## Narrative

Grade 6 begins with a unit on reasoning about area and understanding and applying concepts of surface area. It is common to begin the year by reviewing the arithmetic learned in previous grades, but starting instead with a mathematical idea that students haven’t seen before sets up opportunities for students to surprise the teacher and themselves with the connections they make. Instead of front-loading review and practice from prior grades, these materials incorporate opportunities to practice elementary arithmetic concepts and skills through warm-ups, in the context of instructional tasks, and in practice problems as they are reinforcing the concepts they are learning in the unit.

One of the design principles of these materials is that students should encounter plenty of examples of a mathematical or statistical idea in various contexts before that idea is named and studied as an object in its own right. For example, in the first unit, students will generalize arithmetic by writing simple expressions like $$\frac12 bh$$ and $$6s^2$$ before they study algebraic expressions as a class of objects in the sixth unit. Sometimes this principle is put into play several units before a concept is developed more fully, and sometimes in the first several lessons of a unit, where students have a chance to explore ideas informally and concretely, building toward a more formal and abstract understanding later in the unit.

### Unit 1: Area and Surface Area

Work with area in grade 6 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students’ work with geometric measurement began with length and continued with area. Students learned to “structure two-dimensional space,” that is, to see a rectangle with whole-number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares. In grade 3, students distinguished between perimeter and area. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths.

In grade 6, students extend their reasoning about area to include shapes that are not composed of rectangles. Doing this draws on abilities developed in earlier grades to compose and decompose shapes, for example, to see a rectangle as composed of two congruent right triangles. Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them (MP1), students build on these abilities and their knowledge of areas of rectangles to find the areas of polygons by decomposing and rearranging them to make figures whose areas they can determine (MP7). They learn strategies for finding areas of parallelograms and triangles, and use regularity in repeated reasoning (MP8) to develop formulas for these areas, using geometric properties to justify the correctness of these formulas. They use these formulas to solve problems. They understand that any polygon can be decomposed into triangles, and use this knowledge to find areas of polygons. Students find the surface areas of polyhedra with triangular and rectangular surfaces. They study, assemble, and draw nets for polyhedra and use nets to determine surface areas. Throughout, they discuss their mathematical ideas and respond to the ideas of others (MP3, MP6).

Because grade 6 students will be writing algebraic expressions and equations involving the letter $$x$$ and $$x$$ is easily confused with $$\times$$, these materials use the “dot” notation, e.g., $$2 \boldcdot 3$$, for multiplication instead of the “cross” notation, e.g., $$2 \times 3$$. The dot notation will be new for many students, and they will need explicit guidance in using it.

Many of the lessons in this unit ask students to work on geometric figures that are not set in a real-world context. This design choice respects the significant intellectual work of reasoning about area. Tasks set in real-world contexts that involve areas of polygons are often contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have an opportunity at the end of the unit to tackle a real-world application (MP2, MP4).

In grade 6, students are likely to need physical tools in order to check that one figure is an identical copy of another where “identical copy” is defined as follows:

One figure is an identical copy of another if one can be placed on top of the other so that they match up exactly.

In grade 8, students will understand “identical copy of” as “congruent to” and understand congruence in terms of rigid motions, that is, motions such as reflection, rotation, and translation. In grade 6, students do not have any way to check for congruence except by inspection, but it is not practical to cut out and stack every pair of figures one sees. Tracing paper is an excellent tool for verifying that figures “match up exactly,” and students should have access to this and other tools at all times in this unit. Thus, each lesson plan suggests that each student should have access to a geometry toolkit, which contains tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles. Providing students with these toolkits gives opportunities for students to develop abilities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools. In this grade, all figures are drawn and labeled so that figures that look congruent actually are congruent; in later grades when students have the tools to reason about geometric figures more precisely, they will need to learn that visual inspection is not sufficient for determining congruence. Also note that all arguments laid out in this unit can (and should) be made more precise in later grades, as students’ geometric understanding deepens.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as comparing, explaining, and describing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Compare

• geometric patterns and shapes (Lesson 1)
• strategies for finding areas of shapes (Lesson 3) and polygons (Lesson 11)
• the characteristics of prisms and pyramids (Lesson 13)
• the measures and units of 1-, 2-, and 3-dimensional attributes (Lesson 16)
• representations of area and volume (Lesson 17)

Explain

• how to find areas by composing (Lesson 3)
• strategies used to find areas of parallelograms (Lesson 4) and triangles (Lesson 8)
• how to determine the area of a triangle using its base and height (Lesson 9)
• strategies to find surface areas of polyhedra (Lesson 14)

Describe

• observations about decomposition of parallelograms (Lesson 7)
• information needed to find the surface area of rectangular prisms (Lesson 12)
• the features of polyhedra and their nets (Lesson 13)
• the features of polyhedra (Lesson 15)
• relationships among features of a tent and the amount of fabric needed for the tent (Lesson 19)

In addition, students are expected to justify claims about the base, height, or area of shapes, generalize about the features of parallelograms and polygons, interpret relevant information for finding the surface area of rectangular prisms, and represent the measures and units of 2- and 3-dimensional figures. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.1.1 area
region

plane
gap
6.1.2 compose
decompose

rearrange
two-dimensional
strategy
6.1.4 parallelogram
opposite (sides or angles)
6.1.5 base (of a parallelogram or triangle)
height

corresponding
expression
represent
6.1.6 horizontal
vertical
6.1.7 identical parallelogram
6.1.8 base (of a parallelogram or triangle)
height
compose
decompose

rearrange
6.1.9 opposite vertex
6.1.10 vertex
edge
6.1.11 polygon horizontal
vertical
6.1.12 face
surface area
area
region
6.1.13 polyhedron
net
prism
pyramid
base (of a prism or pyramid)

three-dimensional
polygon
vertex
edge
face
6.1.15 prism
pyramid
6.1.16 volume
appropriate
quantity
two-dimensional
three-dimensional
6.1.17 squared
cubed
exponent

edge length
6.1.18 value (of an expression) squared
cubed
net
6.1.19 estimate
description
surface area
volume

