SEVENTH GRADE > 7.RP.1 > TEACHER GUIDE

## TEACHER GUIDE TO CLARIFICATION

###### 7.RP.1

Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½ / ¼ miles per hour, equivalently 2 miles per hour.

###### Compute Unit Rates Associated with Ratios of Fractions

At this level students are now ready to compute unit rates with rational numbers. Students will need ample time to practice and reason through different ways to approach solving ratios with fractions. Check out the HOT questions for examples of reasoning to find unit rates.

Subtle difference between rate, ratio and proportion

Ratios arise in situations with two or more quantities. Some authors distinguish ratios from rates, using the term “ratio” when units are the same and “rate” when units are different; others use ratio to encompass both kinds of situations. The Standards use ratios in the second sense, applying it to situations in which units are the same as well as to situations when units are different.

A quotient (or fraction) is sometimes called the value of the ratio. In everyday language, the word “ratio” sometimes refers to the value of a ratio.

Ratios have associated rates.

The unit rate is the numerical part of the rate.

Ratio notation should be distinct from fraction notation.

Proportional Relationships involve collections of pairs of measurements in equivalent ratios.

Definitions

Ratio: A pair of nonnegative numbers, A:B, where both are not zero, and describes a relationship between the quantities.

Rate: Indicates how many units of one quantity there are for every 1 unit of the second quantity.

Unit Rate: The numeric value of the rate, e.g. in the rate 2.5 mph, the unit rate is 2.5.

Rate unit: The unit of measure of the rate. (e.g. in the rate 2.5 mph, the rate unit is miles per hour)

Equivalent Ratios: Ratios that have the same value. Percent: Percent of a quantity is a rate per 100.

Associated Ratios: Ratios that are related, e.g. if the ratio of the number of boys to the number of girls is 1:2, we can also determine the ratio of the number of girls to the total number of children is 2:3. We can further determine the ratio of the number of girls to the number of boys is 2:1.

Ratio Table: A table listing pairs of numbers that represent equivalent ratios.

Instructional Strategies:

Building from the development of rate and unit concepts in Grade 6, applications now need to focus on solving unit-rate problems with more sophisticated numbers: fractions per fractions.

Proportional relationships are further developed through the analysis of graphs, tables, equations and diagrams. Ratio tables serve a valuable purpose in the solution of proportional problems. This is the time to push for a deep understanding of what a representation of a proportional relationship looks like and what the characteristics are: a straight line through the origin on a graph, a “rule” that applies for all ordered pairs, an equivalent ratio or an expression that describes the situation, etc.

This is not the time for students to learn to cross multiply to solve problems.

This is NOT TIME FOR STUDENTS TO LEARN TO CROSS MULTIPLY TO SOLVE PROBLEMS.

This is an opportunity for students to reason about proportional raltionships with rational numbers.

Although algorithms provide efficient means for finding solutions, the cross-product algorithm commonly used for solving proportions will not aid in the development of proportional reasoning. Delaying the introduction of rules and algorithms will encourage thinking about multiplicative situations instead of indiscriminately applying rules.

Because percents have been introduced as rates in Grade 6, the work with percents should continue to follow the thinking involved with rates and proportions. Solutions to problems can be found by using the same strategies for solving rates, such as looking for equivalent ratios or based upon understandings of decimals. Previously, percents have focused on “out of 100”; now percents above 100 are encountered.

Providing opportunities to solve problems based within contexts that are relevant to seventh graders will connect meaning to rates, ratios and proportions.

Examples include: researching newspaper ads and constructing their own question(s), keeping a log of prices (particularly sales) and determining savings by purchasing items on sale, timing students as they walk a lap on the track and figuring their rates, creating open-ended problem scenarios with and without numbers to give students the opportunity to demonstrate conceptual understanding, inviting students to create a similar problem to a given problem and explain their reasoning.

Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: melisa@ksu.edu.

Retrieved from: Math Flipbooks.

Multiple strategies and visual tools can be used for students to compute and reason about ratios with rational numbers. Here are a few examples:

* Ratio Table

* Double Number Line Diagram

* Graph

* Tape Diagram

Tara is traveling 120 miles. Tara will use 2/3 of a gallon of gas and it will take her 1/12 of a day. How many gallons of gas does she use if she traveled a whole day?

Gallons of gas per day = 8 gallons of gas per day

A Ratio Table

A Double Number Line Diagram

A Tape Diagram

Think about this multiplicatively. 2/3 twelve times

If each unit represents 2/3 I will split them all in half to have one-thirds and then go through and shade groups of 3 to find out how many gallons I used. 8 groups of 3 so 8 gallons

Another Tape Diagram Example

Tape Diagram create equivalent units

A cookie recipe calls for ¾ cups of flour for every ½ stick of butter

How much butter is needed for every 1 cup of flour? For every 1 cup of flour 4/6 or 2/3 of a stick of butter is used

How much flour is needed for every stick of butter? For every stick of butter 1 ½ cups of flour is used.

Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee. (Summer 2012).

Retrieved December 4, 2014, from http://www4.uwm.edu/cclm/PDFs/7RP1-CCLM-chiarelli.pdf

In Grade 7, students extend their reasoning about ratios and proportional relationships in several ways. Students use ratios in cases that involve pairs of rational number entries, and they compute associated unit rates. They identify these unit rates in representations of proportional relationships. They work with equations in two variables to represent and analyze proportional relationships. They also solve multi-step ratio and percent problems, such as problems involving percent increase and decrease.

