This task was developed by high school and postsecondary mathematics and design/pre-construction educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

This task was developed by high school and postsecondary mathematics and health sciences educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning - and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses.

Unit 2: Introducing Ratios
Lesson 4: Color Mixtures

This is the second of two lessons that help students make sense of equivalent ratios through physical experiences. In this lesson, students mix different numbers of batches of a recipe for green water by combining blue and yellow water (created ahead of time with food coloring) to see if they produce the same shade of green. They also change the ratio of blue and yellow water to see if it changes the result. The activities here reinforce the idea that scaling a recipe up (or down) requires scaling the amount of each ingredient by the same factor (MP7). Students continue to use discrete diagrams as a tool to represent a situation.

For students who do not see color, the lesson can be adapted by having students make batches of dough with flour and water. 1 cup of flour to 5 tablespoons of water makes a very stiff dough, and 1 cup of flour to 6 tablespoons of water makes a soft (but not sticky) dough. In this case, doubling a recipe yields dough with the same tactile properties, just as doubling a colored-water recipe yields a mixture with the same color. The invariant property is stiffness rather than color. The principle that equivalent ratios yield products that are identical in some important way applies to both types of experiments.

Unit 2: Introducing Ratios
Lesson 10: Comparing Situations by Examining Ratios

In previous lessons, students learned that if two situations involve equivalent ratios, we can say that the situations are described by the same rate. In this lesson, students compare ratios to see if two situations in familiar contexts involve the same rate. The contexts and questions are:

Two people run different distances in the same amount of time. Do they run at the same speed?
Two people pay different amounts for different numbers of concert tickets. Do they pay the same cost per ticket?
Two recipes for a drink are given. Do they taste the same?
In each case, the numbers are purposely chosen so that reasoning directly with equivalent ratios is a more appealing method than calculating how-many-per-one and then scaling. The reason for this is to reinforce the concept that equivalent ratios describe the same rate, before formally introducing the notion of unit rate and methods for calculating it. However, students can use any method. Regardless of their chosen approach, students need to be able to explain their reasoning (MP3) in the context of the problem.

Unit 3: Unit Rates and Percentages
Lesson 5: Comparing Speeds and Prices

Previously, students found and used rates per 1 to solve problems in a context. This lesson is still about contexts, but it's more deliberately working toward the general understanding that when two ratios are associated with the same rate per 1, then they are equivalent ratios. Therefore, to determine whether two ratios are equivalent, it is useful to find and compare their associated rates per 1. In this lesson, we also want students to start to notice that dividing one of the quantities in a ratio by the other is an efficient way to find a rate per 1, while attending to the meaning of that number in the context (MP2).

Calculating rates per 1 is also a common way to compare rates in different situations. For example, suppose we find that one car is traveling 30 miles per hour and another car is traveling 40 miles per hour. The different rates tell us not only that the cars are traveling at different speeds, but which one is traveling faster. Similarly, knowing that one grocery store charges $1.50 per item while another charges $1.25 for the same item allows us to select the better deal even when the stores express the costs with rates such as “2 for $3” or “4 for $5.”

Unit 2: Introducing Ratios
Lesson 9: Constant Speed

In the previous lesson, students used the context of shopping to explore how equivalent ratios and ratios involving one can be used to find unknown amounts. In this lesson, they revisit these ideas in a new context—constant speed—and through concrete experiences. Students measure the time it takes them to travel a predetermined distance—first by moving slowly, then quickly—and use it to calculate and compare the speed they traveled in meters per second.

Here, double number lines are used to represent the association between distance and time, and to convey the idea of constant speed as a set of equivalent ratios (e.g., 10 meters traveled in 20 seconds at a constant speed means that 0.5 meters is traveled in 1 second, and 5 meters is traveled in 10 seconds). Students come to understand that, like price, speed can be described using the terms per and at this rate.

The idea of a constant speed relating the quantities of distance and time is foundational for the later, more abstract idea of a constant rate, and is important in the development of students’ ability to reason abstractly about quantities (MP2).

Unit 3: Unit Rates and Percentages
Lesson 4: Converting Units

In grade 4, students began converting units of measurements by multiplying. The work in grade 5 expanded to include conversion by dividing, but was still restricted to units within the same measurement system. In this lesson, students progress to converting units that may be in different systems of measurement, using ratio reasoning and recently-learned strategies such as double number lines, tables, and multiplication or division of unit rates.

Unit 2: Introducing Ratios
Lesson 7: Creating Double Number Line Diagrams

In this lesson, students create double number line diagrams from scratch. They see that it is important to use parallel lines, equally-spaced tick marks, and descriptive labels. They are also introduced to using the word "per" to refer to how much of one quantity there is for every one unit of the other quantity.

Double number lines are included in the first few activity statements to help students find an equivalent ratio involving one item or one unit. In later activities and lessons, students make their own strategic choice of an appropriate representation to support their reasoning (MP5). Regardless of method, students indicate the units that go with the numbers in a ratio, in both verbal statements and diagrams.

Note that students are not expected to use or understand the term "unit rate" in this lesson.

Our team of experts is excited to share with you their favorite tips and tricks about how to access and use Census Bureau Data.

So we created the Data Gems: a series of "how-to" videos available for data users who are looking for an easy and quick way to enhance their knowledge of Census data.

They will introduce you to various concepts and techniques to improve your ability to navigate our website and use our data-access tools.

We hope you find these Gems valuable! Drop us a line at census.academy@census.gov and let us know what you think!

Unit 2: Introducing Ratios
Lesson 5: Defining Equivalent Ratios

Previously, students understood equivalent ratios through physical perception of different batches of recipes. 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 a and 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.

In this activity , Harry gives an overview of working with fractions, decimals and percents, focusing on how to convert between them. Also included: the mini-game Stop the Pop, a comic video and quiz-show questions.

In this activity, Harry explains key concepts about ratio and proportion. Also included: the mini-game Sleuths on the Loose, a comic video about proportions in "body math" and quiz-show questions.

Unit 3: Unit Rates and Percentages
Lesson 7: Equivalent Ratios Have the Same Unit Rates

The purpose of this lesson is to make it explicit to students that equivalent ratios have the same unit rates. For instance, students can see that the ratios 10:4 15:6, and 20:8 all have unit rates of 2/5 and 5/2. Interpreted in a context, this might mean, for example, that no matter how many ounces of raisins are purchased in bulk and how much is paid, the price per ounce will always match the $0.40 per ounce rate marked on the price label.

This understanding gives new insights as students reason with tables. Up to this point, students have often been reasoning about the relationship from row to row, understanding that the rows contain equivalent ratios and the values in any row can be found by multiplying both quantities in another row by a scale factor. Here students see that they can also reason across columns, because the unit rate is the factor that relates the values in one column to those in the other (MP8). In grade 7, students will call the unit rate the constant of proportionality and write equations of the form y=kx to characterize these relationships.

Later in the lesson, students practice using unit rates and tables of equivalent ratios to find unknown quantities and compare rates in context.