This unit introduces students to ratios, rates, and percentages. It builds on previous experiences students had with relating two quantities, such as converting measurements and finding equivalent fractions starting in grade 3, multiplicative comparison in grade 4, and interpreting multiplication as scaling in grade 5. The work prepares students to reason about proportional relationships, constants of proportionality, and percent increase and decrease later in this course.

The first half of the unit focuses on ratios and equivalent ratios. Students learn that a ratio is an association between two quantities, for instance, “There are 3 pencils for every 2 erasers.”  They use sentences, drawings, or discrete diagrams to represent ratios that describe collections of objects and recipes.

The second half of the unit focuses on rates and percentages. Students begin by recalling what they know about standard units of measurement—the attributes that they measure and their relative sizes. They use ratios and rates to reason about measurements and to convert between units of measurement.

Next, students learn about unit rates. They see that there are two unit rates—\(\frac{a}{b}\) and \(\frac{b}{a}\)—associated with any ratio \(a:b\) and interpret them in context. Students practice finding unit rates and using them to solve various problems.

Students then use their understanding of ratios and rates to make sense of percentages. Just as a unit rate can be interpreted in context as a rate per 1, a percentage can be interpreted in context as a rate per 100.  Students see that tables and double number line diagrams are also helpful for reasoning about percentages.

In these materials, a “quantity” is a measurement that can be specified by a number and a unit, for instance, 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. In this unit, the fractions \(\frac{a}{b}\) and \(\frac{b}{a}\) are never called “ratios,” and the meanings of these fractions in contexts are very carefully developed before they are identified as “unit rates” for the ratio \(a:b\). For example, the word “per” is used with students in interpreting a unit rate in context, as in “\$3 per ounce,” and the phrase “at the same rate” is used to signify a situation characterized by equivalent ratios. Later in the unit, students learn then 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” are not used in this unit. A “proportional relationship” is a collection of equivalent ratios, which will be studied in later units. In high school—after their study of ratios, rates, and proportional relationships—students can discard the term “unit rate” and refer to \(a\) to \(b\), \(a:b\), and \(\frac{a}{b}\) all as “ratios.”

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as interpreting, explaining, and representing. 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

• Different representations of ratios (Lessons 1 and 4).
• Situations involving equivalent ratios (Lessons 6 and 7).
• Tables of equivalent ratios (Lessons 9 and 10).
• Unit rates in different contexts (Lesson 18).
• A context in which identifying a unit rate is helpful (Lesson 20).
• Diagrams used to represent percentages (Lessons 21 and 22).
• Situations involving measurement, rates, and cost (Lesson 28).

Explain

• Reasoning about equivalence (Lesson 2).
• Reasoning about equivalent rates (Lesson 8).
• Reasoning with reference to tables (Lessons 9 and 10).
• Reasoning with reference to tape diagrams (Lessons 12 and 13).
• Reasoning for estimating and sorting measurements (Lesson 14).
• Reasoning about relative sizes of units of measurement (Lesson 15).
• Reasoning for comparing rates (Lesson 17).
• Reasoning about percentages (Lessons 21 and 22).

Represent

• Ratio associations (Lesson 1).
• Doubling and tripling of quantities in a ratio (Lesson 2).
• Equivalent ratios (Lessons 5 and 6).
• Ratios and total amounts (Lessons 12 and 13).
• Measurement unit conversions as equivalent ratios (Lesson 15).
• Percentages using diagrams (Lessons 21 and 22).

In addition, students are expected to justify whether ratios are or aren't equivalent and why information is needed to solve a ratio problem. Students also have opportunities to generalize about equivalent ratios, unit rates, and percentages from multiple contexts and with reference to benchmark percentages, tape diagrams, and other mathematical representations.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

This unit develops students’ understanding of division of fractions by fractions and operations on whole numbers and decimals. This work draws on students’ prior knowledge of multiplication, division, and the relationship between the two. It also builds on place value, properties of operations, and the relationship between addition and subtraction.

