Grade 6 Scope and Sequence

Course Narrative

The course begins with a study of area and surface area concepts. This work sets the tone for later units that use area models for arithmetic using rational numbers. Next, students begin study of ratios, rates, and percentages with an introduction using representations such as number line diagrams, tape diagrams, and tables. Student understanding of these concepts expands by exploring fraction and decimal representations of rational numbers. They explore sums, differences, products, and quotients using intuitive methods and efficient algorithms. Next, students are introduced to equations and expressions including finding solutions for linear equations in one variable and basic equations involving exponents. Student understanding of ratios and rates combined with a basic understanding of equations leads students to study proportional relationships with special emphasis on circumference and area of a circle as an example and nonexample of proportional relationships. This is followed by looking at percentage concepts and applications such as sales tax, tipping, and markup. They learn about rational numbers less than zero expanding their understanding of arithmetic to negative numbers. A brief study of data and statistics concludes the new concepts in the course. The last unit offers students an optional opportunity to synthesize their learning from the year using a number of different applications.

Dependency Diagrams

In the unit dependency chart, an arrow indicates that a particular unit is designed for students who already know the material in a previous unit. Reversing the order would have a negative effect on mathematical or pedagogical coherence. Examples:

• There is an arrow from 6.2 to 6.6, because students are expected to use their knowledge of contexts involving ratios (from 6.2) to write and solve equations representing such contexts (in 6.6).

• There is an arrow from 7.4 to 7.8, because students are expected to use their skills in representing percentages (from 7.4) when solving problems about probability (in 7.8).

• There is an arrow from 8.3 to 8.6, because students are expected to use their skills in writing and interpreting an equation that represents a line (from 8.3) to interpret the parameters in an equation that represents a line that fits a scatter plot (in 8.6).

Unit Overviews

Unit 1: Areas and Surface Areas

Work with area in this course 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 this course, 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 students will be writing algebraic expressions and equations involving the letter and is easily confused with , these materials use the “dot” notation, for example, , for multiplication instead of the “cross” notation like . 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).

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:

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 a later course, 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 this course, 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 course, all figures are drawn and labeled so that figures that look congruent actually are congruent; in later courses 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 courses, as students’ geometric understanding deepens.

Unit 2: Ratios, Rates, and Percentages

Work with ratios in this course 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, for example, 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 this course with the study of ratios.

Starting in grade 3, students worked with relationships that can be expressed in terms of ratios and rates (for example, 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 and associated with a ratio for non-zero whole numbers and .

In this unit, students learn that a ratio is an association between two quantities such as, “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). Students find the two values and that are associated with the ratio , 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 meters per 1 second and a pace of seconds per 1 meter.

Students learn that one of the two values ( or ) 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 and are equivalent, then . The values and 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, for example, 2 cups to 20 cups and 10 cups to 100 cups.

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 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 can be made by multiplying both and 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 subsequent courses. 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 later. (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.

Students should internalize the meaning of important benchmark percentages, for example, they should connect “75% of a number” with “ 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 cups of juice

• 250% of 2 cups of juice as cups of juice

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, for example, 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 and are never called ratios. Ratios of the form or (which are equivalent to ) are highlighted as useful but and are not identified as unit rates for the ratio 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.

The terms proportion and proportional relationship are not used anywhere in this unit. A proportional relationship is a collection of equivalent ratios, and such collections are examined further in later units. In high school—after their study of ratios, rates, and proportional relationships—students discard the term “unit rate,” referring to to , , and as “ratios.”

Unit 3: Fractions and Decimals

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)

At the beginning of the unit, students consider how the relative sizes of numerator and denominator affect the size of their quotient.

The second section of the unit focuses on equal groups and comparison situations. It begins with partitive and quotitive situations represented 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.” Building on this and their understanding that (from grade 4), students understand that dividing by a fraction is the same as multiplying by its reciprocal .

