One persistent and common mistake students start to make when they begin to change subtraction problems into "adding the opposite" problems is that they also change the addition into subtracting the opposite. This lesson is meant to clarify the reason we are looking at this relationship and why it is helpful to know that "adding the opposite" is equivalent to subtracting. I tell them that the only reason we are even paying attention to this fact is that it is useful in making the math easier for us to do mentally. The warm up shows students the specific problems we run into when we subtract that don't happen when we add. I tell them that the subtraction is much more confusing than the addition, so even though they are changing the addition into an equivalent subtraction problem they are getting it wrong because they are changing it into a more difficult problem. The whole point of knowing this fact is so that we can change the more difficult subtraction into easier addition problems and not have to do the subtraction.
The key to the whole lesson is for students to see that subtraction is much more difficult than addition. I want them to be motivated to change the subtraction and do the easier equivalent addition problem, but not change the addition into the harder subtraction problem.
There are five addition and subtraction problems in the warm up which they spend a few minutes solving together.
1. -5 + (-4) = 2. -2 + 6 = 3. -3 - (-2) 4. -3 - (+10) = 5. 6 - (-4) =
Before we go over them I tell students that they are about to see why we run in to trouble with subtraction problems. I tell them it is the same trouble Mathmaster chef has when there are no cubes to take out of the pot. (Mathmaster Chef series)
Starting with number one I ask for the answer, which I think most students will know is -9. I ask how they know this will be negative. Students may say that negatives are being added to negatives making more and more negatives. I ask how they know to add the 5 and the 4. Students may say that there is no canceling when there are only negatives so it just increases the number of negatives.
After they give the answer of 4 for number 2 I ask again for their explanations of how they know it will be positive and how they know to subtract the 2 and the 6. They may say that we are adding more positives than negatives (which I model with symbols) or that we go further in the positive direction on the number line. The know to subtract, because the canceling takes away all the negatives and some of the positives.
For number 3 model the three negatives and then point out that this problem is not going to let us add, but remove symbols. I ask what I can do to take away 2 negatives (scribble them out). The answer is -1, not so bad. But for number 4 I model the three negatives and ask what we are being asked to take away (10 positives or hot cubes). I point out that here is where we run into the same problem that Mathmaster Chef had. We are being asked to subtract something that isn't there! There aren't any positives to take away or scribble out. I remind them they are Mathmasters and ask what we can do instead of taking away hot cubes (put in cold cubes). Then I ask how to change the problem so we are putting in cold cubes and model by adding 10 more negatives to our diagram. I follow the same questioning for number 5. I remind them that we don't run into this trouble with addition, so we never want to make those any harder by changing them.
Students work collaboratively in their math family groups to solve subtraction problems on their personal white boards. They all hold up their boards at the same time on a count of three. That way I am sure not to miss anyone and no one can opt out.
-3 - (+5)
-2 - (-10)
-5 + (+10)
2 - (-4)
-4 + (-5)
My goal if for them to change the subtraction, but not the addition. I am also checking for other persistent mistakes. One mistake I may see is changing subtraction correctly, but not to the opposite or changing the first number. If I see something like -3 - (+5) = -3 + 5 I may ask if adding positives is equivalent to subtracting positives or I may relate it to the context of hot and cold cubes (Mathmaster Chef series). If they change the first number I tell them that is not the number being taken away. I may even some mistakes in addition once they've changed it. I circlulate as they work and give individual feedback. I may only get to half of the class during each problem so I alternate sides or check in with the kids who have been struggling with this topic.
The third problem is already addition and I expect many of them to change it. As I circulate I ask them if that is the easiest way to do the problem. I tell them to think about the two ways we could do this problem and choose the easier way. Then I ask them why they don't change this one. Students may say that it is already addition or that it is already the easy one. When we get to the last one I don't expect so many of them to be fooled.
I tell them they can get "two for one" on their homework two for one tonight by solving the simpler addition problems and then matching them to the subtraction problem that would give the same solution.