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

The purpose of the Day 2 activities is to reinforce key concepts from Day 1. Students will also be introduced to a new multiplication scenario where a negative integer is multiplied by a negative integer. Students will engage in a math talk and will have an opportunity to complete independent purposeful practice. The math talk and purposeful practice serve to develop a deeper understanding of the following big ideas.

• 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;
• A negative integer multiplied by a negative integer has a product that is positive;
• 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.

In today’s math talk, students will observe changes on a thermometer. They will be asked to write as many equations as they can to describe the change.

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

Then ask students:

At midnight, the temperature was 0 degrees.
The temperature continued to change by -3 degrees every hour.
What was the temperature 4 hours later?
Write an equation to represent this scenario.

From the animation, students will see that with every passing hour, the temperature drops by 3 degrees (or the change in temperature is -3 degrees per hour).

With this, students might create a table to track the number of hours and the current temperature beginning at 0 hours (12:00 am) and 0 degrees, then 1 hour (1:00 am) and -3 degrees, and so on.

Some students may recognize that they can simply multiply the rate of -3 degrees per hour by 4 hours giving a result of -12 degrees Celsius after the 4th hour.

Facilitator note:

The purpose of this first scenario is to solidify the following big ideas.

• 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.

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

Then ask students:

By midnight, the temperature had reached 0 degrees.
It has been decreasing at a rate of -2 degrees per hour for the past 5 hours.
What was the temperature 5 hours ago?
Write an equation to represent this scenario.

While writing an equation initially may not feel attainable for some students, the hope is that students would recognize that we are going back in time (i.e.: negative hours).

Encourage students to have a conversation with their neighbours to ensure they are making sense of the description of this context. It is likely that students will recognize that if the temperature is decreasing by 2 degrees every hour, if we “rewind the clock” (i.e.: remove hours), the temperature should be higher than it is now.

If 1 hour in the future would mean the temperature drops by 2 degrees, going 1 hour in the past would mean the temperature increases by 2 degrees.

Some students might recognize that 1 hour ago, the temperature would be 2 degrees higher. 2 hours ago, the temperature would be 4 degrees higher. And so on.

Helping students to recognize what operation they are using (repeated addition or multiplication), we can craft an equation that multiplies the negative rate by the negative number of groups (or hours).

Facilitator note:

In this scenario, going backwards in time is one negative, and the drop in temperature is another.The purpose of this scenario is to solidify the following big ideas.

• Multiplication can be understood as repeated equal groups;
• A negative integer (-5 hours) multiplied by a negative integer (-2 degrees per hour) has a product that is positive (10 degrees).

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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.

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