### Unit 2: Introducing Ratios

Work with ratios in grade 6 draws on earlier work with numbers and operations. In elementary school, students worked to understand, represent, and solve arithmetic problems involving quantities with the same units. In grade 4, students began to use two-column tables, e.g., to record conversions between measurements in inches and yards. In grade 5, they began to plot points on the coordinate plane, building on their work with length and area. These early experiences were a brief introduction to two key representations used to study relationships between quantities, a major focus of work that begins in grade 6 with the study of ratios.

Starting in grade 3, students worked with relationships that can be expressed in terms of ratios and rates (e.g., conversions between measurements in inches and in yards), however, they did not use these terms. In grade 4, students studied multiplicative comparison. In grade 5, they began to interpret multiplication as scaling, preparing them to think about simultaneously scaling two quantities by the same factor. They learned what it means to divide one whole number by another, so they are well equipped to consider the quotients $$\frac{a}{b}$$ and $$\frac{b}{a}$$ associated with a ratio $$a : b$$ for non-zero whole numbers $$a$$ and $$b$$.

In this unit, students learn that a ratio is an association between two quantities, e.g., “1 teaspoon of drink mix to 2 cups of water.” Students analyze contexts that are often expressed in terms of ratios, such as recipes, mixtures of different paint colors, constant speed (an association of time measurements with distance measurements), and uniform pricing (an association of item amounts with prices).

One of the principles that guided the development of these materials is that students should encounter examples of a mathematical concept in various contexts before the concept is named and studied as an object in its own right. The development of ratios, equivalent ratios, and unit rates in this unit and the next unit is in accordance with that principle. In this unit, equivalent ratios are first encountered in terms of multiple batches of a recipe and “equivalent” is first used to describe a perceivable sameness of two ratios, for example, two mixtures of drink mix and water taste the same or two mixtures of red and blue paint are the same shade of purple. Building on these experiences, students analyze situations involving both discrete and continuous quantities, and involving ratios of quantities with units that are the same and that are different. Several lessons later, equivalent acquires a more precise meaning (MP6): All ratios that are equivalent to $$a:b$$ can be made by multiplying both $$a$$ and $$b$$ by the same non-zero number (note that students are not yet considering negative numbers).

This unit introduces discrete diagrams and double number line diagrams, representations that students use to support thinking about equivalent ratios before their work with tables of equivalent ratios.

Initially, discrete diagrams are used because they are similar to the kinds of diagrams students might have used to represent multiplication in earlier grades. Next come double number line diagrams. These can be drawn more quickly than discrete diagrams, but are more similar to tables while allowing reasoning based on the lengths of intervals on the number lines. After some work with double number line diagrams, students use tables to represent equivalent ratios. Because equivalent pairs of ratios can be written in any order in a table and there is no need to attend to the distance between values, tables are the most flexible and concise of the three representations for equivalent ratios, but they are also the most abstract. Use of tables to represent equivalent ratios is an important stepping stone toward use of tables to represent linear and other functional relationships in grade 8 and beyond. Because of this, students should learn to use tables to solve all kinds of ratio problems, but they should always have the option of using discrete diagrams and double number line diagrams to support their thinking.

When a ratio involves two quantities with the same units, we can ask and answer questions about ratios of each quantity and the total of the two. Such ratios are sometimes called “part-part-whole” ratios and are often used to introduce ratio work. However, students often struggle with them so, in this unit, the study of part-part-whole ratios occurs at the end. (Note that tape diagrams are reserved for ratios in which all quantities have the same units.) The major use of part-part-whole ratios occurs with certain kinds of percentage problems, which comes in the next unit.

On using the terms ratio, rate, and proportion. In these materials, a quantity is a measurement that is or can be specified by a number and a unit, e.g., 4 oranges, 4 centimeters, “my height in feet,” or “my height” (with the understanding that a unit of measurement will need to be chosen). The term ratio is used to mean an association between two or more quantities and the fractions $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are never called ratios. Ratios of the form $$1 : \frac{b}{a}$$ or $$\frac{a}{b} : 1$$ (which are equivalent to $$a : b$$) are highlighted as useful but $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are not identified as unit rates for the ratio $$a : b$$ until the next unit. However, the meanings of these fractions in contexts is very carefully developed. The word “per” is used with students in interpreting a unit rate in context, as in “\\$3 per ounce,” and “at the same rate” is used to signify a situation characterized by equivalent ratios.

In the next unit, students learn the term “unit rate” and that if two ratios $$a : b$$ and $$c : d$$ are equivalent, then the unit rates $$\frac{a}{b}$$ and $$\frac{c}{d}$$ are equal.

The terms proportion and proportional relationship are not used anywhere in the grade 6 materials. A proportional relationship is a collection of equivalent ratios, and such collections are objects of study in grade 7. In high school—after their study of ratios, rates, and proportional relationships—students discard the term “unit rate,” referring to $$a$$ to $$b$$, $$a:b$$, and $$\frac{a}{b}$$ as “ratios.”

