Discuss the important ways that listeners contribute to mathematical discussions during Ways of Thinking presentations. Then use ratio and rate reasoning to solve a problem about ingredients in a stew.Key ConceptsStudents find the unit rate of a ratio situation.Goals and Learning ObjectivesContribute as listeners during the Ways of Thinking discussion.Understand the concept of a unit rate that is associated with a ratio.Use rate reasoning to solve real-world problems.

Students use ratio cards to find equivalencies and form partnerships for the week. As a class, students discuss and decide on classroom norms.Give each student a ratio card. Instruct students to find a classmate whose card has a ratio that is equivalent to theirs. Classmates with equivalent ratios are now partners for the week. With the class, discuss and decide on classroom norms, or rules. Tell students how to access the application they will use this year.Key ConceptsStudents understand that ratio relationships are multiplicative. They use ratio tables to show ratio relationships.Goals and Learning ObjectivesDistinguish between ratio tables and tables that do no show equivalent ratios.Understand how ratio tables are used to solve ratio problems.Use the basic features of the application.Create and understand the classroom norms.Use mathematical reasoning to justify an answer.

Review the multiplication property of equality. Demonstrate the use of “ask myself” questions to understand a problem before solving it. Have students discuss the strategies that they can use when they feel stuck on a problem. Direct partners to solve a problem using a ratio table and equations, and then justify their solution in a presentation using the multiplication property of equality. Have each student write a Summary of the Mathematics in the lesson and work together to create a classroom summary.Key ConceptsStudents use the multiplication property of equality to justify their solution to a ratio problem.Goals and Learning ObjectivesBefore starting to work on a problem, make sense of the problem by using “ask myself” questions.Persevere in solving a problem even when feeling stuck.Solve a ratio problem using two different strategies.Link arithmetic and algebraic methods to solve a ratio problem.Use the multiplication property of equality to solve ratio problems

Students discuss classroom routines and expectations, work with partners to present their work matching different representations of a ratio situation, and then prepare math summaries.Introduce classroom routines and expectations prior to the full mathematics lesson. Ask students to discuss how to clearly present their work to their classmates. Model an example of partner work, and then have students work with their partners to match different representations of a ratio situation. Read and discuss a Summary of the Math, and then have students write Reflections about their wonderings.Key ConceptsStudents match a data card with its corresponding ratio, decimal, fraction, percent, and description of the relationship in words. Students construct viable arguments for their matches and critique the reasoning of their partner and other classmates.Goals and Learning ObjectivesDescribe the classroom routines and expectations.Consider how to present work clearly to classmates.Collaborate with a partner.Critique a partner’s reasoning.Connect different representations of a ratio situation.

Review the ways classroom habits and routines can strengthen students’ mathematical character. Explain what a Gallery is and how to choose a Gallery problem to solve. Direct students to choose one of three Gallery problems that introduce the unit’s technology resources. The three Gallery problems combine working with ratios and rates with the application resources available with this unit.Key ConceptsStudents understand that a Gallery gives them a choice of problems to solve. Students think about the features of the problems to use when choosing a problem. Students know how to work on a Gallery problem and present a solution.Goals and Learning ObjectivesKnow how to choose a problem from a Gallery.

Proportional Relationships

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Understand what a rate and ratio are.
Make a ratio table.
Make a graph using values from a ratio table.

Lesson Flow

Students start the unit by predicting what will happen in certain situations. They intuitively discover they can predict the situations that are proportional and might have a hard time predicting the ones that are not. In Lessons 2–4, students use the same three situations to explore proportional relationships. Two of the relationships are proportional and one is not. They look at these situations in tables, equations, and graphs. After Lesson 4, students realize a proportional relationship is represented on a graph as a straight line that passes through the origin. In Lesson 5, they look at straight lines that do not represent a proportional relationship. Lesson 6 focuses on the idea of how a proportion that they solved in sixth grade relates to a proportional relationship. They follow that by looking at rates expressed as fractions, finding the unit rate (the constant of proportionality), and then using the constant of proportionality to solve a problem. In Lesson 8, students fine-tune their definition of proportional relationship by looking at situations and determining if they represent proportional relationships and justifying their reasoning. They then apply what they have learned to a situation about flags and stars and extend that thinking to comparing two prices—examining the equations and the graphs. The Putting It Together lesson has them solve two problems and then critique other student work.

Gallery 1 provides students with additional proportional relationship problems.

The second part of the unit works with percents. First, percents are tied to proportional relationships, and then students examine percent situations as formulas, graphs, and tables. They then move to a new context—salary increase—and see the similarities with sales taxes. Next, students explore percent decrease, and then they analyze inaccurate statements involving percents, explaining why the statements are incorrect. Students end this sequence of lessons with a formative assessment that focuses on percent increase and percent decrease and ties it to decimals.

