problem solving: multiplying and dividing rational...

– grade 7 • Teacher Guide

Problem Solving: Multiplying and Dividing Rational Numbers

LAUNCH (7 MIN) _____________________________________________________________ Before

• Why does the diagram use a negative value to represent New Orleans? Why wouldn’t you use a positive value?

During • How can you find the distance the dog digs in one hour?

After • How did you know whether to add or subtract the distance the dog digs?

PART 1 (8 MIN) ______________________________________________________________ Jay Says (Screen 1) Use the Jay Says button to point out that the frequency reveals the

number of times each temperature occurs. To make this point, you might want to have students add up the frequencies to show that the sum is equal to the total number of days (3 + 1 + 3 + 2 + 3 + 2 = 14).

While solving the problem • How do you know the number of values you should have for the temperatures? • How do you find the mean of a set of data?

After showing the solution • In the previous lesson, you learned that the formula for converting from Celsius to

Fahrenheit is F = 9

5 C + 32 . How could you find the corresponding mean temperature in

degrees Fahrenheit?

PART 2 (8 MIN) _______________________________________________________________ Before solving the problem

• How do you know which operation to perform first to simplify an expression with more than one operation? What two methods are shown here?

While solving the problem • If the problem did not specify, how can you tell that at least one student made an error?

PART 3 (10 MIN) ______________________________________________________________ Jay Says (Screen 1) Use the Jay Says button to explain why students may see squirrels during

the winter even though this problem states that some squirrels hibernate.

Before solving part (a) • How do you know whether there is a proportional relationship between two quantities?

After solving part (a) • Why would it be useful to know that the relationship is proportional?

After solving part (b) • Why is the change in weight represented with a negative number?

CLOSE AND CHECK (7 MIN) ______________________________________________ • Describe how a table of rational numbers showing a proportional relationship can help you

solve a problem. • Why is it helpful to recognize different relationships when solving problems?


Problem Solving: Multiplying and Dividing Rational Numbers

LESSON OBJECTIVES 1. Solve real-world and mathematical problems involving the four operations with

rational numbers. 2. Solve multi-step real-life and mathematical problems posed with positive and

negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically.

FOCUS QUESTION What types of problems can you solve using operations on rational numbers?

MATH BACKGROUND Prior to this lesson, students learned to perform the four operations on rational numbers and learned different strategies to simplify expressions containing rational numbers and multiple operations. Students have also learned about ratios, which can be written as rational numbers to describe real-world quantities.

In this lesson, students combine the skills they learned in previous lessons and topics to solve problems that contain rational numbers. Students analyze data to find measures of center, analyze errors in evaluating expressions, and solve multistep rate problems, all by operating with rational numbers. As students complete this lesson, they should recognize that there are several strategies for working with rational numbers. Because most real-world quantities are not whole numbers, students will solve problems containing rational numbers in most topics in this course and in high school.

In the next topic, Decimals and Percents, students will again encounter rational numbers in real-world situations. They can use the skills they learned in this topic to help them work more with rational numbers in decimal form and when solving equations and inequalities in future topics.

LAUNCH (7 MIN) ____________________________________________________ Objective: Solve a real-world problem involving rational numbers and more than one operation.

Author Intent Students apply what they know about operations with rational numbers to solve a real-world problem involving rates. There are different ways to write an expression that describes the relationship between sea level and the dog’s elevation. Students can write an expression that describes the elevation in terms of sea level or use equivalent rates to find the unknown elevation.

Questions for Understanding Before

• Why does the diagram use a negative value to represent New Orleans? Why wouldn’t you use a positive value? [Sample answer: Zero represents sea level, so a negative value is used to represent a distance below sea level. You would not use a positive value because there is also an elevation that is 6.6 ft above sea level.]

During • How can you find the distance the dog digs in one hour? [Sample answer: I can

write the rate that the dog digs and find an equivalent rate with a time of 1 hour.]

Problem Solving: Multiplying and Dividing Rational Numbers continued


After • How did you know whether to add or subtract the distance the dog digs?

[Sample answer: I can either add a negative number to indicate digging down, or I can subtract a positive distance. Either way, the new elevation should be less than 6.6 ft.]

Solution Notes Students who write an expression should understand that they can multiply the rate by the number of 10-minute intervals. Some will subtract the total distance the dog digs, while other students will add the distance and use a negative rate to indicate the direction the dog is digging.

If students use repeated addition to find the elevation, encourage them to use multiplication to write an algebraic expression that can be used for times other than 1 hour.

Students can also use equivalent rates to find the distance the dog digs in one hour. Since

!0.4 ft10 min =

!2.4 ft60 min they can add －2.4 to －6.6.

