Bianca visits a bike shop and learns how bicycle gears work in this Cyberchase video segment.

The CyberSquad tests which broom can travel the furthest in five seconds in this video from Cyberchase.

This lesson unit is intended to help sixth grade teachers assess how well students are able to: Analyze a realistic situation mathematically; construct sight lines to decide which areas of a room are visible or hidden from a camera; find and compare areas of triangles and quadrilaterals; and calculate and compare percentages and/or fractions of areas.

During this two-day lesson, students work with a partner to create and implement a problem-solving plan based on the mathematical concepts of rates, ratios, and proportionality. Students analyze the relationship between different-sized gummy bears to solve problems involving size and price.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Helping students develop and refine these problem solving skills:Creating a problem solving plan and implementing their plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in a real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Use ratios.Write and solve proportions.Create rate tables to organize data and make predictionsUse multiple representations—including tables, graphs, and equations—to organize and communicate data.Articulate strategies, thought processes, and approaches to solving a problem and defend why the solution is reasonable.

In this lesson designed to enhance literacy skills, students examine how to use fractions to measure and help conserve freshwater resources.

This problem, the third in a series of tasks set in the context of a class election, is more than just a problem of computing the number of votes each person receives. In fact, that isnŐt enough information to solve the problem. One must know how many votes it takes to make one half of the total number of votes. Although the numbers are easy to work with, there are enough steps and enough things to keep track of to lift the problem above routine.

Students focus on interpreting, creating, and using ratio tables to solve problems. They also relate ratio tables to graphs as two ways of representing a relationship between quantities.Key ConceptsRatio tables and graphs are two ways of representing relationships between variable quantities. The values shown in a ratio table give possible pairs of values for the quantities represented and define ordered pairs of coordinates of points on the graph representing the relationship. The additive and multiplicative structure of each representation can be connected, as shown: Goals and Learning ObjectivesComplete ratio tables.Use ratio tables to compare ratios and solve problems.Plot values from a ratio table on a graph.Understand the connection between the structure of ratio tables and graphs.

Students use double number lines to model relationships and to solve ratio problems.Key ConceptsDouble number line diagrams are useful for visualizing ratio relationships between two quantities. They are best used when the quantities have different units. (The unit rate appears paired with 1.) Double number line diagrams help students more easily “see” that there are many equivalent forms of the same ratio.Goals and Learning ObjectivesUnderstand double number line diagrams as a way to visually compare two quantities.Use double number line diagrams to solve ratio problems.

Rate

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Understand quantity as a number used with a unit of measurement.
Solve problems involving quantities such as distances, intervals of time, liquid volumes, masses of objects, and money, and with the units of measurement for these quantities.
Understand that a ratio is a comparison of two quantities.
Write ratios for problem situations.
Make and interpret tables, graphs, and diagrams.
Write and solve equations to represent problem situations.

Lesson Flow

In this unit, students will explore the concept of rate in a variety of contexts: beats per minute, unit prices, fuel efficiency of a car, population density, speed, and conversion factors. Students will write and refine their own definition for rate and then use it to recognize rates in different situations. Students will learn that every rate is paired with an inverse rate that is a measure of the same relationship. Students will figure out the logic of how units are used with rates. Then students will represent quantitative relationships involving rates, using tables, graphs, double number lines, and formulas, and they will see how to create one such representation when given another.

During this two-day lesson, students work with a partner to create and implement a problem-solving plan based on the mathematical concepts of rates, ratios, and proportionality. Students analyze the relationship between different-sized gummy bears to solve problems involving size and price.Key ConceptsThroughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Help students develop and refine these problem-solving skills:Creating a problem solving plan and implementing their plan systematicallyPersevering through challenging problems to find solutionsRecalling prior knowledge and applying that knowledge to new situationsMaking connections between previous learning and real-world problemsCommunicating their approaches with precision and articulating why their strategies and solutions are reasonableCreating efficacy and confidence in solving challenging problems in a real worldGoals and Learning ObjectivesCreate and implement a problem-solving plan.Organize and interpret data presented in a problem situation.Analyze the relationship between two variables.Use ratios.Write and solve proportions.Create rate tables to organize data and make predictions.Use multiple representations—including tables, graphs, and equations—to organize and communicate data.Articulate strategies, thought processes, and approaches to solving a problem, and defend why the solution is reasonable.

In this interactive activity adapted from Annenberg Learner's Teaching Math Grades 6–8, explore some of the ways graphs can represent mathematical data contained in a story.

Learn about the dynamic relationships between a jet engine's heat loss, surface area, and volume in this video adapted from Annenberg Learner's Learning Math: Patterns, Functions, and Algebra.

This lesson introduces the concept of a glide ratio and encourages students to use appropriate tools strategically (Mathematical Practice 5). Students use tape diagrams, double number lines, ratio tables, graphs, and equations to represent glide ratios.Key ConceptsA glide ratio for an object or an organism in flight is the ratio of forward distance to vertical distance (in the absence of power and wind). For a given object or organism that glides, this ratio has a constant value and is treated as a feature of the object or organism.Goals and Learning ObjectivesUnderstand the concept of a glide ratio.Make connections within and between different ways of representing ratios.

This is the fourth in a series of tasks about ratios set in the context of a classroom election. What makes this problem interesting is that the number of voters is not given. This information isnŐt necessary, but at first glance some students may believe it is.

Students watch a video in which a double number line is used to solve a problem about getting the right amount of protein mix. Using the double number line is an example of modeling with mathematics, which is Mathematical Practice 4.Key ConceptsA double number line shows corresponding values for two variable quantities with a constant ratio between them. Each pair of tick marks that go together shows a ratio equivalent to all of the other ratios between corresponding tick marks.Goals and Learning ObjectivesWatch an example of students using mathematics to model a relationship between quantities (MP4).Use a double number line to solve a problem.Use a double number line to deepen understanding of equivalence in the context of a relationship between quantities with a constant ratio.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features and/or concepts and will help them to pay close attention to salient information.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is the first and most basic problem in a series of seven problems, all set in the context of a classroom election. Every problem requires students to understand what ratios are and apply them in a context. The problems build in complexity and can be used to highlight the multiple ways that one can reason about a context involving ratios.

Students focus on interpreting, creating, and using ratio tables to solve problems.Key ConceptsA ratio table shows pairs of corresponding values, with an equivalent ratio between each pair. Ratio tables have both an additive and a multiplicative structure:Goals and Learning ObjectivesComplete ratio tables.Use ratio tables to solve problems.

In this lesson, they work with equivalent ratios more abstractly, both in the context of recipes and in the context of abstract ratios of numbers. They understand and articulate that all ratios that are equivalent to a:b can be generated by multiplying both aand b by the same number (MP6).By connecting concrete quantitative experiences to abstract representations that are independent of a context, students develop their skills in reasoning abstractly and quantitatively (MP2). They continue to use diagrams, words, or a combination of both for their explanations. The goal in subsequent lessons is to develop a general definition of equivalent ratios.