qm - alexalex.state.al.us/ccrs/sites/alex.state.al.us.ccrs/files/qm3 6-12 pp.pdf · between finite...

QM #2 Next Steps Participants should do three things before the next session:

• Decide which task you will implement in your class, solve task, andanticipate possible student solution paths. • Implement task in the classroom (monitor, select, and sequence)• Bring student work samples (student’s solution path) to share with yourgroup.

Anticipating In the Anticipating phase, the teacher predicts during planning how students will respond to a particular mathematic task. He or she considers possible alternative strategies that students may use, as well as potential misconceptions that may surface.

Monitoring Monitoring occurs as the teacher circulates during the Explore of the lesson. He or she listens to student discussions while asking clarifying questions and posing new challenges.

Selecting This Selecting phase involves choosing the strategies for students that occur during the Explore that will best illustrate and promote the desired understanding of the Focus Question.

Sequencing In Sequencing, before the Summarize, the teacher plans which students will share their thinking, and in what order, so as to maximize the quality of the discussion.

Connecting Finally, the Connecting practice which occurs during the Summarize involves posing questions that enable students to understand and use each other’s strategies, and to connect the Problem’s Big Idea to previous learning. Connecting practice may also foreshadow future learning

Jour

nal R

efle

ctio

n U

sing

you

r stu

dent

wor

k sa

mpl

es, i

ndiv

idua

lly c

onsi

der t

he fo

llow

ing

and

then

pre

pare

to sh

are

your

thin

king

.

How

did

und

erst

andi

ng th

e Al

gebr

a Pr

ogre

ssio

n im

pact

inst

ruct

ion?

Giv

e sp

ecifi

c ex

ampl

es w

ith re

spec

t to:

help

ing

stud

ents

mak

e m

athe

mat

ical

con

nect

ions

w

ithin

the

Alg

ebra

Pr

ogre

ssio

n

wor

king

with

stru

gglin

g st

uden

ts to

dev

elop

con

cept

ual

unde

rsta

ndin

g of

the

Alg

ebra

Pr

ogre

ssio

n

how

impl

emen

ting

this

task

hel

p yo

u fo

rmat

ivel

y as

sess

stud

ents

un

ders

tand

ing/

mis

conc

eptio

ns

how

you

revi

sed

your

in

stru

ctio

n as

a re

sult

of y

our

lear

ning

on

the

Alg

ebra

Pr

ogre

ssio

n

CCRS

– M

athe

mat

ics

Lear

ning

Pro

gres

sion

s

CCRS

– M

athe

mat

ics Q

uart

erly

Mee

ting

#3 Feb

ruary

2015

Van Hiele Levels of Geometric Reasoning Level 1:______________________ Description: Suggestions for instruction:

Level 2: _______________________ Description: Suggestions for instruction:




Van Hiele’s Levels of Geometric Reasoning

Van Hiele Level

The way that students reason about shapes and other geometric ideas:

A student at this level might say:

An activity at this level might be:

Rigor

The shape is a rectangle because it looks like a box.

Two points determine a line is a postulate.

The figure above is a square because it has four equal sides and four right angles.

Work is in different geometric or axiomatic systems. Most likely geometry is being studied in a college level course.

Construct geometric shapes using pattern blocks.

Geometric figures are recognized/analyzed by their properties or components.

Geometric shapes are recognized on the basis of their physical appearance as a whole.

Informal Deduction

Using models, list the properties of the figure.

Deduction

Working on a geoboard, change a quadrilateral to a trapezoid, trapezoid to parallelogram, parallelogram to rectangle . . . What was required in each transformation?

All squares are rectangles because a square is a rectangle with consecutive sides congruent.

Construct proofs using postulates or axioms and definitions. A typical high school geometry course should be taught at this level.

Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p, as all lines in elliptic geometry intersect. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. Name and explain one of those properties.

Figure ABCD is a parallelogram. Discuss what you know about this figure. Write a problem in “If … then …” form on this figure.

Van Hiele’s Levels of Geometric Reasoning

Visualization Analysis The sum of the angles of any triangle is always greater than 180°.

Geometric relationships are recognized between and among properties of shapes or classes of shapes and logical arguments are followed using such properties.

Anticipation Guide

Geometry

a) In the first blank, mark “+” if you agree with the statement or “o” if you disagree with the statement.

b) Next, when directed to do so, discuss the statements in your group. c) In the second blank, the class should discuss and decide on a “+” or a “o” for each blank.

6 – 8 will read the Overview, the critical areas, and the standards for grades 6 - 8. The high school will read the Conceptual Category: Geometry, the Geometry Overview, and all the Geometry standards. After reading, in the second blank, write whether you agree or disagree. If you disagree, change the statement so that you will agree with it and cite where the information came from in the document

6th grade 1. ___ ___ Decomposing a cylinder to find the surface area is critical in the 6th grade.

7th grade 2. ___ ___ Students solve real-world and mathematical problems involving area, surface

area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms

8th grade 3. ___ ___ Students write a proof for the Pythagorean Theorem.

High School 4. ___ ___ Although there are many types of geometry, school mathematics in Alabama is

devoted primarily to plane Euclidean geometry which is characterized by the Parallel Postulate.

5. ___ ___ The concepts of congruence, similarity , and symmetry can be understood from the perspective of geometric transformation.

6. ___ ___ In the approach used in the CCRS, two geometric figures are defined to be congruent if a two column proof can be constructed.

2013 Alabama Course of Study: Mathematics i

GRADE 5 OVERVIEW

Grade 5 content is organized into five domains of focused study as outlined below in the column to the left. The Grade 5 domains listed in bold print on the shaded bars are Operations and Algebraic Thinking, Number and Operations in Base Ten, Number and Operations – Fractions, Measurement and Data, and Geometry. Immediately following the domain and enclosed in brackets is an abbreviation denoting the domain. Identified below each domain are the clusters that serve to group related content standards. AllGrade 5 content standards, grouped by domain and cluster, are located on the pages that follow.

The Standards for Mathematical Practice are listed below in the column to the right. These mathematical practice standards should be incorporated into classroom instruction of the content standards.

Content Standard Domains and Clusters Standards for Mathematical Practice

Operations and Algebraic Thinking [OA]• Write and interpret numerical expressions.• Analyze patterns and relationships.

Number and Operations in Base Ten [NBT]• Understand the place value system.• Perform operations with multi-digit whole

numbers and with decimals to hundredths.

Number and Operations – Fractions [NF]• Use equivalent fractions as a strategy to add

and subtract fractions.• Apply and extend previous understandings

of multiplication and division to multiply and divide fractions.

Measurement and Data [MD]• Convert like measurement units within a given

measurement system.• Represent and interpret data.• Geometric measurement: understand concepts

of volume and relate volume to multiplicationand to addition.

Geometry [G]• Graph points on the coordinate plane to solve

real-world and mathematical problems.• Classify two-dimensional figures into

categories based on their properties.

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

2013 Alabama Course of Study: Mathematics 39

GRADE 5

In Grade 5, instructional time should focus on three critical areas. These areas are (1) developing fluency with addition and subtraction of fractions and developing understanding of the multiplication of fractions and of division of fractions in limited cases, such as unit fractions divided by whole numbers and whole numbers divided by unit fractions; (2) extending division to two-digit divisors, integrating decimal fractions into the place value system, developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume. Important information regarding these three critical areas of instruction follows:

(1) Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. However, this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.

(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. Students apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations and make reasonable estimates of their results. Students use the relationship between decimals and fractions as well as the relationship between finite decimals and whole numbers, as for example, a finite decimal multiplied by an appropriate power of 10 is a whole number, to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

(3) Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. Students select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real-world and mathematical problems.


5th

c. Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. [5-NF7c]

Examples: How much chocolate will each person get if 3 people share 1

2lb of

chocolate equally? How many 1

3-cup servings are in 2 cups of raisins?

Measurement and Data

Convert like measurement units within a given measurement system.

18. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real-world problems. [5-MD1]

Represent and interpret data.

19. Make a line plot to display a data set of measurements in fractions of a unit ( 12

, 14

, 18

).

Use operations on fractions for this grade to solve problems involving information presented in line plots. [5-MD2]

Example: Given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

20. Recognize volume as an attribute of solid figures, and understand concepts of volume measurement. [5-MD3]

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. [5-MD3a]

b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. [5-MD3b]

21. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. [5-MD4]

22. Relate volume to the operations of multiplication and addition, and solve real-world and mathematical problems involving volume. [5-MD5]

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. [5-MD5a]

b. Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems. [5-MD5b]


5th

c. Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the nonoverlapping parts, applying this technique to solve real-world problems. [5-MD5c]

Geometry

Graph points on the coordinate plane to solve real-world and mathematical problems.

23. Use a pair of perpendicular number lines, called axes, to define a coordinate system with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). [5-G1]

24. Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. [5-G2]

Classify two-dimensional figures into categories based on their properties.

25. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. [5-G3]

Example: All rectangles have four right angles, and squares are rectangles, so all squares have four right angles.

26. Classify two-dimensional figures in a hierarchy based on properties. [5-G4]


GRADE 6 OVERVIEW

Grade 6 content is organized into five domains of focused study as outlined below in the column to the left. The Grade 6 domains listed in bold print on the shaded bars are Ratios and Proportional Relationships, The Number System, Expressions and Equations, Geometry, and Statistics and Probability. Immediately following the domain and enclosed in brackets is an abbreviation denoting the domain. Identified below each domain are the clusters that serve to group related content standards. All Grade 6 content standards, grouped by domain and cluster, are located on the pages that follow.



Ratios and Proportional Relationships [RP]• Understand ratio concepts and use ratio

reasoning to solve problems.

The Number System [NS]• Apply and extend previous understanding of

multiplication and division to divide fractionsby fractions.

• Compute fluently with multi-digit numbers and find common factors and multiples.

• Apply and extend previous understanding of numbers to the system of rational numbers.

Expressions and Equations [EE]• Apply and extend previous understanding of

arithmetic to algebraic expressions.• Reason about and solve one-variable

equations and inequalities.• Represent and analyze quantitative

relationships between dependent and independent variables.

Geometry [G]• Solve real-world and mathematical problems

involving area, surface area, and volume.

Statistics and Probability [SP]• Develop understanding of statistical

variability.• Summarize and describe distributions.










GRADE 6

In Grade 6, instructional time should focus on four critical areas. These areas are (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. Important information regarding these four critical areas of instruction follows:

(1) Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from and extending pairs of rows or columns in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. They solve a wide variety of problems involving ratios and rates.

(2) Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. They use these operations to solve problems. Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular, negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane.

(3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. They know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. They construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations such as 3x = y to describe relationships between quantities.

(4) Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. They recognize that a measure of variability, the interquartile range or mean absolute deviation, can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students learn to describe and summarize numerical data sets, including identifying clusters, peaks, gaps, and symmetry, with consideration to the context in which the data were collected.

Students in Grade 6 also build on their elementary school work with area by reasoning about relationships among shapes to determine area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss,


6th

develop, and justify formulas for areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They prepare for work on scale drawings and constructions in Grade 7 by drawing polygons in the coordinate plane.