Students begin by exploring meanings of division and the relationship between the quantities in division situations. They recall that we can think of dividing as finding an unknown factor in a multiplication equation. In situations involving equal-size groups, division can be used to answer two questions: “How many groups?” and “How much in each group?” Students investigate ways to answer those two questions.

Students then apply their insights to generalize the process of finding quotients. They notice regularity: Dividing a number by a fraction \(\frac{a}{b}\) is the same as multiplying that number by \(\frac{b}{a}\). Students go on to use this algorithm to solve problems about geometric figures that have fractional length, area, or volume measurements and to compute unit rates of fractions.

Next, students revisit addition and subtraction of decimals, using both concrete representations and numerical calculations. They also investigate various ways to find the product of two decimals: using decimal fractions, using diagrams and partial products, and reasoning about the relationship between a decimal and a related whole number.

The next section focuses on division. Students have an opportunity to use base-ten blocks or diagrams to represent division of multi-digit numbers before exploring other numerical methods, such as using partial quotients and long division. Students progress through calculations of increasing complexity. They first divide whole numbers that give a whole-number quotient, and then divide whole numbers with a (terminating) decimal quotient. Next, they divide a decimal by a whole number, and finally a decimal by a decimal.

Mai’s diagram for \(62 \div 5\)

 

Lin’s calculation for \(62 \div 5\)

A deeper understanding of multiplication, division, and ways to represent them will support students in reasoning about writing and solving variable equations later in the course.

A note about diagrams:

Because tape diagrams are a flexible tool for illustrating and reasoning about division of fractions, they are the primary representation used in this unit. Students may, however, create other representations to support their reasoning.

A note about materials:

Base-ten blocks and paper versions of them will be useful throughout the unit. Consider preparing commercially produced base-ten blocks, if available, or printing representations of base-ten units on card stock, cutting them out, and organizing them for easy reuse.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as interpreting, representing, 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 and Represent

• Situations involving division (Lessons 2, 6, 9, and 12).
• Base-ten diagrams showing addition or subtraction of decimals (Lesson 14).
• Area diagrams showing products of decimals (Lesson 16).
• Calculations showing partial quotients or steps in long division (Lessons 18 and 19).
• Base-ten diagrams representing division of a whole number or a decimal by a whole number (Lessons 19).
• Situations involving measurement constraints (Lesson 22).

Explain

• How to create and make sense of division diagrams (Lesson 3).
• How to represent division situations (Lesson 6).
• How to find unknown lengths (Lesson 11).
• Processes of estimating and finding costs (Lesson 13).
• Approaches to adding and subtracting decimals (Lesson 15).
• Methods for multiplying decimals (Lesson 17).
• A plan for optimizing costs (Lesson 22).
• Reasoning about relationships among measurements (Lesson 23).

Compare

• Verbal and numerical division representations (Lessons 4 and 5).
• Representations of division (Lesson 10).
• Base-ten diagrams with numerical calculations (Lesson 15).
• Methods for multiplying decimals (Lesson 16).
• Methods for finding quotients (Lessons 18 and 19).
• Measurements of two- and three-dimensional objects (Lesson 23).

In addition, students are expected to critique the reasoning of others about division situations and representations, generalize about multiplication and division, and justify strategies for finding sums, differences, products, and quotients.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students apply their understanding of arithmetic to reason about algebraic expressions and equations.

In the first section, students work with equations of the form \(x+p=q\) and \(px=q\) where \(p\) and \(q\) are positive rational numbers. They use tape diagrams and hanger diagrams to reason about the meaning of equations, and to develop an understanding that to solve an equation is to find a value that would make the equation true. Students end the section by identifying, interpreting, and writing equations to represent and solve real-world problems.

In the second section, students write algebraic expressions and evaluate them for given values. They identify and write equivalent expressions, reasoning using diagrams, the distributive property, and other properties of operations.

The third section is all about exponents. First, students learn to use exponents ² and ³ to express areas of squares and volumes of cubes and their units. Next, they write expressions with a whole-number exponent and a base that may be a whole number, a fraction, or a variable. They analyze such expressions for equivalence, as well as use the conventional order of operations to evaluate them. Students also identify solutions to simple exponential equations.