The third section focuses on applying fraction division to solve problems, including finding 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. 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 then switches to focus on decimals. 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 section 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 several lessons focus 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 and differences 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, or vice versa. Next, they examine “vertical calculations,” representations of calculations with symbols that show one operand above the other, with the sum or difference written below. They check each vertical calculation by representing it with base-ten diagrams and see the advantages of using efficient algorithms.

n this section, students also 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 fifth 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 fractions and decimals to solve problems set in real-world contexts. These require students to interpret diagrams, formulate appropriate equations that use the four operations, and to interpret results of calculations in the contexts from which they arose (MP2). The last 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.

Unit 4: Equations and Expressions

Students begin the unit by working with linear equations that have single occurrences of one variable, e.g., and . 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 where and 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 and 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 , e.g., , and simple quadratic and cubic equations, e.g., .

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).

Unit 5: Proportional Relationships

In this unit, students develop the idea of a proportional relationship based on the idea of equivalent ratios in an earlier unit. Proportional relationships prepare the way for the study of linear functions in later courses.

In a table of equivalent ratios, a multiplicative relationship between the pair of rows is given by a scale factor. By contrast, the multiplicative relationship between the columns is given by a unit rate. Every number in the second column is obtained by multiplying the corresponding number in the first column by one of the unit rates, and every number in the first column is obtained by multiplying the number in the second column by the other unit rate. The relationship between pairs of values in the two columns is called a proportional relationship, the unit rate that describes this relationship is called a constant of proportionality, and the quantity represented by the right column is said to be proportional to the quantity represented by the left. (Although a proportional relationship between two quantities represented by and is associated with two constants of proportionality, and , throughout the unit, the convention is if and are, respectively, in the left and right columns of a table, then is the constant of proportionality for the relationship represented by the table.)

For example, if a person runs at a constant speed and travels 12 miles in 2 hours, then the distance traveled is proportional to the time elapsed, with constant of proportionality 6, because . The time elapsed is proportional to distance traveled with constant of proportionality , because . Students learn that any proportional relationship can be represented by an equation of the form where is the constant of proportionality, that its graph lies on a line through the origin that passes through Quadrant I, and that the constant of proportionality indicates the steepness of the line. By the end of the unit, students should be able to easily work with common contexts associated with proportional relationships (such as constant speed, unit pricing, and measurement conversions) and be able to determine whether or not a relationship is proportional.

Because this unit focuses on understanding what a proportional relationship is, how it is represented, and what types of contexts give rise to proportional relationships, the contexts have been carefully chosen. The first tasks in the unit employ contexts that should be familiar to students from earlier in this course, such as servings of food, recipes, constant speed, and measurement conversion. These contexts are revisited throughout the unit as new aspects of proportional relationships are introduced. The tasks in this unit avoid discussion of measurement error and statistical variability, which will be addressed in later units.

In the second half of the unit, students extend their knowledge of circles and geometric measurement, applying their knowledge of proportional relationships to the study of circles. They also extend their earlier work with perimeters of polygons to circumferences of circles, and recognize that the circumference of a circle is proportional to its diameter, with constant of proportionality . They encounter informal derivations of the relationship between area, circumference, and radius.

The sections begin with activities designed to help students come to a more precise understanding of the characteristics of a circle (MP6): a “circle” is the set of points that are equally distant from a point called the “center”; the diameter of a circle is a line segment that passes through its center with endpoints on the circle; the radius is a line segment with one endpoint on the circle and one endpoint at the center. Students identify these characteristics in a variety of contexts (MP2). They use compasses to draw circles with given diameters or radii, and to copy designs that involve circles. Using their newly gained familiarity with circumference and diameter, students measure circular objects, investigating the relationship between measurements of circumference and diameter by making tables and graphs.

Then students focus their attention on the area of circles. Students encounter informal derivations of the fact that the area of a circle is equal to times the square of its radius. One derivation involves dissecting a disk into sectors and rearranging them to form a shape that approximates a parallelogram of height and width . A second argument involves considering a disk as formed of concentric rings, “cutting” the rings with a radius, and “opening” the rings to form a shape that approximates an isosceles triangle of height and base .

Optional lessons at the end of the unit let students select and use formulas for the area and circumference of a circle to solve abstract and real-world problems that involve calculating lengths and areas as well as to examine situations using proportional relationships.