Students have ample opportunities to check, deepen, and apply their understanding of proportional relationships, including percents, with the selection of problems in Gallery 2.

Students analyze the graph of a proportional relationship in order to find the approximate constant of proportionality, to write the related formula, and to create a table of values that lie on the graph.Key ConceptsThe constant of proportionality determines the steepness of the straight-line graph that represents a proportional relationship. The steeper the line is, the greater the constant of proportionality.On the graph of a proportional relationship, the constant of proportionality is the constant ratio of y to x, or the slope of the line.A proportional relationship can be represented in different ways: a ratio table, a graph of a straight line through the origin, or an equation of the form y = kx, where k is the constant of proportionality.Goals and Learning ObjectivesIdentify the constant of proportionality from a graph that represents a proportional relationship.Write a formula for a graph that represents a proportional relationship.Make a table for a graph that represents a proportional relationship.Relate the constant of proportionality to the steepness of a graph that represents a proportional relationship (i.e., the steeper the line is, the greater the constant of proportionality).

Students connect percent to proportional relationships in the context of sales tax.Key ConceptsWhen there is a constant tax percent, the total cost for items purchase—including the price and the tax—is proportional to the price.To find the cost, c , multiply the price of the item, p, by (1 + t), where t is the tax percent, written as a decimal: c = p(1 + t).The constant of proportionality is (1 + t) because of the structure of the situation:c = p + tp = p(1 + t).Because of the distributive property, multiplying the price by (1 + t) means multiplying the price by 1, then multiplying the price by t, and then taking the sum of these products.Goals and Learning ObjectivesFind the total cost in a sales tax situation.Understand that a proportional relationship only exists between the price of an item and the total cost of the item if the sales tax is constant.Find the constant of proportionality in a sales tax situation.Make a graph of an equation showing the relationship between the price of an item and the total amount paid.

Students create equations, tables, and graphs to show the proportional relationships in sales tax situations.Key ConceptsThe quantities—price, tax, and total cost—can each be known or unknown in a given situation, but if you know two quantities, you can figure out the missing quantity using the structure of the relationship among them.If either the price or the total cost are unknown, you can write an equation of the form y = kx, with k as the known value (1 + tax), and solve for x or y.If the tax is the unknown value, you can write an equation of the form y = kx and solve for k, and then subtract 1 from this value to find the tax (as a decimal value).Building a general model for the relationship among all three quantities helps you sort out what you know and what you need to find out.Goals and Learning ObjectivesMake a table to organize known and unknown quantities in a sales tax problem.Write and solve an equation to find an unknown quantity in a sales tax problem.Make a graph to represent a table of values.Determine the unknown amount—either the price of an item, the amount of the sales tax, or the total cost—in a sales tax situation when given the other two amounts.

Lesson OverviewStudents calculate the constant of proportionality for a proportional relationship based on a table of values and use it to write a formula that represents the proportional relationship.Key ConceptsIf two quantities are proportional to one another, the relationship between them can be defined by a formula of the form y = kx, where k is the constant ratio of y-values to corresponding x-values. The same relationship can also be defined by the formula x=(1k)y , where 1k is now the constant ratio of x-values to y-values.Goals and Learning ObjectivesDefine the constant of proportionality.Calculate the constant of proportionality from a table of values.Write a formula using the constant of proportionality.

Students write the relationship between two fractions as a unit rate and use unit rates and the constant of proportionality to solve problems involving proportional relationships.Key ConceptsIn situations where there is a constant rate involved, the unit rate is a constant of proportionality between the two variable quantities and can be used to write a formula of the form y = kx.A given constant rate can be simplified to find the unit rate by expressing its value with a denominator of 1.The ratios of two fractions can be expressed as a unit rate.Goals and Learning ObjectivesExpress the ratios of two fractions as a unit rate.Understand that when a constant rate is involved, the unit rate is the constant of proportionality.Use the unit rate to write and solve a formula of the form y = kx.

Students look at the relationship between the number of flags manufactured and the stars on the flag and determine whether it represents a proportional relationship.Key ConceptsThe form of the equation of a proportional relation is y = kx, where k is the constant of proportionality.A graph of a proportional relationship is a straight line that passes through the origin.The constant of a proportionality in a graph of a proportional relationship is the constant ratio of y to x (the slope of the line).Goals and Learning ObjectivesIdentify the constant of proportionality in a proportional relationship based on a real-world problem situation.Write a formula using the constant of proportionality.Analyze a graph of a proportional relationship.Make a graph and determine if it represents a proportional relationship.Identify the constant of proportionality in a graph of a proportional relationship.

Allow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionSolving Percent ProblemsStudents understand the structure of percent problems by analyzing many problems.Running a Clothing StoreStudents help the owner of a clothing store determine how to get the greatest profit.Less FatStudents determine the percentage of fat in whole milk.10% MoreStudents evaluate three statements from Huey, Dewey, and Louie and determine which statement is correct.Free SpaceStudents determine which of two hard drives has the most free space.