In the last two sections, students analyze real-world relationships between two quantities where one quantity depends on the other. They use tables, graphs, and equations to represent and reason about such relationships.

The work here prepares students to work with proportional relationships in a later unit, as well as to solve equations that are more complex in grade 7.

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 5).
• Numerical expressions involving exponents (Lesson 12).
• Different representations of the same relationship between quantities (Lesson 17).

Describe

• 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 using tables, graphs, and equations (Lesson 16).

Explain

• The meaning of a solution using hanger diagrams (Lesson 3).
• How to solve an equation (Lesson 4).
• How to determine whether two expressions are equivalent, including with reference to diagrams (Lesson 7).
• Strategies for determining whether expressions are equivalent (Lesson 14).

In addition, students are expected to compare descriptions of situations, expressions, equations, diagrams, tables, and graphs. They generalize about properties of operations and strategies for solving equations. Students also justify claims about equivalent expressions and justify reasoning when evaluating expressions.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students develop the idea of a proportional relationship. They work with proportional relationships that are represented in tables, as equations, and on graphs. This builds on previous work with equivalent ratios and helps prepare students for the study of linear functions in later courses.

In a table of equivalent ratios, a multiplicative relationship between a pair of rows is given by a scale factor, while the multiplicative relationship between the columns is given by a unit rate. Students learn that the relationship between pairs of values in the two columns is called a "proportional relationship," and the unit rate that describes this relationship is called a "constant of proportionality." Students use equations of the form \(y = kx\) to represent proportional relationships and solve problems. They determine whether given tables and equations could represent a proportional relationship.

Then students investigate graphs of proportional relationships. They recognize that the graph of a proportional relationship is a straight line through \((0, 0)\). They interpret points on the graph, including the point \((1, k)\). Here is an example of a graph, an equation, and a table that all represent the same proportional relationship.

Three representations of the linear function y = 7 fourths times x. Horizontal axis scale is 0 to 8 by 1’s. Vertical axis scale is 0 to 10 by 1’s. The graph of the function has the points (1 comma the fraction 7 over 4) and (4 comma 7). The table has rows (x comma y), (0 comma 0), (1 comma the fraction 7 over 4), (3 comma the fraction 21 over 4), (4 comma 7).  

Next, students apply their knowledge of proportional relationships to the context of measuring circles. This builds on students’ work from previous grades with perimeter and area of polygons. Students will build on this work in later courses when they study the volume of spheres, cylinders, and cones.

The terms "center," "radius," "diameter," and "circumference" are introduced. Then students investigate the relationship between circumference and diameter and see that it is a proportional relationship. They apply this relationship to solve problems. Next, students explore the area of circular regions. They see an informal derivation that shows where the formula \(A = \pi r^2\) comes from and then use this formula to solve problems.

A picture of three different circular objects. The leftmost object is a wagon wheel with a measuring tool starting from one point on the wheel, goes through the wheel center to a point on the other side of the wheel. The center object is a plane propellor with three identical propellor blades. A measuring tool starts from the center of the propellor and goes to the end of the blade. The third object is of a sliced orange. A measuring tool goes around the entire circular region of the orange.

A note on using the terms "ratio," "proportional relationship," and "unit rate":

In these materials, the term "ratio" is used to mean a type of association between two or more quantities. A quantity is a measurement that can be specified by a number and a unit, for example 4 oranges, 4 centimeters, or “my height in feet.” 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 \(\frac{a}{b}\) and \(\frac{b}{a}\) are never called ratios. The fractions \(\frac{a}{b}\) and \(\frac{b}{a}\) are identified as “unit rates” for the ratio \(a : b\). In high school—after the study of ratios, rates, and proportional relationships—students discard the term “unit rate” and start referring to \(a\) to \(b\), \(a:b\), and \(\frac{a}{b}\) as “ratios.”

In grades 6–8, 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 \(\frac{\text{mi}}{\text{hr}}\) waits for high school—except for the special cases of area and volume.