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as interpreting, explaining, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret

• ratio notation (Lesson 1)
• different representations of ratios (Lesson 6)
• situations involving equivalent ratios (Lesson 8)
• situations with different rates (Lesson 9)
• tables of equivalent ratios (Lessons 11 and 12)
• questions about situations involving ratios (Lesson 17)

Explain

• features of ratio diagrams (Lesson 2)
• reasoning about equivalence (Lesson 4)
• reasoning about equivalent rates (Lesson 10)
• reasoning with reference to tables (Lesson 14)
• reasoning with reference to tape diagrams (Lesson 15)

Compare

• situations with and without equivalent ratios (Lesson 3)
• representations of ratios (Lessons 6 and 13)
• situations with different rates (Lessons 9 and 12)
• situations with same rates and different rates (Lesson 10)
• representations of ratio and rate situations (Lesson 16)

In addition, students are expected to describe and represent ratio associations, represent doubling and tripling of quantities in a ratio, represent equivalent ratios, justify whether ratios are or aren't equivalent and why information is needed to solve a ratio problem, generalize about equivalent ratios and about the usefulness of ratio representations, and critique representations of ratios.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.2.1 ratio
___ to ___
___ for every ___
6.2.2 diagram
6.2.3 recipe
batch
same taste
ratio
___ to ___
___ for every ___
6.2.4 mixture
same color
batch
6.2.5 equivalent ratios
6.2.6 double number line diagram
tick marks
representation
diagram
6.2.7 per
6.2.8 unit price
how much for 1
at this rate
double number line
6.2.9 meters per second
constant speed
6.2.10 same rate equivalent ratios
6.2.11 table
row
column
6.2.14 calculation per
table
6.2.15 tape diagram
parts
suppose
6.2.16 tape diagram

### Unit 3: Unit Rates and Percentages

In the previous unit, students began to develop an understanding of ratios and rates. They started to describe situations using terms such as “ratio,” “rate,” “equivalent ratios,” “per,” “constant speed,” and “constant rate” (MP6). They understood specific instances of the idea that $$a : b$$ is equivalent to every other ratio of the form $$sa : sb$$, where $$s$$ is a positive number. They learned that “at this rate” or “at the same rate” signals a situation that is characterized by equivalent ratios. Although the usefulness of ratios of the form $$\frac{a}{b} : 1$$ and $$1 : \frac{b}{a}$$ was highlighted, the term “unit rate” was not introduced.

In this unit, students find the two values $$\frac{a}{b}$$ and $$\frac{b}{a}$$ that are associated with the ratio $$a : b$$, and interpret them as rates per 1. For example, if a person walks 13 meters in 10 seconds at a constant rate, that means they walked at a speed of $$\frac{13}{10}$$ meters per 1 second and a pace of $$\frac{10}{13}$$ seconds per 1 meter.

Students learn that one of the two values ($$\frac{a}{b}$$ or $$\frac{b}{a}$$) may be more useful than the other in reasoning about a given situation. They find and use rates per 1 to solve problems set in contexts (MP2), attending to units and specifying units in their answers. For example, given item amounts and their costs, which is the better deal? Or given distances and times, which object is moving faster? Measurement conversions provide other opportunities to use rates.

Students observe that if two ratios $$a : b$$ and $$c : d$$ are equivalent, then $$\frac{a}{b} = \frac{c}{d}$$. The values $$\frac{a}{b}$$ and $$\frac{c}{d}$$ are called unit rates because they can be interpreted in the context from which they arose as rates per unit. Students note that in a table of equivalent ratios, the entries in one column are produced by multiplying a unit rate by the corresponding entries in the other column. Students learn that “percent” means “per 100” and indicates a rate. Just as a unit rate can be interpreted in context as a rate per 1, a percentage can be interpreted in the context from which it arose as a rate per 100. For example, suppose a beverage is made by mixing 1 cup of juice with 9 cups of water. The percentage of juice in 20 cups of the beverage is 2 cups and 10 percent of the beverage is juice. Interpreting the 10 as a rate: “there are 10 cups of juice per 100 cups of beverage” or, more generally, “there are 10 units of juice per 100 units of beverage.” The percentage—and the rate—indicate equivalent ratios of juice to beverage, e.g., 2 cups to 20 cups and 10 cups to 100 cups.

In this unit, tables and double number line diagrams are intended to help students connect percentages with equivalent ratios, and reinforce an understanding of percentages as rates per 100. Students should internalize the meaning of important benchmark percentages, for example, they should connect “75% of a number” with “$$\frac{3}{4}$$ times a number” and “0.75 times a number.” Note that 75% (“seventy-five per hundred”) does not represent a fraction or decimal (which are numbers), but that “75% of a number” is calculated as a fraction of or a decimal times the number.

