTS based on VDOE ESS lessons) may be found on the Elementary Math Resources VISION site: http://loudounvision.net/ Search: Math—Elementary Resources Enrollment key: MATH (all caps)
The First 20 Days Classroom Routines: Establishing a Mathematics Classroom Community
Overview: The mini lessons included in this guide are intended to be used in conjunction with your first unit of study. The daily 10-15 minute lessons will help
you set routines, develop references for students, establish protocols, and create norms for an engaging math classroom community. The lessons may be
modified or extended based on students’ need or grade level. The routines, protocols, and experiences should be revisited throughout the school year in order
to maintain a productive math community.
Goals:
Build a classroom community of learners
Support students’ understanding of math content by establishing guidelines related to the VA process goals (problem solving, communication,
reasoning, connections, and representation).
Develop routines that will help students become reflective problem solvers and engage in a rigorous study of mathematics.
Background: This guide is based on a document developed by Austin Independent School District. Their document was modeled after the First 20 Days of Independent Reading by Fountas & Pinnell. Many of the suggested routines will also connect to other effective protocols used in Being a Writer and Responsive Classroom. This guide was adapted from a resource created by Arlington Public Schools.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 1 Management: Classroom Procedures/ Community Guidelines
Establish routines, procedures, and student expectations for daily math lessons.
Students develop criteria for a “Being a Mathematician” chart that will be posted in the classroom. Students understand that the information posted in the classroom will be a valuable reference for them.
Develop a “Being a Mathematician” anchor chart to which students can refer. The chart should have less than 6 criteria to be effective and manageable. Example behaviors:
• Remain on task • Participate/stay engaged • Listen actively • Discuss math ideas • Treat materials with respect • Always try your best
*Brainstorm the list with the students
Chart paper, Markers Have a discussion about routines and procedures with the students. This is a good time to have students talk about expectations for engaging in classroom discussions and completing their work.
Day 2 Management: Mathematical Tools VA Process Goals: Problem Solving & Representation
Mathematicians can utilize math tools to help them solve problems.
Tools are a valuable resource for mathematicians. Students are aware of the tools that are available in the classroom.
Brainstorm a list of mathematical tools and discuss how they can be used and stored. Add additional information to the “Being a Mathematician” chart about placing materials in their proper storage containers and location after use. Examples: Base ten blocks Cubes Number cubes Hundreds chart Two-colored counters
Emphasize how and why materials are to be used during math instruction.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 3 Math Talk/ Classroom Discourse VA Process Goal: Communication
Mathematicians communicate orally about their work. Norms for classroom discussions need to be established in order to engage in respectful discourse and have equitable participation.
In order to communicate and learn from each other, mathematicians must listen to, as well as speak, with their classmates. We will function as a respectful classroom community in order to learn.
Create an anchor chart for “Norms for a Math Discussion” or “Rights and Obligations During Discussions” Example norms include: Speak respectfully Take turns (equitable participation) Give others time to think Eyes on the speaker
Norms may be similar to those you establish in other content areas. These established routines should be revisited all year long.
Day 4 Math Talk/ Classroom Discourse VA Process Goal: Communication
Mathematicians communicate orally about their work. Different talk moves can be used while facilitating classroom discussions. Students learn content through the process goal of communication.
Math can be more rigorous when you communicate with others. There are sentence starters that can be used to help one engage in discussions.
Post and discuss Talk Moves to encourage students to share their thinking. Identify 1 or 2 moves to begin the year with (based on your first units of study).
- Talk bubbles or Talk move sticks
Introduce talk moves
- Turn and Talk (also called partner talk, or think-pair-share)
- Say More: You ask an individual student to expand on what he or she said
- Revoicing (also called verify and clarify)
- Repeat - Agree/Disagree and why?
Encourage students to speak in complete thoughts when communicating orally. The utilization and introduction of talk moves is a continuous process. This day is one way to introduce moves, but it should be ongoing.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 5 Collaboration (Game) VA Process Goal: Communication
Mathematicians can work collaboratively while playing a game in order to learn important math concepts.
Students understand that they can work with others to explore math content. Cooperation is a key component of working with a partner.
Establish rules for working with a partner while playing a math game. Try the “Say it to Play it” guideline: When playing a game in partners, the students must state their move and/or provide an explanation for why they are playing that move (Ex: In the game Compare, a student may say “9 is greater than 5, so I win the cards”).
Post rules and directions for engaging in a game with a partner. Consider utilizing a fact fluency game for this mini lesson.
Rules and clear directions will help make group work successful. After the mini lesson, have students practice a game during the math lesson for the day.
Day 6 Collaborative/ Independent Work (Rotations) VA Process Goal: Communication
Mathematicians can explore/ engage in a variety of experiences within a math period. Work may be collaborative or independent.
In order to have a variety of activities during a math block, it is important to be mindful of procedures, noise level, expectations, etc.
Review procedures for moving around the classroom to different centers Consider utilizing visual time reminders Use cues for sound control/reminders
Post clear directions at independent centers. Provide a materials checklist.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 7 Real Life Connections to Math VA Process Goal: Connections
Mathematicians make connections between math ideas and the world around them.
Math connects to other content areas/disciplines (i.e. Science). Students relate math to the world around them.
Brainstorm a list of math concepts that relate to the real world. Consider using the following discussion prompts: Where in the world do you see numbers? When do you use math in your everyday life?
Chart paper Calendar / Daily Schedule
Consider connecting this discussion to everyday events in their life.
Day 8 Representing Thinking VA Process Goals: Representation, Communication
Mathematicians can represent ideas in multiple ways. Mathematicians use words to explain their thinking. Mathematicians can explain their thinking verbally or in writing in order to process information.
Students will become more familiar with ways they can represent math ideas. Students can show their math thinking in written words.
In order to fully communicate their understanding, mathematicians may provide written explanations of their reasoning.
Brainstorm ways that students can represent their thinking. Ex: Pictures/drawing Words Numbers Symbols Manipulative models
Utilize sentence frames: “This is a ______________. It is a ______ because it ______________. “ This example shows a picture, numbers, and a written explanation.
Encourage students to show math concepts in a variety of ways. Encourage students to write about their understanding or show their thinking using words, pictures, numbers, etc.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 9 Recording & Reflecting in Math
VA Process Goal: Communication
Mathematicians keep a record of their daily experiences (i.e. math game).
Students will understand how to utilize a recording sheet or guide as they play a game or solve a problem. Students will record and reflect upon their work to communicate their understanding in writing.
Example of Game Recording Sheet:
Introduce a Recording Sheet as a student tool.
Day 10 Academic Language of Math VA Process Goal: Communication
Specialized language is used in math. Mathematical language can be modeled and explicitly taught.
Students will develop an understanding of specific math terminology. Conceptual understanding is developed as students use math terminology.
Post examples of key vocabulary terms with visual examples.
Math Word Wall, Word Banks, VDOE Vocabulary Cards http://www.doe.virginia.gov/instruction/mathematics/resources/vocab_cards/index.shtml The vocabulary terms introduced are then posted for class reference.
New vocabulary should be explicitly introduced and utilized within daily lessons. This is a continuous routine/ element for all units of study.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 11 Vocabulary Development VA Process Goals: Representation, Communication, Connections
Mathematicians use a variety of strategies to build vocabulary.
Students will utilize a tool to reinforce their math vocabulary.
Select model to implement with students (i.e. Frayer).
Student math journal VDOE Math Vocabulary Cards Frayer Model
Students can utilize math journals to keep a record of math vocabulary. Their journals can also serve as a valuable resource in addition to the Word Wall or class references (see Day 10).
Day 12 Math Strategies VA Process Goals: Problem Solving, Representation, Connections
A variety of strategies can be used to solve problems and explore mathematical concepts.
Students develop a repertoire of strategies. Students see connections between different strategies used to solve problems.
Build or add to a strategy wall showing models of strategies for various skills or concepts.
Anchor charts can be developed for a wide variety of strategies depending on the grade level. Examples are shown to the left.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 13 Connections VA Process Goals: Connections, Communication
Mathematicians make and recognize connections among mathematical ideas.
Students understand that they can make connections among math ideas. Math can be related to the world outside the classroom.
Discussion questions: How is that answer like the one you modeled yesterday? Where have you seen that before?
Consider having students glue question/ comment starters in the back of their math journal. They can refer to it during class discussions.
Day 14 Justification VA Process Goals: Reasoning, Representation
Mathematicians verify their thinking by showing it multiple ways.
Students will develop a deeper understanding of content when asked to justify their thinking.
Create an anchor chart that depicts ways that students can justify their thinking.
Justify means: explain, defend,
describe, prove, give reasons, show
you understand, validate…
Using verbal explanation first can help facilitate written justification.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 15 Problem Solving Strategies VA Process Goals: Problem Solving, Communication
Mathematicians choose from a variety of strategies to solve problems.
Students have a resource of strategies to help them solve problems. Sample strategies: -find a pattern -estimate and check -make an organized list -draw a diagram -write an equation -work backward -solve a simpler problem -read a table/chart
Introduce problem solving strategies (a variety of strategies can be used). Explain that the different strategies can be used to help students with problem solving. Choose 1 strategy to explain/highlight for the mini lesson. You will continue to model/introduce/use the strategies throughout the year.
Students can create their own problem solving strategy icons or bookmarks as well as refer to a class anchor chart of strategies.
During classroom instruction, teachers can engage students in discourse about their problem solving strategy.
Day 16 Problem Solving Protocol VA Process Goals: Problem Solving, Communication
There are processes that can be used to help solve problems.
Students will be introduced to a problem solving protocol. Students will become familiar with the protocol steps.
Develop and post a problem solving protocol.
Post the protocol in the classroom for student reference.
Consider trying a problem as a class to model how the protocol is used. The emphasis should be on the steps, so it may be easiest to select content that is readily accessible to all learners.
Step 1: Read and quietly think on your own – release your pencils. Step 2: Talk about the problem. What is your plan to solve? Pick your strategy. Step 3: Share your strategy. Step 4: Solve the problem and communicate your thinking.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 17 Rubric Familiarization VA Process Goals: Reasoning, Connections
There are tools mathematicians use to monitor and assess their work or behavior.
Students understand how to use a rubric to assess themselves/ their work.
Create a class rubric that is not math related. The topic should be something relevant to an everyday student activity in the classroom or school. Examples include: Lunchroom behavior Morning routine Dismissal Cubbie/desk organization
Day 18 Reflection/ Self-Monitoring VA Process Goal: Reasoning
Mathematicians modify their work as needed.
Students reflect upon and revise their work to demonstrate their full understanding.
Introduce a criteria chart and rubric for self- monitoring of work.
Sample Rubric:
Rubric & Problem Solving Protocol Create an anchor chart with “How to Self-Correct or Modify Your Work”
Help students develop a clear understanding of the criteria and how upcoming math tasks will be scored. Emphasize how this is similar to the revisions they do during the writing process.
Mini Lesson Key Ideas Essential Understandings
Anchor Charts/Supports Resources Teacher Notes
Day 19 Collaboration (Task) VA Process Goal: Communication
Mathematicians can work collaboratively on a problem solving task to learn important math concepts.
Students understand that they can work with others to solve problems and learn new information.
Review roles that pairs or small groups should follow/hold when working together on a task. Examples: Materials manager Recorder Reporter Time keeper
Develop an anchor chart with roles/procedures for group work on a task/problem. Self-assess/reflect upon collaborative work experiences. Students can use the problem solving protocol together (See Day 16).
Save time at the end of the lesson to debrief the experience. What went well? What could be improved next time they are working in a group?
Day 20
Process Goals
VA Process Goals: Problem Solving, Reasoning, Communication, Connections, & Representation
“The content of the mathematics standards is intended to support the following five process goals for students: *becoming mathematical problem solvers *communicating mathematically *reasoning mathematically *making mathematical connections and *using mathematical representations to model and interpret practical situations.”
-2009 Mathematics Standards of Learning
Student-friendly process goals poster (can be a poster for the classroom and/or a small version can be taped to desks or in math journals) Process Goals bookmark (click on picture to the left to access the file for the poster and bookmark)
Students should be engaged in process goals throughout every mathematical task and lesson throughout the year.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
VDOE Technical Assistance Document
to be used in conjunction with the VDOE Curriculum Framework (click title above to link to document)
Virginia Mathematics Standards of Learning
Curriculum Framework 2009 Introduction
The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity the content that all teachers should teach and all students should learn. Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for each standard. The Curriculum Framework is divided into two columns: Essential Understandings and Essential Knowledge and Skills. The purpose of each column is explained below. Essential Understandings This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the Standards of Learning. Essential Knowledge and Skills Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. It is meant to be the key knowledge and skills that define the standard. The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills from Standards of Learning presented in previous grades as they build mathematical expertise.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.1
The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
The structure of the Base-10 number system is
based upon a simple pattern of tens in which each
place is ten times the value of the place to its
right. This is known as a ten-to-one place value
relationship.