A note on using the term "circle":

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

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as comparing, 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:

Compare

• Approaches to solving problems involving proportional relationships (Lesson 3).
• Proportional relationships with nonproportional relationships (Lesson 5).
• Tables, descriptions, and graphs representing the same situations (Lesson 7).
• Graphs of proportional relationships (Lesson 8).
• The relationships of square diagonals and perimeters to square diagonals and areas (Lesson 10).
• The relationships of diameters and circumferences to diameters and areas (Lesson 15).

Justify

• Reasoning about circumference and perimeter (Lesson 13).
• Estimates for the areas of circles (Lesson 15).
• Reasoning about areas of curved figures (Lesson 16).
• Whether or not a relationship is proportional (Lesson 17).
• Reasoning about the cost of stained-glass windows (Lesson 20).

Generalize

• About proportional relationships (Lesson 1).
• About equations that represent proportional relationships (Lesson 2).
• About categories for sorting circles (Lesson 11).
• About the relationships between circumference and diameter (Lesson 12).

In addition, students are expected to explain how to determine whether or not a relationship is proportional, how to use different approximations of \(\pi\), how to find the area of composite shapes, and how to compare and represent situations with different constants of proportionality. Students are also asked to interpret situations involving proportional relationships, floor plans and maps, situations involving circles, and situations involving circumference and area.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students deepen their understanding of proportional relationships and percentages. They solve multi-step problems and work with situations that involve fractional amounts. This builds on the work students did in previous units with ratios, rates, percentages, and proportional relationships. Students will build on this work in high school with exponential functions representing compounded percent increase and decrease.

Students begin the unit by revisiting proportional relationships, but this time the given values are fractional amounts. To determine the constant of proportionality, students must compute the quotient of two fractions. Students also make sense of situations where an increase or decrease is expressed as a fraction of the initial amount. They create diagrams and apply the distributive property to generate expressions that represent these situations. They also use long division to write fractions as decimals, including their first introduction to repeating decimals.

Next, students make sense of situations where an increase or decrease is expressed as a percentage of the initial amount. They continue creating diagrams and writing equations to represent the situations. They solve for any one of the three quantities—the initial amount, the final amount, or the percentage of the change—given the other two quantities. They also reason about fractional percentages.

Then students apply percent increase and decrease to solve problems in a variety of real-world situations, such as tax, tip, interest, markup, discount, depreciation, and commission. Lastly, students make sense of situations where the difference between a correct measurement and an incorrect measurement is expressed as a percentage of the correct amount.

\(y = x + \frac14 x\)

\(y = (1 + \frac14)x\)

\(y = \frac54 x\)

\(y = (1 + 0.25)x\)

\(y = 1.25x\)

“a 25% increase”

​​​​​

\(y = x - \frac14 x\)

\(y = (1 - \frac14)x\)

\(y = \frac34 x\)

\(y = (1 - 0.25)x\)

\(y = 0.75x\)

“a 25% decrease”

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as interpreting, explaining, and representing. 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

• Concrete problems involving percent increase and decrease (Lesson 5).
• Problems involving sales tax and tip (Lesson 8).
• Concrete situations involving percent error (Lesson 11).

Explain

• How to solve concrete and abstract problems involving an amount plus (or minus) a fraction of that amount (Lesson 2).
• How to solve percent change problems (Lesson 4).
• Strategies for solving percent problems with fractional percentages (Lesson 7).
• How to measure lengths and interpret measurement error (Lesson 11).
• Strategies for solving percent error problems (Lesson 11).

Represent

• Situations involving percent increase and decrease (Lessons 6 and 12).
• Situations from the news involving percent change (Lesson 13).

In addition, students are expected to compare decimal and fraction representations, compare representations of an increase (or decrease) of an amount by a fraction or decimal, generalize about using constants of proportionality to solve problems efficiently and about relationships with percent increase and decrease, and justify why specific information is needed to solve percent change problems.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students learn about negative numbers and ways to represent them on a number line and the coordinate plane. They perform operations on rational numbers, which are all numbers that can be written as a positive or negative fraction or zero.