Students will:

Ratios and Proportional Relationships

Understand ratio concepts and use ratio reasoning to solve problems.

1. Understand the concept of a ratio, and use ratio language to describe a ratio relationship between two quantities. [6-RP1]

Examples: “The ratio of wings to beaks in the bird house at the zoo was 2:1 because for every2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

2. Understand the concept of a unit rate a

bassociated with a ratio a:b with b

language in the context of a ratio relationship. [6-RP2]Examples: “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3

4cup of

flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)

3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. [6-RP3]

a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. [6-RP3a]

b. Solve unit rate problems including those involving unit pricing and constant speed.[6-RP3b]

Example: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30100

times the

quantity); solve problems involving finding the whole, given a part and the percent.[6-RP3c]

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. [6-RP3d]


6th

14. Apply the properties of operations to generate equivalent expressions. [6-EE3]Example: Apply the distributive property to the expression 3(2 + x) to produce the equivalent

expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

15. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). [6-EE4]

Example: The expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y represents.

Reason about and solve one-variable equations and inequalities.

16. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. [6-EE5]

17. Use variables to represent numbers, and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or, depending on the purpose at hand, any number in a specified set. [6-EE6]

18. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers. [6-EE7]

19. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. [6-EE8]

Represent and analyze quantitative relationships between dependent and independent variables.

20. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.[6-EE9]

Example: In a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Geometry

Solve real-world and mathematical problems involving area, surface area, and volume.

21. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. [6-G1]


6th

22. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. [6-G2]

23. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. [6-G3]

24. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. [6-G4]

Statistics and Probability

Develop understanding of statistical variability.

25. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6-SP1]

Example: “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

26. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. [6-SP2]

27. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. [6-SP3]

Summarize and describe distributions.

28. Display numerical data in plots on a number line, including dot plots, histograms, and box plots. [6-SP4]

29. Summarize numerical data sets in relation to their context, such as by: [6-SP5]a. Reporting the number of observations. [6-SP5a]b. Describing the nature of the attribute under investigation, including how it was measured

and its units of measurement. [6-SP5b]c. Giving quantitative measures of center (median and/or mean) and variability (interquartile

range and/or mean absolute deviation) as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. [6-SP5c]

d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. [6-SP5d]


GRADE 7 OVERVIEW

Grade 7 content is organized into five domains of focused study as outlined below in the column to the left. The Grade 7 domains listed in bold print on the shaded bars are Ratios and Proportional Relationships, The Number System, Expressions and Equations, Geometry, and Statistics and Probability. Immediately following the domain and enclosed in brackets is an abbreviation denoting the domain. Identified below each domain are the clusters that serve to group related content standards. All Grade 7 content standards, grouped by domain and cluster, are located on the pages that follow.



Ratios and Proportional Relationships [RP]• Analyze proportional relationships and use

them to solve real-world and mathematical problems.

The Number System [NS]• Apply and extend previous understandings of

operations with fractions to add, subtract, multiply, and divide rational numbers.

Expressions and Equations [EE]• Use properties of operations to generate

equivalent expressions.• Solve real-life and mathematical problems

using numerical and algebraic expressions and equations.

Geometry [G]• Draw, construct, and describe geometrical

figures and describe the relationships between them.

• Solve real-life and mathematical problems involving angle measure, area, surface area,and volume.

Statistics and Probability [SP]• Use random sampling to draw inferences

about a population.• Draw informal comparative inferences about

two populations.• Investigate chance processes and develop, use,

and evaluate probability models.










GRADE 7

In Grade 7, instructional time should focus on four critical areas. These areas are (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples. Important information regarding these four critical areas of instruction follows:

(1) Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. They use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.

(2) Students develop a unified understanding of number, recognizing fractions, decimals that have a finite or a repeating decimal representation, and percents as different representations of rational numbers. They extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction and multiplication and division. By applying these properties and by viewing negative numbers in terms of everyday contexts, such as amounts owed or temperatures below zero, students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. Students use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

(3) Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity in Grade 8, they reason about relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

(4) Students build on their previous work with single data distributions to compare two data distributions and address questions about differences between populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences.


7th

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

9. Solve multistep real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate, and assess the reasonableness of answers using mental computation and estimation strategies. [7-EE3]

Examples: If a woman making $25 an hour gets a 10% raise, she will make an additional 1

10

of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3

4inches long in the center of a door that is 27 1

2inches wide, you will

need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

10. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. [7-EE4]

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p,q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. [7-EE4a]

Example: The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p,q, and r are specific rational numbers. Graph the solution set of the inequality, and interpret it in the context of the problem. [7-EE4b]

Example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Geometry

Draw, construct, and describe geometrical figures and describe the relationships between them.

11. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. [7-G1]

12. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. [7-G2]

13. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. [7-G3]


7th

Solve real-world and mathematical problems involving angle measure, area, surface area, and volume.

14. Know the formulas for the area and circumference of a circle, and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. [7-G4]

15. Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. [7-G5]

16. Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. [7-G6]


Use random sampling to draw inferences about a population.

17. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7-SP1]

18. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7-SP2]

Example: Estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

Draw informal comparative inferences about two populations.

19. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. [7-SP3]

Example: The mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

20. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. [7-SP4]

Example: Decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.


GRADE 8 OVERVIEW

Grade 8 content is organized into five domains of focused study as outlined below in the column to the left. The Grade 8 domains listed in bold print on the shaded bars are The Number System, Expressions and Equations, Functions, Geometry, and Statistics and Probability. Immediately following the domain and enclosed in brackets is an abbreviation denoting the domain. Identified below each domain are the clusters that serve to group related content standards. All Grade 8 content standards, grouped by domain and cluster, are located on the pages that follow.



The Number System [NS]• Know that there are numbers that are not

rational, and approximate them by rational numbers.

Expressions and Equations [EE]• Work with radicals and integer exponents.• Understand the connections among

proportional relationships, lines, and linear equations.

• Analyze and solve linear equations and pairs of simultaneous linear equations.

Functions [F]• Define, evaluate, and compare functions.• Use functions to model relationships between

quantities.

Geometry [G]• Understand congruence and similarity using

physical models, transparencies, or geometrysoftware.

• Understand and apply the Pythagorean Theorem.

• Solve real-world and mathematical problemsinvolving volume of cylinders, cones, and spheres.

Statistics and Probability [SP]• Investigate patterns of association in bivariate

data.










GRADE 8

In Grade 8, instructional time should focus on three critical areas. These areas are (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; and (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence and understanding and applying the Pythagorean Theorem. Important information regarding these three critical areas of instruction follows:

(1) Students use linear equations and systems of linear equations to represent, analyze, and solve a

variety of problems. Students recognize equations for proportions such as y

x= m or y = mx as

special linear equations such as y = mx + b, understanding that the constant of proportionality, m,is the slope, and the graphs are lines through the origin. They understand that the slope, m, of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A. Students also use a linear equation to describe the association between two quantities in bivariate data such as arm span versus height for students in a classroom. At this grade, fitting the model and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship, such as slope and y-intercept, in terms of the situation.

Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. They solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.

(2) Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. Students can translate among representations and partial representations of functions, while noting that tabular and graphical representations may be partial representations, and they can describe how aspects of the function are reflected in the different representations.

(3) Students use ideas about distance and angles, including how they behave under translations, rotations, reflections, and dilations and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. They show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.


8th

13. Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. [8-F3]

Example: The function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4), and (3,9), which are not on a straight line.

Use functions to model relationships between quantities.

14. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y)values, including reading these from a table or from a graph. Interpret the rate of change andinitial value of linear function in terms of the situation it models and in terms of its graph or a table of values. [8-F4]

15. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. [8-F5]

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software.

16. Verify experimentally the properties of rotations, reflections, and translations: [8-G1]a. Lines are taken to lines, and line segments are taken to line segments of the same length.

[8-G1a]b. Angles are taken to angles of the same measure. [8-G1b]c. Parallel lines are taken to parallel lines. [8-G1c]

17. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8-G2]

18. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. [8-G3]

19. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. [8-G4]

20. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. [8-G5]

Example: Arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give argument in terms of transversals why this is so.


8th

Understand and apply the Pythagorean Theorem.

21. Explain a proof of the Pythagorean Theorem and its converse. [8-G6]

22. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. [8-G7]

23. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.[8-G8]

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

24. Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems. [8-G9]


Investigate patterns of association in bivariate data.

25. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. [8-SP1]

26. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. [8-SP2]

27. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. [8-SP3]

Example: In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

28. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. [8-SP4]

Example: Collect data from students in your class on whether or not they have a curfew on school nights, and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?


CONCEPTUAL CATEGORY: GEOMETRY

An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts. These include interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material.

Although there are many types of geometry, school mathematics is devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates). Euclidean geometry is characterized most importantly by the Parallel Postulate that through a point not on a given line there is exactly one parallel line. Spherical geometry, in contrast, has no parallel lines.

During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms.

The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, andcombinations of these, all of which are here assumed to preserve distance and angles, and therefore shapes generally. Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent.

In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. For triangles, congruence means the equality of all corresponding pairs of sides and all corresponding pairs of angles. During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures.

Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of “same shape” and “scale factor” developed in the middle grades. These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent.

The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean Theorem, are fundamental in many real-world and theoretical situations. The Pythagorean Theorem is generalized to nonright triangles by the Law of Cosines. Together, the Law of Sines and the Law of Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that side-side-angle is not a congruence criterion.

Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Just as the number line associates numbers with locations in one dimension, a pair of perpendicular axes associates pairs of numbers with locations in two dimensions. Geometric transformations of the graphs of equations correspond to algebraic changes in their equations.


Dynamic geometry environments provide students with experimental and modeling tools. These tools allow them to investigate geometric phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.

Connections to Equations

The correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.


GEOMETRY OVERVIEW

The Geometry conceptual category focuses on the six domains listed below in the column to the left. The Geometry domains listed in bold print on the shaded bars are Congruence; Similarity, Right Triangles, and Trigonometry; Circles; Expressing Geometric Properties With Equations; Geometric Measurement and Dimension; and Modeling With Geometry. Immediately following the domain and enclosed in brackets is an abbreviation denoting the domain. Identified below each domain are the clusters that serve to group related content standards.



Congruence [G-CO]• Experiment with transformations in the plane.• Understand congruence in terms of rigid motions.• Prove geometric theorems.• Make geometric constructions.

Similarity, Right Triangles, and Trigonometry [G-SRT]

• Understand similarity in terms of similarity transformations.

• Prove theorems involving similarity.• Define trigonometric ratios and solve problems

involving right triangles.• Apply trigonometry to general triangles.

Circles [G-C]• Understand and apply theorems about circles.• Find arc lengths and areas of sectors of circles.

Expressing Geometric Properties With Equations [G-GPE]

• Translate between the geometric description and the equation for a conic section.

• Use coordinates to prove simple geometric theorems algebraically.