A decimal point separates the whole number
places from the places less than one. Place values
extend infinitely in two directions from a decimal
point. A number containing a decimal point is
called a decimal number or simply a decimal.
To read decimals,
– read the whole number to the left of the
decimal point, if there is one;
– read the decimal point as “and”;
– read the digits to the right of the decimal
point just as you would read a whole
number; and
– say the name of the place value of the digit in
the smallest place.
Decimals may be written in a variety of forms:
– Standard: 23.456
– Written: Twenty-three and four hundred fifty-
six thousandths
– Expanded: (2 10) + (3 1) + (4 0.1) +
(5 0.01) + (6 0.001)
To help students identify the ten-to-one place
value relationship for decimals through
thousandths, use Base-10 manipulatives, such as
place value mats/charts, decimal squares, Base-10
blocks, and money.
All students should
Understand that decimals are rounded in a
way that is similar to the way whole numbers
are rounded.
Understand that decimal numbers can be
rounded to estimate when exact numbers are
not needed for the situation at hand.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Round decimal numbers to the nearest whole number,
tenth, or hundredth.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.1
The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Decimals can be rounded to the nearest whole
number, tenth or hundredth in situations when
exact numbers are not needed.
Strategies for rounding decimal numbers to the
nearest whole number, tenth and hundredth are as
follows:
– Look one place to the right of the digit to
which you wish to round.
– If the digit is less than 5, leave the digit in the
rounding place as it is, and change the
digits to the right of the rounding place to
zero.
– If the digit is 5 or greater, add 1 to the digit in
the rounding place and change the digits to
the right of the rounding place to zero.
Create a number line that shows the decimal that
is to be rounded.
The position of the decimal will help children
conceptualize the number’s placement relative for
rounding. An example is to round 5.747 to the
nearest hundredth:
5.74 5.747 5.75
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.2
The student will
a) recognize and name fractions in their equivalent decimal form and vice versa; and
b) compare and order fractions and decimals in a given set from least to greatest and greatest to least.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Students should recognize, name, and focus on
finding equivalent decimals of familiar fractions
such as halves, fourths, fifths, eighths, and tenths.
Students should be able to determine equivalent
relationships between decimals and fractions with
denominators up to 12.
Students should have experience with fractions
such as1
8, whose decimal representation is a
terminating decimal (e. g., 1
8= 0.125) and with
fractions such as 2
9, whose decimal
representation does not end but continues to
repeat (e. g., 2
9= 0.222…). The repeating
decimal can be written with ellipses (three dots)
as in 0.222… or denoted with a bar above the
digits that repeat as in 0 .2 .
To help students compare the value of two
decimals through thousandths, use manipulatives,
such as place value mats/charts, 10-by-10 grids,
decimal squares, Base-10 blocks, meter sticks,
number lines, and money.
All students should
Understand the relationship between
fractions and their decimal form and vice
versa.
Understand that fractions and decimals can
be compared and ordered from least to
greatest and greatest to least.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Represent fractions (halves, fourths, fifths, eighths, tenths,
and twelfths) in their equivalent decimal form and vice
versa.
Recognize and name equivalent relationships between
decimals and fractions with denominators up to 12.
Compare and order from least to greatest and greatest to
least a given set of no more than five numbers written as
decimals, fractions, and mixed numbers with denominators
of 12 or less.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
A procedure for comparing two decimals by
examining may include the following:
– Line up the decimal numbers at their decimal
points.
– Beginning at the left, find the first place value
where the digits are different.
– Compare the digits in this place value to
determine which number is greater (or
which is less).
– Use the appropriate symbol > or < or the
words greater than or less than to compare
the numbers in the order in which they are
presented.
– If both numbers are the same, use the symbol
= or words equal to.
Two numbers can be compared by examining
place value and/or using a number line.
Decimals and fractions represent the same
relationships; however, they are presented in two
different formats. Decimal numbers are another
way of writing fractions. Base-10 models (e.g.,
10-by-10 grids, meter sticks, number lines,
decimal squares, money) concretely relate
fractions to decimals and vice versa.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.3
The student will
a) identify and describe the characteristics of prime and composite numbers; and
b) identify and describe the characteristics of even and odd numbers.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A prime number is a natural number that has
exactly two different factors, one and the number
itself.
A composite number is a natural number that has
more than two different factors.
The number 1 is neither prime nor composite
because it has only one factor, itself.
The prime factorization of a number is a
representation of the number as the product of its
prime factors. For example, the prime
factorization of 18 is 2 3 3.
Prime factorization concepts can be developed by
using factor trees.
Prime or composite numbers can be represented
by rectangular models or rectangular arrays on
grid paper. A prime number can be represented by
only one rectangular array (e.g., 7 can be
represented by a 7 1 and a 1 x 7). A composite
number can always be represented by more than
two rectangular arrays (e.g., 9 can be represented
by a 9 1, a 1 x 9, or a 3 3).
Divisibility rules are useful tools in identifying
prime and composite numbers.
Students should use manipulatives (e.g., Base-10
blocks, cubes, tiles, hundreds board, etc.) to
explore and categorize numbers into groups of
odd or even.
All students should
Understand and use the unique characteristics
of certain sets of numbers, including prime,
composite, even, and odd numbers.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Identify prime numbers less than or equal to 100.
Identify composite numbers less than or equal to 100.
Explain orally and in writing why a number is prime or
composite.
Identify which numbers are even or odd.
Explain and demonstrate with manipulatives, pictorial
representations, oral language, or written language why a
number is even or odd.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.3
The student will
a) identify and describe the characteristics of prime and composite numbers; and
b) identify and describe the characteristics of even and odd numbers.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Students should use rules to categorize numbers
into groups of odd or even. Rules can include:
– An odd number does not have 2 as a factor or
is not divisible by 2.
– The sum of two even numbers is even.
– The sum of two odd numbers is even.
– The sum of an even and an odd is odd.
– Even numbers have an even number or zero in
the ones place.
– Odd numbers have an odd number in the ones
place.
– An even number has 2 as a factor or is
divisible by 2.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.4
The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication,
and division with and without remainders of whole numbers.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
An example of an approach to solving problems
is Polya’s four-step plan:
– Understand: Retell the problem; read it twice;
take notes; study the charts or diagrams;
look up words and symbols that are new.
– Plan: Decide what operation(s) to use and
what sequence of steps to use to solve the
problem.
– Solve: Follow the plan and work accurately.
If the first attempt doesn’t work, try
another plan.
– Look back: Does the answer make sense?
Estimation gives a rough idea of an amount.
Strategies such as front-end, rounding, and mental
computation may be used to estimate addition,
subtraction, multiplication, and division of whole
numbers.
Examples of problems to be solved by using
estimation strategies are encountered in shopping
for groceries, buying school supplies, budgeting
allowance, and sharing the cost of a pizza or the
prize money from a contest.
Estimation can be used to check the
reasonableness of the results.
All students should
Understand the meaning of mathematical
operations and how these operations relate to
one another when creating and solving
single-step and multistep word problems.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Select appropriate methods and tools from among paper
and pencil, estimation, mental computation, and
calculators according to the context and nature of the
computation in order to compute with whole numbers.
Create single-step and multistep problems involving the
operations of addition, subtraction, multiplication, and
division with and without remainders of whole numbers,
using practical situations.
Estimate the sum, difference, product, and quotient of
whole number computations.
Solve single-step and multistep problems involving
addition, subtraction, multiplication, and division with
and without remainders of whole numbers, using paper
and pencil, mental computation, and calculators in which
– sums, differences, and products will not exceed five
digits;
– multipliers will not exceed two digits;
– divisors will not exceed two digits; or
– dividends will not exceed four digits.
Use two or more operational steps to solve a multistep
problem. Operations can be the same or different.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.5
The student will
a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with
only one nonzero digit); and
b) create and solve single-step and multistep practical problems involving decimals.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Addition and subtraction of decimals may be
investigated using a variety of models (e.g., 10-by-10
grids, number lines, money).
Decimal computation uses similar procedures as
those developed for whole number computation and
applies them to decimal place values, giving careful
attention to the placement of the decimal point in the
solution.
Multiplication of decimals follows the same
procedure as multiplication of whole numbers. The
only difference is that a decimal point must be
correctly placed in the product giving careful
attention to the placement of the decimal point in the
solution.
The product of decimals is dependent upon the two
factors being multiplied.
In cases where an exact product is not required, the
product of decimals can be estimated using strategies
for multiplying whole numbers, such as front-end
and compatible numbers, or rounding. In each case,
the student needs to determine where to place the
decimal point to ensure that the product is
reasonable.
Division is the operation of making equal groups or
shares. When the original amount and the number of
shares are known, divide to find the size of each
share. When the original amount and the size of each
share are known, divide to find the number of shares.
Both situations may be modeled with Base-10
manipulatives.
All students should
Use similar procedures as those developed for
whole number computation and apply them to
decimal place values, giving careful attention to
the placement of the decimal point in the
solution.
Select appropriate methods and tools from
among paper and pencil, estimation, mental
computation, and calculators according to the
context and nature of the computation in order to
compute with decimal numbers.
Understand the various meanings of division and
its effect on whole numbers.
Understand various representations of division,
i.e.,
dividend divisor = quotient
quotient
divisor dividend
dividend
divisor = quotient.
The student will use problem solving, mathematical
communication, mathematical reasoning,
connections, and representations to
Determine an appropriate method of calculation to
find the sum, difference, product, and quotient of
two numbers expressed as decimals through
thousandths, selecting from among paper and
pencil, estimation, mental computation, and
calculators.
Estimate to find the number that is closest to the
sum, difference, and product of two numbers
expressed as decimals through thousandths.
Find the sum, difference, and product of two
numbers expressed as decimals through
thousandths, using paper and pencil, estimation,
mental computation, and calculators.
Determine the quotient, given a dividend expressed
as a decimal through thousandths and a single-digit
divisor. For example, 5.4 divided by 2 and 2.4
divided by 5.
Use estimation to check the reasonableness of a
sum, difference, product, and quotient.
Create and solve single-step and multistep
problems.
A multistep problem needs to incorporate two or
more operational steps (operations can be the same
or different).
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.5
The student will
a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with
only one nonzero digit); and
b) create and solve single-step and multistep practical problems involving decimals.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
The fair-share concept of decimal division can be
modeled, using manipulatives (e.g., Base-10 blocks).
Division with decimals is performed the same way as
division of whole numbers. The only difference is the
placement of the decimal point in the quotient.
The quotient can be estimated, given a dividend
expressed as a decimal through thousandths (and no
adding of zeros to the dividend during the division
process) and a single-digit divisor.
Estimation can be used to check the reasonableness
of a quotient.
Division is the inverse of multiplication; therefore,
multiplication and division are inverse operations.
Terms used in division are dividend, divisor, and
quotient.
dividend divisor = quotient
quotient
divisor ) dividend
There are a variety of algorithms for division such as
repeated multiplication and subtraction. Experience
with these algorithms may enhance understanding of
the traditional long division algorithm.
A multistep problem needs to incorporate no more
than two operational steps (operations can be the
same or different).
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.6
The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed
numbers and express answers in simplest form.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A fraction can be expressed in simplest form
(simplest equivalent fraction) by dividing the
numerator and denominator by their greatest
common factor.
When the numerator and denominator have no
common factors other than 1, then the fraction is
in simplest form.
Fractions having like denominators means the
same as fractions having common denominators.
Equivalent fractions name the same amount. To
find equivalent fractions, multiply or divide the
numerator and denominator by the same nonzero
number.
Addition and subtraction with fractions and
mixed numbers can be modeled using a variety of
concrete materials and pictorial representations as
well as paper and pencil.
To add, subtract, and compare fractions and
mixed numbers, it often helps to find the least
common denominator. The least common
denominator (LCD) of two or more fractions is
the least common multiple (LCM) of the
denominators.
To add or subtract with fractions having the same
or like denominators, add or subtract the
numerators and write in simplest form.
All students should
Develop and use strategies to estimate and
compute addition and subtraction of
fractions.
Understand the concept of least common
multiple and least common denominator as
they are important when adding and
subtracting fractions.
Understand that a fraction is in simplest form
when its numerator and denominator have no
common factors other than 1. The numerator
can be greater than the denominator.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Solve single-step and multistep practical problems
involving addition and subtraction with fractions having
like and unlike denominators. Denominators in the
problems should be limited to 12 or less (e.g., 1
5 +
1
4 ) and
answers should be expressed in simplest form.
Solve single-step and multistep practical problems
involving addition and subtraction with mixed numbers
having like and unlike denominators, with and without
regrouping. Denominators in the problems should be
limited to 12 or less, and answers should be expressed in
simplest form.
Use estimation to check the reasonableness of a sum or
difference.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.6
The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed
numbers and express answers in simplest form.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
To add or subtract with fractions that do not have
the same denominator, first find equivalent
fractions with the least common denominator.
Then add or subtract and write the answer in
simplest form.