On using the terms quantity, ratio, proportional relationship, unit rate, and fraction. 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, MP6). The term ratio is used to mean a type of association between two or more quantities. A proportional relationship is a collection of equivalent ratios.

A unit rate is the numerical part of a rate per 1 unit, for example, the 6 in 6 miles per hour. The fractions and are never called ratios. The fractions and are identified as “unit rates” for the ratio . In high school—after their study of ratios, rates, and proportional relationships—students discard the term “unit rate,” referring to to , , and as “ratios.”

In this course, students write rates without abbreviated units, for example as “3 miles per hour” or “3 miles in every 1 hour.” Use of notation for derived units such as waits for high school—except for the special cases of area and volume. Students have worked with area since grade 3 and volume since grade 5. Before this course, they have learned the meanings of such things as sq cm and cu cm. After students learn exponent notation, they also use cm² and cm³.

A fraction is a point on the number line that can be located by partitioning the segment between 0 and 1 into equal parts, then finding a point that is a whole number of those parts away from 0. A fraction can be written in the form or as a decimal.

On using the term circle. Strictly speaking, a circle is one-dimensional—the boundary of a two-dimensional region rather than the region itself. Because students are not yet expected to make this distinction, these materials refer to both circular regions (that is, disks) and boundaries of disks as “circles,” using illustrations to eliminate ambiguity.

Unit 6: Percentage Increase and Decrease

Students began their work with ratios, rates, and unit rates in earlier units, representing them with expressions, tape diagrams, double number line diagrams, and tables. They used these to reason about situations involving color mixtures, recipes, unit price, discounts, constant speed, and measurement conversions. They extended their understanding of rates to include percentages as rates per 100, reasoning about situations involving whole-number percentages.

In previous units, students worked with proportional relationships and constants of proportionality. Although students have learned how to compute quotients of fractions earlier, the previous unit on proportional relationships did not require such calculations, allowing the new concept to be the main focus.

In this unit, students deepen their understanding of ratios,unit rates (also called constants of proportionality), and proportional relationships, using them to solve multi-step problems that are set in a wide variety of contexts that involve fractions and percentages.

In the first section of the unit, students extend their use of ratios and rates to problems that involve computing quotients of fractions, and interpret these quotients in contexts such as running at constant speed (MP2). They use long division to write fractions presented in the form as decimals, like .

The second section of the unit is about percent increase and decrease. Students consider situations for which percentages can be used to describe a change relative to an initial amount—for example, prices before and after a 25% increase. They begin by considering situations with unspecified amounts, like matching tape diagrams with statements such as, “Compared with last year’s strawberry harvest, this year’s strawberry harvest increased by 25%.” They next consider situations with a specified amount and percent change, or with initial and final amounts, using double number line diagrams to find the unknown amount or percent change. Next, they use equations to represent such situations, using the distributive property to show that different expressions for the same amount are equivalent—for example, . So far, percent change in this section has focused on whole-number rates per 100, like 75%. The last lesson asks students to compute fractional percentages of given amounts.

In the third section of the unit, students begin by using their abilities to find percentages and percent rates to solve problems that involve sales tax, tip, discount, markup, markdown, and commission (MP2). The remaining lessons of the section continue the focus on situations that can be described in terms of percentages, but the situations involve error rather than change—describing an incorrect value as a percentage of the correct value rather than describing an initial amount as a percentage of a final amount (or vice versa).

The optional last section of the unit consists of a lesson in which students analyze news items that involve percent increase or decrease. In small groups, students identify important quantities in a situation described in a news item, use diagrams to map the relationship of the quantities, and reason mathematically to draw conclusions (MP4). This is an opportunity to choose an appropriate type of diagram (MP5), to state the meanings of symbols used in the diagram, to specify units of measurement, and to label the diagram accurately (MP6). Each group creates a display to communicate its reasoning and critiques the reasoning shown in displays from other groups (MP3).

These materials follow specific conventions for the use of language around ratios, rates, and proportional relationships. Please see the unit narrative for the previous unit to read about those conventions.

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 extend the operations of addition, subtraction, multiplication, and division from fractions to all rational numbers, written as decimals or in the form .