Geometric Measurement and Dimension [G-GPE]• Explain volume formulas and use them to

solve problems.• Visualize relationships between two-dimensional

and three-dimensional objects.

Modeling With Geometry [G-MG]• Apply geometric concepts in modeling situations.










GEOMETRY

The Geometry course builds on Algebra I concepts and increases students’ knowledge of shapes and their properties through geometry-based applications, many of which are observable in aspects of everyday life. This knowledge helps develop visual and spatial sense and strong reasoning skills. The Geometry course requires students to make conjectures and to use reasoning to validate or negate these conjectures. The use of proofs and constructions is a valuable tool that enhances reasoning skills and enables students to better understand more complex mathematical concepts. Technology should be used to enhance students’ mathematical experience, not replace their reasoning abilities. Because of its importance, this Euclidean geometry course is required of all students receiving an Alabama High School Diploma.

School systems may offer Geometry and Geometry A and Geometry B. Content standards 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 26, 30, 31, 32, 33, and 34 must be taught in the Geometry A course.Content standards 2, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 35, 36, 37, 38, 39, 40, 41, 42, and 43 must be taught in the Geometry B course.

Students will:

GEOMETRY

Congruence

Experiment with transformations in the plane.

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc. [G-CO1]

2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [G-CO2]

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [G-CO3]

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO4]

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence oftransformations that will carry a given figure onto another. [G-CO5]


GEOMETRY

Understand congruence in terms of rigid motions. (Build on rigid motions as a familiar starting point for development of concept of geometric proof.)

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. [G-CO6]

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [G-CO7]

8. Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8]

Prove geometric theorems. (Focus on validity of underlying reasoning while using variety of ways of writing proofs.)

9. Prove theorems about lines and angles. Theorems include vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. [G-CO9]

10. Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180º, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10]

11. Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals. [G-CO11]

Make geometric constructions. (Formalize and explain processes.)

12. Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. [G-CO12]

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO13]


GEOMETRY

Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations.

14. Verify experimentally the properties of dilations given by a center and a scale factor. [G-SRT1]a. A dilation takes a line not passing through the center of the dilation to a parallel line and

leaves a line passing through the center unchanged. [G-SRT1a]b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

[G-SRT1b]

15. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. [G-SRT2]

Example 1:

6 6 12 12

4

8Given the two triangles above, show that they are similar.

12

6

8

4

They are similar by SSS. The scale factor is equivalent.

Example 2:

5 645° 45°

10 12Show that the triangles are similar. Two corresponding sides are proportional and the included angle is congruent. (SAS similarity)

16. Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

Prove theorems involving similarity.

17. Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [G-SRT4]

18. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [G-SRT5]


GEOMETRY

Define trigonometric ratios and solve problems involving right triangles.

19. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles. [G-SRT6]

20. Explain and use the relationship between the sine and cosine of complementary angles. [G-SRT7]

21. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [G-SRT8]

Apply trigonometry to general triangles.

22. (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [G-SRT10]

23. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). [G-SRT11]

Circles

Understand and apply theorems about circles.

24. Prove that all circles are similar. [G-C1]

25. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. [G-C2]

26. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

27. (+) Construct a tangent line from a point outside a given circle to the circle. [G-C4]

Find arc lengths and areas of sectors of circles. (Radian introduced only as unit of measure.)

28. Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. [G-C5]

Expressing Geometric Properties With Equations

Translate between the geometric description and the equation for a conic section.

29. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. [G-GPE1]


GEOMETRY

Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.)

30. Use coordinates to prove simple geometric theorems algebraically. [G-GPE4]Example: Prove or disprove that a figure defined by four given points in the coordinate plane

is a rectangle; prove or disprove that the point (1, 3 ) lies on the circle centered at the origin and containing the point (0, 2).

31. Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). [G-GPE5]

32. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [G-GPE6]

33. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* [G-GPE7]

Use coordinates to prove simple geometric theorems algebraically.

34. Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.

Geometric Measurement and Dimension

Explain volume formulas and use them to solve problems.

35. Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. [G-GMD1]

36. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [G-GMD3]

37. Determine the relationship between surface areas of similar figures and volumes of similar figures.

Visualize relationships between two-dimensional and three-dimensional objects.

38. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. [G-GMD4]

Modeling With Geometry

Apply geometric concepts in modeling situations.

39. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* [G-MG1]

40. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).* [G-MG2]


GEOMETRY

41. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).*[G-MG3]

STATISTICS AND PROBABILITY

Using Probability to Make Decisions

Use probability to evaluate outcomes of decisions. (Introductory; apply counting rules.)

42. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]

43. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7]

Example:

6 units

What is the probability of tossing a penny and having it land in the non-shaded region? Geometric Probability is the Non-Shaded Area divided by the Total Area.

= = or 1


ALGEBRAIC CONNECTIONS

Finance

7. Use analytical, numerical, and graphical methods to make financial and economic decisions, including those involving banking and investments, insurance, personal budgets, credit purchases, recreation, and deceptive and fraudulent pricing and advertising.

Examples: Determine the best choice of certificates of deposit, savings accounts, checking accounts, or loans. Compare the costs of fixed- or variable-rate mortgage loans. Compare costs associated with various credit cards. Determine the best cellular telephone plan for a budget.

a. Create, manually or with technological tools, graphs and tables related to personal finance and economics.

Example: Use spreadsheets to create an amortization table for a mortgage loan or a circle graph for a personal budget.

GEOMETRY

Modeling

8. Determine missing information in an application-based situation using properties of right triangles, including trigonometric ratios and the Pythagorean Theorem.

Example: Use a construction or landscape problem to apply trigonometric ratios and the Pythagorean Theorem.

Symmetry

9. Analyze aesthetics of physical models for line symmetry, rotational symmetry, or the golden ratio.Example: Identify the symmetry found in nature, art, or architecture.

Measurement

10. Critique measurements in terms of precision, accuracy, and approximate error.Example: Determine whether one candidate has a significant lead over another candidate

when given their current standings in a poll and the margin of error.

11. Use ratios of perimeters, areas, and volumes of similar figures to solve applied problems.Example: Use a blueprint or scale drawing of a house to determine the amount of carpet to be

purchased.


Graphing

12. Create a model of a set of data by estimating the equation of a curve of best fit from tables of values or scatter plots.

Examples: Create models of election results as a function of population change, inflation or employment rate as a function of time, cholesterol density as a function of age or weight of a person.

a. Predict probabilities given a frequency distribution.


DISCRETE MATHEMATICS

8. Apply algorithms, including Kruskal’s and Prim’s, relating to minimum weight spanning trees, networks, flows, and Steiner trees.

a. Use shortest path techniques to find optimal shipping routes.

9. Determine a minimum project time using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms.

GEOMETRY

10. Use vertex-coloring techniques and matching techniques to solve application-based problems.Example: Use graph-coloring techniques to color a map of the western states of the United

States so no adjacent states are the same color, including determining the minimum number of colors needed and why no fewer colors may be used.

11. Solve application-based logic problems using Venn diagrams, truth tables, and matrices.


12. Use combinatorial reasoning and counting techniques to solve application-based problems.Example: Determine the probability of a safe opening on the first attempt given the

combination uses the digits 2, 4, 6, and 8 with the order unknown.Answer: The probability of the safe opening on the first attempt is 1

24.

13. Analyze election data to compare election methods and voting apportionment, including determining strength within specific groups.


MATHEMATICAL INVESTIGATIONS

ALGEBRA

5. Identify beginnings of algebraic symbolism and structure through the works of European mathematicians.

a. Create a Fibonacci sequence when given two initial integers. b. Investigate Tartaglia’s formula for solving cubic equations.

6. Explain the development and applications of logarithms, including contributions of John Napier, Henry Briggs, and the Bernoulli family.

7. Justify the historical significance of the development of multiple perspectives in mathematics. Example: Relate the historical development of multiple perspectives to the works of Sir Isaac

Newton and Gottfried Wilhelm von Leibniz in the foundations of calculus.

a. Summarize the significance of René Descartes’ Cartesian coordinate system. b. Interpret the foundation of analytic geometry with regard to geometric curves and algebraic

relationships.

GEOMETRY

8. Solve problems from non-Euclidean geometry, including graph theory, networks, topology, and fractals.

Examples: Observe the figure to the right to determine if it is traversable, and if it is, describe a path that will traverse it.Verify that two objects are topologically equivalent.Sketch four iterations of Sierpínski’s triangle.

9. Analyze works of visual art and architecture for mathematical relationships. Examples: Use Leonardo da Vinci’s Vitruvian Man to explore the golden ratio.

Identify mathematical patterns in Maurits Cornelis Escher’s drawings, including the use of tessellations in art, quilting, paintings, pottery, and architecture.

a. Summarize the historical development of perspective in art and architecture. 10. Determine the mathematical impact of the ancient Greeks, including Archimedes, Eratosthenes,

Euclid, Hypatia, Pythagoras, and the Pythagorean Society. Example: Use Euclid’s proposition to inscribe a regular hexagon within a circle.

a. Construct multiple proofs of the Pythagorean Theorem. b. Solve problems involving figurate numbers, including triangular and pentagonal numbers.

Example: Write a sequence of the first 10 triangular numbers and hypothesize a formula for finding the nth triangular number.

11. Describe the development of mathematical tools and their applications. Examples: Use knotted ropes for counting; Napier’s bones for multiplication; a slide rule for

multiplying and calculating values of trigonometric, exponential, and logarithmic functions; and a graphing calculator for analyzing functions graphically and numerically.


PRECALCULUS

GEOMETRY

Similarity, Right Triangles, and Trigonometry

Apply trigonometry to general triangles.

35. (+) Derive the formula A = ( 1

2)ab sin(C) for the area of a triangle by drawing an auxiliary line

from a vertex perpendicular to the opposite side. (Apply formulas previously derived in Geometry.) [G-SRT9]

Expressing Geometric Properties With Equations

Translate between the geometric description and the equation for a conic section.

36. (+) Derive the equations of a parabola given a focus and directrix. [G-GPE2]

37. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. [G-GPE3]

Explain volume formulas and use them to solve problems.

38. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]


Interpreting Categorical and Quantitative Data

Summarize, represent, and interpret data on a single count or measurement variable.

39. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. (Focus on increasing rigor using standard deviation.) [S-ID2]

40. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (Identify uniform, skewed, and normal distributions in a set of data. Determine the quartiles and interquartile range for a set of data.)[S-ID3]

41. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. [S-ID4]


6 –

8 G

eom

etry

Dom

ain

and

Hig

h Sc

hool

Geo

met

ry C

once

ptua

l Cat

egor

y St

udy

Dire

ctio

ns

Fo

r gra

des 6

, 7, a

nd 8

, the

stan

dard

s for

eac

h cl

uste

r are

giv

en.

Pr

ovid

e a

desc

riptio

n of

how

the

clus

ter i

s uni

que

from

the

othe

rs.