Students begin by considering situations involving temperature or elevation and interpreting what negative numbers mean in those contexts. Previously, when students worked only with nonnegative numbers, magnitude and order were indistinguishable. In this unit, when comparing two signed numbers, students learn to distinguish between the absolute value of a number (magnitude) and a number’s relative position on the number line (order).

• A represents 15 feet.
• B represents \(|15|\) feet.
• ​​​​​​C represents ​\(|\text-4|\) feet.
• D represents -4 feet.

Then students use tables and number line diagrams to represent changes in temperature or elevation. They extend addition and subtraction from fractions to all rational numbers. And they see that \(a - b\) is equivalent to \(a + (\text-b)\).

Then students use ordered pairs to describe pairs of numbers that include negative numbers. In grade 5, they plotted pairs of positive numbers on the coordinate grid. Here, they plot pairs of rational numbers in all four quadrants of the coordinate plane. They interpret the meanings of plotted points in given contexts and use coordinates to calculate horizontal or vertical distances between two points.

Next, students examine multiplication and division. They work with constant velocity, which is a signed number that indicates direction and speed. This allows products of signed numbers to be interpreted in terms of position, direction of movement, and time before or after a specific point. Students use the relationship between multiplication and division to understand how division extends to rational numbers.

Then students work with expressions that use the four operations on rational numbers. They also solve problems that involve interpreting negative numbers in context. They solve linear equations of the form \(p+x=q\) or \(px=q\), where \(p\) and \(q\) are rational numbers. 

A note on using the terms "expression," "equation," and "signed number":

In these materials, an expression is built from numbers, variables, operation symbols (\(+\), \(-\), \(\cdot\), \(\div\)), parentheses, and exponents. (Exponents—in particular, negative exponents—are not a focus of this unit. Students work with integer exponents in a future course and noninteger exponents in high school.) An equation is a statement that two expressions are equal, thus it always has an equal sign. Signed numbers include all rational numbers, written as decimals or in the form \(\frac a b\).

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as interpreting, representing, 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:

Interpret

• Situations involving negative numbers (Lessons 1 and 5).
• Graphs involving positive and negative numbers  (Lesson 12).
• Tables and situations involving signed numbers (throughout unit).

Represent

• Addition of signed numbers on a number line (Lesson 6).
• Situations involving signed numbers (Lessons 7, 10, and 16).
• Changes in elevation (Lesson 10).
• Position, speed, and direction (Lesson 14).

Generalize

• About subtracting and adding signed numbers (Lesson 9).
• About differences and magnitude (Lesson 10).
• About multiplying negative numbers (Lessons 14 and 15).
• About additive and multiplicative inverses (Lesson 20).

In addition, students are expected 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, Students are also expected to explain how to order rational numbers, how to determine distances on the coordinate plane, how to determine changes in temperature, how to find information using inverses, and how to model situations involving signed numbers.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

This unit is a brief overview of some key statistical concepts. First, students learn about populations and study variables associated with a population. They begin by classifying questions as either statistical or non-statistical—based on whether variable data is necessary to answer the question. This leads to further investigation into variability and data displays, such as dot plots and histograms. As students visualize data, they begin to describe the distribution of data more precisely as they work with mean and mean absolute deviation (MAD).

After working with those statistics, students begin to recognize that some distributions are not well-suited to description by mean and MAD. Students are introduced to median, range, and interquartile range as additional measures of center and variability that can be used to describe distributions in some situations. That also leads to the box plot as an additional way to visualize data.