Use a Know-Need-Plan organizer to help students understand that there are two components to solving this problem. They not only need to find the distance the dog digs, but they also need to add that distance to the elevation the dog begins at.

Know: starting elevation is －6.6 ft, and rate is 0.4 ft per 10 minutes Need: the dog’s elevation after 1 hour Plan: Write an expression that relates the elevation of the dog to the digging time. Substitute and evaluate for 1 hour (or 60 minutes).

Connect Your Learning Move to the Connect Your Learning Screen. Make sure students see that what they learned about multiplying and dividing positive and negative rational numbers can be applied to real-life situations. The Launch used elevation, and you may want to ask students what other real-life situations they've seen thus far in the topic. Some examples include glacial advance and retreat, falling stocks, rate of submarine descent, conversion between Celsius and Fahrenheit, and rate of water lowering in a lock.

PART 1 (8 MIN) ______________________________________________________ Objective: Use operations on rational numbers to analyze data.

Author Intent Students use operations with rational numbers to find measures of center. They are given a frequency table and use multiple operations to find means and medians.

Instructional Design You may need to review with students that mean is simply the average of the data values. For the Got It, students should recall that the median of an even number of data values is the mean of the two middle values.

Call on a student to come to the whiteboard and write the temperatures that occurred over the fourteen days. Connect the fourteen temperatures to the temperatures in the table to emphasize how to interpret the frequency.



Questions for Understanding Jay Says (Screen 1) Use the Jay Says button to point out that the frequency

reveals the number of times each temperature occurs. To make this point, you might want to have students add up the frequencies to show that the sum is equal to the total number of days (3 + 1 + 3 + 2 + 3 + 2 = 14).

While solving the problem • How do you know the number of values you should have for the temperatures?

[Sample answer: I can add the frequencies, and 2 weeks should be 14 days.]

• How do you find the mean of a set of data? [Sample answer: Find the sum of the data values and divide by the number of data values.]

After showing the solution • In the previous lesson, you learned that the formula for converting from Celsius

to Fahrenheit is F = 9

5 C + 32 . How could you find the corresponding mean

temperature in degrees Fahrenheit? [Sample answer: Substitute the mean temperature in degrees Celsius and solve for F, which is approximately 27°F.]

Solution Notes Students who want to list all fourteen data values and find the sum may be interested to see the provided solution, which multiplies each data value by its frequency.

Differentiated Instruction For struggling students: Use the Number Line tool in Add & Subtract Integers

mode to remind students how to add and subtract positive and negative numbers.

For advanced students: The word mean has several meanings. While students may think of synonyms, such as intend or hateful, remind them that in mathematics it is synonymous with the word average. Ask students to come up with a way to remember the difference, and share it with the class.

Error Prevention Students may not understand how to read the Frequency column and find the mean of 6 data values. Help them write out each data value so that they see the entire data set.

Got It Notes Students may notice that the table is already in order from least to greatest and that there are 14 daily temperatures, as there were in the Example. Students who understand how to read the frequency table might notice that they can cancel the values of －8°C and 5°C because there are four on each end.

If you show answer choices, consider the following possible student errors:

Students who choose A or B may not know how to read the Frequency column. If students select B or C, they may be trying to find the mean.

PART 2 (8 MIN) ______________________________________________________ Objective: Solve a multi-step real-life problem posed with positive and negative rational numbers.

ELL Support Beginning After discussing the Got It, pair Beginning learners with native speakers and have the native speakers provide an explanation of Renee’s mistake. Then have Beginning learners explain the mistake in their own words. Encourage Beginning learners to ask for assistance when necessary and use words they know to help explain words they



do not know. [Sample: Renee didn’t make 16 minus. A minus times a plus is minus. It is –16

14 .]

Intermediate After discussing the Got It, pair students with native speakers and have the native speakers provide an explanation of Renee’s mistake. Then have Intermediate learners explain the mistake in their own words. Encourage them to use more specific language and ask the native speakers to assist them with vocabulary and pronunciation. [Sample: Renee didn’t write the right sign for 16. The answer is negative, –16

14 .]

Advanced After discussing the Got It, have Advanced learners work with a partner to explain Renee’s mistake. Then have students think of another way to explain the concept. [Sample: Renee didn’t multiply correctly. A negative multiplied by a positive is a negative. The answer is –16

14 because –16 –

14 is –16

14 .]

Author Intent Students analyze the steps for simplifying an expression containing rational numbers and multiple operations. One expression is simplified by following the order of operations, and the other is simplified using the Distributive Property. Both methods contain an error, which students need to identify and correct.