Pr

ovid

e ea

ch d

omai

n an

d cl

uste

r tha

t con

nect

s to

the

dom

ain

cate

gory

that

is b

eing

inve

stig

ated

. (T

he o

verv

iew

for e

ach

grad

e w

ill b

e he

lpfu

l.)

Ex

plai

n ho

w e

ach

clus

ter r

elat

es to

the

van

Hie

le m

odel

of g

eom

etric

reas

onin

g.

6th grade: C

luster: Solve real‐w

orld and m

athematical problems involving area, surface area, and volume.

Stan

dard

s 21

– 24

21

. Fin

d th

e ar

ea o

f rig

ht tr

iang

les,

othe

r tria

ngle

s, sp

ecia

l qua

drila

tera

ls, a

nd p

olyg

ons b

y co

mpo

sing

into

rect

angl

es o

r dec

ompo

sing

into

tria

ngle

s and

oth

er sh

apes

; ap

ply

thes

e te

chni

ques

in th

e co

ntex

t of s

olvi

ng re

al-w

orld

and

mat

hem

atic

al p

robl

ems.

[6-G

1]

22. F

ind

the

volu

me

of a

righ

t rec

tang

ular

pris

m w

ith fr

actio

nal e

dge

leng

ths b

y pa

ckin

g it

with

uni

t cub

es o

f the

app

ropr

iate

uni

t fra

ctio

n ed

ge le

ngth

s, an

d sh

ow th

at

the

volu

me

is th

e sa

me

as w

ould

be

foun

d by

mul

tiply

ing

the

edge

leng

ths o

f the

pris

m. A

pply

the

form

ulas

V =

lwh

and

V =

Bh

to fi

nd v

olum

es o

f rig

ht re

ctan

gula

r pr

ism

s with

frac

tiona

l edg

e le

ngth

s in

the

cont

ext o

f sol

ving

real

-wor

ld a

nd m

athe

mat

ical

pro

blem

s. [6

-G2]

23

. Dra

w p

olyg

ons i

n th

e co

ordi

nate

pla

ne g

iven

coo

rdin

ates

for t

he v

ertic

es; u

se c

oord

inat

es to

find

the

leng

th o

f a si

de jo

inin

g po

ints

with

the

sam

e fir

st c

oord

inat

e or

th

e sa

me

seco

nd c

oord

inat

e. A

pply

thes

e te

chni

ques

in th

e co

ntex

t of s

olvi

ng re

al-w

orld

and

mat

hem

atic

al p

robl

ems.

[6-G

3]

24. R

epre

sent

thre

e-di

men

sion

al fi

gure

s usi

ng n

ets m

ade

up o

f rec

tang

les a

nd tr

iang

les,

and

use

the

nets

to fi

nd th

e su

rfac

e ar

ea o

f the

se fi

gure

s. A

pply

thes

e te

chni

ques

in

the

cont

ext o

f sol

ving

real

-wor

ld a

nd m

athe

mat

ical

pro

blem

s. [6

-G4]

H

ow is

this

clu

ster

uni

que?

M

ajor

con

nect

ions

to o

ther

dom

ains

and

clu

ster

s.

(The

ove

rvie

w fo

r eac

h gr

ade

will

be

help

ful.)

H

ow d

oes t

his c

lust

er re

late

to th

e va

n H

iele

mod

el?

7th g

rade

: C

lust

er:

Dra

w, c

onst

ruct

, and

des

crib

e ge

omet

rica

l fig

ures

and

des

crib

e th

e re

latio

nshi

ps b

etw

een

them

.

Stan

dard

s 11

– 13

11

. Sol

ve p

robl

ems i

nvol

ving

scal

e dr

awin

gs o

f geo

met

ric fi

gure

s, in

clud

ing

com

putin

g ac

tual

leng

ths a

nd a

reas

from

a sc

ale

draw

ing

and

repr

oduc

ing

a sc

ale

draw

ing

at a

diff

eren

t sca

le. [

7-G

1]

12. D

raw

(fre

ehan

d, w

ith ru

ler a

nd p

rotra

ctor

, and

with

tech

nolo

gy) g

eom

etric

shap

es w

ith g

iven

con

ditio

ns. F

ocus

on

cons

truct

ing

trian

gles

from

thre

e m

easu

res o

f an

gles

or s

ides

, not

icin

g w

hen

the

cond

ition

s det

erm

ine

a un

ique

tria

ngle

, mor

e th

an o

ne tr

iang

le, o

r no

trian

gle.

[7-G

2]

13. D

escr

ibe

the

two-

dim

ensi

onal

figu

res t

hat r

esul

t fro

m sl

icin

g th

ree-

dim

ensi

onal

figu

res,

as in

pla

ne se

ctio

ns o

f rig

ht re

ctan

gula

r pris

ms a

nd ri

ght r

ecta

ngul

ar

pyra

mid

s. [7

-G3]

How

is th

is c

lust

er u

niqu

e?

Maj

or c

onne

ctio

ns to

oth

er d

omai

ns a

nd

clus

ters

. H

ow d

oes t

his c

lust

er re

late

to th

e va

n H

iele

m

odel

?

7th g

rade

: C

lust

er:

Solv

e re

al-w

orld

and

mat

hem

atic

al p

robl

ems i

nvol

ving

ang

le m

easu

re, a

rea,

surf

ace

area

, and

vol

ume.

Stan

dard

s 14

– 16

14

. Kno

w th

e fo

rmul

as fo

r the

are

a an

d ci

rcum

fere

nce

of a

circ

le, a

nd u

se th

em to

solv

e pr

oble

ms;

giv

e an

info

rmal

der

ivat

ion

of th

e re

latio

nshi

p be

twee

n th

e ci

rcum

fere

nce

and

area

of a

circ

le. [

7-G

4]

15. U

se fa

cts a

bout

supp

lem

enta

ry, c

ompl

emen

tary

, ver

tical

, and

adj

acen

t ang

les i

n a

mul

tiste

p pr

oble

m to

writ

e an

d so

lve

sim

ple

equa

tions

for a

n un

know

n an

gle

in a

figu

re. [

7-G

5]

16. S

olve

real

-wor

ld a

nd m

athe

mat

ical

pro

blem

s inv

olvi

ng a

rea,

vol

ume,

and

surf

ace

area

of t

wo-

and

thre

e-di

men

sion

al o

bjec

ts c

ompo

sed

of tr

iang

les,

quad

rilat

eral

s, po

lygo

ns, c

ubes

, and

righ

t pris

ms.

[7-G

6]

How

is th

is c

lust

er u

niqu

e?

Maj

or c

onne

ctio

ns to

oth

er d

omai

ns a

nd

clus

ters

.

How

doe

s thi

s clu

ster

rela

te to

the

van

Hie

le

mod

el?

8thg

rade

: C

lust

er:

Und

erst

and

cong

ruen

ce a

nd si

mila

rity

usi

ng p

hysi

cal m

odel

s, tr

ansp

aren

cies

, or

geom

etry

soft

war

e.

Stan

dard

s 16

– 20

16

. Ver

ify e

xper

imen

tally

the

prop

ertie

s of r

otat

ions

, ref

lect

ions

, and

tran

slat

ions

: [8-

G1]

a.

Li

nes a

re ta

ken

to li

nes,

and

line

segm

ents

are

take

n to

line

segm

ents

of t

he sa

me

leng

th. [

8-G

1a]

b.

Ang

les a

re ta

ken

to a

ngle

s of t

he sa

me

mea

sure

. [8-

G1b

] c.

Pa

ralle

l lin

es a

re ta

ken

to p

aral

lel l

ines

. [8-

G1c

] 17

. Und

erst

and

that

a tw

o-di

men

sion

al fi

gure

is c

ongr

uent

to a

noth

er if

the

seco

nd c

an b

e ob

tain

ed fr

om th

e fir

st b

y a

sequ

ence

of r

otat

ions

, ref

lect

ions

, and

tra

nsla

tions

; giv

en tw

o co

ngru

ent f

igur

es, d

escr

ibe

a se

quen

ce th

at e

xhib

its th

e co

ngru

ence

bet

wee

n th

em. [

8-G

2]

18. D

escr

ibe

the

effe

ct o

f dila

tions

, tra

nsla

tions

, rot

atio

ns, a

nd re

flect

ions

on

two-

dim

ensi

onal

figu

res u

sing

coo

rdin

ates

. [8-

G3]

19

. Und

erst

and

that

a tw

o-di

men

sion

al fi

gure

is si

mila

r to

anot

her i

f the

seco

nd c

an b

e ob

tain

ed fr

om th

e fir

st b

y a

sequ

ence

of r

otat

ions

, ref

lect

ions

, tra

nsla

tions

, and

di

latio

ns; g

iven

two

sim

ilar t

wo-

dim

ensi

onal

figu

res,

desc

ribe

a se

quen

ce th

at e

xhib

its th

e si

mila

rity

betw

een

them

. [8-

G4]

20

. Use

info

rmal

arg

umen

ts to

est

ablis

h fa

cts a

bout

the

angl

e su

m a

nd e

xter

ior a

ngle

of t

riang

les,

abou

t the

ang

les c

reat

ed w

hen

para

llel l

ines

are

cut

by

a tra

nsve

rsal

, an

d th

e an

gle-

angl

e cr

iterio

n fo

r sim

ilarit

y of

tria

ngle

s. [8

-G5]

Exam

ple:

Arr

ange

thre

e co

pies

of t

he sa

me

trian

gle

so th

at th

e su

m o

f the

thre

e an

gles

app

ears

to fo

rm a

line

, and

giv

e ar

gum

ent i

n te

rms o

f t

rans

vers

als

why

this

is so

. H

ow is

this

clu

ster

uni

que?

M

ajor

con

nect

ions

to o

ther

dom

ains

an

d cl

uste

rs.

How

doe

s thi

s clu

ster

rela

te to

the

van

Hie

le m

odel

?

8thg

rade

: C

lust

er:

Und

erst

and

and

appl

y th

e Py

thag

orea

n T

heor

em.

Stan

dard

s 21

– 23

21

. Exp

lain

a p

roof

of t

he P

ytha

gore

an T

heor

em a

nd it

s con

vers

e. [8

-G6]

22

. App

ly th

e Py

thag

orea

n Th

eore

m to

det

erm

ine

unkn

own

side

leng

ths i

n rig

ht tr

iang

les i

n re

al-w

orld

and

mat

hem

atic

al p

robl

ems i

n tw

o an

d th

ree

dim

ensi

ons.

[8-G

7]

23. A

pply

the

Pyth

agor

ean

Theo

rem

to fi

nd th

e di

stan

ce b

etw

een

two

poin

ts in

a c

oord

inat

e sy

stem

. [8-

G8]

H

ow is

this

clu

ster

uni

que?

M

ajor

con

nect

ions

to o

ther

dom

ains

an

d cl

uste

rs.

How

doe

s thi

s clu

ster

rela

te to

the

van

Hie

le m

odel

?

8thg

rade

: C

lust

er:

Solv

e re

al-w

orld

and

mat

hem

atic

al p

robl

ems i

nvol

ving

vol

ume

of c

ylin

ders

, con

es, a

nd sp

here

s.