A mixed number has two parts: a whole number
and a fraction. The value of a mixed number is
the sum of its two parts.
To add or subtract with mixed numbers, students
may use a number line, draw a picture, rewrite
fractions with like denominators, or rewrite mixed
numbers as fractions.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.7
The student will evaluate whole number numerical expressions, using the order of operations limited to parentheses, addition,
subtraction, multiplication, and division.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
An expression, like a phrase, has no equal sign.
Expressions are simplified by using the order of
operations.
The order of operations defines the computation
order to follow in simplifying an expression.
The order of operations is as follows:
– First, complete all operations within grouping
symbols. If there are grouping symbols
within other grouping symbols, do the
innermost operation first.
– Second, evaluate all exponential expressions.
– Third, multiply and/or divide in order from
left to right.
– Fourth, add and/or subtract in order from left
to right.
All students should
Understand that the order of operations
describes the order to use to simplify
expressions containing more than one
operation.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Simplify expressions by using the order of operations in a
demonstrated step-by-step approach.
Find the value of numerical expressions, using the order of
operations.
Given an expression involving more than one operation,
describe which operation is completed first, which is
second, etc.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.8
The student will
a) find perimeter, area, and volume in standard units of measure;
b) differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or
volume is appropriate for a given situation;
c) identify equivalent measurements within the metric system;
d) estimate and then measure to solve problems, using U.S. Customary and metric units; and
e) choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Perimeter is the distance around an object. It is a
measure of length. Area is the number of square
units needed to cover a surface. Volume is a
measure of capacity and is measured in cubic
units.
To find the perimeter of any polygon, add the
lengths of the sides.
Students should label the perimeter, area, and
volume with the appropriate unit of linear, square,
or cubic measure.
Area is the number of square units needed to
cover a surface or figure.
Students should investigate, using manipulatives,
to discover the formulas for the area of a square,
rectangle, and right triangle; and volume of a
rectangular solid.
– Area of a rectangle = Length Width
– Area of a square = Side Side
– Area of a right triangle = 1
2 Base Height
– Volume of a rectangular solid = Length x
Width x Height
Length is the distance along a line or figure from
one point to another.
All students should
Understand the concepts of perimeter, area,
and volume.
Understand and use appropriate units of
measure for perimeter, area, and volume.
Understand the difference between using
perimeter, area, and volume in a given
situation.
Understand how to select a measuring device
and unit of measure to solve problems
involving measurement.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Determine the perimeter of a polygon, with or without
diagrams, when
– the lengths of all sides of a polygon that is not a
rectangle or a square are given;
– the length and width of a rectangle are given; or
– the length of a side of a square is given.
Estimate and determine the perimeter of a polygon, and
area of a square, rectangle, and right triangle following the
parameters listed above, using only whole number
measurements given in metric or U.S. Customary units,
and record the solution with the appropriate unit of
measure (e.g., 24 square inches).
Estimate and determine the area of a square, with or
without diagrams, when the length of a side is given.
Estimate and determine the area of a rectangle, with or
without diagrams, when the length and width are given.
Estimate and determine the area of a right triangle, with or
without diagrams, when the base and the height are given.
Differentiate among the concepts of area, perimeter, and
volume.
Develop a procedure for finding volume using
manipulatives (e.g., cubes).
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.8
The student will
a) find perimeter, area, and volume in standard units of measure;
b) differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or
volume is appropriate for a given situation;
c) identify equivalent measurements within the metric system;
d) estimate and then measure to solve problems, using U.S. Customary and metric units; and
e) choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
U.S. Customary units for measurement of length
include inches, feet, yards, and miles. Appropriate
measuring devices include rulers, yardsticks, and
tape measures. Metric units for measurement of
length include millimeters, centimeters, meters,
and kilometers. Appropriate measuring devices
include centimeter ruler, meter stick, and tape
measure.
When measuring with U.S. Customary units,
students should be able to measure to the nearest
part of an inch (1
2 ,
1
4 ,
1
8 ), foot, or yard.
Weight and mass are different. Mass is the
amount of matter in an object. Weight is
determined by the pull of gravity on the mass of
an object. The mass of an object remains the same
regardless of its location. The weight that an
object changes is dependent on the gravitational
pull at its location. In everyday life, most people
are actually interested in determining an object’s
mass, although they use the term weight (e.g.,
“How much does it weigh?” versus “What is its
mass?”).
Appropriate measuring devices to measure mass
in U.S. Customary units (ounces, pounds) and
metric units (grams, kilograms) are balances.
Determine volume in standard units.
Describe practical situations where area, perimeter, and
volume are appropriate measures to use, and justify their
choices orally or in writing.
Identify whether the application of the concept of
perimeter, area, or volume is appropriate for a given
situation.
Identify equivalent measurements within the metric system
for the following:
– length: millimeters, centimeters, meters, and
kilometers;
– mass: grams and kilograms;
– liquid volume: milliliters, and liters.
Solve problems involving measurement by selecting an
appropriate measuring device and a U.S. Customary or
metric unit of measure for the following:
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.8
The student will
a) find perimeter, area, and volume in standard units of measure;
b) differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or
volume is appropriate for a given situation;
c) identify equivalent measurements within the metric system;
d) estimate and then measure to solve problems, using U.S. Customary and metric units; and
e) choose an appropriate unit of measure for a given situation involving measurement using U.S. Customary and metric units.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
U.S. Customary units to measure liquid volume
(capacity) include cups, pints, quarts, and gallons.
Metric units to measure liquid volume (capacity)
include milliliters and liters.
Temperature is measured using a thermometer.
The U.S. Customary unit of measure is degrees
Fahrenheit; the metric unit of measure is degrees
Celsius.
Practical experience measuring familiar objects
helps students establish benchmarks and
facilitates students’ ability to use the units of
measure to make estimates.
– length: part of an inch (1
2 ,
1
4 ,
1
8 ), inches, feet, yards,
millimeters, centimeters, meters, and kilometers;
– weight: ounces, pounds, and tons;
– mass: grams and kilograms;
– liquid volume: cups, pints, quarts, gallons, milliliters,
and liters;
– area: square units; and
– temperature: Celsius and Fahrenheit units.
– Water freezes at 0C and 32F.
– Water boils at 100C and 212F.
– Normal body temperature is about 37C and
98.6F.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.9
The student will identify and describe the diameter, radius, chord, and circumference of a circle.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A circle is a set of points on a flat surface (plane)
with every point equidistant from a given point
called the center.
A chord is a line segment connecting any two
points on a circle. Students will benefit from
understanding that a chord goes from one side of
the circle to the other, but does not need to pass
through the center.
A diameter is a chord that goes through the center
of a circle. The diameter is two times the radius.
A radius is a segment from the center of a circle
to any point on the circle. Two radii end-to-end
form a diameter of a circle.
Circumference is the distance around or perimeter
of a circle. The circumference is about 3 times
larger than the diameter of a circle.
All students should
Understand that a chord is a line segment that
extends between any two unique points of a
circle.
Understand that a diameter is also a special
chord that goes through the center of a circle.
Understand the relationship between the
measures of diameter and radius and the
relationship between the measures of radius
and circumference.
Understand that a radius is a line segment
that extends between the center and the
circumference of the circle.
Understand that the circumference is the
distance around the circle. Perimeter is the
measure of the circumference.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Identify and describe the diameter, radius, chord, and
circumference of a circle.
Describe the relationship between
– diameter and radius;
– diameter and chord;
– radius and circumference; and
– diameter and circumference.
The length of the diameter of a circle is twice the length of
the radius.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.10
The student will determine an amount of elapsed time in hours and minutes within a 24-hour period.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Elapsed time is the amount of time that has
passed between two given times.
Elapsed time can be found by counting on from
the beginning time to the finishing time.
– Count the number of whole hours between
the beginning time and the finishing time.
– Count the remaining minutes.
– Add the hours and minutes. For example, to
find the elapsed time between 10:15
a.m. and 1:25 p.m., count on as follows:
from 10:15 a.m. to 1:15 p.m., count 3
hours;
from 1:15 p.m. to 1:25 p.m., count 10
minutes; and then
add 3 hours to 10 minutes to find the total
elapsed time of 3 hours and 10 minutes.
All students should
Understand that elapsed time can be found by
counting on from the beginning time to the
finishing time.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Determine elapsed time in hours and minutes within a 24-
hour period.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.11
The student will measure right, acute, obtuse, and straight angles.
UNDERSTANDING THE STANDARD
(Background Information for Instructor Use Only) ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Angles are measured in degrees. There are up to
360 degrees in an angle. A degree is 1
360 of a
complete rotation of a full circle. There are 360
degrees in a circle.
To measure the number of degrees in an angle,
use a protractor or an angle ruler.
A right angle measures exactly 90°.
An acute angle measures less than 90°.
An obtuse angle measures greater than 90° but
less than 180°.
A straight angle measures exactly 180°.
Before measuring an angle, students should first
compare it to a right angle to determine whether
the measure of the angle is less than or greater
than 90°.
Students should understand how to work with a
protractor or angle ruler as well as available
computer software to measure and draw angles
and triangles.
All students should
Understand how to measure acute, right,
obtuse, and straight angles.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Identify the appropriate tools (e.g., protractor and
straightedge or angle ruler as well as available software)
used to measure and draw angles and triangles.
Measure right, acute, straight, and obtuse angles, using
appropriate tools, and identify their measures in degrees.
Recognize angle measure as additive. When an angle is
decomposed into nonoverlapping parts, the angle measure
of the whole is the sum of the angle measures of the parts.†
Solve addition and subtraction problems to find unknown
angle measures on a diagram in practical and mathematical
problems, (e.g., by using an equation with a symbol for the
unknown angle measure).†
†Revised March
2011
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.12
The student will classify
a) angles as right, acute, obtuse, or straight; and
b) triangles as right, acute, obtuse, equilateral, scalene, or isosceles.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A right angle measures exactly 90.
An acute angle measures greater than 0 but less
than 90.
An obtuse angle measures greater than 90 but
less than 180.
A straight angle forms an angle that measures
exactly 180°.
A right triangle has one right angle.
An obtuse triangle has one obtuse angle.
An acute triangle has three acute angles (or no
angle measuring 90 or greater).
A scalene triangle has no congruent sides.
An isosceles triangle has two congruent sides.
To facilitate the exploration of relationships, ask
students whether a right triangle can have an
obtuse angle. Why or why not? Can an obtuse
triangle have more than one obtuse angle? Why or
why not? What type of angles are the two angles
other than the right angle in a right triangle? What
type of angles are the two angles other than the
obtuse angle in an obtuse triangle?
All students should
Understand that angles can be classified as
right, acute, obtuse, or straight according to
their measures.
Understand that a triangle can be classified as
either right, acute, or obtuse according to the
measure of its largest angle.
Understand that a triangle can be classified as
equilateral, scalene, or isosceles according to
the number of sides with equal length.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Classify angles as right, acute, straight, or obtuse.
Classify triangles as right, acute, or obtuse.
Classify triangles as equilateral, scalene, or isosceles.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.13
The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will
a) develop definitions of these plane figures; and
b) investigate and describe the results of combining and subdividing plane figures.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A triangle is a polygon with three sides. Triangles
may be classified according to the measure of
their angles, i.e., right, acute, or obtuse. Triangles
may also be classified according to the measure of
their sides, i.e., scalene (no sides congruent),
isosceles (at least two sides congruent) and
equilateral (all sides congruent).
A quadrilateral is a polygon with four sides.
A parallelogram is a quadrilateral in which both
pairs of opposite sides are parallel. Properties of a
parallelogram include the following:
– A diagonal (a segment that connects two
vertices of a polygon but is not a side)
divides the parallelogram into two
congruent triangles.
– The opposite sides of a parallelogram are
congruent.
– The opposite angles of a parallelogram are
congruent.
– The diagonals of a parallelogram bisect each
other. To bisect means to cut a geometric
figure into two congruent halves. A
bisector is a line segment, line, or plane
that divides a geometric figure into two
congruent halves. A sample of a bisected
parallelogram is below.
All students should
Understand that simple plane figures can be
combined to make more complicated figures
and that complicated figures can be
subdivided into simple plane figures.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections and
representation to
Develop definitions for squares, rectangles, triangles,
parallelograms, rhombi, and trapezoids.
Investigate and describe the results of combining and
subdividing plane figures.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.13
The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will
a) develop definitions of these plane figures; and
b) investigate and describe the results of combining and subdividing plane figures.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A rectangle is a parallelogram with four right
angles. Since a rectangle is a parallelogram, a
rectangle has the same properties as those of a
parallelogram.
A square is a rectangle with four congruent sides.
Since a square is a rectangle, a square has all the
properties of a rectangle and of a parallelogram.
A rhombus is a parallelogram with four congruent
sides. Opposite angles of a rhombus are
congruent. Since a rhombus is a parallelogram,
the rhombus has all the properties of a
parallelogram.