Work done in grades 4 and 5 supports learning about the concept of a percentage. In grade 5, students understand why multiplying a given number by a fraction less than 1 results in a product that is less than the original number, and why multiplying a given number by a fraction greater than 1 results in a product that is greater than the original number. This understanding of multiplication as scaling comes into play as students interpret, for example,

• 35% of 2 cups of juice as $$\frac{35}{100} \boldcdot 2$$ cups of juice.
• 250% of 2 cups of juice as $$\frac{250}{100} \boldcdot 2$$ cups of juice.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as interpreting, explaining, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret

• a context in which a identifying a unit rate is helpful (Lesson 1)
• unit rates in different contexts (Lesson 6)
• situations involving constant speed (Lesson 8)
• tape diagrams used to represent percentages (Lesson 12)
• situations involving measurement, rates, and cost (Lesson 17)

Explain

• reasoning for estimating and sorting measurements (Lesson 2)
• reasoning about relative sizes of units of measurement (Lesson 3)
• how to make decisions using rates (Lesson 9)
• reasoning about percentages (Lesson 11)
• strategies for finding missing information involving percentages (Lesson 14)

Justify

• reasoning about equivalent ratios and unit rates (Lesson 7)
• reasoning about finding percentages (Lessons 15 and 16)
• reasoning about costs and time (Lesson 17)

In addition, students have opportunities to generalize about unit ratios, unit rates, and percentages from multiple contexts and with reference to benchmark percentages, tape diagrams, and other mathematical representations. Students can also be expected to describe measurements and observations, describe and compare situations involving percentages, compare speeds, compare prices, and critique reasoning about costs and time.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.3.1 at this rate
6.3.3 order
6.3.5 (good / better / best) deal
rate per 1
unit price
same speed
6.3.6 unit rate gallon
6.3.7 result unit rate
6.3.8 pace speed
6.3.9 meters per second
(good / better / best) deal
6.3.10 percentage
___% of
6.3.11 tick marks
6.3.12 ___% as much tape diagram
___% of
6.3.14 ___% of
6.3.15 regular price
sale price
percentage

### Unit 4: Dividing Fractions

Work with fractions in grade 6 draws on earlier work in operations and algebraic thinking, particularly the knowledge of multiplicative situations developed in grades 3 to 5, and making use of the relationship between multiplication and division. Multiplicative situations include three types: equal groups; comparisons of two quantities; dimensions of arrays or rectangles. In the equal groups and comparison situations, there are two subtypes, sometimes called the partitive and the quotitive (or measurement) interpretations of division. Students are not expected to identify the three types of situations or use the terms “partitive” or “quotitive.” However, they should recognize the associated interpretations of division in specific contexts (MP7).

For example, in an equal groups situation when the group size is unknown, division can be used to answer the question, “How many in each group?” If the number of groups is unknown, division answers the question, “How many groups?” For example, if 12 pounds of almonds are equally shared among several bags:

There are 2 bags. How many pounds in each bag? (partitive)

There are 6 pounds in each bag. How many bags? (quotitive)

In a comparison situation that involves division, the size of one object may be unknown or the relative sizes of two objects may be unknown. For example, when comparing two ropes:

A rope is 12 feet long. It is twice as long as another rope. How long is the second rope? (partitive)

One rope is 12 feet long. One rope is 6 feet long. How many times longer than the second rope is the first rope? (quotitive)

In situations that involve arrays or rectangles, division can be used to find an unknown factor. In an array situation, the unknown is the number of entries in a row or a column; in a rectangle, the unknown is a length or a width measurement. For example, “The area of a rectangle is 12 square feet. One side is 6 feet long. How long is the other side?” If the rectangle is viewed as tiled by an array of 12 unit squares with 6 tiles in each row, this question can seen as asking for the number of entries in each column.

At beginning of the unit, students consider how the relative sizes of numerator and denominator affect the size of their quotient. Students first compute quotients of whole numbers, then—without computing—consider the relative magnitudes of quotients that include divisors which are whole numbers, fractions, or decimals, e.g., “Is $$800 \div \frac{1}{10}$$ larger than or smaller than $$800 \div 2.5$$?”

The second section of the unit focuses on equal groups and comparison situations. It begins with partitive and quotitive situations that involve whole numbers, represented by tape diagrams and equations. Students interpret the numbers in the two situations (MP2) and consider analogous situations that involve one or more fractions, again accompanied by tape diagrams and equations. Students learn to interpret, represent, and describe these situations, using terminology such as “What fraction of 6 is 2?,” “How many 3s are in 12?,” “How many fourths are in 3?,” “is one-third as long as,” “is two-thirds as much as,” and “is one-and-one-half times the size of.”

The third section concerns computing quotients of fractions. Students build on their work from the previous section by considering quotients related to products of numbers and unit fractions, e.g., “How many 3s in 12?” and “What is $$\frac {1}{3}$$ of 12?,” to establish that dividing by a unit fraction $$\frac{1}{b}$$ is the same as multiplying by its reciprocal $$b$$. Building on this and their understanding that $$\frac{a}{b} = a \boldcdot \frac{1}{b}$$ (from grade 4), students understand that dividing by a fraction $$\frac{a}{b}$$ is the same as multiplying by its reciprocal $$\frac{b}{a}$$.

The fourth section returns to interpretations of division in situations that involve fractions. This time, the focus is on using division to find an unknown area or volume measurement. In grade 3, students connected areas of rectangles with multiplication, viewing a rectangle as tiled by an array of unit squares and understanding that, for whole-number side lengths, multiplying the side lengths yields the number of unit squares that tile the rectangle. In grade 5, students extended the formula for the area of rectangles with whole-number side lengths to rectangles with fractional side lengths. For example, they viewed a $$\frac 23$$-by-$$\frac 57$$ rectangle as tiled by 10 $$\frac 13$$-by-$$\frac 17$$ rectangles, reasoning that 21 such rectangles compose 1 square unit, so the area of one such rectangle is $$\frac {1}{21}$$, thus the area of a shape composed of 10 such rectangles is $$\frac {10}{21}$$. In a previous grade 6 unit, students used their familiarity with this formula to develop formulas for areas of triangles and parallelograms. In this unit, they return to this formula, using their understanding of it to extend the formula for the volume of a right rectangular prism (developed in grade 5) to right rectangular prisms with fractional side lengths.