Stan

dard

24

24. K

now

the

form

ulas

for t

he v

olum

es o

f con

es, c

ylin

ders

, and

sphe

res,

and

use

them

to so

lve

real

-wor

ld a

nd m

athe

mat

ical

pro

blem

s. [8

-G9]

H

ow is

this

clu

ster

uni

que?

M

ajor

con

nect

ions

to o

ther

dom

ains

an

d cl

uste

rs.

How

doe

s thi

s clu

ster

rela

te to

the

van

Hie

le m

odel

?

Hig

h Sc

hool

Geo

met

ry C

once

ptua

l Cat

egor

y St

udy

Dire

ctio

ns

Th

e do

mai

ns a

nd c

lust

ers f

or th

e co

ncep

tual

cat

egor

y G

eom

etry

are

giv

en.

Pr

ovid

e a

desc

riptio

n of

how

the

dom

ain

is u

niqu

e fr

om th

e ot

hers

.

Prov

ide

each

dom

ain

and

clus

ter t

hat c

onne

cts t

o th

e do

mai

n ca

tego

ry th

at is

bei

ng in

vest

igat

ed.

(The

Geo

met

ry O

verv

iew

will

be

help

ful.)

Expl

ain

how

eac

h do

mai

n re

late

s to

the

van

Hie

le m

odel

of g

eom

etric

reas

onin

g.

Dom

ain:

Con

grue

nce

[G-C

O]

Clu

ster

s:

Ex

perim

ent w

ith tr

ansf

orm

atio

ns in

the

plan

e.

U

nder

stan

d co

ngru

ence

in te

rms o

f rig

id m

otio

ns.

Pr

ove

geom

etric

theo

rem

s.

Mak

e ge

omet

ric c

onst

ruct

ions

. H

ow is

this

dom

ain

uniq

ue?

Maj

or c

onne

ctio

ns to

oth

er d

omai

ns a

nd c

lust

ers.

(T

he G

eom

etry

Ove

rvie

w w

ill b

e he

lpfu

l.)

How

doe

s thi

s dom

ain

rela

te to

the

van

Hie

le m

odel

?

Dom

ain:

Sim

ilari

ty, R

ight

Tri

angl

es, a

nd T

rigo

nom

etry

[G

-SR

T]

Clu

ster

s:

U

nder

stan

d si

mila

rity

in te

rms o

f sim

ilarit

y tra

nsfo

rmat

ions

.

Prov

e th

eore

ms i

nvol

ving

sim

ilarit

y.

D

efin

e tri

gono

met

ric ra

tios a

nd so

lve

prob

lem

s inv

olvi

ng ri

ght t

riang

les.

A

pply

trig

onom

etry

to g

ener

al tr

iang

les.

How

is th

is d

omai

n un

ique

? M

ajor

con

nect

ions

to o

ther

dom

ains

and

cl

uste

rs.

(The

Geo

met

ry O

verv

iew

will

be

help

ful.)

How

doe

s thi

s dom

ain

rela

te to

the

van

Hie

le m

odel

?

Dom

ain:

Cir

cles

[G-C

]

Clu

ster

s:

U

nder

stan

d an

d ap

ply

theo

rem

s abo

ut c

ircle

s.

Find

arc

leng

ths a

nd a

reas

of s

ecto

rs o

f circ

les.

How

is th

is d

omai

n un

ique

? M

ajor

con

nect

ions

to o

ther

dom

ains

and

cl

uste

rs.

(The

Geo

met

ry O

verv

iew

will

be

help

ful.)

How

doe

s thi

s dom

ain

rela

te to

the

van

Hie

le m

odel

?

Dom

ain:

Exp

ress

ing

Geo

met

ric

Prop

ertie

s With

Equ

atio

ns [G

-GPE

]

Clu

ster

s:

Tr

ansl

ate

betw

een

the

geom

etric

des

crip

tion

and

the

equa

tion

for a

con

ic se

ctio

n.

U

se c

oord

inat

es to

pro

ve si

mpl

e ge

omet

ric th

eore

ms a

lgeb

raic

ally

. H

ow is

this

dom

ain

uniq

ue?

Maj

or c

onne

ctio

ns to

oth

er d

omai

ns a

nd

clus

ters

. (T

he G

eom

etry

Ove

rvie

w w

ill b

e he

lpfu

l.)

How

doe

s thi

s dom

ain

rela

te to

the

van

Hie

le m

odel

?

Dom

ain:

Geo

met

ric

Mea

sure

men

t and

Dim

ensi

on [G

-GPE

]

Clu

ster

s:

Ex

plai

n vo

lum

e fo

rmul

as a

nd u

se th

em to

solv

e pr

oble

ms.

V

isua

lize

rela

tions

hips

bet

wee

n tw

o-di

men

sion

al a

nd th

ree-

dim

ensi

onal

obj

ects

. H

ow is

this

dom

ain

uniq

ue?

Maj

or c

onne

ctio

ns to

oth

er d

omai

ns a

nd

clus

ters

. (T

he G

eom

etry

Ove

rvie

w w

ill b

e he

lpfu

l.)

How

doe

s thi

s dom

ain

rela

te to

the

van

Hie

le m

odel

?

Dom

ain:

Mod

elin

g W

ith G

eom

etry

[G-M

G]

Clu

ster

s:

A

pply

geo

met

ric c

once

pts i

n m

odel

ing

situ

atio

ns.

How

is th

is d

omai

n un

ique

? M

ajor

con

nect

ions

to o

ther

dom

ains

and

clu

ster

s.

(The

Geo

met

ry O

verv

iew

will

be

help

ful.)

How

doe

s thi

s dom

ain

rela

te to

the

van

Hie

le m

odel

?

Journal Reflection How did exploring the College and Career Ready standards for Geometry

and van Hiele’s model deepen your understanding of the flow of the

CCRS math standards?

How might understanding a mathematical progression impact instruction? Give specific examples with respect to:

* collaboratively planning lessons,* helping students make mathematical connections,* working with struggling students,

working with advanced students, and*using formative assessment and revising instruction*

Provided with a list of standards that are related to 2‐D and 3‐D geometry, discuss the following: 1. What relevant changes occur from grade to grade?

Consider both content and practices.2. How should the teacher accommodate students with different entry points?

Students that have not mastered the previous grade’s standard(s).

Students that have mastered the previous grade’s standard(s).

Grade or

Course

Sample learning progression: 2D and 3D Geometry

K

19. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).[K-G3] 20. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations,using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices or “corners”), and other attributes (e.g., having sides of equal length). [K-G4]

1st

20. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, andquarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”) [1-G2]

2nd 24. Recognize and draw shapes having specified attributes such as a given number of angles or agiven number of equal faces. (Sizes are compared directly or visually, not compared by measuring.) Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. [2-G1]

3rd

24. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) mayshare attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. [3-G1]

4th 26. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular andparallel lines. Identify these in two-dimensional figures. [4-G1]

5th 26. Classify two-dimensional figures in a hierarchy based on properties. [5-G4]

6th

22. Find the volume of a right rectangular prism with fractional edge lengths by packing it withunit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. [6-G2]

7th 13. Describe the two-dimensional figures that result from slicing three-dimensional figures, as inplane sections of right rectangular prisms and right rectangular pyramids. [7-G3]

8th 24. Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solvereal-world and mathematical problems. [8-G9]

HS

Geometry Course 39. Use geometric shapes, their measures, and their properties to describe objects (e.g., modelinga tree trunk or a human torso as a cylinder).* [G-MG1] Precalculus Course 38. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of asphere and other solid figures. [G-GMD2]

Illustrative Mathematics

6.G Banana Bread

Alignments to Content Standards

Alignment: 6.G.A.2

Tags

• This task is not yet tagged.

Leo's recipe for banana bread won't fit in his favorite pan. The batter fills the 8.5 inch by 11 inchby 1.75 inch pan to the very top, but when it bakes it spills over the side. He has another panthat is 9 inches by 9 inches by 3 inches, and from past experience he thinks he needs about aninch between the top of the batter and the rim of the pan. Should he use this pan?

http://www.illustrativemathematics.org

Commentary

The purpose of this task is two-fold. One is to provide students with a multi-step probleminvolving volume. The other is to give them a chance to discuss the difference between exactcalculations and their meaning in a context. It is important to note that students could arguethat whether the new pan is appropriate depends in part on how accurate Leo's estimate for theneeded height is.

Solutions

Solution: Solution

In order to find out how high the batter will be in the second pan, we must first find out thetotal volume of the batter that the recipe makes. We know that the recipe fills a pan that is8.5 inch by 11 inch by 1.75 inches. We can calculate the volume of the batter multiplyingthe length, the width, and the height:

We know that the batter will have the same volume when we pour it into the new pan.When the batter is poured into the new pan, we know that the volume will be where is the height of the batter in the pan. We already know that , so:

Therefore, the batter will fill the second pan about 2 inches high. Since the pan is 3 incheshigh, there is nearly an inch between the top of the batter and the rim of the pan, so it willprobably work for the banana bread (assuming that Leo is right that that an inch of space isenough).

6.G Banana Bread is licensed by Illustrative Mathematics under a Creative CommonsAttribution-NonCommercial-ShareAlike 4.0 International License

VV

= 8.5 in × 11 in × 1.75 in= 163.625 in3

9 × 9 × hh V = 163.625 in3

V163.625 in3

163.625 in3

163.625 in3

81 in2

2.02 in

= l × w × h= 9 in × 9 in × h= 81 × h in2

= h

≈ h

http://www.illustrativemathematics.org/illustrations/657


http://creativecommons.org/licenses/by-nc-sa/4.0/deed.en_US


7.G Cube Ninjas!


Alignment: 7.G.A.3

Tags


Imagine you are a ninja that can slice solid objects straight through. You have a solid cube infront of you. You are curious about what 2-dimensional shapes are formed when you slice thecube. For example, if you make a slice through the center of the cube that is parallel to one ofthe faces, the cross section is a square:

For each of the following slices,

(i) describe using precise mathematical language the shape of the cross section.

(ii) draw a diagram showing the cross section of the cube.

a. A slice containing edge AC and edge EG


b. A slice containing the vertices C, B, and G.

C. A slice containing the vertex A, the midpoint of edge EG, and the midpoint of edge FG.

Commentary

The purpose of this task is to have students explore various cross sections of a cube and useprecise language to describe the shape of the resulting faces. The drawings can be either 2- or 3-dimensional; some students may not have experience in drawing 3-D projections, in which casea 2-D drawing of the shape of the slice would be appropriate.

The ability to visualize and understand the relationships between 3-dimensional and 2-dimensional objects is useful in many areas. Architects need to relate 3-dimensional buildings to2-dimensional blueprints. Physicians need to relate 2-dimensional x-rays and MRI’s to 3-dimensional bodies. Engineers need to relate 2-dimensional schematics to 3-dimensionalstructures. Many people find such visualization challenging, often because they have had fewopportunities to practice. Depending on the students, teachers may want to provide some toolsto support this task. Clay can be made into cubes and sliced neatly with plastic knives, dentalfloss, or wire. Paper models can be cut and taped together. Isometric dot paper can also aid withdrawings. There are free CAD programs such as Google sketchup that allow students to buildand manipulate virtual models, but this goes beyond the intended purpose of this task.