Box plots and dot plots for two sets of data: "pug weights in kilograms” and "beagle weights in kilograms". The numbers 6 through 11 are indicated and there are tick marks midway between each indicated number. Each box plot is above it's corresponding dot plot. The approximate data for the box plot for "pug weights in kilograms" are as follows: Minimum value, 6. Maximum value, 8. Q1, 6.5. Q2, 7. Q3, 7.3. The approximate data for the dot plot for "pug weights in kilograms" are as follows: 6 kilograms, 1 dot. 6.2 kilograms, 2 dots. 6.4 kilograms, 2 dots. 6.6 kilograms, 2 dots. 6.8 kilograms, 2 dots. 7 kilograms, 3 dots. 7.2 kilograms, 3 dots. 7.4 kilograms, 1 dot. 7.6 kilograms, 2 dots. 7.8 kilograms, 1 dot. 8 kilograms, 1 dot. The approximate data for the box plot for "beagle weights in kilograms" are as follows: Minimum value, 9. Maximum value, 11. Q1, 9.6. Q2, 10. Q3, 10.5. The approximate data for the dot plot for "beagle weights in kilograms" are as follows: 9 kilograms, 1 x. 9.2 kilograms, 2 x's. 9.4 kilograms, 1 x. 9.6 kilograms, 3 x's. 9.8 kilograms, 1 x. 10 kilograms, 3 x's. 10.2 kilograms, 3 x's. 10.4 kilograms, 1 x. 10.6 kilograms, 2 x's. 10.8 kilograms, 2 x's. 11 kilograms, 1 x.  

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. They explore the connection between probability and results of repeated experiments, ways to examine the sample space for more complex experiments, and simulating experiments. 

Note that the introduction of mean absolute deviation is used as an introductory model for understanding variability. Although standard deviation is more mathematically useful, its calculation and meaning may be difficult for students at this level without an understanding of normal distributions. In later courses, when student understanding of variability and their exposure to additional distributions is expanded, students will learn about standard deviation and evolve their understanding away from mean absolute deviation.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as comparing, interpreting, 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:

Compare

• Questions that produce numerical and categorical data (Lesson 1).
• Dot plots and histograms (Lesson 3).
• Features and distributions of data sets (Lessons 4 and 5).
• Measures of center with samples (Lesson 9).
• Sampling methods (Lesson 10).
• Methods for writing sample spaces (Lesson 15).

Interpret

• Dot plots (Lessons 2 and 5).
• Histograms (Lesson 3).
• Mean of a data set (Lesson 4).
• Five-number summaries and box plots (Lesson 7).
• Situations involving populations and samples (Lesson 8).
• Situations involving sample spaces and probability (Lesson 16).

Justify

• Reasoning for matching data sets to questions (Lesson 1).
• Reasoning about mean and median (Lesson 6).
• Which samples are or are not representative of a larger population (Lesson 9).
• Which samples correspond with different populations (Lesson 11).
• Whether situations are surprising and possible (Lesson 14).

In addition, students are expected to represent data using dot plots, histograms, five-number summaries, and box plots, and to represent probabilities using sample spaces. Students also have opportunities to use language to describe features of a data set, describe patterns observed in repeated experiments, and explain how to use a simulation to answer questions about the situation.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this optional unit, students use concepts and skills from previous units to solve problems. The first section explores a variety of different contexts, such as population density, Fermi problems, measurement error, and energy usage. The second section has five lessons about different systems of voting. In the last section, students build a trundle wheel and design a five-kilometer race course.

All related standards in this unit have been addressed in prior units. These sections provide an optional opportunity for students to go more deeply and make connections between domains.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as critiquing, comparing, 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:

Critique

• Reasoning about Fermi problems (Lesson 1).
• Peer reasoning about percent error in length, area, and volume measurement (Lesson 7).
• Claims about percentages (Lesson 10).
• Reasoning about the fairness of voting systems (Lesson 13).
• Peer methods of measuring distance (Lesson 15).

Compare

• Sources of energy (Lessons 2 and 3).
• Rectangles and fractions (Lesson 5).
• Regions with different population densities (Lesson 8).
• Voting systems (Lesson 11).
• Advantages and disadvantages of different methods (Lesson 16).

Justify

• Reasoning about Fermi problems (Lesson 1).
• Reasoning about the fairness of voting systems (Lessons 11 and 13).

In addition, students are also expected to describe distributions of voters and methods for measuring distance, including how to build and use a trundle wheel. Students also have opportunities to interpret and represent characteristics of the world population and generalize about decomposition of area and numbers

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.