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)
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:
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).
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).
Slides 21 and 22 ask students for multiple ways to make a given temperature change. They are asked to explain how they know and why they are equivalent. Students are told that the Mathmaster Chef does not like drawing diagrams for his assistant chefs and he wants to find a shorter way of leaving them directions. Students are askedto help him figure out in their groups how to represent what's happening with the cubes mathematically. I remind them that I am not asking for a representation of the answer, but the problem. These will take a while to go over. Slide 21 asks for ways to decrease the temperature 3 degrees. There are many ways to do this and 2 or three ways to represent each way mathematically. For example, if they remove 2 hot cubes and add 1 cold cube they could represent it with - 2 + (-1) or -1 - 2 or -1 - (+2), etc. Students may express being confused by one or more of the representations, because they aren't sure if it works. I go back and ask how the removal of 2 hot cubes is represented in each one and how the addition of one cold cube is represented. Someone one may say that the number could be confused for a different instruction. I ask if they think the assistants might do something different than remove 2 hot cubes and add 1 cold cube? I ask what they might mistake the directions to mean. This will show them that it turns out the same anyway. This is hugely eye opening for students and is well worth the time to work through it. I expect it to prevent the question that I get asked so often "is this a minus or a negative symbol?" Now I can just tell them to try it both ways and see if they can tell. It turns out the same either way since they are equivalent.
The last slide for this lesson tells students that Mathmaster Chef gave directions to remove cubes from the pot to adjust the temperature, but the pot is empty, there are no cubes to remove. Students are asked what they could do instead to get the same result. I have students explain how they know they will get the same result for all of the methods they come up with. This helps students see adding the opposite as an equivalent alternative. Students are asked to explain how they know they will get the right answer. Now when they get stuck on problems like 5 - (-2) or 2-5 I can ask what Mathmaster chef could do instead or I could just remind them to think about the hot and cold cubes and look for an equivalent alternative.