A trapezoid is a quadrilateral with exactly one
pair of parallel sides. The parallel sides are called
bases, and the nonparallel sides are called legs. If
the legs have the same length, then the trapezoid
is an isosceles trapezoid.
Two or more figures can be combined to form a
new figure. Students should be able to identify the
figures that have been combined.
The region of a polygon may be subdivided into
two or more regions that represent figures.
Students should understand how to divide the
region of a polygon into familiar figures.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.14
The student will make predictions and determine the probability of an outcome by constructing a sample space.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Probability is the chance of an event occurring.
The probability of an event occurring is the ratio of
desired outcomes to the total number of possible
outcomes. If all the outcomes of an event are equally
likely to occur, the probability of the event =
number of favorable outcomes
total number of possible outcomes.
The probability of an event occurring is represented by a
ratio between 0 and 1. An event is “impossible” if it has
a probability of 0 (e.g., the probability that the month of
April will have 31 days). An event is “certain” if it has a
probability of 1 (e.g., the probability that the sun will
rise tomorrow morning).
When a probability experiment has very few trials, the
results can be misleading. The more times an experiment
is done, the closer the experimental probability comes to
the theoretical probability (e.g., a coin lands heads up
half of the time).
Students should have opportunities to describe in
informal terms (i.e., impossible, unlikely, as likely as
unlikely, as likely as, equally likely, likely, and certain)
the degree of likelihood of an event occurring. Activities
should include practical examples.
For any event such as flipping a coin, the equally likely
things that can happen are called outcomes. For
example, there are two equally likely outcomes when
flipping a coin: the coin can land heads up, or the coin
can land tails up.
A sample space represents all possible outcomes of an
experiment. The sample space may be organized in a
list, chart, or tree diagram.
All students should
Understand that the basic concepts of
probability can be applied to make
predictions of outcomes of simple
experiments.
Understand that a sample space represents all
possible outcomes of an experiment.
The student will use problem solving, mathematical
communication, mathematical reasoning,
connections, and representations to
Construct a sample space, using a tree diagram to
identify all possible outcomes of a single event.
Construct a sample space, using a list or chart to
represent all possible outcomes of a single event.
Predict and determine the probability of an outcome
by constructing a sample space. The sample space
will have a total of 24 or less possible outcomes.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.14
The student will make predictions and determine the probability of an outcome by constructing a sample space.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Tree diagrams show all possible outcomes in a sample
space. The Fundamental Counting Principle describes
how to find the number of outcomes when there are
multiple choices. For example, how many different
outfit combinations can you make from 2 shirts (red and
blue) and 3 pants (black, white, khaki)? The sample
space displayed in a tree diagram would show that there
are 2 3 = 6 (Fundamental Counting Principle) outfit
combinations: red-black; red-white; red-khaki; blue-
black; blue-white; blue-khaki.
A spinner with eight equal-sized sections is equally
likely to land on any one of the sections, three of which
are red, three green, and two yellow. Have students write
a problem statement involving probability, such as,
“What is the probability that the spinner will land on
green?”
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.15
The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots
and line graphs.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
The emphasis in all work with statistics should be
on the analysis and the communication of the
analysis, rather than on a single correct answer.
Data analysis should include opportunities to
describe the data, recognize patterns or trends,
and make predictions.
Statistical investigations should be active, with
students formulating questions about something
in their environment and finding quantitative
ways to answer the questions.
Investigations can be brief class surveys or more
extended projects taking many days.
Through experiences displaying data in a variety
of graphical representations, students learn to
select an appropriate representation.
Line graphs are used to show how two continuous
variables are related. Line graphs may be used to
show how one variable changes over time. If one
variable is not continuous, then a broken line is
used. By looking at a line graph, it can be
determined whether the variable is increasing,
decreasing, or staying the same over time.
– The values along the horizontal axis represent
continuous data on a given variable,
usually some measure of time (e.g., time in
years, months, or days). The data presented
on a line graph is referred to as
“continuous data” because it represents
data collected over a continuous period of
time.
All students should
Understand how to interpret collected and
organized data.
Understand that stem-and-leaf plots list data
in a meaningful array. It helps in finding
median, modes, minimum and maximum
values, and ranges.
Understand that line graphs show changes
over time.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Formulate the question that will guide the data collection.
Collect data, using observations (e.g., weather),
measurement (e.g., shoe sizes), surveys (e.g., hours
watching television), or experiments (e.g., plant growth).
Organize the data into a chart, table, stem-and-leaf plots,
and line graphs.
Display data in line graphs and stem-and-leaf plots.
Construct line graphs, labeling the vertical axis with equal
whole number, decimal, or fractional increments and the
horizontal axis with continuous data commonly related to
time (e.g., hours, days, months, years, and age). Line
graphs will have no more than six identified points along a
continuum for continuous data (e.g., the decades: 1950s,
1960s, 1970s, 1980s, 1990s, and 2000s).
Construct a stem-and-leaf plot to organize and display
data, where the stem is listed in ascending order and the
leaves are in ascending order, with or without commas
between leaves.
Title the given graph or identify the title.
Interpret the data in a variety of forms (e.g., orally or in
written form).
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
– The values along the vertical axis are the
scale and represent the frequency with
which those values occur in the data set.
The values should represent equal
increments of multiples of whole numbers,
fractions, or decimals depending upon the
data being collected. The scale should
extend one increment above the greatest
recorded piece of data.
– Each axis should be labeled and the graph
should have a title.
– A line graph tells whether something has
increased, decreased, or stayed the same
with the passage of time. Statements
representing an analysis and interpretation
of the characteristics of the data in the
graph should be included (e.g., trends of
increase and/or decrease, and least and
greatest). A broken line is used if the data
collected is not continuous data (such as
test scores); a solid line is used if the data
is continuous (such as height of a plant).
Stem-and-leaf plots allow the exact values of data
to be listed in a meaningful array. Data covering a
range of 25 numbers are best displayed in a stem-
and-leaf plot and are utilized to organize
numerical data from least to greatest, using the
digits of the greatest to group data.
– The data is organized from least to greatest.
– Each value should be separated into a stem
and a leaf [e.g., two-digit numbers are
separated into stems (tens) and leaves
(ones)].
– The stems are listed vertically from least to
greatest with a line to their right. The
leaves are listed horizontally, also from
least to greatest, and can be separated by
spaces or commas. Every value is recorded
regardless of the number of repeats.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.15
The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots
and line graphs.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
– A key is often included to explain how to
read the plot.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.16
The student will
a) describe mean, median, and mode as measures of center;
b) describe mean as fair share;
c) find the mean, median, mode, and range of a set of data; and
d) describe the range of a set of data as a measure of variation.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
Statistics is the science of conducting studies to
collect, organize, summarize, analyze, and draw
conclusions from data.
A measure of center is a value at the center or
middle of a data set. Mean, median, and mode
are measures of center.
The mean, median, and mode are three of the
various ways that data can be analyzed.
Mean represents a fair share concept of the data.
Dividing the data constitutes a fair share. This is
done by equally dividing the data points. This
should be demonstrated visually and with
manipulatives. The arithmetic way is to add all of
the data points then divide by the number of data
points to determine the average or mean.
The median is the piece of data that lies in the
middle of the set of data arranged in order.
The mode is the piece of data that occurs most
frequently in the data set. There may be one, more
than one, or no mode in a data set. Students
should order the data from least to greatest so
they can better find the mode.
The range is the spread of a set of data. The range
of a set of data is the difference between the
greatest and least values in the data set. It is
determined by subtracting the least number in the
data set from the greatest number in the data set.
An example is ordering test scores from least to
greatest: 73, 77, 84, 87, 89, 91, 94. The greatest
All students should
Understand that mean, median, and mode are
described as measures of center.
Understand that mean, median, and mode are
three of the various ways that data can be
described or summarized.
Understand that mean as fair share is
described as equally dividing the data set or
the data set has already been divided equally.
Understand how to find the mean, median,
and mode of a set of data as measures of
center.
Understand values in the context of other
characteristics of the data in order to best
describe the results.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Describe and find the mean of a group of numbers
representing data from a given context as a measure of
center.
Describe and find the median of a group of numbers
representing data from a given context as a measure of
center.
Describe and find the mode of a group of numbers
representing data from a given context as a measure of
center.
Describe mean as fair share.
Describe and find the range of a group of numbers
representing data from a given context as a measure of
variation.
Describe the impact on measures of center when a single
value of a data set is added, removed, or changed.†
†Revised March 2011
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.16
The student will
a) describe mean, median, and mode as measures of center;
b) describe mean as fair share;
c) find the mean, median, mode, and range of a set of data; and
d) describe the range of a set of data as a measure of variation.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
score in the data set is 94 and the least score is 73,
so the least score is subtracted from the greatest
score or 94 - 73 = 21. The range of these test
scores is 21.
Students need to learn more than how to identify
the mean, median, mode, and range of a set of
data. They need to build an understanding of what
the number tells them about the data, and they
need to see those values in the context of other
characteristics of the data in order to best describe
the results.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.17
The student will describe the relationship found in a number pattern and express the relationship.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
There are an infinite number of patterns.
The simplest types of patterns are repeating
patterns. In such patterns, students need to
identify the basic unit of the pattern and repeat it.
Growing patterns are more difficult for students
to understand than repeating patterns because not
only must they determine what comes next, they
must also begin the process of generalization.
Students need experiences with growing patterns.
Sample numerical patterns are
6, 9, 12, 15, 18, ;
5, 7, 9, 11, 13, ;
1, 2, 4, 7, 11, 16, ;
2, 4, 8, 16, 32, ;
32, 30, 28, 26, 24…; and
1, 5, 25, 125, 625,.
An expression, like a phrase, has no equal sign.
When the pattern data are expressed in a T-
table, an expression can represent that data. An example is:
X Y
6 9
7 10
11 14
15 18
This example defines the relationship as x + 3.
Expressions are simplified by using the order of
operations.
A verbal quantitative expression involving one
operation can be represented by a variable
All students should
Understand that patterns and functions can be
represented in many ways and described
using words, tables, and symbols.
Understand the structure of a pattern and how
it grows or changes using concrete materials
and calculators.
Understand that mathematical relationships
exist in patterns.
Understand that an expression uses symbols
to define a relationship and shows how each
number in the list, after the first number, is
related to the preceding number.
Understand that expressions can be numerical
or variable or a combination of numbers and
variables.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Describe numerical and geometric patterns formed by
using concrete materials and calculators.
Describe the relationship found in patterns, using words,
tables, and symbols to express the relationship.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.17
The student will describe the relationship found in a number pattern and express the relationship.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
expression that describes what is going on.
Numbers are used when they are known;
variables are used when the numbers are
unknown. For example, “a full box of cookies and
four extra” can be represented by b + 4; “three
full boxes of cookies” by 3b; “a full box of
cookies shared among four” by b
4 .
A mathematical expression contains a variable or
a combination of variables, numbers, and/or
operation symbols and represents a mathematical
relationship. An expression cannot be solved.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.18
The student will
a) investigate and describe the concept of variable;
b) write an open sentence to represent a given mathematical relationship, using a variable;
c) model one-step linear equations in one variable, using addition and subtraction; and
d) create a problem situation based on a given open sentence, using a single variable.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A variable is a symbol that can stand for an unknown
number or object.
A variable expression is like a phrase: as a phrase
does not have a verb, so an expression does not have
an equals sign (=).
A verbal expression involving one operation can be
represented by a variable expression that describes
what is going on. Numbers are used when they are
known; variables are used when the numbers are
unknown. For example, “a full box of cookies and
four extra” can be represented by b + 4; “three full
boxes of cookies” by 3b; “a full box of cookies
shared among four” by b
4 .
An open sentence contains a variable and an equals
sign (=). For example, the sentence, “A full box of
cookies and four extra equal 24 cookies.” can be
written as b + 4 = 24, where b stands for the number
of cookies in one full box. “Three full boxes of
cookies equal 60 cookies.” can be written as 3b = 60.
Another example of an open sentence is b + 3 = 23
and represents the answer to the word problem,
“How many cookies are in a box if the box plus three
more equals 23 cookies, where b stands for the
number of cookies in the box?
All students should
Understand that a variable is a symbol that can
stand for an unknown number or object.
Understand that a variable expression is a
variable or combination of variables, numbers,
and symbols that represents a mathematical
relationship.
Understand that verbal expressions can be
translated to variable expressions.
Understand that an open sentence has a variable
and an equal sign (=).
Understand that problem situations can be
expressed as open sentences.
The student will use problem solving, mathematical
communication, mathematical reasoning,
connections, and representations to
Describe the concept of a variable (presented as
boxes, letters, or other symbols) as a representation
of an unknown quantity.
Write an open sentence with addition, subtraction,
multiplication, or division, using a variable to
represent a missing number.
Model one-step linear equations using a variety of
concrete materials such as colored chips on an
equation mat or weights on a balance scale.
