# Cooking with Mathmaster Chef (Day 3 of 4)

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## Objective

SWBAT explain the equivalence of subtraction and adding the opposite. SWBAT represent the mathematics symbolically.

#### Big Idea

Students will become familiar with mathematical models for equivalent methods of solving integer addition and subtraction.

## Intro & Rationale

This lesson continues to highlight the equivalence of subtracting and adding opposites. It also transitions students into using mathematical representations for the problem and not just the result. Before I ask students to solve a symbolic problem I ask them to create the symbolic problem they have already solved  with a more concrete model. I think this gives them a better understanding of what the abstract mathematical model means and forces them to take the time to make sense of it. (mp1)

## Warm Up

20 minutes

Before we start working on the warmup I ask students if they were able to make the target numbers from our homework consecutive sums negative (-4, 4, 8, 16, 32). I expect someone will have been able to make -4, 4 and maybe 8. I have them show -4: -4+-3+-2+-1+0+1+2+3. I draw their attention to the part of the expression that cancels to zero. When they see the same pattern for 4, I let them try with 8, which is a long expression. Then I ask if they want to find 16 or 32. Because these result in really long expressions I suspect they will begin to generalize. I have them describe the pattern of adding all the opposites below or above the target number to make zero so the only number left is the target number. I would ask what property (additive identity) they used here from earlier Number Talk lessons on number properties (Let's Talk addition).

Students begin working on slide 18 from our Mathmaster Chef Hot and Cold cubes.

Which of the follwing will decrease the temperature of the chef's pot 5 degrees: taking out 5 hot cubes, adding 5 cold cubes, taking out 5 cold cubes, and adding 5 hot cubes.

I circulate to see who is noticing there are two possibilities. I point out to the class that some students say there are two ways to do it and ask who would agree. If several students raise their hands we move forward. They explain which two will work and why.

I underline "decrease temperature 5 degrees" and ask how to represent this mathematically. (-5) Then I underline "take away 5 hot cubes"& "add 5 cold cubes" and ask how they might represent these mathematically with symbols and not words:

•  -(+5)
• +(-5)

The next question I ask them to discuss in their math family is "why does it make sense that -(+5) and +(-5) are equivalent?" I tell them to use the idea of hot and cold cubes or to use the number line to help them explain. (white boards may help). I circulate to help choose students to share explanations and representations. Some explanations may be that -(+5) is like taking out hot cubes and +(-5) is like adding cold cubes, which both result in -5 (5 degree temperature decrease). They may use a number line or use + and - symbols.

I use the same questioning process on slide 19 which illustrates the equivalence of +(+4) and -(-4).

## Powerpoint Exploration

34 minutes

Slide 20 from our powerpoint Hot and Cold cubes asks what the temperature of the pot is with 3 hot and 3 cold cubes in it. (0) Students are asked how they could add cubes but still maintain a temperature of zero. I get a suggestion from each math family group. This reinforces the idea that, at any time, they can add "neutral pairs" without changing the value. Some students may even point out the connection to the Identity property of addition learned in an earlier Number Talk lesson (Let's talk addition).