Instructional Design Call on a pair of students to come to the whiteboard and circle the error in each student’s work. Have them correctly evaluate the expression using both methods and discuss as a class how the error affected the answer.

Questions for Understanding Before solving the problem

• How do you know which operation to perform first to simplify an expression with more than one operation? What two methods are shown here? [Sample answers: You can follow the order of operations, or you can use properties of algebra to rewrite the expression. Paula uses the order of operations, while Aaron first rewrites the expression using the Distributive Property.]

While solving the problem • If the problem did not specify, how can you tell that at least one student made an

error? [Sample answer: The values do not match. The value should be the same no matter which method you use.]

Solution Notes The provided solution corrects Paula’s work, using the order of operations. Get students to also show how to solve correctly using the Distributive Property.

Invite students to make a list of errors they commonly make when evaluating expressions like this one. Comparing their work to the work of Paula and Aaron will help them recognize and avoid common mistakes and develop comfort with multiple methods.

Got It Notes Encourage students to evaluate the expression first on their own and then compare their work to Renee’s in order to find her error.


Students who choose A did not use the reciprocal when rewriting division as multiplication. If students select C, they are not following the order of operations.



Students who choose D may be trying to use the Distributive Property in a situation that does not apply. Point out that －8 is not being multiplied to both terms, only to

12 .

Got It 2 Notes This problem recalls information from the Example. Students previously corrected Paula’s work, which did not follow the order of operations. Advise students that although they have inserted a pair of parentheses to make Paula’s work correct, the work is still incorrect in the context of the Example. Adding a pair of parentheses usually does not result in an equivalent expression.

PART 3 (10 MIN) _____________________________________________________ Objective: Solve a mathematical problem involving more than one operation with rational numbers.

Author Intent Students write an equation that describes a proportional relationship shown in a table. They use the proportional relationship, which involves rational numbers, to find an unknown value.

Questions for Understanding Jay Says (Screen 1) Use the Jay Says button to explain why students may see

squirrels during the winter even though this problem states that some squirrels hibernate.

Before solving part (a) • How do you know whether there is a proportional relationship between two

quantities? [If the ratios of corresponding values are equivalent, there is a proportional relationship.]

After solving part (a) • Why would it be useful to know that the relationship is proportional? [Sample

answer: You can use the equation to find other equivalent ratios.]

After solving part (b) • Why is the change in weight represented with a negative number? [The negative

shows that the weight was lost, not gained.]

Solution Notes Students can show that the ratios are equivalent to demonstrate the proportional relationship between days of hibernation and change in weight. From this, they can determine the constant of proportionality and use that constant to find other values that fit this relationship.

Students might assume that the relationship is proportional and write the equation y = mx. If they substitute one row into the equation, they can find m and check that every other row satisfies this equation. That would prove that the quantities have a proportional relationship.

Alternatively, students might use the Data and Graphs tool to graph each row of the table as a point. When they find that the graph is a straight line, they will confirm that the relationship is proportional.



Got It Notes Students should assume that the relationship is proportional. The numbers in this problem actually allow students to just as easily find equivalent ratios to complete the table as it does to write an equation and find the constant of proportionality.


If students choose A or C, they may have found the correct constant of proportionality but reversed the quantities at some point. Students who choose B are using a constant of proportionality that is 10 times the correct constant of proportionality.

CLOSE AND CHECK (7 MIN) _____________________________________

Focus Question Sample Answer Sample: You can use operations with rational numbers to analyze data that include negative values, such as temperature data, and to model proportional relationships with a negative rate of change, such as the position of a retreating glacier over time.

Focus Question Notes Look for student answers that state that they can solve any problem with positive or negative fractions, decimals, integers, or mixed numbers. Now that they know the order of operations, they can even solve problems that contain multiple operations. Students may list some of the real-life problems from this topic: problems involving elevation in reference to sea level, glacial advance and retreat, falling stocks, rate of submarine descent, conversion between Celsius and Fahrenheit, rate of water lowering in a lock, mean and median temperatures, and rate of animal weight loss during hibernation.

Essential Question Connection This lesson addresses the portion of the Essential Question that asks what models and relationships help you make sense of multiplying and dividing positive and negative rational numbers. Listen for students to describe proportional relationships, or how the relationship between distance and time help you solve rate problems.

• Describe how a table of rational numbers showing a proportional relationship can help you solve a problem. [Sample answer: If the ratio is consistent throughout the rows of the table, you know that the relationship is proportional. You can then use this ratio to find a value that would appear in another, missing row of the table.]

• Why is it helpful to recognize different relationships when solving problems? [Sample answer: If you recognize different relationships, you can plan a solution path, and sometimes you can approach a problem in more than one way.]

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