Patterns in Subtraction
Lesson 14 of 24
Objective: SWBAT recognize equivalent addition and subtraction problems.
This context is the best I have found for helping my students understand why subtracting is equivalent to adding the opposite! Now that students are beginning to understand the equivalence of sums and differences I want them to view this pattern as a tool for them to choose when they see the opportunity arise. So often students will only change subtraction problems into the equivalent addition when they are told to, or they ask permission. This tells me one of two things, either they don't really understand why they are equivalent or they don't see it as something that is useful in making the math easier to do, but as a rule to follow.
In their homework they are given an option. Do all the problems or find the six that you can do that will give you the answers to all twelve. Many students may not notice this since a lot of them don't read directions, but once we go over it and they realize they could have used the equivalence in a way that made their homework easier.
The warmup shows four sentence frames that relate to the context learned in earlier lessons (Mathmaster Chef series):
Putting in __________ cubes increases temperature.
Taking out ___________ cubes increases temperature.
Putting in __________ cubes decreases temperature.
Taking out ___________ cubes decreases temperature.
Then it asks students to explain why it makes sense that taking out one type is equivalent to putting in the opposite type.
The bottom part of the warm up is covered, because I want them to do something other than solve them.
This first part of the warm up emphasizes the reason subtracting is equivalent to adding the opposite, because both have the same effect. This is often left out of textbooks. I expect their explanation to say something like "taking out one type does the same thing (increase or decrease) as putting in the opposite type". If students are having trouble with this I would ask them:
"If our recipe tells us to decrease the temperature by taking out hot cubes, but there are no hot cubes to take out, what could we do instead to decrease the temperature?" (add cold cubes)
Once this is clear I uncover the four subtraction problems at the bottom
-2 - (-3) -5 - (+2) 3 - (-6) -4 - 6
I ask students which two are taking away cold cubes. For each one I ask them
"If there are no cold cubes to take away what could we do instead?" (add hot cubes)
For the first one I make the change -2 + (+3) and ask them "is that what it would look like if we added hot cubes instead?" For the second one I ask them to tell me what it would look like if we added the hot cubes instead and make the changes as they suggest them. I follow the same questioning for the last two. I want them to understand this as an option and a tool for them to choose. I have so often seen students in the past ask if they are allowed to change the subtraction on certain problems. What this tells me is that they don't view it as a useful tool for them to use, but a rule for them to follow only when they are told.
For these I ask students not to solve the problems, but just to change them into the simpler equivalent addition problem. As we go through them I relate each problem to the context of hot and cold cubes to reinforce their equivalence. "adding cold cubes is the same as taking away hot cubes", etc.
I start with subtracting negatives because they are the easiest to change:
-3 - (-7) +4 - (-3) 5 - (-2)
If students change the first number I tell them that is not the one being subtracted but the one being put in. Then we move on to subtracting positives since the sign may or may not be there. When the positive sign is there students are sometimes reticent to scribble it out and when there isn't a sign there they often don't change it.
-3 - (+4) 2 - (+5) -3 - 3
I expect some students may change to addition, but may not change to the opposite. In this case I ask them if taking way hot cubes it the same as adding hot cubes. I may need to ask them to think about what we, as Mathmaster Chefs now, can do instead of taking out hot cubes. When they tell me we can put in cold cubes I go back to the +4 and ask how we change those to cold cubes (make it negative). I may need to do this several times. Since the last one doesn't show the sign on the positive three I expect to have to ask them what kind of three it is in order for them to realize they have to make it negative and I may do another example like it.
For their homework they only need to recognize the equivalent problems, not actually change them so if they are still making some mistakes today that's okay.
For the remainder of the time I want them to work together on their homework subtraction patterns. This is the first time they are doing mixed addition and subtraction that are not in separate sections. I also have not given them number lines and I suspect many students will try without them.
As students are working I circulate to see who is using model and who might be trying them mentally and also to look for a couple of mistakes. Some students may be adding addends with opposite signs (-3 + 5 = 8) and subtracting addends with same signs (-3 + -5 = -2). I would ask these students to show me with a number line or other symbols so they can see that when signs are opposite they cancel out which takes away from the addends and when signs are the same they add on to what was already there. I also expect students to add for the subtraction problems (-2 - +5 = 3) instead of subtract. I point out that we are not adding the hot cubes but taking them away and work with them on the number line. I would also have them reread the directions which tell them they can look for the problems that give answers to other problems. If they want to do less work they will have to look for the equivalent expressions.