The unit ends with two lessons in which students use what they have learned about working with fractions (including the volume formula) to solve problems set in real-world contexts, including a multi-step problem about calculating shipping costs. These require students to formulate appropriate equations that use the four operations or draw diagrams, and to interpret results of calculations in the contexts from which they arose (MP2).

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as interpreting, representing, justifying, and explaining. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret and represent

• situations involving division (Lessons 2, 3, 12, and 16)
• situations involving measurement constraints (Lesson 17)

Justify

• reasoning about division and diagrams (Lessons 4 and 5)
• strategies for dividing numbers (Lesson 11)
• reasoning about volume (Lesson 15)

Explain

• how to create and make sense of division diagrams (Lesson 6)
• how to represent division situations (Lesson 9)
• how to find missing lengths (Lesson 14)
• a plan for optimizing costs (Lesson 17)

In addition, students are expected to critique the reasoning of others about division situations and representations, and make generalizations about division by comparing and connecting across division situations, and across the representations used in reasoning about these situations. The Lesson Syntheses in Lessons 2 and 12 offer specific disciplinary language that may be especially helpful for supporting students in navigating the language of important ideas in this unit.
The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.4.1 divisor
dividend
quotient
6.4.2 equation
interpretation
How many groups of ___?
How many ___ in each group?
6.4.3 unknown
equal-sized
6.4.4 whole
6.4.5 relationship
6.4.6 equal-sized
6.4.7 times as ___
fraction of ___
6.4.8 container unknown
fraction of ___
6.4.9 whole
6.4.10 reciprocal
observations
times as ___
numerator
denominator
6.4.11 evaluate
6.4.13 gaps
6.4.14 packed
6.4.17 assumption packed

### Unit 5: Arithmetic in Base Ten

By the end of grade 5, students learn to use efficient algorithms to fluently calculate sums, differences, and products of multi-digit whole numbers. They calculate quotients of multi-digit whole numbers with up to four-digit dividends and two-digit divisors. These calculations use strategies based on place value, the properties of operations, and the relationship between multiplication and division. Grade 5 students illustrate and explain these calculations with equations, rectangular arrays, and area diagrams.

In grade 5, students also calculate sums, differences, products, and quotients of decimals to hundredths, using concrete representations or drawings, and strategies based on place value, properties of operations, and the relationship between addition and subtraction. They connect their strategies to written methods and explain their reasoning.

In this unit, students learn an efficient algorithm for division and extend their use of other base-ten algorithms to decimals of arbitrary length. Because these algorithms rely on the structure of the base-ten system, students build on the understanding of place value and the properties of operations developed during earlier grades (MP7).

The unit begins with a lesson that revisits sums and differences of decimals to hundredths, and products of a decimal and whole number. The tasks are set in the context of shopping and budgeting, allowing students to be reminded of appropriate magnitudes for results of calculations with decimals.

The next section focuses on extending algorithms for addition, subtraction, and multiplication, which students used with whole numbers in earlier grades, to decimals of arbitrary length.

Students begin by using “base-ten diagrams,” diagrams analogous to base-ten blocks for ones, tens, and hundreds. These diagrams show, for example, ones as large squares, tenths as rectangles, hundredths as medium squares, thousandths as small rectangles, and ten-thousandths as small squares. These are designed so that the area of a figure that represents a base-ten unit is one tenth of the area of the figure that represents the base-ten unit of next highest value. Thus, a group of 10 figures that represent 10 like base-ten units can be replaced by a figure whose area is the sum of the areas of the 10 figures.

Students first calculate sums of two decimals by representing each number as a base-ten diagram, combining representations of like base-ten units and replacing representations of 10 like units by a representation of the unit of next highest value, e.g., 10 rectangles compose 1 large square. Next, they examine “vertical calculations,” representations of calculations with symbols that show one summand above the other, with the sum written below. They check each vertical calculation by representing it with base-ten diagrams. This is followed by a similar lesson on subtraction of decimals. The section concludes with a lesson designed to illustrate efficient algorithms and their advantages, and to promote their use.

The third section, multiplication of decimals, begins by asking students to estimate products of a whole number and a decimal, allowing students to be reminded of appropriate magnitudes for results of calculations with decimals. In this section, students extend their use of efficient algorithms for multiplication from whole numbers to decimals. They begin by writing products of decimals as products of fractions, calculating the product of the fractions, then writing the product as a decimal. They discuss the effect of multiplying by powers of 0.1, noting that multiplying by 0.1 has the same effect as dividing by 10. Students use area diagrams to represent products of decimals. The efficient multiplication algorithms are introduced and students use them, initially supported by area diagrams.

In the fourth section, students learn long division. They begin with quotients of whole numbers, first representing these quotients with base-ten diagrams, then proceeding to efficient algorithms, initially supporting their use with base-ten diagrams. Students then tackle quotients of whole numbers that result in decimals, quotients of decimals and whole numbers, and finally quotients of decimals.