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., 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.

In the second section of the unit, students extend addition and subtraction from fractions to all rational numbers. They begin by considering how changes in temperature and elevation can be represented—first with tables and number line diagrams, then with addition and subtraction expressions and equations. Initially, physical contexts provide meanings for sums and differences that include negative numbers. Students work with numerical addition and subtraction expressions and equations, becoming more fluent in computing sums and differences of signed numbers. Using the meanings that they have developed for addition and subtraction of signed numbers, they write equivalent numerical addition and subtraction expressions, e.g., and ; and they write different equations that express the same relationship.

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 fourth section of the unit focuses on multiplication and division. It begins with problems about position, direction, constant speed, and constant velocity in which students represent quantities with number line diagrams and tables of numerical expressions with signed numbers. This allows products of signed numbers to be interpreted in terms of position and direction, using the understanding that numbers that are additive inverses are located at the same distance but opposite sides of the starting point. These examples illustrate how multiplication of how multiplication of fractions extends to rational numbers. In the third lesson, students use the relationship between multiplication and division to understand how division extends to rational numbers. In the process of solving problems set in contexts (MP4), they write and solve multiplication and division equations.

In the fifth section of the unit, students work with expressions that use the four operations on rational numbers. They extend their use of the “next to” notation to include negative numbers and products of numbers, e.g., writing and rather than and . They extend their use of the fraction bar to include variables as well as numbers, writing as well as . Students begin working with linear equations in one variable that have rational number coefficients. The focus of this section is representing situations with equations (MP4) and what it means for a number to be a solution for an equation, rather than methods for solving equations.

Note. In these materials, an expression is built from numbers, variables, operation symbols (, , , ), parentheses, and exponents. (Exponents—in particular, negative exponents—are not a focus of this unit.) An equation is a statement that two expressions are equal, thus always has an equal sign. Signed numbers include all rational numbers, written as decimals or in the form .

Unit 8: Data Sets and Distributions

This unit is meant only as an overview of some key statistical concepts. Students are introduced to visual representations of data and their distributions, ways to quantify measures of center and measures of variability, and sampling from a population when access to all of the relevant data is not possible. An optional section introducing the basics of probability is also included.

The first section of the unit moves student thinking from line plots, familiar from as early as grade 2, to dot plots as a way of visualizing data. Students describe the shape of the distributions they see paying close attention to a typical value and the spread of the data. Then, students are introduced to histograms as another way to visualize a distribution of data in which numerical data is grouped together with other values nearby.

The next section helps students quantify their descriptions of distributions by giving them ways to quantify measures of center (mean and median) to describe typical values as well as measures of variability (mean absolute deviation and interquartile range) to describe the spread of data. While students are shown how the values are calculated, the focus of work in this section should be on understanding the values in context. Students then work to decide which measure of center and associated measure of variability are most appropriate to use to describe typical values for a given distribution.

Then, students consider what they can do if access to all of the relevant data is not possible. Students examine methods of sampling to decide which methods are more likely to produce representative samples from the population and how using random processes can help. Next, students see that some variability is expected in measures of center from different samples within the same population, but that some inferences about the population may be possible.

The unit concludes with an optional section exploring probability. Students are introduced to probability as a way to quantify how likely an event is to happen, the connection between probability and results of repeated experiments, ways to examine the sample space for more complex experiments, and simulating experiments with easier to use experiments.

Unit 9: Putting It All Together

This optional unit consists of thirteen lessons. Each of the first five lessons is independent of the others, requiring only the mathematics of the previous units. The second and third sections each consist of four lessons that build on one another within the section.

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 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).

Lessons four and five are connected thematically through calculating and estimating length, area, and volume measurements introducing considerations of measurement error. The activities in these lessons can stand on their own, though.

The second section explores 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 examines population density using proportional relationships, equations, and graphs. The last 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.

The last section asks students to design a five-kilometer race course for their school using their knowledge of measurements and drawing maps. They select appropriate tools and methods for measuring their school campus, build a trundle wheel and use it to make measurements, make a scale drawing of the course on a map or a satellite image of the school grounds, and describe the number of laps, start, and finish of the race.