Create and write a word problem to match a given
open sentence with a single variable and one
operation.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.18
The student will
a) investigate and describe the concept of variable;
b) write an open sentence to represent a given mathematical relationship, using a variable;
c) model one-step linear equations in one variable, using addition and subtraction; and
d) create a problem situation based on a given open sentence, using a single variable.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
At this level, discuss how the symbol used to
represent multiplication can often be confused with
the variable x. Students can minimize this confusion
by using parentheses [e.g., 4(x) = 20 or 4x = 20] or a
small dot raised off the line to represent
multiplication [4 • x = 20].
By using story problems and numerical sentences,
students begin to explore forming equations and
representing quantities using variables.
An open sentence containing a variable is neither true
nor false until the variable is replaced with a number.
Mathematics Standards of Learning Curriculum Framework 2009: Grade 5
5.19
The student will investigate and recognize the distributive property of multiplication over addition.
UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only)
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
The distributive property states that multiplying a
sum by a number gives the same result as
multiplying each addend by the number and then
adding the products (e.g.,
3(4 + 5) = 3 x 4 + 3 x 5,
5 x (3 + 7) = (5 x 3) + (5 x 7); or
(2 x 3) + (2 x 5) = 2 x (3 + 5).
The distributive property can be used to simplify
expressions (e.g., 9 x 23 = 9(20+3) =180+ 27 = 207;
or 5 x 19 = 5(10 + 9) = 50 + 45 = 95).
All students should
Understand that the distributive property
states that multiplying a sum by a number
gives the same result as multiplying each
addend by the number and then adding the
products.
Understand that using the distributive
property with whole numbers helps with
understanding mathematical relationships.
Understand when and why the distributive
property is used.
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Investigate and recognize the distributive property of
whole numbers, limited to multiplication over addition
using diagrams and manipulatives.
Investigate and recognize an equation that represents the
distributive property, when given several whole number
equations, limited to multiplication over addition.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Learning Progressions The following pages are the Learning Progressions for the curriculum. More information about the Learning Progressions can be found on VISION. The Grading and Assessment, Module 3: Learning Progressions is about what Learning Progressions are, how they were developed, and how they are used to support instruction and build student learning.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.1 SOL 5.1: The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth. Learning Target: I can round decimal numbers expressed through thousandths to the nearest whole number, tenth, or hundredth, and represent my thinking using symbols, pictures, numbers, and words.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can compare and contrast the accuracy of rounding compared to other forms of estimations (compatible number, front-end estimation, etc.) and describe an example from real-life in which rounding would be appropriate.
Proficient
I can round decimal numbers expressed through thousandths to the nearest whole number, tenth, or hundredth, and represent my thinking using symbols, pictures, numbers, and words.
Intermediate
I can determine whether a number is closer to one landmark number or another using a model (for example: on a number line, visualizing that 571 is closer to 600 than 500).
Beginner
I can round whole numbers to the nearest ten, hundred, thousand, ten thousand, and hundred thousand and recognize a model for rounding.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.2 SOL 5.2: The student will
a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least.
Learning Target: I can recognize and name fractions in their equivalent decimal form and compare and order fractions and decimals using models
and multiple strategies.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can compare and contrast multiple strategies for comparing and ordering fractions and decimals and develop generalizations about my strategies.
Proficient
I can recognize and name fractions in their equivalent decimal form and compare and order fractions and
decimals using models and multiple strategies.
Intermediate
I can recognize and name fractions in their equivalent decimal form using models and compare fractions to fractions, decimals to decimals, and fractions to decimals.
Beginner
I can identify equivalent fractions and equivalent decimals and order a set of fractions and/or order a set of decimals.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.3 SOL 5.3: The student will
a) identify and describe the characteristics of prime and composite numbers; and b) identify and describe the characteristics of even and odd numbers.
Learning Target: I can demonstrate examples of prime, composite, even, and odd numbers using models, numbers, and words.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create examples and non-examples of prime, composite, even, and odd numbers and compare and contrast the rules for each category of numbers.
Proficient
I can demonstrate examples of prime, composite, even, and odd numbers using models, numbers, and words.
Intermediate
I can recognize and sort examples of prime, composite, odd, and even numbers.
Beginner
I can understand that numbers can be categorized into different groups based on rules.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.4 SOL 5.4: The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and
division with and without remainders of whole numbers. Learning Target: I can create, estimate, and solve addition, subtraction, multiplication, and division problems that have two or more steps involved in order to find the answer.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create multi-step problems involving more than one operation and compare and contrast strategies used to solve problems.
Proficient
I can create, estimate, and solve addition, subtraction, multiplication, and division problems that have two or
more steps involved in order to find the answer.
Intermediate
I can estimate and solve addition, subtraction, multiplication, and division problems that can be solved in one step using one strategy.
Beginner
I can understand that different computation strategies will help me to estimate and solve problems.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.5 SOL 5.5: The student will
a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and
b) create and solve single-step and multistep practical problems involving decimals. Learning Target: I can investigate, create, and solve single and multi-step problems involving decimal operations (addition, subtraction,
multiplication, and division; decimals through thousandths and divisors with only one nonzero digit).
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can compare and contrast strategies for solving multi-step problems as well as identify similarities and differences between whole number operations and decimal operations.
Proficient
I can investigate, create, and solve single and multi-step problems involving decimal operations (addition,
subtraction, multiplication, and division; decimals through thousandths and divisors with only one nonzero digit).
Intermediate
I can investigate and estimate problems involving decimal operations using decimal models (number lines, 10x10 grids, etc.).
Beginner
I can solve single and multi-step problems involving whole number operations.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.6 SOL 5.6: The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form. Learning Target: I can develop strategies and use them to compute the sum or difference of fractions and mixed numbers in practical single and
multi-step problems.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create single and multi-step problems involving the sum and difference of fractions and compare and contrast my solution strategy with a classmate’s strategy.
Proficient
I can develop strategies and use them to compute the sum or difference of fractions and mixed numbers in
practical single and multi-step problems.
Intermediate
I can use a strategy to compute the sum or difference of fractions and mixed numbers.
Beginner
I can use models and strategies to identify equivalent fractions.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.7 SOL 5.7: The student will evaluate whole number numerical expressions, using the order of operations limited to parentheses, addition,
subtraction, multiplication, and division. Learning Target: I can simplify expressions with more than two operations using the order of operations and explain each step.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create expressions with more than two operations and provide a proof using models and numbers.
Proficient
I can simplify expressions with more than two operations using the order of operations and explain each step.
Intermediate
I can simplify expressions with one or two inverse operations (limited to only addition/subtraction or multiplication/division).
Beginner
I can understand that expressions do not contain an equal sign and must be solved in a particular order.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.8 SOL 5.8: The student will
a) find perimeter, area, and volume in standard units of measure; b) differentiate among perimeter, area, and volume, and identify whether the application of the concept of perimeter, area, or volume is appropriate
for a given situation; c) identify equivalent measurements within the metric system; d) estimate and then measure to solve problems, using US Customary and metric units; and e) choose an appropriate unit of measure for a given situation involving measurement using US Customary and metric units.
Learning Target: The student will, given a problem situation, decide if the problem requires perimeter, area, and/or volume and estimate and then measure, using appropriate units, to solve the problem.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can make connections between the metric measures and the base 10 system of numeration.
Proficient
I can, given a problem situation, decide if the problem requires perimeter, area, and/or volume and estimate and then measure, using appropriate units and tools, to solve the problem.
Intermediate I will investigate volume of solids and areas of polygons, using manipulatives to develop formulas for area of each, and find perimeter as well.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Beginning I can identify equivalent measurements between units in the metric system; and estimate and then use US Customary and metric units to measure length, liquid volume, and mass, area and perimeter.
LP 5.9 SOL 5.9: The student will identify and describe the diameter, radius, chord, and circumference of a circle. Learning Target: I can identify and describe the relationship between the measures of the parts of a circle (radius, diameter, chord, and circumference).
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create a model of a circle when given clues about a circle without using formulas (for example: Draw a circle whose circumference is 18 centimeters and label each part of a circle with its measurement).
Proficient
I can identify and describe the relationship between the measures of the parts of a circle (radius, diameter, chord, and
circumference).
Intermediate
I can use strategies to measure parts of a circle in order to compare them to other parts.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Beginner
I can identify the parts of a circle when given a drawing or model.
LP 5.10 SOL 5.10: The student will determine an amount of elapsed time in hours and minutes within a 24-hour period. Learning Target: I can determine the elapsed time between two events within a 24-hour period.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can solve practical problems that use elapsed time, expressed in a variety of structures (ie: We arrived in New York City at 2:45 p.m. and the car ride lasted 5 hours and 47 minutes. What time did the car ride begin?).
Proficient I can determine the elapsed time between two events within a 24-hour period.
Intermediate I can “count on” from the beginning time to the finish time to find the elapsed time.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Beginning I can determine elapsed time in hours and minutes within a 12-hour period.
LP 5.11 SOL 5.11: The student will measure right, acute, obtuse, and straight angles. Learning Target: I can use a protractor or angle ruler to measure an angle in degrees and solve for a missing angle using addition or subtraction.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create angles that fit given guidelines (ie: draw an angle that is 75 degrees) and find angles to measure in real life.
Proficient
I can use a protractor or angle ruler to measure an angle in degrees and solve for a missing angle using addition or
subtraction.
Intermediate
I can compare an angle to a right angle to estimate its measure and identify the angle as right, acute, obtuse, or straight.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Beginner
I can recognize that a protractor or angle ruler are tools to measure angels and that the unit for the measure is degrees.
LP 5.12 SOL 5.12: The student will classify
a) angles as right, acute, obtuse, or straight; and b) triangles as right, acute, obtuse, equilateral, scalene, or isosceles.
Learning Target: I can sort angles by their measures (acute, right, obtuse, or straight) and classify triangles by their characteristics (acute,
right, obtuse, equilateral, scalene, or isosceles.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create angles and triangles that have certain characteristics (ie: draw an acute, equilateral triangle) and find examples of specific triangles in real life.
Proficient
I can sort angles by their measures (acute, right, obtuse, or straight) and classify triangles by their characteristics
(acute, right, obtuse, equilateral, scalene, or isosceles).
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Intermediate
I can measure lengths of sides and measures of angles.
Beginner
I can sort angles and triangles by the way that they look.
LP 5.13 SOL 5.13: The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will
a) develop definitions of these plane figures; and b) investigate and describe the results of combining and subdividing plane figures.
Learning Target: I can identify plane figures based on their definitions and describe how shapes change when they are combined with other shapes or divided into smaller parts.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can compare and contrast shapes based on their characteristics, create new shapes given certain parameters (ie: create a parallelogram using rhombuses), and justify my work.
Proficient
I can identify plane figures based on their definitions and describe how shapes change when they are combined
with other shapes or divided into smaller parts.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Intermediate
I can match plane figures to their description. I can put shapes together to make new shapes.
Beginner
I can sort plane figures based on how they look.
LP 5.14 SOL 5.14: The student will make predictions and determine the probability of an outcome by constructing a sample space. Learning Target: I can make predictions and determine the probability of an outcome by using tools (spinners, number cubes, etc.) and models
(sample space, number line, tree diagram, chart, etc.).
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can compare and contrast different tools and models to represent the probability of a simple event and create events that would fit positions on a number line from 0 to 1.
Proficient
I can make predictions and determine the probability of an outcome by using tools (spinners, number cubes, etc.) and models
(sample space, number line, tree diagram, chart, etc.).
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Intermediate
I can conduct simple experiments, identify how many outcomes are possible, and organize my results.
Beginner
I can identify the probability of an event as impossible, less likely, equally likely, more likely, or certain and label a number line in equal parts to represent the probability (0=impossible, 1=certain).
LP 5.15 SOL 5.15: The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots and line graphs. Learning Target: The student will interpret data organized in line graphs and stem-and-leaf plots and answer experimental questions quantitatively.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can justify a data display strategy.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Proficient
I can interpret data organized in line graphs and stem-and-leaf plots and answer experimental questions quantitatively.
Intermediate I can organize my data into tables, charts, line graphs, and stem-and-leaf plots.
Beginning Given a problem situation, I can formulate a question that will guide my data collection.
LP 5.16 SOL 5.16: The student will
a) describe mean, median, and mode as measures of center; b) describe mean as fair share; c) find the mean, median, mode, and range of a set of data; and d) describe the range of a set of data as a measure of variation.
Learning Target: I can calculate measures of center and range and describe them as measures of center and variation.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Advanced Proficient
I can use the measures of center and variation to understand other characteristics of my data set.
Proficient I can calculate measures of center and range and describe them as measures of center and variation.
Intermediate I can explore the mean as a description of “equally dividing” and range as variation.
Beginning I can understand that mean, median, and mode are measures of center and describe the center or middle of a data set.
LP 5.17 SOL 5.17: The student will describe the relationship found in a number pattern and express the relationship. Learning Target: I can describe the mathematical relationships found in patterns using symbols.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Advanced Proficient
I can justify my representation of a mathematical relationship in symbols with manipulatives and words in an organized way.