The unit ends with two lessons in which students use calculations with decimals to solve problems set in real-world contexts. These require students to interpret diagrams, and to interpret results of calculations in the contexts from which they arose (MP2). The second lesson draws on work with geometry and ratios from previous units. Students fold papers of different sizes to make origami boxes of different dimensions, then compare the lengths, widths, heights, and surface areas of the boxes.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as explaining, interpreting and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Explain

• processes of estimating and finding costs (Lesson 1)
• approaches to adding and subtracting decimals (Lesson 4)
• reasoning about products and quotients involving powers of 10 (Lesson 5)
• methods for multiplying decimals (Lesson 8)
• reasoning about relationships among measurements (Lesson 15)

Interpret

• representations of decimals (Lesson 2)
• base ten diagrams showing addition/subtraction of decimals (Lesson 3)
• area diagrams showing products of decimals (Lesson 7)
• base ten diagrams and long division when the quotient is a decimal value (Lesson 11)

Compare

• base ten diagrams with numerical calculations (Lesson 4)
• methods for multiplying decimals (Lesson 6)
• base ten diagrams showing quotients with partial quotient method (Lesson 9)
• previously studied methods for finding quotients with long division (Lesson 10)

In addition, students are expected to describe decimal values up to hundredths, generalize about multiplication by powers of 10 and about decimal measurements, critique approaches to operations on decimals, and justify strategies for finding quotients with reference to base-ten diagrams and more efficient algorithms.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.5.1 digits
budget
at least
6.5.2 base-ten diagram
bundle
vertical calculation
place value
digits
6.5.3 unbundle
6.5.4 method
6.5.5 powers of 10 product
decimal point
6.5.7 partial products method
6.5.9 partial quotients remainder
6.5.10 long division divisor
6.5.13 long division
6.5.14 precision
accuracy
6.5.15 operation

### Unit 6: Expressions and Equations

Students begin the unit by working with linear equations that have single occurrences of one variable, e.g., $$x + 1 = 5$$ and $$4x = 2$$. They represent relationships with tape diagrams and with linear equations, explaining correspondences between these representations. They examine values that make a given linear equation true or false, and what it means for a number to be a solution to an equation. Solving equations of the form $$px = q$$ where $$p$$ and $$q$$ are rational numbers can produce complex fractions (i.e., quotients of fractions), so students extend their understanding of fractions to include those with numerators and denominators that are not whole numbers.

The second section introduces balanced and unbalanced “hanger diagrams” as a way to reason about solving the linear equations of the first section. Students write linear equations to represent situations, including situations with percentages, solve the equations, and interpret the solutions in the original contexts (MP2), specifying units of measurement when appropriate (MP6). They represent linear expressions with tape diagrams and use the diagrams to identify values of variables for which two linear expressions are equal. Students write linear expressions such as $$6w - 24$$ and $$6(w - 4)$$ and represent them with area diagrams, noting the connection with the distributive property (MP7). They use the distributive property to write equivalent expressions.

In the third section of the unit, students write expressions with whole-number exponents and whole-number, fraction, or variable bases. They evaluate such expressions, using properties of exponents strategically (MP5). They understand that a solution to an equation in one variable is a number that makes the equation true when the number is substituted for all instances of the variable. They represent algebraic expressions and equations in order to solve problems. They determine whether pairs of numerical exponential expressions are equivalent and explain their reasoning (MP3). By examining a list of values, they find solutions for simple exponential equations of the form $$a = b^x$$, e.g., $$2^x = 32$$, and simple quadratic and cubic equations, e.g., $$64 = x^3.$$

In the last section of the unit, students represent collections of equivalent ratios as equations. They use and make connections between tables, graphs, and linear equations that represent the same relationships (MP1).

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as interpreting, describing and explaining. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret

• tape diagrams involving letters that stand for numbers (Lesson 1)
• the parts of an equation (Lesson 2)
• descriptions of situations (Lesson 6)
• numerical expressions involving exponents (Lesson 13)
• different representations of the same relationship between quantities (Lesson 17)

Describe

• how parts of an equation represent parts of a story (Lesson 2)
• solutions to equations (Lesson 2)
• stories represented by given equations (Lesson 5)
• patterns of growth that can be represented using exponents (Lesson 12)
• relationships between independent and dependent variables (Lesson 16)

Explain

• the meaning of a solution using hanger diagrams (Lesson 3)
• how to solve an equation (Lesson 4)
• how to use equations to solve percent problems (Lesson 7)
• how to determine whether two expressions are equivalent, including with reference to diagrams (Lesson 8)
• strategies for determining whether expressions are equivalent (Lesson 13)
• the process of evaluating variable exponential expressions (Lesson 15)

In addition, students are expected to compare equations with balanced hanger diagrams and with descriptions of situations, represent quantities with mathematical expressions, generalize about equivalent numerical expressions using rectangle diagrams and the distributive property, justify claims about equivalent variable expressions using rectangle diagrams and the distributive property, and justify reasoning when evaluating and comparing numerical expressions with exponents.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.6.1 value (of a variable) operation
6.6.2 variable
coefficient
solution to an equation

true equation / false equation
value (of a variable)
6.6.3 each side
balanced hanger
6.6.4 solve (an equation) each side
6.6.6 equation
6.6.7 true equation / false equation
6.6.8 equivalent expressions
6.6.9 term
distributive property
area as a product
area as a sum
6.6.12 to the power
6.6.13 base (of an exponent) to the power
exponent
6.6.14 solution to an equation
6.6.16 independent variable
dependent variable
variable
relationship
6.6.17 coordinate plane
coordinates
6.6.18 horizontal axis
vertical axis
plot

### Unit 7: Rational Numbers

In this unit, students are introduced to signed numbers and plot points in all four quadrants of the coordinate plane for the first time. They work with simple inequalities in one variable and learn to understand and use “common factor,” “greatest common factor,” “common multiple,” and “least common multiple.”