Note: We kept the language of the task simple and in line with the wording of the standard,referring to “slices” instead of “planar slices” or “planar cross-sections”. Some students may askabout or want to use non-planar slices. This is an opportunity to attend to the precision of thelanguage, and indicate that in this task “slices” means “planar slices”. The use of non-planarslices changes the nature, complexity, and mathematics of this task.

Solutions

Solution: Solution

a. A rectangle.

Comment: In Grade 7, students have not had the Pythagorean theorem, so they are notexpected to determine that the rectangle has sides of 1 and . They should, however, notethat the diagonal of a face is longer than the edge, hence the cross section is not a square.

b. An equilateral triangle.

2‾‾√

Comment: Because the three sides of the cross-section are all the diagonals of the faces,they have the same length. As noted in part A, the students are not expected to determinethe length to be , but they are expected to note that all three lengths are the same,hence the triangle is equilateral.

c. An irregular pentagon.

Comment: This one is by far the most challenging of the three. A key insight is to figure outwhere the plane intersects the edges CF and BE. Many people have a great deal of difficultyvisualizing the shape. However, students using clay or paper models can figure it out.Although it is not expected for 7th grade students to be able to determine the exactcoordinates (it ends up that the intersections are ⅔ the way up the edge), students canestimate that it is more than halfway, and be able to make an argument that the resultingshape has five sides and not all the sides have the same length, hence it is an irregularpentagon.

7.G Cube Ninjas! is licensed by Illustrative Mathematics under a Creative CommonsAttribution-NonCommercial-ShareAlike 4.0 International License

2‾‾√





8.G Shipping Rolled Oats


Alignment: 8.G.C.9

Tags


Rolled oats (dry oatmeal) come in cylindrical containers with a diameter of 5 inches and aheight of 9 inches. These containers are shipped to grocery stores in boxes. Each shipping boxcontains six rolled oats containers. The shipping company is trying to figure out the dimensionsof the box for shipping the rolled oats containers that will use the least amount of cardboard.They are only considering boxes that are rectangular prisms so that they are easy to stack.

a. What is the surface area of the box needed to ship these containers to the grocery store that uses theleast amount of cardboard?

b. What is the volume of this box?

12


Commentary

Students should think of different ways the cylindrical containers can be set up in a rectangularbox. Through the process, students should realize that although some setups may seemdifferent, they result in a box with the same volume. In addition, students should come to therealization (through discussion and/or questioning) that the thickness of a cardboard box is verythin and will have a negligible effect on the calculations.

Given the different setups, students will generate the dimensions of the boxes and calculate thedifferent surface areas. To accomplish this, students will build on their work in geometry fromprevious grades. Using this understanding, students can find the area of all six faces and thenadd them together. This work should lead students to realize that each face has another that isidentical in measure which could help in the development of the surface area formula for arectangular prism. In addition, there is an opportunity to discuss how it is possible for thevolume for two boxes to be the same even though the surface area of the boxes is different.

The teacher may want to provide students with a table in order to organize their work. Inaddition, students may benefit from manipulatives in order to model the problem. This can bedone with actual rolled oats containers in a whole class setting or with small cylinders in groups(cylindrical blocks or foam shapes).

Submitted by Kevin Rothman from Heritage Middle School, Newburgh Enlarged City SchoolDistrict, for the fourth Illustrative Mathematics Task Writing Contest 2012/02/13.</p

Solutions

Solution: Solution

a. i.

Setup of box Dimensions1 cylinder wide by 1 cylinder high by 6 cylinderslong

5 in. wide by 9.5 in. high by 30 in.long

At first students may find the sum of all the areas of the faces.

Sum of the areas of the 6 faces:

Using the surface area formula:

Students may continue to use the sum of all the areas of the faces to find the

Front: (5)(9.5)Back: (5)(9.5)

= 47.5 in2

= 47.5 in2Top: (5)(30)

Bottom: (5)(30)= 150 in2

= 150 in2Right side: (9.5)(30)

Left side: (9.5)(30)= 285 in2

= 285 in2

SA = 47.5 + 47.5 + 150 + 150 + 285 + 285 = 965 in2

SASASA

= 2wl + 2lh + 2wh= 2(5)(30) + 2(30)(9.5) + 2(5)(9.5)= 965 in2

surface area which gives the same value as using the surface area formula which isshown below.

ii.

Setup of box Dimensions2 cylinders wide by 1 cylinder high by 3 cylinderslong


iii.

Setup of box Dimensions1 cylinder wide by 2 cylinders high by 3 cylinderslong

5 in. wide by 19 in. high by 15 in.long

iv.

Setup of box Dimensions1 cylinder wide by 3 cylinders high by 2 cylinderslong


v.

Setup of box Dimensions1 cylinder wide by 6 cylinders high by 1 cylinders long 5 in. wide by 57 in. high by 5 in. long

The best solution is a setup of 2 cylinders wide by 1 cylinder high by 3 cylinders long.This is the same as 3 cylinders wide by 1 cylinder high by 2 cylinders long.

b. The volume of this and all of the boxes needed to ship the containers is .

8.G Shipping Rolled Oats is licensed by Illustrative Mathematics under a Creative

SASASA

= 2wl + 2lh + 2wh= 2(10)(15) + 2(15)(9.5) + 2(10)(9.5)= 775 in2

SASASA

= 2wl + 2lh + 2wh= 2(5)(15) + 2(15)(19) + 2(5)(19)= 910 in2

SASASA

= 2wl + 2lh + 2wh= 2(5)(10) + 2(10)(28.5) + 2(5)(28.5)= 955 in2

SASASA

= 2wl + 2lh + 2wh= 2(5)(5) + 2(5)(57) + 2(5)(57)= 1190 in2

1425 in3




Commons Attribution-NonCommercial-ShareAlike 4.0 International License


G-GMD Use Cavalieri’s Principle to Compare Aquarium Volumes


Alignment: G-GMD.A.2Alignment: G-MG.A.1

Tags

Tags: MP 4

The management of an ocean life museum will choose to include either Aquarium A or AquariumB in a new exhibit.

Aquarium A is a right cylinder with a diameter of 10 feet and a height of 5 feet. Covering thelower base of Aquarium A is an “underwater mountain” in the shape of a 5-foot-tall right cone.This aquarium would be built into a pillar in the center of the exhibit room.

Aquarium B is half of a 10-foot-diameter sphere. This aquarium would protrude from the ceilingof the exhibit room.

a. How many cubic feet of water will Aquarium A hold?

b. For each aquarium, what is the area of the water’s surface when filled to a height of feet?

c. Use your results from parts (a) and (b) and Cavalieri's principle to find the volume ofAquarium B.

h


Therefore, the horizontal distance between the cone and the aquarium wall at height is also .

Commentary

This task presents a context that leads students toward discovery of the formula for calculatingthe volume of a sphere. Students who are given this task must be familiar with the formula forthe volume of a cylinder, the formula for the volume of a cone, and Cavalieri’s principle.

Specific radius/diameter measurements help to make the procedure of analyzing cross-sectionsless daunting, but even so, finding expressions to represent areas of varying cross-sections is avery new and challenging skill for many students. For this reason, assigning this task within asetting where students can receive guidance from one another and from the instructor is asensible alternative to using this task strictly as an assessment tool.

Submitted by Aaron Stidham to the Fourth illustrative Mathematics task writing contest.

Solutions

Solution: Solution

a. The number of cubic feet of water that Aquarium A will hold is found by subtractingthe volume of the cone from the volume of the cylinder:

so Aquarium A can hold about 262 cubit feet of water.

b. The area of the water’s surface at height can be derived by examining cross-sections. The radius and height of the cylinder and cone are both 5 feet, so thevertical cross-section through the vertex of the cone shown below shows that thecross section of the water forms an isosceles right triangle:

Volume of Cylinder − Volume of Cone = Maximum Volume of Water for Aquarium A

π × (radius × (height) − × π × (radius × (height))2 13 )2 = × π × (radius × (height)2

3 )2

= × π × × 523 52

= ≈ 262250π3

h

hh

The area of the water’s surface at height is found by subtracting the area of theinner circle, with radius , from the area of the outer circle, with radius 5.

Momentarily, the horizontal distance between the curved wall and the sphere’s centerat height is denoted by . But can be expressed in terms of , since and bothrepresent leg lengths of a right triangle with hypotenuse 5:

When Aquarium A is filled to a height of feet, the area of the water’s surface is square feet.

h5– h

Area of Outer Circle − Area of Inner Circle = Area of Water’s Surface at Height h

π × − π × (5 − h52 )2 = 25π − π(25 − 10h + )h2

= 25π − 25π + 10πh − π )h2

= 10πh − πh2

h10πh − πh2

h x x h x 5– h

+ (5– hx2 )2

+ 25– 10h +x2 h2

x2

x

= 25= 25= 10h– h2

= 10h– h2‾ ‾‾‾‾‾‾‾√

In this case, the area of the water’s surface at height is simply the area of a circlewhose radius is :

c. Cavalieri’s principle states that two 3-dimensional solids of the same height are equalin volume if the areas of their horizontal cross-sections at every height are equal.Since both aquariums stand 5 feet tall, and since the area of the water’s surface whenfilled to a height of feet is the same for each aquarium, the volumes of water mustbe equal when both aquariums are filled to full capacity according to this principle.Therefore, no further calculations are necessary to determine that Aquarium B willhold cubic feet of water, the very same volume that Aquarium A will hold.

G-GMD Use Cavalieri’s Principle to Compare Aquarium Volumes is licensed byIllustrative Mathematics under a Creative Commons Attribution-NonCommercial-

ShareAlike 4.0 International License

h10h– h2‾ ‾‾‾‾‾‾‾√

Area = π( 10h– h2‾ ‾‾‾‾‾‾‾√ )2

= π(10h– )h2

= 10πh − πh2

h

h

≈ 262250π3




EQ

uIP

Ru

bri

c f

or

Les

so

ns

& U

nit

s:

Math

em

ati

cs

Gra

de:

Ma

the

ma

tic

s L

es

so

n/U

nit

Tit

le:

Ove

rall R

ati

ng

:

Th

e EQ

uIP

ru

bri

c is

der

ived

fro

m t

he

Tri-

Sta

te R

ub

ric

an

d t

he

colla

bo

rati

ve d

evel

op

men

t p

roce

ss le

d b

y M

ass

ach

use

tts,

New

Yo

rk, a

nd

Rh

od

e Is

lan

d a

nd

fa

cilit

ate

d b

y A

chie

ve.

This

ver

sio

n o

f th

e EQ

uIP

ru

bri

c is

cu

rren

t a

s o

f 0

6-1

5-1

3.