Proficient I can describe the mathematical relationships found in patterns using words, numbers, and symbols.
Intermediate I can investigate the structure of patterns and how they grow or change, using concrete materials and calculators.
Beginning I can understand that patterns and functions can be represented with words, numbers, and symbols.
LP 5.18 SOL 5.18: The student will
a) investigate and describe the concept of variable; b) write an open sentence to represent a given mathematical relationship, using a variable; c) model one-step linear equations in one variable, using addition and subtraction; and d) create a problem situation based on a given open sentence, using a single variable.
Learning Target: I can describe and write an open sentence (including a variable) to represent a mathematical relationship. I can then model solving an open sentence (limited to addition or subtraction) and create a problem situation based on a given open sentence.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can compare and contrast a variety of models for solving open sentences and explain why a variable could have many solutions in an expression but one solution in an equation (at the 5th grade level).
Proficient
I can describe and write an open sentence (including a variable) to represent a mathematical relationship. I can then model solving an open sentence (limited to addition or subtraction) and create a problem situation based on a
given open sentence.
Intermediate
I can write an open sentence (including a variable) to represent a mathematical relationship. I can model solving an open sentence (limited to addition or subtraction).
Beginner
I can understand that a variable is a symbol that stands for an unknown number or object.
Loudoun County Public Schools Mathematics Office – Department of Curriculum and Instruction Grade 5
LP 5.19 SOL 5.19: The student will investigate and recognize the distributive property of multiplication over addition. Learning Target: I can recognize the distributive property of multiplication over addition, identify examples of the property, and model the property using pictures, numbers, and words.
Learning Progression The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations at each level of the learning progression.
Advanced Proficient
I can create equations that demonstrate the properties (identity, commutative, associative, and distributive) and compare and contrast the properties.
Proficient
I can recognize the distributive property of multiplication over addition, identify examples of the property, and model
the property using pictures, numbers, and words.
Intermediate
I can match models and equations with the property that they demonstrate (identity, commutative, associative, and/or distributive).
Beginner
I understand that each side of an equation represents an equivalent value and that properties are true regardless of the numbers.
Grade 5 Math Intervention Ideas
Unit 2 – Whole Number Operations & Applications
Hands-On Standards Book Grades 3-4 Every Day Counts Partner Games
Number and Operations Lesson 13 Measurement Lessons 4-6
Game #1: Prime or Not? Game #2: Multiple Rally Game # 3: Target Products Game #10: Target Quotients Game #14: Factor Search Game #15: Divisibility Search
Hands-On Standards Book Grades 5-6
Number and Operations Lessons 9, 15 Algebra Lessons 5-8 Measurement Lesson 2
Unit 3 – Patterns & Properties
Hands-On Standards Book Grades 3-4 Every Day Counts Partner Games
Algebra Lessons 2, 9
Hands-On Standards Book Grades 5-6
Algebra Lessons 3, 4, 9
Unit 4 – Comparing & Applying Rational Number Concepts
Hands-On Standards Book Grades 3-4 Every Day Counts Partner Games
Number and Operations Lessons 15, 17-18, 21-23 Data Analysis and Probability Lessons 6-8
Game #1: Make One With Decimals Game #4: Get Them In Order Decimals Game #5: Make One With Fractions Game #6: Fraction/Decimal Concentration Game #9: From Here to There Decimals Game #13: Fraction/Decimal Match Up Game #16: Get Them In Order Fractions Game #12: Go For Zero With Decimals Game #17: Decimal Number Maker Game #18: From Here to There Fractions Game #19: Target Fractions
Hands-On Standards Book Grades 5-6
Number and Operations Lessons 1, 4-6
Unit 5 – Rational Number Operations & Measurement Applications
Hands-On Standards Book Grades 3-4 Every Day Counts Partner Games
Number and Operations Lessons 20, 24 Measurement Lessons 3, 10-11
Game #12: Go For Zero With Decimals Game #20: Ready, Set, Fractions
Hands-On Standards Book Grades 5-6
Number and Operations Lessons 11-13, 17 Measurement Lessons 1, 4, 6
Unit 6 – Classifying & Subdividing Plane Geometric Figures
Hands-On Standards Book Grades 3-4 Every Day Counts Partner Games
Geometry Lessons 3, 8
Hands-On Standards Book Grades 5-6
Geometry Lessons 1-4
Unit 7 – Data & Statistics
Hands-On Standards Book Grades 3-4 Every Day Counts Partner Games
Data Analysis and Probability Lessons 1, 5
Hands-On Standards Book Grades 5-6
Data Analysis and Probability Lessons 1-3
Resources available in all LCPS Elementary Schools
Hands-On Standards books Every Day Counts Partner Games
NCSM Great Tasks K-5 (available in all LCPS Elementary Schools)
VA SOL Alignment
Kindergarten Math
Domino Addition and Subtraction
Launch
SOL K.2 The student, given a set containing 15 or fewer concrete objects, will a) tell how many are in the set by counting the number of objects orally; b) write the numeral to tell how many are in the set; and c) select the corresponding numeral from a given set of numerals.
Activity
SOL K.1 The student, given two sets, each containing 10 or fewer concrete objects, will identify and describe one set as having more, fewer, or the same number of members as the other set, using the concept of one-to-one correspondence.
Counting Sheep
Launch
SOL K.2 The student, given a set containing 15 or fewer concrete objects, will a) tell how many are in the set by counting the number of objects orally; d) write the numeral to tell how many are in the set; and e) select the corresponding numeral from a given set of numerals.
Activity SOL K.3 The student, given an ordered set of ten objects and/or
pictures, will indicate the ordinal position of each object, first through tenth, and the ordered position of each object.
How Big is Your Foot?
Launch & Activity
SOL K.10 The student will compare two objects or events, using direct comparisons or nonstandard units of measure, according to one or more of the following attributes: length (shorter, longer), height (taller, shorter), weight (heavier, lighter), temperature (hotter, colder). Examples of nonstandard units include foot length, hand span, new pencil, paper clip, and block.
1st Grade Math
Bunny Hip Hop
Launch
SOL 1.1 The student will a) count from 0 to 100 and write the corresponding numerals;
and b) group a collection of up to 100 objects into tens and ones and
write the corresponding numeral to develop an understanding of place value.
Activity
SOL 1.2 The student will count forward by ones, twos, fives, and tens to 100 and backward by ones from 30.
When does it Happen?
Launch & Activity
SOL 1.8 The student will tell time to the half-hour, using analog and digital clocks.
Ten is our Friend!
Launch
SOL 1.5 The student will recall basic addition facts with sums to 18 or less and the corresponding subtraction facts.
Activity SOL 1.6 The student will create and solve one-step story and picture
problems using basic addition facts with sums to 18 or less and the corresponding subtraction facts.
2nd Grade Math
Creative Cards
Launch & Activity
SOL 2.16 The student will identify, describe, compare, and contrast plane and solid geometric figures (circle/sphere, square/cube, and rectangle/rectangular prism).
Piggy Banks
Launch & Activity
SOL 2.10 The student will a) count and compare a collection of pennies, nickels, dimes, and
quarters whose total value is $2.00 or less; and b) correctly use the cent symbol (¢), dollar symbol ($), and
decimal point (.).
Show What You Know!
Launch
SOL 2.2 The student will a) identify the ordinal positions first through twentieth, using an
ordered set of objects; and b) write the ordinal numbers.
Activity
SOL 2.8 The student will create and solve one- and two-step addition and subtraction problems, using data from simple tables, picture graphs, and bar graphs.
SOL 2.9 The student will recognize and describe the related facts that represent and describe the inverse relationship between addition and subtraction.
Pies for Sale
Launch SOL 2.19 The student will analyze data displayed in picture graphs,
pictographs, and bar graphs.
Activity
SOL 2.17 The student will use data from experiments to construct picture graphs, pictographs, and bar graphs.
SOL 2.19 The student will analyze data displayed in picture graphs, pictographs, and bar graphs.
3rd Grade Math
Playful Puppies
Launch
SOL 3.10 The student will a) measure the distance around a polygon in order to determine
perimeter; and b) count the number of square units needed to cover a given
surface in order to determine area.
SOL 3.20 The student will a) investigate the identity and the commutative properties for
addition and multiplication; and b) identify examples of the identity and commutative properties
for addition and multiplication.
Activity
SOL 3.5 The student will recall multiplication facts through the twelves table, and the corresponding division facts.
SOL 3.6 The student will represent multiplication and division, using area, set, and number line models, and create and solve problems that involve multiplication of two whole numbers, one factor 99 or less and the second factor 5 or less.
SOL 3.10 The student will a) measure the distance around a polygon in order to determine
perimeter; and b) count the number of square units needed to cover a given
surface in order to determine area.
SOL 3.20 The student will a) investigate the identity and the commutative properties for
addition and multiplication; and b) identify examples of the identity and commutative properties
for addition and multiplication.
Correcting the Calculator
Launch & Activity
SOL 3.1 The student will a) read and write six-digit numerals and identify the place value
and value of each digit; b) round whole numbers, 9,999 or less, to the nearest ten,
hundred, and thousand; and c) compare two whole numbers between 0 and 9,999, using
symbols (>, <, or =) and words (greater than, less than, or equal to).
SOL 3.2 The student will recognize and use the inverse relationships between addition/subtraction and multiplication/division to complete basic fact sentences. The student will use these relationships to solve problems.
Fraction Reactions
Launch & Activity
SOL 3.3 The student will a) name and write fractions (including mixed numbers)
represented by a model; b) model fractions (including mixed numbers) and write the
fractions’ names; and c) compare fractions having like and unlike denominators, using
words and symbols (>, <, or =).
4th Grade Math
Bugs, Giraffes, Elephants, and More
Launch
SOL 4.2 The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions; and c) identify the division statement that represents a fraction.
SOL 4.7 The student will a) estimate and measure length, and describe the result in
both metric and U.S. Customary units; and b) identify equivalent measurements between units within
the U.S. Customary system (inches and feet; feet and yards; inches and yards; yards and miles) and between units within the metric system (millimeters and centimeters; centimeters and meters; and millimeters and meters).
SOL 4.14 The student will collect, organize, display, and interpret data from a variety of graphs.
Activity
SOL 4.14 The student will collect, organize, display, and interpret data from a variety of graphs.
Does it Make Sense?
Launch
SOL 4.3 The student will a) read, write, represent, and identify decimals expressed
through thousandths; b) round decimals to the nearest whole number, tenth, and
hundredth; c) compare and order decimals; and
SOL 4.4 The student will a) estimate sums, differences, products, and quotients of whole
numbers; b) add, subtract, and multiply whole numbers; c) divide whole numbers, finding quotients with and without
remainders; and d) solve single-step and multistep addition, subtraction, and
multiplication problems with whole numbers.
Activity
SOL 4.4 The student will a) estimate sums, differences, products, and quotients of whole
numbers; b) add, subtract, and multiply whole numbers; c) divide whole numbers, finding quotients with and without
remainders; and d) solve single-step and multistep addition, subtraction, and
multiplication problems with whole numbers.
The Bigger Half
Launch & Activity
SOL 4.2 The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions; and c) identify the division statement that represents a fraction.
Harry’s Hike
Launch
SOL 4.2 The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions; and c) identify the division statement that represents a fraction.
Activity
SOL 4.5 The student will a) determine common multiples and factors, including least
common multiple and greatest common factor; b) add and subtract fractions having like and unlike
denominators that are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fractions, using common multiples and factors;
c) add and subtract with decimals; and d) solve single-step and multistep practical problems involving
addition and subtraction with fractions and with decimals.
5th Grade Math
Packed Parking
Launch
SOL 5.4 The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers.
Activity
SOL 5.5 The student will a) find the sum, difference, product, and quotient of two numbers
expressed as decimals through thousandths (divisors with only one nonzero digit); and
b) create and solve single-step and multistep practical problems involving decimals.
SOL 5.6 The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form.
Finding Fractions
Launch
SOL 5.2 The student will a) recognize and name fractions in their equivalent decimal form
and vice versa; and b) compare and order fractions and decimals in a given set from
least to greatest and greatest to least.
Activity
SOL 5.17 The student will describe the relationship found in a number pattern and express the relationship.
SOL 5.18 The student will a) investigate and describe the concept of variable; b) write an open sentence to represent a given mathematical
relationship, using a variable; c) model one-step linear equations in one variable, using addition
and subtraction; and d) create a problem situation based on a given open sentence,
using a single variable.
SOL 5.19 The student will investigate and recognize the distributive property of multiplication over addition.
Varying Volumes
Launch & Activity
SOL 5.8 The student will a) find perimeter, area, and volume in standard units of measure; b) differentiate among perimeter, area, and volume and identify
whether the application of the concept of perimeter, area, or volume is appropriate for a given situation;
c) identify equivalent measurements within the metric system; d) estimate and then measure to solve problems, using U.S.