The first section of the unit introduces signed numbers. Students begin by considering examples of positive and negative temperatures, plotting each temperature on a vertical number line on which 0 is the only label. Next, they consider examples of positive and negative numbers used to denote height relative to sea level. In the second lesson, they plot positive and negative numbers on horizontal number lines, including “opposites”—pairs of numbers that are the same distance from zero. They use “less than,” “greater than,” and the corresponding symbols to describe the relationship of two signed numbers, noticing correspondences between the relative positions of two numbers on the number line and statements that use these symbols, e.g., $$0.8 > \text- 1.3$$ means that 0.8 is to the right of -1.3 on the number line. Students learn that the sign of a number indicates whether the number is positive or negative, and that zero has no sign. They learn that the absolute value of a number is its distance from zero, how to use absolute value notation, and that opposites have the same absolute value because they have the same distance from zero.

Previously, when students worked only with non-negative numbers, magnitude and order were indistinguishable: if one number was greater than another, then on the number line it was always to the right of the other number and always farther from zero. In comparing two signed numbers, students distinguish between magnitude (the absolute value of a number) and order (relative position on the number line), distinguishing between “greater than” and “greater absolute value,” and “less than” and “smaller absolute value.”

Students examine opposites of numbers, noticing that the opposite of a negative number is positive.

The second section of the unit concerns inequalities. Students graph simple inequalities in one variable on the number line, using a circle or disk to indicate when a given point is, respectively, excluded or included. In these materials, inequality symbols in grade 6 are limited to < and > rather than $$\leq$$ and $$\geq.$$ However, in this unit students encounter situations when they need to represent statements such as $$2 < x$$ or $$2 = x.$$

Students represent situations that involve inequalities, symbolically and with the number line, understanding that there may be infinitely many solutions for an inequality. They interpret and graph solutions in contexts (MP2), understanding that some results do not make sense in some contexts, and thus the graph of a solution might be different from the graph of the related symbolic inequality. For example, the graph describing the situation “A fishing boat can hold fewer than 9 people” omits values other than the whole numbers from 0 to 8, but the graph of $$x < 8$$ includes all numbers less than 8. Students encounter situations that require more than one inequality statement to describe, e.g., “It rained for more than 10 minutes but less than 30 minutes” ($$t > 10$$ and $$t < 30$$, where $$t$$ is the amount of time that it rained in minutes) but which can be described by one number line graph.

The third section of the unit focuses on the coordinate plane. In grade 5, students learned to plot points in the coordinate plane, but they worked only with non-negative numbers, thus plotted points only in the first quadrant. In a previous unit, students again worked in the first quadrant of the coordinate plane, plotting points to represent ratio and other relationships between two quantities with positive values. In this unit, students work in all four quadrants of the coordinate plane, plotting pairs of signed number coordinates in the plane. They understand that for a given data set, there are more and less strategic choices for the scale and extent of a set of axes. They understand the correspondence between the signs of a pair of coordinates and the quadrant of the corresponding point. They interpret the meanings of plotted points in given contexts (MP2), and use coordinates to calculate horizontal and vertical distances between two points.

The last section of the unit returns to consideration of whole numbers. In the first lesson, students are introduced to “common factor” and “greatest common factor,” and solve problems that illustrate how the greatest common factor of two numbers can be used in real-world situations, e.g., determining the largest rectangular tile with whole-number dimensions that can tile a given rectangle with whole-number dimensions. The second lesson introduces “common multiple” and “least common multiple,” and students solve problems that involve listing common multiples or identifying common multiples of two or more numbers. In the third and last lesson, students solve problems that revisit situations similar to those in the first two lessons and identify which of the new concepts is involved in each problem. This lesson includes two optional classroom activities.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as describing, interpreting, justifying, and generalizing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe and interpret

• situations involving negative numbers (Lesson 1)
• features of a number line (Lessons 2, 4 and 6)
• situations involving elevation (Lesson 7)
• situations involving minimums and maximums (Lesson 8)
• points on a coordinate plane (Lessons 11 and 14)
• situations involving factors and multiples (Lesson 18)

Justify

• reasoning about magnitude (Lesson 3)
• reasoning about a situation involving negative numbers (Lesson 5)
• reasoning about solutions to inequalities (Lesson 9)
• that all possible pairs of factors have been identified (Lesson 16)

Generalize

• the meaning of integers for a specific context (Lesson 5)
• understandings of solutions to inequalities (Lesson 9)
• about the relationships between shapes (Lesson 10)
• about greatest common factors (Lesson 16)
• about least common multiples (Lesson 17)

In addition, students are expected to critique the reasoning of others, represent inequalities symbolically and in words, and explain how to order rational numbers and how to determine distances on the coordinate plane. Students also have opportunities to use language to compare magnitudes of positive and negative numbers, compare features of ordered pairs, and compare appropriate axes for different sets of coordinates.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.7.1 positive number
negative number

temperature
degrees Celsius
elevation
sea level
number line
below zero
6.7.2 opposite (numbers)
rational number

location
distance (away) from zero
6.7.3 sign
inequality
closer to 0
farther from 0
greater than
less than
6.7.4 from least to greatest temperature
elevation
sea level
6.7.5 positive change
negative change
context
6.7.6 absolute value positive number
negative number

distance (away) from zero
6.7.7 closer to 0
farther from 0
6.7.8 maximum
minimum
6.7.9 requirement
solution to an inequality
6.7.10 unbalanced hanger inequality
$$x$$-coordinate
$$y$$-coordinate
6.7.12 (line) segment axis
6.7.13 degrees Fahrenheit degrees Celsius
6.7.14 absolute value
$$x$$-coordinate
$$y$$-coordinate
6.7.16 common factor
greatest common factor (GCF)
factor
6.7.17 common multiple
least common multiple (LCM)
multiple