V

iew

Cre

ati

ve C

om

mo

ns

Att

rib

uti

on

3.0

Un

po

rted

Lic

ense

at

htt

p:/

/cre

ati

veco

mm

on

s.o

rg/l

icen

ses/

by/

3.0

/. E

du

cato

rs m

ay

use

or

ad

ap

t. If

mo

dif

ied

, ple

ase

att

rib

ute

EQ

uIP

an

d r

e-ti

tle.

I. A

lign

me

nt

to t

he

De

pth

o

f th

e C

CSS

II

.Key

Sh

ifts

in t

he

CC

SSII

I.In

stru

ctio

nal

Su

pp

ort

sIV

.Ass

essm

en

t

The

less

on

/un

it a

lign

s w

ith

th

e le

tter

an

d s

pir

it o

f th

e C

CSS

:

oTa

rget

s a

set

of

grad

e-

leve

l CC

SS m

ath

emat

ics

stan

dar

d(s

) to

th

e fu

lld

epth

of

the

stan

dar

ds

for

teac

hin

g an

d le

arn

ing.

oSt

and

ard

s fo

rM

ath

emat

ical

Pra

ctic

eth

at a

re c

entr

al t

o t

he

less

on

are

iden

tifi

ed,

han

dle

d in

a g

rad

e-ap

pro

pri

ate

way

, an

d w

ell

con

nec

ted

to

th

e co

nte

nt

bei

ng

add

ress

ed

.

oP

rese

nts

a b

alan

ce o

fm

ath

emat

ical

pro

ced

ure

san

d d

eep

er c

on

cep

tual

un

der

stan

din

g in

her

ent

inth

e C

CSS

.

The

less

on

/un

it r

efle

cts

evid

ence

of

key

shif

ts t

ha

t a

re r

efle

cted

in t

he

CC

SS:

oFo

cus:

Les

son

s an

d u

nit

s ta

rget

ing

the

maj

or

wo

rk o

f th

e gr

ade

pro

vid

e an

esp

ecia

lly in

-dep

th t

reat

men

t, w

ith

esp

ecia

lly h

igh

exp

ecta

tio

ns.

Les

son

s an

d u

nit

s ta

rget

ing

sup

po

rtin

g w

ork

of

the

grad

e h

ave

visi

ble

co

nn

ecti

on

to

th

e m

ajo

r w

ork

of

the

grad

ean

d a

re s

uff

icie

ntl

y b

rief

. Le

sso

ns

and

un

its

do

no

t h

old

stu

den

tsre

spo

nsi

ble

fo

r m

ater

ial f

rom

late

r gr

ades

.o

Co

he

ren

ce:

The

con

ten

t d

evel

op

s th

rou

gh r

easo

nin

g ab

ou

t th

en

ew c

on

cep

ts o

n t

he

bas

is o

f p

revi

ou

s u

nd

erst

and

ings

. Wh

ere

app

rop

riat

e, p

rovi

des

op

po

rtu

nit

ies

for

stu

den

ts t

o c

on

nec

tkn

ow

led

ge a

nd

ski

lls w

ith

in o

r ac

ross

clu

ster

s, d

om

ain

s an

dle

arn

ing

pro

gres

sio

ns.

oR

igo

r: R

equ

ire

s st

ud

ents

to

en

gage

wit

h a

nd

dem

on

stra

tech

alle

ngi

ng

mat

he

mat

ics

wit

h a

pp

rop

riat

e b

alan

ce a

mo

ng

the

follo

win

g:−

Ap

plic

atio

n:

Pro

vid

es o

pp

ort

un

itie

s fo

r st

ud

ents

to

ind

epen

den

tly

app

ly m

ath

emat

ical

co

nce

pts

in r

eal-

wo

rld

si

tuat

ion

s an

d s

olv

e ch

alle

ngi

ng

pro

ble

ms

wit

h p

ersi

sten

ce,

cho

osi

ng

and

ap

ply

ing

an a

pp

rop

riat

e m

od

el o

r st

rate

gy t

o

new

sit

uat

ion

s.

−

Co

nce

ptu

al U

nd

ers

tan

din

g: D

evel

op

s st

ud

ents

’ co

nce

ptu

al

un

der

stan

din

g th

rou

gh t

asks

, bri

ef p

rob

lem

s, q

ues

tio

ns,

m

ult

iple

rep

rese

nta

tio

ns

and

op

po

rtu

nit

ies

for

stu

den

ts t

o

wri

te a

nd

sp

eak

abo

ut

thei

r u

nd

erst

and

ing.

−

Pro

ced

ura

l Ski

ll an

d F

lue

ncy

: E

xpec

ts, s

up

po

rts

and

pro

vid

es

guid

elin

es f

or

pro

ced

ura

l ski

ll an

d f

luen

cy w

ith

co

re

calc

ula

tio

ns

and

mat

hem

atic

al p

roce

du

res

(wh

en c

alle

d f

or

in

the

stan

dar

ds

for

the

grad

e)

to b

e p

erfo

rmed

qu

ickl

y an

d

accu

rate

ly.

The

less

on

/un

it is

res

po

nsi

ve t

o v

ari

ed s

tud

ent

lea

rnin

g n

eed

s:

oIn

clu

des

cle

ar a

nd

su

ffic

ien

t gu

idan

ce t

o s

up

po

rt t

each

ing

and

lear

nin

g o

f th

eta

rget

ed s

tan

dar

ds,

incl

ud

ing,

wh

en a

pp

rop

riat

e, t

he

use

of

tech

no

logy

an

dm

edia

.

oU

ses

and

en

cou

rage

s p

reci

se a

nd

acc

ura

te m

ath

emat

ics,

aca

dem

ic la

ngu

age,

term

ino

logy

an

d c

on

cret

e o

r ab

stra

ct r

epre

sen

tati

on

s (e

.g.,

pic

ture

s, s

ymb

ols

,ex

pre

ssio

ns,

eq

uat

ion

s, g

rap

hic

s, m

od

els)

in t

he

dis

cip

line.

oEn

gage

s st

ud

ents

in p

rod

uct

ive

stru

ggle

th

rou

gh r

ele

van

t, t

ho

ugh

t-p

rovo

kin

gq

ues

tio

ns,

pro

ble

ms

and

tas

ks t

hat

sti

mu

late

inte

rest

an

d e

licit

mat

he

mat

ical

thin

kin

g.

oA

dd

ress

es

inst

ruct

ion

al e

xpe

ctat

ion

s an

d is

eas

y to

un

der

stan

d a

nd

use

.

oP

rovi

de

s ap

pro

pri

ate

leve

l an

d t

ype

of

scaf

fold

ing,

dif

fere

nti

atio

n, i

nte

rven

tio

nan

d s

up

po

rt f

or

a b

road

ran

ge o

f le

arn

ers.

−

Sup

po

rts

div

erse

cu

ltu

ral a

nd

lin

guis

tic

bac

kgro

un

ds,

inte

rest

s an

d s

tyle

s.

−

Pro

vid

es

extr

a su

pp

ort

s fo

r st

ud

ents

wo

rkin

g b

elo

w g

rad

e le

vel.

−

Pro

vid

es

exte

nsi

on

s fo

r st

ud

en

ts w

ith

hig

h in

tere

st o

r w

ork

ing

abo

vegr

ade

leve

l.

A u

nit

or

lon

ger

less

on

sh

ou

ld:

oR

eco

mm

end

an

d f

acili

tate

a m

ix o

f in

stru

ctio

nal

ap

pro

ach

es f

or

a va

riet

y o

fle

arn

ers

such

as

usi

ng

mu

ltip

le r

epre

sen

tati

on

s (e

.g.,

incl

ud

ing

mo

del

s, u

sin

g a

ran

ge o

f q

ues

tio

ns,

ch

ecki

ng

for

un

der

stan

din

g, f

lexi

ble

gro

up

ing,

pai

r-sh

are)

.

oG

rad

ual

ly r

emo

ve s

up

po

rts,

req

uir

ing

stu

den

ts t

o d

emo

nst

rate

th

eir

mat

hem

atic

al u

nd

erst

and

ing

ind

epen

den

tly.

oD

emo

nst

rate

an

eff

ecti

ve s

equ

ence

an

d a

pro

gres

sio

n o

f le

arn

ing

wh

ere

the

con

cep

ts o

r sk

ills

adva

nce

an

d d

eep

en o

ver

tim

e.

oEx

pec

t, s

up

po

rt a

nd

pro

vid

e gu

idel

ines

fo

r p

roce

du

ral s

kill

and

flu

ency

wit

hco

re c

alcu

lati

on

s an

d m

ath

em

atic

al p

roce

du

res

(wh

en c

alle

d f

or

in t

he

stan

dar

ds

for

the

grad

e) t

o b

e p

erfo

rmed

qu

ickl

y an

d a

ccu

rate

ly.

The

less

on

/un

it r

egu

larl

y a

sses

ses

wh

eth

er s

tud

ents

are

ma

ster

ing

st

an

da

rds-

ba

sed

co

nte

nt

an

d

skill

s:

oIs

des

ign

ed t

o e

licit

dir

ect

,o

bse

rvab

le e

vid

ence

of

the

deg

ree

to w

hic

h a

stu

den

t ca

nin

dep

end

entl

y d

emo

nst

rate

the

targ

eted

CC

SS.

oA

sses

ses

stu

den

t p

rofi

cien

cyu

sin

g m

eth

od

s th

at a

reac

cess

ible

an

d u

nb

iase

d,

incl

ud

ing

the

use

of

grad

e-

leve

l lan

guag

e in

stu

den

tp

rom

pts

.

oIn

clu

des

alig

ned

ru

bri

cs,

answ

er k

eys

and

sco

rin

ggu

idel

ines

th

at p

rovi

de

suff

icie

nt

guid

ance

fo

rin

terp

reti

ng

stu

den

tp

erfo

rman

ce.

A u

nit

or

lon

ger

less

on

sh

ou

ld:

oU

se v

arie

d m

od

es

of

curr

icu

lum

-em

bed

ded

asse

ssm

ents

th

at m

ay in

clu

de

pre

-, f

orm

ativ

e, s

um

mat

ive

and

sel

f-as

sess

men

tm

easu

res.

Rat

ing:

3

2

1

0

R

atin

g:

3

2

1

0

Rat

ing:

3

2

1

0

R

atin

g:

3

2

1

0

EQ

uIP

Ru

bri

c f

or

Les

so

ns

& U

nit

s:

Math

em

ati

cs

Dir

ect

ion

s: T

he

Qu

alit

y R

evie

w R

ub

ric

pro

vid

es c

rite

ria

to d

ete

rmin

e th

e q

ual

ity

and

alig

nm

ent

of

less

on

s an

d u

nit

s to

th

e C

om

mo

n C

ore

Sta

te S

tan

dar

ds

(CC

SS)

in o

rder

to

: (1

) Id

enti

fy e

xem

pla

rs/

mo

del

s fo

r te

ach

ers’

use

wit

hin

an

d a

cro

ss

stat

es; (

2)

pro

vid

e co

nst

ruct

ive

crit

eria

-bas

ed f

eed

bac

k to

dev

elo

per

s; a

nd

(3

) re

view

exi

stin

g in

stru

ctio

nal

mat

eria

ls t

o d

eter

min

e w

hat

rev

isio

ns

are

nee

ded

.

