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11.2

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

MLR7: Compare and Connect

Materials

None

Activity Narrative

In this activity, students rewrite multiplication equations as division equations and use their understanding of multiplication and division with integers to solve a problem in context. Students use concrete examples and their prior experiences to more precisely articulate a rule for the sign of a quotient given the signs of the dividend and divisor (MP6).

Monitor for students who identify and describe the rule clearly.

Launch

Remind students that we can rearrange division equations to be multiplication equations, and vice versa. It may be useful to demonstrate with positive numbers if students struggle to recall this. For example, ask how we could rewrite as a multiplication equation. Students can also reference the multiplication table from a previous lesson, if needed.

Arrange students in groups of 2. Give students 4 minute of quiet work time followed by 2 minutes of partner discussion, then follow with a whole-class discussion.

MLR7 Compare and Connect. After all strategies used to determine where Han and Clare are in the last question have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What did the approaches have in common? How were they different?” “What kinds of additional details or language helped you understand the displays?” and “Were there any additional details or language that you have questions about?”
Advances: Representing, Conversing

Action and Expression: Internalize Executive Functions. To support organization, provide students with a graphic organizer that lists the multiplication equations and has space for students to write the associated division equation.
Supports accessibility for: Language, Organization

Activity

None

Find the unknown value in each equation.
Rewrite each unknown factor problem (the last four equations of the previous problem) as a division problem.
Complete the sentences. Be prepared to explain your reasoning.
1. A positive number divided by a positive number equals a _______________________.
2. A positive number divided by a negative number equals a _______________________.
3. A negative number divided by a positive number equals a _______________________.
4. A negative number divided by a negative number equals a _______________________.
Han and Clare walk towards each other at a constant rate, meet up, and then continue past each other in opposite directions. We will call the position where they meet up 0 feet and the time when they meet up 0 seconds.
- Han’s velocity is 4 feet per second.
- Clare’s velocity is -5 feet per second.
1. Where is each person 10 seconds before they meet up?
2. When is each person at the position -10 feet from the meeting place?

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

The purpose of this discussion is for students to articulate their understanding about the sign of a quotient given the signs of the dividend and divisor. Note that students do not need to use these terms—they can give examples or use informal language. Begin by inviting students to share a multiplication problem and its corresponding division problem. Then consider discussing the following questions:

"What patterns do you notice between the multiplication equation and the division equation you wrote?" (The numbers used in both stay the same. The division equations all start with 12 or -12.)
"What patterns do you notice between the signs of each number in the division equations?" (When the two numbers being divided are both positive or both negative, the answer is positive. When the two numbers being divided have different signs, the answer is negative.)
"How do these patterns compare to what we saw with multiplication?" (The patterns for the sign of the answer in both multiplication and division problems are the same.)

11.3

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students multiply and divide rational numbers to represent and solve problems in the new context of a drilling rig. They use multiplication and division of negative numbers to work with a relationship that has a negative constant of proportionality. Students use what they know about the structure of proportional relationships to help them represent this situation with a graph (MP7).

In the digital version of the activity, students use an applet to plot points on a graph. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.

Launch

Remind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.

Ask students what they remember about proportional relationships. Students may say that:

They are often represented with an equation in the form .
The constant of proportionality, often called , is the change in for a change by 1 in .
A graph representing a proportional relationship is a line through (0,0) and .

Arrange students in groups of 3 during the discussion.

Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language

Activity Synthesis

The goal of this discussion is for students to share their reasoning. Ask students to share their solutions with each other in groups of 3 and work to come to an agreement.

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Lesson 11

Dividing Rational Numbers

Warm-up

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Standards Alignment

Building On

Addressing

7.NS.A.2.b

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Standards Alignment

Building On

Addressing

7.NS.A.2.b

Instructional Routines

Materials

Activity Narrative

Launch

Activity Synthesis

Lesson Synthesis

Student Lesson Summary

Standards Alignment

Building On

7.NS.A.2

Addressing

Building Toward

7.EE.B.4.a

Activity

Student Response

Building on Student Thinking

Building Toward

Building Toward