### Unit 8: Data Sets and Distributions

In this unit, students learn about populations and study variables associated with a population. They understand and use the terms “numerical data,” “categorical data,” “survey” (as noun and verb), “statistical question,” “variability,” “distribution,” and “frequency.” They make and interpret histograms, bar graphs, tables of frequencies, and box plots. They describe distributions (shown on graphical displays) using terms such as “symmetrical,” "peaks," “gaps,” and “clusters.” They work with measures of center—understanding and using the terms “mean,” “average,” and “median.” They work with measures of variability—understanding and using the terms “range,” ”mean absolute deviation” or MAD, “quartile,” and “interquartile range” or IQR. They interpret measurements of center and variability in contexts.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as justifying, representing, and interpreting. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Justify

• reasoning for matching data sets to questions (Lesson 2)
• reasoning about dot plots (Lesson 3)
• reasoning about mean and median (Lesson 13)
• reasoning about changes in mean and median (Lesson 14)
• reasoning about which information is needed (Lesson 17)
• which summaries and graphs best represent given data sets (Lesson 18)

Represent

• data using dot plots (Lessons 3 and 4)
• data using histograms (Lesson 7)
• mean using bar graphs (Lesson 9)
• data with five number summaries (Lesson 15)
• data using box plots (Lesson 16)

Interpret

• dot plots (Lessons 4 and 11)
• histograms (Lessons 6 and 18)
• mean of a data set (Lesson 9)
• five number summaries (Lesson 15)
• box plots (Lesson 16)

In addition, students are expected to critique the reasoning of others, describe how quantities are measured, describe and compare features and distributions of data sets, generalize about means and distances in data sets, generalize categories for sorting data sets, and generalize about statistical questions. Students are also expected to use language to compare questions that produce numerical and categorical data, compare dot plots and histograms, and compare histograms and bar graphs.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.8.1 numerical data
categorical data
dot plot
6.8.2 statistical question
variability
6.8.3 distribution
frequency
bar graph
6.8.4 typical
6.8.5 center
variability
6.8.6 histogram
bins
distribution
center
6.8.7 statistical question
6.8.8 symmetrical
peak
cluster
unusual value
numerical data
categorical data

gap
6.8.9 average
mean

fair share
6.8.10 measure of center
balance point
symmetrical
mean
typical
6.8.13 median measure of center
6.8.14 peak
cluster
unusual value
6.8.15 range
quartile
interquartile range (IQR)

five-number summary
minimum
maximum
6.8.16 box plot
whisker
median
interquartile range (IQR)
6.8.17 range
quartile
6.8.18 dot plot
histogram
box plot

### Unit 9: Putting it All Together

This optional unit consists of six lessons. Each of the first three lessons is independent of the others, requiring only the mathematics of the previous units. The last three lessons build on each other.

The first lesson concerns Fermi problems—problems that require making rough estimates for quantities that are difficult or impossible to measure directly (MP4). The three problems in this lesson involve measurement conversion and calculation of volumes and surface areas of three-dimensional figures or the relationship of distance, rate, and time.

The second lesson involves finding approximately equivalent ratios for groups from two populations, one very large (the population of the world) and one comparatively small (a 30-student class). Students work with percent rates that describe subgroups of the world population, e.g., about 59% of the world population lives in Asia. Using these rates, which include numbers expressed in the form $$\frac a b$$ or as decimals, they determine, for example, the number of students who would live in Asia—“if our class were the world” (MP2). Because students choose their own methods to determine these numbers, possibly making strategic use of benchmark percentages or spreadsheets (MP5), there is an opportunity for them to see correspondences between approaches (MP1). Because the size of the world population and its subgroups are estimates, and because pairs of values in ratios may both be whole numbers, considerations of accuracy may arise (MP6).

The third lesson is an exploration of the relationship between the greatest common factor of two numbers, continued fractions, and decomposition of rectangles with whole-number side lengths, providing students an opportunity to perceive this relationship through repeated reasoning (MP8) and to see correspondences between two kinds of numerical relationships, and between numerical and geometric relationships (MP1).

The remaining three lessons explore the mathematics of voting (MP2, MP4). In some activities, students chose how to assign votes and justify their choices (MP3). The first of these lessons focuses on proportions of voters and votes cast in elections in which there are two choices. It requires only the mathematics of the previous units, in particular, equivalent ratios, part–part ratios, percentages, unit rates, and, in the final activity, the concept of area. The second of these lessons focuses on methods for voting when there are more than two choices: plurality, runoff, and instant runoff. They compute percentages, finding that different voting methods have different outcomes. The third of these lessons focuses on representation in the case when voters have two choices. It’s not always possible to have the same number of constituents per representative. How can we fairly share a small number of representatives? Students again compute percentages to find outcomes.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as critiquing, justifying, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Critique

• reasoning about Fermi problems (Lesson 1)
• claims about percentages (Lesson 4)
• reasoning about the fairness of voting systems (Lesson 6)

Justify

• reasoning about Fermi problems (Lesson 1)
• reasoning about the fairness of voting systems (Lessons 5 and 6)

Compare

• rectangles and fractions (Lesson 3)
• voting systems (Lesson 5)

In addition, students are expected to interpret and represent characteristics of the world population, describe distributions of voters, and generalize about decomposition of area and numbers.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson new terminology
receptive productive
6.9.3 mixed number
6.9.4 in favor
majority

6.9.5 plurality
runoff
majority
6.9.6 in all
fair