Ste

p 1

– R

evi

ew

Mat

eri

als

Rec

ord

th

e gr

ade

and

tit

le o

f th

e le

sso

n/u

nit

on

th

e re

cord

ing

form

.

Scan

to

see

wh

at t

he

less

on

/un

it c

on

tain

s an

d h

ow

it is

org

aniz

ed.

R

ead

key

mat

eria

ls r

elat

ed t

o in

stru

ctio

n, a

sses

smen

t an

d t

each

er g

uid

ance

.

Stu

dy

and

wo

rk t

he

task

th

at s

erve

s as

th

e ce

nte

rpie

ce f

or

the

less

on

/un

it, a

nal

yzin

g th

e co

nte

nt

and

mat

hem

atic

al p

ract

ices

th

e ta

sks

req

uir

e.St

ep

2 –

Ap

ply

Cri

teri

a in

Dim

en

sio

n I:

Alig

nm

en

t

Iden

tify

th

e gr

ade-

leve

l CC

SS t

hat

th

e le

sso

n/u

nit

tar

gets

.

Clo

sely

exa

min

e th

e m

ater

ials

th

rou

gh t

he

“len

s” o

f ea

ch c

rite

rio

n.

In

div

idu

ally

ch

eck

each

cri

teri

on

fo

r w

hic

h c

lear

an

d s

ub

stan

tial

evi

den

ce is

fo

un

d.

Id

enti

fy a

nd

rec

ord

inp

ut

on

sp

ecif

ic im

pro

vem

ents

th

at m

igh

t b

e m

ade

to m

eet

crit

eria

or

stre

ngt

hen

alig

nm

ent.

En

ter

you

r ra

tin

g 0

– 3

fo

r D

imen

sio

n I:

Alig

nm

ent.

No

te: D

imen

sio

n I

is n

on

-neg

oti

ab

le.

In o

rder

fo

r th

e re

view

to

co

nti

nu

e, a

ra

tin

g o

f 2

or

3 is

req

uir

ed. I

f th

e re

view

is d

isco

nti

nu

ed, c

on

sid

er g

ener

al f

eed

ba

ck t

ha

t m

igh

t b

e g

iven

to

dev

elo

per

s/te

ach

ers

reg

ard

ing

nex

t st

eps.

St

ep

3 –

Ap

ply

Cri

teri

a in

Dim

en

sio

ns

II –

IV

C

lose

ly e

xam

ine

the

less

on

/un

it t

hro

ugh

th

e “

len

s” o

f ea

ch c

rite

rio

n.

R

eco

rd c

om

men

ts o

n c

rite

ria

met

, im

pro

vem

ents

nee

ded

an

d t

hen

rat

e 0

– 3

.W

hen

wo

rkin

g in

a g

rou

p, i

nd

ivid

ua

ls m

ay

cho

ose

to

co

mp

are

ra

tin

gs

aft

er e

ach

dim

ensi

on

or

del

ay

con

vers

ati

on

un

til e

ach

per

son

ha

s ra

ted

an

d r

eco

rded

th

eir

inp

ut

for

the

rem

ain

ing

Dim

ensi

on

s II

– IV

. St

ep

4 –

Ap

ply

an

Ove

rall

Ra

tin

g an

d P

rovi

de

Su

mm

ary

Co

mm

en

ts

R

evie

w r

atin

gs f

or

Dim

ensi

on

s I –

IV a

dd

ing/

clar

ifyi

ng

com

men

ts a

s n

eed

ed.

W

rite

su

mm

ary

com

men

ts f

or

you

r o

vera

ll ra

tin

g o

n y

ou

r re

cord

ing

shee

t.

Tota

l dim

ensi

on

rat

ings

an

d r

eco

rd o

vera

ll ra

tin

g E,

E/I

, R, N

– a

dju

st a

s n

eces

sary

.If

wo

rkin

g in

a g

rou

p, i

nd

ivid

ua

ls s

ho

uld

rec

ord

th

eir

ove

rall

rati

ng

pri

or

to c

on

vers

ati

on

. St

ep

5 –

Co

mp

are

Ove

rall

Ra

tin

gs a

nd

De

term

ine

Ne

xt S

tep

s

N

ote

th

e ev

iden

ce c

ited

to

arr

ive

at f

inal

rat

ings

, su

mm

ary

com

men

ts a

nd

sim

ilari

ties

an

d d

iffe

ren

ces

amo

ng

rate

rs.

Rec

om

me

nd

nex

t st

eps

for

the

less

on

/un

it a

nd

pro

vid

e re

com

men

dat

ion

s fo

r im

pro

vem

ent

and

/or

rati

ngs

to

dev

elo

per

s/te

ach

ers.

Ad

dit

ion

al G

uid

ance

on

Dim

en

sio

n II

: Sh

ifts

- W

hen

co

nsi

der

ing

Focu

s it

is im

po

rtan

t th

at le

sso

ns

or

un

its

targ

etin

g ad

dit

ion

al a

nd

su

pp

ort

ing

clu

ster

s ar

e su

ffic

ien

tly

bri

ef –

th

is e

nsu

res

that

stu

den

ts w

ill s

pen

d t

he

stro

ng

maj

ori

ty o

f th

e ye

ar o

n m

ajo

r w

ork

of

the

grad

e. S

ee t

he

K-8

Pu

blis

her

s C

rite

ria

fo

r th

e C

om

mo

n C

ore

Sta

te S

tan

da

rds

in M

ath

ema

tics

, par

ticu

larl

y p

ages

8-9

fo

r fu

rth

er in

form

atio

n o

n t

he

focu

s cr

iter

ion

wit

h r

esp

ect

to m

ajo

r w

ork

of

the

grad

e at

w

ww

.co

rest

and

ard

s.o

rg/a

sset

s/M

ath

_Pu

blis

her

s_C

rite

ria_

K-8

_Su

mm

er%

20

20

12

_FIN

AL.

pd

f. W

ith

res

pec

t to

Co

her

ence

it is

imp

ort

ant

that

th

e le

arn

ing

ob

ject

ives

are

lin

ked

to

CC

SS c

lust

er h

ead

ings

(se

e w

ww

.co

rest

and

ard

s.o

rg/M

ath

).

Rat

ing

Sca

les

R

ati

ng

fo

r D

imen

sio

n I:

Alig

nm

ent

is n

on

-neg

oti

ab

le a

nd

req

uir

es a

ra

tin

g o

f 2

or

3.

If r

ati

ng

is 0

or

1 t

hen

th

e re

view

do

es n

ot

con

tin

ue.

Rat

ing

Sca

le f

or

Dim

en

sio

ns

I, II

, III

, IV

:

3: M

eets

mo

st t

o a

ll o

f th

e cr

iter

ia in

th

e d

ime

nsi

on

2

: Mee

ts m

any

of

the

crit

eria

in t

he

dim

ensi

on

1: M

eets

so

me

of

the

crit

eria

in t

he

dim

ensi

on

0

: Do

es n

ot

mee

t th

e cr

ite

ria

in t

he

dim

ensi

on

Ove

rall

Rat

ing

for

the

Le

sso

n/U

nit

:

E: E

xem

pla

r –

Alig

ne

d a

nd

mee

ts m

ost

to

all

of

the

crit

eria

in d

ime

nsi

on

s II

, III

, IV

(to

tal 1

1 –

12

) E/

I: E

xem

pla

r if

Imp

rove

d –

Alig

ned

an

d n

eed

s so

me

imp

rove

men

t in

on

e o

r m

ore

dim

en

sio

ns

(to

tal 8

– 1

0)

R:

Rev

isio

n N

eed

ed

– A

lign

ed p

arti

ally

an

d n

eed

s si

gnif

ican

t re

visi

on

in o

ne

or

mo

re d

imen

sio

ns

(to

tal 3

– 7

) N

: N

ot

Rea

dy

to R

evie

w –

No

t al

ign

ed a

nd

do

es n

ot

mee

t cr

iter

ia (

tota

l 0 –

2)

De

scri

pto

rs f

or

Dim

en

sio

ns

I, II

, III

, IV

:

3: E

xem

plif

ies

CC

SS Q

ual

ity

- m

eets

th

e st

and

ard

des

crib

ed b

y cr

iter

ia in

th

e d

imen

sio

n, a

s ex

pla

ine

d in

cr

iter

ion

-bas

ed o

bse

rvat

ion

s.

2: A

pp

roac

hin

g C

CSS

Qu

alit

y -

mee

ts m

any

crit

eria

bu

t w

ill b

enef

it f

rom

re

visi

on

in o

ther

s, a

s su

gges

ted

in

crit

erio

n-b

ased

ob

serv

atio

ns.

1: D

eve

lop

ing

tow

ard

CC

SS Q

ual

ity

- n

eed

s si

gnif

ican

t re

visi

on

, as

sugg

este

d in

cri

teri

on

-bas

ed

ob

serv

atio

ns.

0

: No

t re

pre

sen

tin

g C

CSS

Qu

alit

y -

do

es n

ot

add

ress

th

e cr

iter

ia in

th

e d

imen

sio

n.

De

scri

pto

r fo

r O

vera

ll R

atin

gs:

E: E

xem

plif

ies

CC

SS Q

ual

ity –

Alig

ned

an

d e

xem

plif

ies

the

qu

alit

y st

and

ard

an

d e

xem

plif

ies

mo

st o

f th

e cr

iter

ia a

cro

ss D

imen

sio

ns

II, I

II, I

V o

f th

e ru

bri

c.

E/I:

Ap

pro

ach

ing

CC

SS Q

ua

lity –

Alig

ned

an

d e

xem

plif

ies

the

qu

alit

y st

and

ard

in s

om

e d

ime

nsi

on

s b

ut

will

ben

efit

fro

m s

om

e re

visi

on

in

oth

ers

.

R:

De

velo

pin

g to

war

d C

CSS

Qu

alit

y –

Alig

ned

par

tial

ly a

nd

ap

pro

ach

es t

he

qu

alit

y st

and

ard

in s

om

e d

ime

nsi

on

s an

d n

eed

s si

gnif

ica

nt

revi

sio

n

in o

ther

s.

N:

No

t re

pre

sen

tin

g C

CSS

Qu

alit

y –

No

t al

ign

ed a

nd

do

es n

ot

add

ress

cri

teri

a.

http://www.corestandards.org/assets/Math_Publishers_Criteria_K-8_Summer%202012_FINAL.pdf

http://www.corestandards.org/Math

Step Back – Reflection Questions

What are the benefits of considering coherence when designing learning experiences for students?

How can understanding learning progressions support increased focus of grade level instruction?

How do the learning progressions allow teachers to support students with unfinished learning?

qm - alexalex.state.al.us/ccrs/sites/alex.state.al.us.ccrs/files/qm3 6-12 pp.pdf · between finite...

Documents