Customary and metric units; and e) choose an appropriate unit of measure for a given situation
involving measurement using U.S. Customary and metric units.
SOL 5.13 The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will
a) develop definitions of these plane figures; and b) investigate and describe the results of combining and
subdividing plane figures.
Location, Location, Location
Launch
SOL 5.17 The student will describe the relationship found in a number pattern and express the relationship.
Activity
SOL 5.18 The student will a) investigate and describe the concept of variable; b) write an open sentence to represent a given mathematical
relationship, using a variable; c) model one-step linear equations in one variable, using addition
and subtraction; and d) create a problem situation based on a given open sentence,
using a single variable.
SOL 5.19 The student will investigate and recognize the distributive property of multiplication over addition.
Mathematics Literature Connections Organized by Curriculum Units
Grade K Math Literature Connections
Unit 2: Counting
One, Two, Skip a Few: First Number Rhymes by Roberta Arenson
98,99,100! Ready or Not, Here I come! by Marilyn Bums and Teddy Slater
Unit 3: Comparing Sets
20 Hungry Piggies: A Number Book by Trudy Harris
Ten Little Rubber Ducks by Eric Carle
Ten Little Caterpillars by Bill Martin Jr
Henry the Fourth by Stuart J. Murphy
One Monday Morning by Uri Shulevitz
The Napping House by Audrey Wood
Tally O’Malley by Stewart J. Murphy
So you want to be President? by Judith St. George
The Great Graph Contest by Loreen Leedy
Unit 4: Geometry & Sorting
Dave’s Down-to-Earth Rock Shop by Stuart J. Murphy
Unit 5: Shapes in Space
Twizzlers Shapes and Patterns by Jerry Pallotta
Unit 6: Geometry & Fractions
Give Me Half by Stuart J. Murphy
Full House by Dayle Ann Dodds
Unit 7: Measuring My World
Measuring Up by J.E. Osborne
Dumpling Soup by Jama Kim Rattigan
How Big is a Foot by Rolf Myller
Big and Little by Steven Jenkins
Time to… by Bruce McMillan
Telling Time: How to Tell Time on Digital and Analog Clocks by Jules Older
Telling Time with Big Mama Cat by D. Harper
Biggest, Strongest, Fastest by Steven Jenkins
Inch by Inch by Leo Lionni
Before and After: A Book of Nature Timescapes by Jan Thornhill
Unit 8: Skip Counting & Money
Arctic Fives Arrive by Elinor J. Pinczes
26 Letters and 99 Cents by Tana Hoban
Unit 9: Combining & Separating
More or Less by Stuart J. Murphy
Animals on Board by Stuart J. Murphy
A Quarter from the Tooth Fairy by Caren Holtzman
Grade 1 Math Literature Connections
Unit 2: Sorting, Ordering, & Patterns
Twizzlers Shapes and Patterns by Jerry Pallotta
Unit 3: Developing a Base Ten System
Moira’s Birthday by Robert Munsch
Something Good by Robert Munsch
Is It Larger? Is It Smaller? By T. Hoban
One Hundred Hungry Ants by Elinor J. Pinczes
Ten Sly Piranhas: A Counting Story in Reverse by William Wise
How Many, How Many, How Many by Rick Walton
98, 99, 100! Ready or Not, Here I Come! By Marilyn Burns and Teddy Slater
Stay in Line by Teddy Slater
Unit 4: Geometry & Fractions
Three Pigs, One Wolf, and Seven Magic Shapes by Grace Maccarone
Flat Stanley by J. Brown
The Shapes We Eat by Simone T. Ribke
Give Me Half! By Stuart J. Murphy
Gator Pie by L. Mathews
Unit 5: Time & Fractions
Give Me Half by Stuart J. Murphy
Telling Time: How to Tell Time on Digital and Analog Clock by Jules Older
Before and After: A Book of Nature Timescapes by Jan Thornhill
Unit 6: Working With Data
Probably Pistachio by Stuart J. Murphy
So You Want to be President? By Judith St. George
The Great Graph Contest by Loreen Leedy
Ready, Set, Hop! By Stuart J. Murphy
Bunches and Bunches of Bunnies by Mathews and Bassett
Unit 7: Combining & Separating
Rooster’s Off to See the World by Eric Carle
Round Trip by A. Jonas
Lemonade For Sale by Stuart J. Murphy
Unit 8: Measuring My World
How Do You Measure Weight? by Thomas K. and Heather Adamson
The Greedy Triangle by Marilyn Burns
Dumpling Soup by Jama Kim Rattigan
How Big is a Foot? by Rolf Myller
Big and Little by Steven Jenkins
Biggest, Strongest, Fastest by Steven Jenkins
Inch by Inch by Leo Lionni
More or Less by Stuart J. Murphy
Best Bug Parade by Stuart J. Murphy
Me and the Measure of Things by J. Sweeney
Unit 9: Applying Place Value
Shoes, Shoes, Shoes by A. Morris
Unit 10: Whole Number Computation
Animals on Board by Stuart J. Murphy
Elevator Magic by Stuart J. Murphy
Ten Black Dots by Donald Crew
Rooster’s Off to See the World by Eric Carle
Elevator Magic by Stuart J. Murphy
How High Can a Dinosaur Count? by V. Fisher
Unit 11: Skip Counting & Money
The Penny Pot by Stuart J. Murphy
Grade 2 Math Literature Connections
Unit 2: Extending Place Value
The Crayon Counting Book by Pam Munoz
Underwater Counting: Even Numbers by Jerry Pallotta
Unit 3: Computational Fluency
Growing Patterns: Fibonacci Numbers in Nature by S.G. and R.P. Campbell
Mission: Addition by Loreen Leedy
Each Orange Had 8 Slices: A Counting Book by Paul Giganti
Elevator Magic by Stuart J. Murphy
Unit 4: Applying Place Value to Computation/Problem Solving
Great Estimations by Bruce Goldstone
How Many Seeds in a Pumpkin? By Margaret McNamara and G. Brian Karas
How Many Feet? How Many Tails? A Book of Math Riddles by Marilyn Burns
Sam and the Lucky Money by K. Chinn
Balancing Act by Ellen Stoll Walsh
Betcha by Stuart J. Murphy
Unit 5: Probability & Data
Frog and Toad are Friends by A. Lobel
Polar Bear Math: Learning About Fractions from Klondike and Snow by Nagda and Bickel
Get Up and Go! By Stuart J. Murphy
Unit 6: Data & Problem Solving
So You Want to be President? By Judith St. George
Unit 7: Time & Temperature
Telling Time: How to Tell Time on Digital and Analog Clock by Jules Older
Before and After: A Book of Nature Timescapes by Jan Thornhill
Why Mosquitoes Buzz in People’s Ears: A West African Tale by V. Aardema
The Grouchy Lady Bug by Eric Carle
Chimp Math: Learning About Time from a Baby Chimpanzee by Nagda and Bickel
What Time Is It? A Book of Math Riddles by Sheila Keenan
Unit 8: Geometry & Fractions
Eating Fractions by Bruce McMillan
Give Me Half by Stuart J. Murphy
Full House by Dayle Ann Dodds
The Patchwork Quilt by Valerie Flournoy
Unit 9: Measuring My World
Inch by Inch by Leo Lionni
How Big is a Foot? By Rolf Myller
Dumpling Soup by Jama Kim Rattigan
Big and Little by Steven Jenkins
Biggest, Strongest, and Fastest by Steven Jenkins
More or Less by Stuart J. Murphy
Unit 10: Skip Counting & Money
Jelly Beans for Sale by Bruce McMillan
The Penny Pot by Stuart J. Murphy
The Coin Counting Book by Rosanne Lanczak Williams
Once Upon a Dime by Nancy Kelly Allen
Grade 3 Math Literature Connections
Unit 2: Place Value
Many Is How Many? By Illa Pondendorf
A Light in the Attic (“How Many, How Much” and “Overdues”) by Shel Silverstein
Counting on Frank by Rod Clement
How Much Is a Million? by David M. Schwartz
If You Made a Million by David M. Schwartz
Moira’s Birthday by Robert Munsch
Something Good by Robert Munsch
Unit 3: Computation With Whole Numbers (addition/subtraction)
Ten Black Dots by Donald Crews
Dealing with Addition Lynette Long
One Duck Stuck by Phyllis Root
One Gorilla by Atsuko Morozumi
A Three Hat Day by Laura Geringer
Unit 4: Money
Alexander, Who Used To Be Rich Last Sunday by Judith Viorst
Penny: The Forgotten Coin by Denise Brenna-Nelson
The Coin Counting Book by Rozanne Lanczak Williams
The Penny Pot by Stuart Murphy
Pigs Will Be Pigs: Fun With Math and Money by Amy Axelrod
Unit 5: Computation With Whole Numbers (multiplication/division)
Amanda Bean’s Amazing Dream by Cindy Neuschwander
A Remainder of One (*extension) by Elinor J. Pinczes
One Hundred Angry Ants by Elinor J. Pinczes
2 x 2 = Boo by Loreen Leedy
7 x 9 Trouble by Claudia Mills
Too Many Kangaroo Things to Do by Stuart Murphy
Divide and Ride by Stuart Murphy
Bananas Jacqueline Farmer
Centipede’s 100 Shoes by Tony Ross
Ten Times Better by Richard Michelson
Unit 6: Patterns & Data
Emma’s Christmas by Irene Trivias
The Doorbell Rang by Pat Hutchins
One Hundred Angry Ants by Elinor Pinczes
She Came Bringing Me That Little Baby Girl by Eloise Greenfield
Knots on a Counting Rope by Bill Martin Jr.
Berries, Nuts, and Seeds by Diane L. Burns
Lemonade for Sale by Stuart Murphy
Tiger Math: Learning to Graph from a Baby Tiger by Ann Whitehead Nagda
Grapes of Math by Greg Tang
The Quilting Bee by Gail Gibbons
Two Ways to Count to Ten: A Liberian Folktale by Ruby Dee
Unit 7: Geometry
The Important Book by Margaret Wise Brown
Three Pigs, One Wolf, and Severn Magic Shapes by Grace Maccarone
Pablo’s Tree Pat Mora
If You Were a Polygon Marcie Aboff
It Looked Like Spilt Milk by Charles G. Shaw
Mummy Math by Cindy Neuschwander
Shape Up by David Adler
A Cloak for the Dreamer by Aileen Friedman
Unit 8: Fractions, Probability, & Measurement (length) / Unit 9: Computation With Fractions
Eating Fractions by Bruce McMillan
Seven Little Hippos by Mike Thaler
Shoes, Shoes, Shoes by Ann Morris
Biggest, Strongest, Fastest Steve Jenkins
The Wolf’s Chicken Stew Keiko Kasza
A Very Improbably Story: A Math Adventure by Edward Einhorn
The Thirteen Days of Halloween Carool Greene
The Doorbell Rang Pat Hutchins
Whole-y Cow, Fractions are Fun! by Taryn Souders
Apple Fractions by Jerry Pallotta
The Hershey’s Milk Chocolate Bar Fractions BookU by Jerry Pallotta
Fraction Action by Loreen Leedy
Unit 10: Elapsed Time and Temperature
Telling Time: How to Tell Time on Digital and Analog Clocks by Jules Older
What Time is it, Mr. Crocodile? By Judy Sierra
Chimp Math by Ann Whitehead Nagda
Unit 11: Measurement
Spaghetti and Meatballs for All by Marilyn Burns
Perimeter, Area, and Volume David A. Adler
Pastry School in Paris Cindy Neuschwander
Measuring Penny (length) by Loreen Leedy
Biggest, Strongest, Fastest by Steve Jenkins
Is a Blue Whale the Biggest Thing There Is? by Robert E. Wells
Polly’s Pen Pal by Stuart L. Murphy
Spaghetti and Meatballs for All by Marilyn Burns
Room for Ripley by Stuart Murphy
Grade 4 Math Literature Connections
Unit 2: Number Sense: Whole Numbers
A Million Fish…More or Less by Patricia C. McKissack
Unit 3: Whole Number Operations & Applications (adding & subtracting)
Math Curse by Jon Scieszka and Lane Smith
The $1.00 Word Riddle Book by Marilyn Burns
Esio Trot by Roald Dahl
From Seashells to Smart Cards: Money and Currency (everyday economics) by Ernestine
Giesecke
Anno’s Magic Seeds by Mitsumasa Anno
Equal Shmequal by Virginia Kroll
Unit 4: Whole Number Operations & Applications (multiplication & division)
The King’s Chessboard by David Birch
The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan
Math Curse by Jon Scieszka and Lane Smith
Hottest, Coldest, Highest, Deepest by Steve Jenkins
In the Next Three Seconds…Predictions for the Millennium by Comp. Rowland Morgan
Ten Times Better by Richard Michelson
Two Ways to Count to Ten: A Liberian Folktale by Ruby Dee
A Remainder of One by Elinor J. Pinczes
Counting on Frank by Rod Clement
Unit 5: Data & Statistics
The Great Graph Contest by Loreen Leedy
Unit 6: Number Sense: Rational Numbers
The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan
One Riddle, One Answer by Lauren Thompson
Icebergs and Glaciers by Seymour Simon
Tiger Math: Learning to Graph from a Baby Tiger by Ann W. Nagda and Cindy Bickel
Unit 7: Rational Number Operations
Jump, Kangaroo Jump (Math Start) by Stuart Murphy and Kevin O’Malley
Pizza Counting by Christina Dobson
Piece=Part=Portion by Gifford and Thaler
Fractions=Trouble! By Claudia Mills
Unit 8: Probability & Data Using Rational Numbers
Jumanji by Chris Van Allsburg
A Very Improbable Story by Edward Einhorn and Adam Gustavson
Pigs at Odds by Amy Axelrod and Sharon Nally
Unit 9: Patterns & Measurement
G is for Googol: A Math Alphabet Book by David M. Schwartz
How Much, How Many, How Far, How Heavy, How Long, How Tall Is 1000? by Helen
Nolan
Icebergs and Glaciers by Seymour Simon
If You Hopped Like a Frog by David M. Schwartz
Is a Blue Whale the Biggest Thing There Is? By Robert E. Wells
Biggest, Strongest, Fastest by Steve Jenkins
Unit 10: Plane Geometry & Transformations
Marvelous Math by Lee Bennett Hopkins
The Warlord’s Puzzle by Virginia Walton Pilegard
Shape Up! Fun with Triangles and Other Polygons by David Adler and Nancy Tobin
Spaghetti and Meatballs for All! by Marilyn Burns and Debbie Tilley
Chickens on the Move (Math Matters!) by Pamela Pollack
Grade 5 Math Literature Connections
Unit 2
A Remainder of One by Elinor Pinczes
My Even Day by Doris Fisher
The Grapes of Math by Greg Tang
Math Appeal by Greg Tang
Among the Odds and Evens by Prescilla Turner
Spaghetti and Meatballs for All by Marilyn Burns
Unit 3
Germs Make Me Sick by Melvin Berger
Bats on Parade by Kathi Appelt
Unit 4
Alexander Who Used to Be Rich Last Sunday by Judith Viorst
Fraction Fun by David Adler
Unit 5
Measuring Penny by Loreen Leedy
Alexander Who Used to Be Rich Last Sunday by Judith Viorst
Counting On Frank by Rod Clements
How Long? How Wide? by Brian Cleary
Millions to Measure by David Schwartz
Fractions, Decimals, and Percents by David Adler
Unit 6
The Greedy Triangle by Marilyn Burns
Sir Cumference and the Dragon of Pi by Cindy Neuschwander
Unit 7
Chimp Math, Tiger Math, Polar Bear Math, and Cheetah Math (series) by Anne Nagda
A More Perfect Union by Betsy Maestro
Model Performance Indicator Information for Curriculum Guides
Embedded in the LCPS curriculum guides are sample Model Performance Indicator (MPI) tables (below).
