Task Teacher Guide

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In This Task…

Students will model multiplication and division contexts involving positive and negative integers in order to solve.

Intentionality…

In this task, students will represent multiplication as repeated equal groups of a single quantity (or quota). In today’s context, the repeated equal group is a negative rate of feet per hour. Students will then use quotative division to determine how many groups of the negative rate are in a total quantity. Some of the big ideas revealed in today’s lesson include:

• Multiplication can be understood as repeated equal groups;
• The first factor can be understood as the number of groups and the second factor as the size of the groups;
• A positive integer multiplied by a negative integer has a product that is negative;
• Quotative division is when the total quota is known (the dividend), and the number per group or the rate (the divisor) is known;
• Quotative division reveals the number of copies or iterations of a rate that can be derived from the overall quota (the dividend);
• A negative number divided by a negative number has a quotient that is positive.

Spark

What Do You Notice? What Do You Wonder?

Show students the following video:

Then, ask students:

What do you notice?
What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Replaying the video can be helpful here.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group.

Some of the noticing and wondering that came up in a class recently included:

• I notice dirt
• I notice someone digging.
• I notice the dig 4 feet down every hour.
• I wonder how long they will dig.
• I wonder how deep they will go.
• I wonder why they are digging.
• I notice there is a scale.

At this point, you can answer any notices and wonders that you can cross off the list right away. Confirm for students that this person is digging a hole for a new well. He appears to be digging at a rate of -4 feet per hour.

Estimation: Prompt

Next, share the following visual prompt briefly and ask the following question:

If digging continues at the same rate for 6 hours, how would you describe the depth of the hole?

Give students an opportunity to turn and talk. We want students to communicate that the depth can be described as -24 feet. The purpose of this estimation is to build confidence with the use of negative integers to describe measurements.

Estimation: Reveal

Share the following image revealing the total depth after 6 hours of digging at a rate of -4 feet per hour.

Share the following prompt:

How could we represent this situation as an equation?

The purpose of this estimation is to help students see that this scenario can be represented as a multiplication.

For example,

6 hours x (-4 feet/hour) = -24 feet

or

6 x (-4) = -24

Here is a visual of this thinking:

Sense Making

Crafting A Productive Struggle: Prompt

Since you have already taken some time to set the context for this problem and student curiosity is already sparked, we have them in a perfect spot to help push their thinking further and fuel sense making.

Share the following prompt.

The well needs to be -56 feet to reach groundwater. Assuming he continues digging at a rate of -4feet/hour, how long will it take to dig this well?

During Moves

While Students Are Productively Struggling…

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Student Approach #1: Using Repeated Addition on a Double Numberline

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Student Approach #2: Using Repeated Subtraction on a Double Number Line

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Student Approach #3: Using Partial Products

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Next Moves

Consolidation: Making Connections

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Reveal

Show students the following reveal video:

Reflect

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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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.

Download Editable/Printable Handout

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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.