Equivalent Expressions

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SWBAT explain why a single number line can represent both an addition and a subtraction problem.

Big Idea

Students will see that adding the opposite is equivalent to subtracting.

Intro & Rationale

Subtracting integers can be a really complex idea for students and it can conflict with their prior knowledge and sense of positive number subtraction. It gets especially confusing for them when we tell them that subtraction is actually equivalent to addition. For concepts that create this kind of conflict I think students need to be shown in all different ways and have multiple experiences. This lesson relates the positive and negative numbers back to the context of hot and cold cubes learned in earlier lessons (Mathmaster Chef series) and uses the number line to show equivalence. Students work together to sort equivalent expressions together. They must engage in argumentation in order to come to agreement and convince each other. As I circulate during the activity I am encouraging and facilitating the argumentation process. I ask if they have doubts, if they can show me what they mean on a number line, etc. I highlight argumentation as it happens by saying things like "Mason is trying to prove to Brandon that two expressions should go together", "this group is talking about hot and cold cubes", etc. (mp3)

Warm up

15 minutes

The first part of the warmup relates back to the context of hot and cold cubes from earlier lessons (Mathmaster Chef series). It tells students that we are asked to decrease the temperature in the pot by taking out hot cubes, but there are no hot cubes to take out. Students are asked what we can do instead. I tell them that they are now the Mathmasters and don't need to be told what to do anymore. They know by now that putting in cold cubes would have the same effect.

The next part of the warm up tells students that some of our assistants don't believe this will always work. Students are told to show all the assistants in their group an equivalent way to get the same result for three subtraction problems and to explain why it makes sense that we can do this. When we go over the three problems I relate them to the earlier context of hot and cold cubes to help them make sense of them (mp1).

For +4 - (-5) We go over what it means with hot and cold cubes, "put in 4 hot cubes and take out 5 cold cubes". Then I ask them "if there are no cold cubes to take out what can we do instead?" (add hot cubes) I ask them what the expression (recipe) would look like if we added hot cubes instead. (+4 + 5). Then I ask students to explain or show why this will work to give us the same result. I expect students to say that taking out cold cubes and adding hot cubes both increase the temperature. I ask if anyone used a number line to show how this works. If not, I ask what it would look like on a numbe line. They should be able to show that taking away negatives moves in the same direction on the numberline as adding positives.

I follow the same type of questioning for the remaining two problems: -2 - 6 and 5 - 7. Both of these can be tricky because they don't show the positive sign for the integer being subtracted. I may have to ask them to clarify "what kind of cubes are these?" If they  say cold I would circle the number and ask what type of number we use to represent cold cubes (negative). I ask "if the number isn't negative then it doesn't represent cold cubes".


30 minutes

This is a sorting activity that students do in pairs or in trios. Each pair or trio is given a set of 18 cards matching equivalent expressions. Each card either has a subtraction problem, and addition problem, or a number line. Students are told to sort them into categories of equivalence. All the cards with an equal value go together. 

As I circulate I will aslo ask the partners if there are any expressions they are unsure about. Sometimes their partner puts one in a category and they don't really agree. I will ask one of them what it is that makes them unsure and ask what the other partner(s) think of that. Often this will prompt one of them to start trying to prove it with some modeling or an explanation. But a lot of times students, especially those who feel academically weaker than their partner(s), will not challenge a peer's ideas. Asking about their doubts is a good way of helping them learn to dissagree respectfully and argue productively.

When a few pairs or trios finish early I choose a category and ask them to explain how the same number line can work for all of the problems in that category. I tell them to be ready to share with the class. I will also ask someone to sort my cards under the document camera. Then I ask these students to explain to the class using my cards how they number line can work for all the problems. I am hoping that for expressions like -3 + 5 & -3 + (+5) they will just say they are the same thing and then proceed to show how the number line works for these and for -3 - (-5) because when we use the number line to take away negatives or to add positives we are moving in the same direction. They can also contextualize their explanation saying that the move in the same direction reflects the same resulting temperature change. 

Flash Cards

9 minutes

I use recycled manila folders as whole class flash cards. All students stand behind their desks. When I hold up a flash card no one raises their hands or shouts out. I call on one student and expect their answer in 1 second or less. This forces them to get the answer ready in their heads, so, theoretically they should all be doing every problem mentally. When a student gets the answer right they sit down and are done (but usually they keep doing the problems). When a student gets the answer wrong or doesn't know I go to another student. Once a students has answered it correctly I go back to the original student and explain it to them or have another student explain it to them. For these subtraction flash cards I give them a choice to give me the final answer or to give me the equivalent addition problem. This way two students can sit down on one card. Asking them to solve these mentally involves a little more accountability then having them work on boards.

When they are all seated I pass out the homework equivalent subtraction.