These tables will be useful as you differentiate instruction for all of your learners, but they are especially
helpful for English Language Learners. Below are frequently asked questions about MPI.
What is a Model Performance Indicator (MPI)?
An MPI is a tool that can be used to show examples of how language is processed or produced within a
particular context, including the language with which students may engage during classroom instruction and
assessment.
Each MPI contains three main parts:
Language Function: The first part of an MPI, this shows how students are processing/producing
language at each level of language proficiency
Content Stem: This will remain consistent throughout an MPI strand and should reflect the knowledge
and skills of the state’s content standards
Support: The final part of an MPI, this highlights the differentiation that should be incorporated for
students at each language level by suggesting appropriate instructional supports for students at each
level of language proficiency
The samples provided also include an example context for language use that provides a brief descriptor of the
activity or task in which students would be engaged, while the inclusion of topic-related language helps to
support the emphasis on imbedding academic language instruction into our content-area teaching practices.
How can these sample MPIs help me?
Educators can use MPI strands in several ways:
to align students’ performance to levels of language development
as a tool for creating language objectives/targets that will help extend students’ level of language
proficiency
as a means for differentiating instruction that incorporates the language of the content area in a way that
meets the needs of students’ levels of language proficiency
An MPI strand helps illustrate the progression of language development from one proficiency level to the next
within a particular context. As these strands are examples, they represent one of many possibilities; therefore,
they can be transformed in order to be made more relevant to the individual classroom context.
Where can I get more information about WIDA, MPIs, etc.?
See My Learning Plan for several WIDA training modules
Introduction to the WIDA ELD Standards
Transforming the WIDA ELD Standards
Interpreting the WIDA ACCESS Score Report
The information above was adapted from the 2012 Amplification of the English Development Standards Kindergarten-Grade 12 resource guide and can be accessed at www.wida.us
SOL Strand and Bullet: 5.14 The student will make predictions and determine the probability of an outcome by constructing a sample space.
Example Context for Language Use: The student will work in small groups or with a partner to create a tree diagram, list, or chart to show all
possible outcomes of an event (e.g., ordering pizza with the following choices: 2 types of crust, 3 toppings, and 4 sides).
COGNITIVE FUNCTION: Students at all levels of English language proficiency will EVALUATE all possible outcomes of a simple event in a
sample space with 24 or less possible outcomes.
SP
EA
KIN
G
Level 1
Entering
Level 2
Emerging
Level 3
Developing
Level 4
Expanding
Level 5
Bridging
Lev
el 6-R
each
ing
Identify all possible
outcomes of a simple
event as a class using
photos or illustrations
Discuss all possible
outcomes of a simple
event with a small group
using oral sentence
starters
Predict all possible
outcomes of a simple
event with a partner using
a graphic organizer (e.g.,
tree diagram)
Explain all possible
outcomes of a simple
event and the related
sample space to a partner
Defend outcome
predictions of a simple
event and the related
sample space to a
partner
WR
ITIN
G
Create a sample space of
all possible outcomes of
a simple event using a
tree diagram, list, or chart
with a partner
Describe all possible
outcomes of a simple
event in a tree diagram
using sentence starters
with a small group
Predict in a written list all
possible outcomes of a
simple event with a
partner
Explain in complete
sentences all possible
outcome predictions of a
simple event and the
related sample space in a
math journal
Defend in complete
sentence all outcome
predictions of a simple
event and the related
sample space in a math
journal
TOPIC-RELATED LANGUAGE: Students at all levels of English language proficiency interact with grade level words and expressions
such as: predict, prediction, outcome, probability, possible, sample space, tree diagram, list, chart, certain, likely, equally likely, unlikely,
impossible, fraction, numerator, denominator, identify, discuss, predict, explain, defend, create, and describe.
SOL Strand and Bullet: 5.15 The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-
leaf plots and line graphs.
Example Context for Language Use: Students will collect data on the daily temperature lows or highs for one month. Students will work in small
groups or with a partner to choose the best way to organize the data (e.g., stem-and-leaf plot or line graph). Students will represent the temperature
data on a stem-and-leaf plot and the temperature changes through time on a line graph.
COGNITIVE FUNCTION: Students at all levels of English language proficiency will ANALYZE representations of data in various forms.
LIS
TE
NIN
G
Level 1
Entering
Level 2
Emerging
Level 3
Developing
Level 4
Expanding
Level 5
Bridging
Lev
el 6-R
each
ing
Follow oral directions to
represent data in various
forms (e.g., stem-and-
leaf plot, line graph)
using illustrated
examples in small groups
using L1 or L2
Follow oral directions to
represent data in various
forms using illustrated
examples in small
groups
Identify the best
representation of data to
use for data gathered (e.g.,
stem-and-leaf plot, line
graph) based on teacher-
modeled simple oral
directions with a partner
Identify the best
representation of data to
use for data gathered
(e.g., stem-and-leaf plot,
line graph) based on
teacher-modeled multi-
step oral directions with a
partner
Recommend the best
representation for each
type of data gathered
(e.g., stem-and-leaf
plot, line graph) based
on oral discourse with a
partner
SP
EA
KIN
G
Verbally identify
characteristics of
representations of data
(e.g., stem-and-leaf plot,
line graph) using
illustrated word banks in
small groups
Formulate questions
about the representation
of data using
illustrations and oral
sentence frames in small
groups
Verbally Formulate
questions about the
representation of data
using illustrated examples
with a partner
Explain the preferred
form for the
representation of data
(e.g., stem-and-leaf plot
or line graph) to a partner
Present completed
representations of data
to the class with a
partner
RE
AD
ING
Associate representations
of data with their name
(e.g., stem-and-leaf plot,
line graph) using
illustrated text, labeled
examples, and illustrated
word banks in small
groups
Develop an
understanding of various
forms for representation
of data (e.g., stem-and-
leaf plot, line graph)
using illustrated text and
labeled examples in a
small group
Compare various forms of
representation of data
using illustrated text with
a partner
Analyze various forms of
representation of data
(e.g., stem-and-leaf plot,
line graph) for accuracy
with a partner
Evaluate another
group’s representation
of data with a partner
Lev
el 6-R
each
ing
WR
ITIN
G
Write labels on
representations of data
(e.g., titles on stem-and-
leaf plot or line graph)
using examples and
illustrated word banks in
small groups
Describe types of
representations of data
using labeled examples
and written sentence
frames with a partner
Compare types of
representations of data
using labeled examples
and written sentence
frames with a partner
Create a representation of
data (e.g., stem-and-leaf
plot, line graph) using
labeled examples
Explain findings from
representations of data
in a math journal
TOPIC-RELATED LANGUAGE: Students at all levels of English language proficiency interact with grade level words and expressions
such as: data, survey, problem situation, collect, organize, interpret, analyze, stem-and-leaf plot, line graph, identify, explain, formulate,
present, associate, develop, compare, analyze, evaluate, describe, create
SOL Strand and Bullet: 5.16 The student will
a) describe mean, median, and mode as measures of center;
b) describe mean as fair share;
c) find the mean, median, mode, and range of a set of data; and
d) describe the range of a set of data as a measure of variation.
Example Context for Language Use: The student will work independently or with a partner to find the mean, median, mode and range of the ages
of his/her classmates.
COGNITIVE FUNCTION: Students at all levels of English language proficiency will EVALUATE measures of center and measure of variation
of gathered data.
LIS
TE
NIN
G
Level 1
Entering
Level 2
Emerging
Level 3
Developing
Level 4
Expanding
Level 5
Bridging
Lev
el 6-R
each
ing
Identify mean, median,
mode and range of
gathered data based on
online videos
(e.g., study jams or
Brainpop) in L1 or L2
Distinguish between
mean, median, mode
and range for gathered
data based on online
videos
(e.g., study jams or
Brainpop) in L1 or L2
Follow oral directions to
find the mean, median,
mode and range of
gathered data using a
template with a partner
(e.g., record classmates’
ages and find measures of
center and variation)
Apply knowledge of
mean, median, mode and
range of gathered data to
check work with a
partner
Compare and contrast
the mean, median,
mode and range of
gathered data with
those of another
partnership
(e.g., “I got _____.
What did you get?”
SP
EA
KIN
G
Describe mean, median,
mode and range of
gathered data using
illustrated word banks
and following teacher
model
Describe the mean,
median, mode and range
of gathered data with a
partner using oral
sentence frames
Justify the mean, median,
mode and range of
gathered data using a
graphic organizer with a
partner
Formulate and answer
questions on mean,
median, mode and range
of gathered data with a
partner
Compare and contrast
the mean, median,
mode and range of
gathered data with
those of another
partnership
(e.g., “I got _____.
What did you get?”
RE
AD
ING
Identify mean, median,
mode and range of
gathered data using a
table and illustrated text
with a partner
Distinguish between
mean, median, mode
and range of gathered
data from online videos
(e.g., study jams or
Brainpop)
Compare mean, median,
mode and range of
gathered data with a
partner
Analyze for accuracy the
completed mean, median,
mode and range of
gathered data with a
partner
Evaluate another
partnership’s mean,
median, mode and
range of gathered data
with a partner
Lev
el 6-
Rea
chin
g
WR
ITIN
G
Record mean, median,
mode and range of
gathered data using a
written model and a
graphic organizer
(e.g., foldable) with a
partner
Record mean, median,
mode and range of
gathered data using a
written model and a
graphic organizer
(e.g., foldable)
Solve mean, median,
mode and range of
gathered data using a
graphic organize
Explain mean, median,
mode and range of
gathered data using a
template or graphic
organizer
Summarize findings
from mean, median,
mode and range of
gathered data in a math
journal
TOPIC-RELATED LANGUAGE: Students at all levels of English language proficiency interact with grade level words and expressions
such as: mean, median, mode, measures of center, range, measure of variation, data, set, fair share, identify, distinguish, apply, compare and
contrast, describe, justify, formulate, identify, analyze, evaluate, record, solve, explain, summarize