Students explore the idea that not all straight lines are proportional by comparing a graph representing a stack of books with a graph representing a stack of cups. They recognize that all proportional relationships are represented as a straight line that passes through the origin.Key ConceptsNot all graphs of straight lines represent proportional relationships.There are three ways to tell whether a relationship between two varying quantities is proportional:The graph of the relationship between the quantities is a straight line that passes through the point (0, 0).You can express one quantity in terms of the other using a formula of the form y = kx.The ratios between the varying quantities are constant.Goals and Learning ObjectivesUnderstand when a graph of a straight line is and when it is not a proportional relationship.Recognize that a proportional relationship is shown on a graph as a straight line that passes through the origin (0, 0).Make a table of values to represent two quantities that vary.Graph a table of values representing two quantities that vary.Describe what each variable and number in a formula represents.

Students are given a collection of statements that are incorrect. Their task is to construct arguments about why the statements are flawed and then correct the flawed statements.Key ConceptsPercent change is a rate of change of an original amount.In two situations with the same percent change but different original amounts, the percent amount will be different because the percent amount depends directly on the original amount. For example: 50% of 20 is 10. 50% of 10 is 5.Similarly, in two situations with the same amount of increase but different original amounts, the percent change of each amount is different. For example: Suppose two amounts increase by $5. If one original amount is $20, the increase is 25%. If the other original amount is $25, the increase is 20%.Goals and Learning ObjectivesIdentify errors in reasoning in percent situations.Use examples to explain why the reasoning is incorrect.

Students determine whether a relationship between two quantities that vary is a proportional relationship in three different situations: the relationship between the dimensions of the actual Empire State Building and a miniature model of the building; the relationship between the distance and time to travel to an amusement park; and the relationship between time and temperature at an amusement park.Key ConceptsWhen the ratio between two varying quantities remains constant, the relationship between the two quantities is called a proportional relationship. For a ratio A:B, the proportional relationship can be described as the collection of ratios equivalent to A:B, or cA:cB, where c is positive.Goals and Learning ObjectivesIdentify proportional relationships.Explain why a situation represents a proportional relationship or why it does not.Determine missing values in a table of quantities based on a proportional relationship.

Students interpret verbal descriptions of situations and determine whether the situations represent proportional relationships.Key ConceptsIn a proportional relationship, there has to be some value that is constant.There are some relationships in some situations that can never be proportional.Goals and Learning ObjectivesIdentify verbal descriptions of situations as being proportional relationships or notUnderstand that some relationships can never be proportionalUnderstand that for two variable quantities to be proportional to one another, something in the situation has to be constant

Students represent and solve percent decrease problems.Key ConceptsWhen there is a percent decrease between a starting amount and a final amount, the relationship can be represented by an equation of the form y = kx where y is the final amount, x is the starting amount, and k is the constant of proportionality, which is equal to 1 minus the percent change, p, represented as a decimal: k = 1 – p, so y = (1 – p)x.The constant of proportionality k has the value it does—a number less than 1—because of the way the distributive property can be used to simplify the expression for the starting amount decreased by a percent of the starting amount: x – x(p) = x(1 – p).Goals and Learning ObjectivesDetermine the unknown amount—either the starting amount, the percent change, or the final amount—in a percent decrease situation when given the other two amounts.Make a table to represent a percent decrease problem.Write and solve an equation to represent a percent decrease problem.

Students represent and solve percent increase problems.Key ConceptsWhen there is a percent increase between a starting amount and a final amount, the relationship can be represented by an equation of the form y = kx where y is the final amount, x is the starting amount, and k is the constant of proportionality, which is equal to 1 plus the percent change, p, represented as a decimal: k = 1 + p, so y = (1 + p)x.The constant of proportionality k has the value it does—a number greater than 1—because of the way the distributive property can be used to simplify the expression for the starting amount increased by a percent of the starting amount: x + x(p) = x(1 + p).Goals and Learning ObjectivesDetermine the unknown amount—either the starting amount, the percent change, or the final amount—in a percent increase situation when given the other two amounts.Make a table to represent a percent increase problem.Write and solve an equation to represent a percent increase problem.

Students continue to explore the three relationships from the previous lessons: Comparing Dimensions, Driving to the Amusement Park, and Temperatures at the Amusement Park. They graph the three situations and realize that the two proportional relationships form a straight line, but the time and temperature relationship does not.Key ConceptsA table of values that represent equivalent ratios can be graphed in the coordinate plane. The graph represents a proportional relationship in the form of a straight line that passes through the origin (0, 0). The unit rate is the slope of the line.Goals and Learning ObjectivesRepresent relationships shown in a table of values as a graph.Recognize that a proportional relationship is shown on a graph as a straight line that passes through the origin (0, 0).