Task Teacher Guide

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Through this math talk and purposeful practice, students are encouraged to model the behaviour of multiplying and dividing positive and negative integers. All division scenarios today will be partitive.

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The purpose of the Day 4 activities is to reinforce key concepts from Day 3. Students will continue to explore examples of partitive division involving a negative and positive integer. Students will engage in a math talk that will serve to reinforce the relationship between multiplication and division. They will independently complete purposeful practice to develop a deeper understanding of the following big ideas.

• Partitive division is when the total quota is known (the dividend), and the number of parts or groups (the divisor) is known;
• Partitive division reveals a rate;
• In partitive division, the dividend and the divisor often have different units;
• A negative number divided by a positive number has a quotient that is negative.

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Math Talk

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In today’s math talk, students explore the inverse relationship between multiplication and division. Present each sentence one at a time. Throughout the math talk, continue leveraging the context from Day 3; a team digging for fossils. All division sentences should be viewed as partitive division (fair share). 

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Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

A team of 6 workers are digging for a dinosaur fossil.
The dig will require the hole height compared to ground level to change by -12 feet per worker to reach the fossil.
What will the height at the bottom of the hole be compared to ground level?
Use a number line and write an equation to represent this scenario.

The goal of this particular prompt is to get students thinking about the result when you multiply a positive number by a negative. In this case, the number of groups or iterations is positive and the rate is negative. Therefore, we are iterating a negative rate and the result should be a negative number. As workers repeatedly dig 12 feet further into the ground, the height at the bottom of the hole is more negative than it was the worker before.

6 people x -12 feet per person = -72 feet

And

-72 feet ÷ 6 people = -12 feet per person

The next image makes it clear that the length (or height) of the hole is 72 feet into the ground or -72 feet relative to ground level.

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Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

A team of 8 workers are digging for a dinosaur fossil.
The dig will require the hole height compared to ground level to change by -7 feet per worker to reach the fossil.
What will the height at the bottom of the hole be compared to ground level?
Use a number line and write an equation to represent this scenario.

Here we reiterate the behaviour of multiplication when multiplying a positive by a negative – again, a positive number of groups or iterations and a negative rate.

8 people x (-7 feet per person) = -56 feet

And

-56 feet ÷ 8 people = -7 feet per person

Again, ensure students are understanding why this is true by reiterating the number of positive iterations where -7 feet per worker is removed from the hole. By repeating this process, it makes sense that the hole gets deeper and thus the measure compared to ground level heads further in the negative direction.

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Purposeful Practice

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[accordion-item title=”Multiplying and Dividing Integers”]

Login/Join to access one (1) additional Math Talk Prompt as well as the Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem-based units.

We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

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Resources and Downloads

Explore The Entire Unit of Study

This Make Math Moments Task was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, and extensions to elicit and emerge mathematical models and strategies.

Click the links at the top of this task to head to the other related lessons created for this unit of study.