This lesson is meant to help students transition from the concrete model to a mathematical model by presenting them with the mathematical model and asking them to interpret it based on the context. Adding and subtracting integers can be very confusing for students because it seems to contradict their prior knowledge and experience with addition and subtraction. Suddenly we are subtracting when the problem says to add or adding when the problem says to subtract. Suddenly numbers aren't always getting bigger when we add or smaller when we subtract. Many models we use with middle school students are not as accessible as the hot and cold cubes, because they require experiences that students may not have had, like borrowing and lending. This context helps them make sense of the math.
The warmup helps students to both transition from the concrete to the abstract by using the context to make sense of the mathematical model and also to reinforce the equivalence of subtracting and adding the opposite for students who may be ready. I also want to encourage choice making, so they realize they have some autonomy and flexibility in problem solving.
Slide 24 of the powerpoint Hot and Cold cubes is the warm up for students today. It asks what the Mathmaster Chef could want his assistant to do with hot and cold cubes if the only notes he left were:
+2 - 3 - 3 + 2 - 1 - 2
As I circulate I encourage students to write out what the directions our chef left could be telling us to do with cubes.
I look to see if students come up with different ways of interpreting the directions, either within the same math family group or between the groups. I am hoping they will so that I can ask them if we need to ask Mathmaster Chef for clarification on his directions. I would remind them that Mathmaster Chef is very tempermental and yells at his assistants when they can't figure things out on their own, but he also yells if they get the temperature wrong, so we have to be careful.
Slides 25 - 27 follow with more expressions. Ask students if we need to ask for clarification of the directions since there are sometimes more than one interpretation? Whatever they answer, ask them to explain why or why not. Hopefully, one of them will say that we get the same temperature change either way, so it doesn't matter which one we do. If not, I encourage them to find the temperature change either way. When they see that it results in the same temperature change I ask what they think Mathmaster Chef's reaction would be if we asked for clarification? I will point out that assistants like us can make some decisions on their own. As long as they are interpreting the numeric expression in at least one way using hot and cold cubes, they will be fine.
Slide 28 of the powerpoint Hot and Cold cubes asks what Mathmaster Chef wants his assistant to do if his directions are 2 - (-2). Students should only interpret this in one way "add two hot cubes and take out 2 cold cubes" They will likely try to interpret it in another way since the three before had multiple methods. As they work on this I circulate to highlight argumentation at the math family groups. Even though there is only one interpretation students may find an equivalent method of adding 2 hot cubes and 2 more hot cubes instead. If they don't I will ask if there is another way the assistant could get the same result.
Follow the same type of questioning for slides 29 and 30. Both of these do have multiple interpretations. I would ask how they know that each interpretation is equivalent and which they might choose to do and why.
The last slide is very open ended and gives rise to many different methods for reaching a temperature decrease of 8. I want them to work in groups so that they can see different methods and ideas. The slide tells them the chef added some cubes, removed some cubes, then added some more cubes, and asks what he might have done. The question is purposely vague. They might come up with a way for the chef to add, subtract, then add cubes to reach -8 or they might choose their own equiv alent method as an option for the chef. After students share multiple methods I ask what they would do to reinforce autonomy and choice. Autonomy and choice are so important because students taught traditionally wait for all the directions to be given by the teacher, but it's only when they are forced to make choices that they truly become flexible with the math and utilize the relationships they have uncovered.