At this grade, students will also work with ratios specified by rational numbers, such as 3/4 cups flour for every 1/2 stick butter. 7.RP.1

Students continue to use ratio tables, extending this use to finding unit rates.

Common Core Standards Writing Team. (2011, December 26).

Progressions for the Common Core State Standards in Mathematics(draft). 6-7, Ratios and Proportional Relationships. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.

These progression documents explain the ways in which students will be applying and extending their reasoning about ratios and proportions. This standard takes it deeper than simply solving for a unit rate. This material will need to be supplemented by the teacher as many current textbooks do not ask questions that involve such proportional reasoning. Also, continue to expose students to percent problems involving percent increase and percent decrease.

Strategies for solving Ratio problems with rational numbers

Notice how this question is getting at the reasoning of proportions. Not just solve for a unit rate…

Allow students to share how they thought about solving this.

Common Core Standards Writing Team. (2011, December 26).

Progressions for the Common Core State Standards in Mathematics(draft). 6-7, Ratios and Proportional Relationships. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.

###### Coherence and Connections: Need to Know

Grade Below

Grade Level

Grade Above

6.RP.2

7.RP.1

7.RP.2

The Progression of the Ratio and Proportion Domain starts in grade 6 and ends in grade 7. These concepts need to be fully developed through discovery learning and various strategies. Now is not the time to lay down algorithms, but to develop this number sense.

Examples of Key Advances from Grade 6 to Grade 7

In grade 6, students learned about negative numbers and the kinds of quantities they can be used to represent; they also learned about absolute value and ordering of rational numbers, including in real-world contexts. In grade 7, students will add, subtract, multiply, and divide within the system of rational numbers.

Students grow in their ability to analyze proportional relationships. They decide whether two quantities are in a proportional relationship (7.RP.2a); they work with percents, including simple interest, percent increase and decrease, tax, markups and markdowns, gratuities and commission, and percent error (7.RP.3); they analyze proportional relationships and solve problems involving unit rates associated with ratios of fractions (e.g., if a person walks 1/2 mile in each 1/4 hour, the unit rate is the complex fraction ½ / ¼ miles per hour or 2 miles per hour) (7.RP.1); and they analyze proportional relationships in geometric figures (7.G.1).

PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 4). (2014, December).

Retrieved January 2015, from http://www.parcconline.org/resources/educator-resources/model-content-frameworks/mathematics-model-content-framework

PARCC Evidence Tables http://www.parcconline.org/assessments/test-design/mathematics/math-test-specifications-documents

Evidence

Statement Key

Evidence Statement Text

Clarifications

**MP**

7.RP.1

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

i) Tasks have a real-world context.

ii) Tasks do not assess unit conversions.

2,4,6

7.C.4

Base explanations/reasoning on a coordinate plane diagram (whether provided in the prompt or constructed by the student in her response). Content Scope: Knowledge and skills articulated in 7.RP.A

i) Tasks use only coordinates in Quadrant 1 and use only a positive constant of proportionality.

2,3,5,6

Calculator

yes

yes

###### HOT Questions

1. Lauren bikes 1 miles in hour. What is her rate of speed in miles per hour? Below is one way to solve. Show another way.

Ratio Table

Example Answers – using proportional reasoning

I could use a tape diagram. If she bikes 1 miles in one-fifth of an hour then I could add 1 5 times. I could think of it in terms of the whole numbers and fractions.

Example Answer using proportional Reasoning

Allow for multiple strategies and show case the different ways students reason proportionally

See sample strategies below the HOT questions

2a. Billy can write 2 ¼ pages in ¾ of an hour. How many pages can he write in an hour?

3 Pages

b. How long will it take Billy to write 15 pages?

5 hours

3. You can buy ¾ foot of taffy for $2.50 or 2 ½ feet for $4.50. Which is the better deal?

2 ½ feet for $4.50

4. Sally is selling lemonade for $1.75 a cup. It will be 95 degrees on Saturday and Sally will be able to increase her price of lemonade by 20%. How much will a cup of lemonade on Saturday cost?

$2.10

2a

3 pages per hour

2a

I can also see it pictorially as well, ¾ four times gives me 12 fourths which is 3 wholes

2b

which equals 5 hours

Using what I know from 2a. He can write 3 pages in one hour so he can write 15 pages in 5 hours.

2b

3

better deal

X = $3.33 x = 1.80

1.75 ÷ 5 = 0.35 I choose to break it up into 5 so I could get 20%’s.

So 1.75 + 0.35 = $2.10

4

4

###### Additional Resources

Performance Task from Inside Mathematics

http://www.insidemathematics.org/assets/common-core-math-tasks/mixing%20paints.pdf

MARS Tasks – Mathshell

http://map.mathshell.org/materials/download.php?fileid=816

Illustrated Mathematics

Track Practice – Ratio Tables

https://www.illustrativemathematics.org/illustrations/82

Sale – Percent

https://www.illustrativemathematics.org/illustrations/114

Cooking with the whole cup – rational numbers and unit rate

https://www.illustrativemathematics.org/illustrations/470

Engage NY Module

https://www.engageny.org/sites/default/files/resource/attachments/math-g7-m4-teacher-materials.pdf