How Do You Know?
Lesson 10 of 24
Objective: SWBAT explain and use evidence to support their explanation of when a sum will be positive, negative, or zero.
The purpose of this lesson is to slow students down and force them to think about the problem and make sense of it before jumping straight to a solution. If students don't take the time to make sense of integer addition they will not recognize how different it is from "positive only" addition. If students rush through and use their prior knowledge from positive number addition they will make mistakes because integer addition does not follow the same patterns. Numbers don't always get bigger when we add integers. Sometimes we add the numbers and sometimes we subtract them to find the sum. Students need to make sense of these patterns either by modeling or using a familiar context to help them see and explain the patterns.
This warmup gives students integer addition problems and asks how they know each will be zero, positive, or negative.
I ask how students know that -8 + 8 and 100 + (-100) will both equal zero. I am looking for as many explanations as possible. Some explanations may include:
- "putting in the same amount of hot and cold cubes will make the temperature zero" (Mathmaster Chef series)
- "if you go a certain amount in one direction on the number line and then back the same amount in the other direction you land back at zero"
- "opposite numbers cancel each other out"
Students may need a reminder that these are called "opposites" if some of them are still calling them "the same but with different signs". I like to write an example of opposites on the board and ask kids for words to describe them underneath, like left & right (number line), negative & positive, cold & hot (cubes) and ask what would describe all these pairs of words. (opposites) I definitely want to recognize that element of sameness when we discuss distance or quantity, but I want them to call these numbers opposites and relate them to distance on a number line.
I ask how students know that -5 + (+10) and 100 + (-1) will both be positive. Students may say that:
- "more positives are being added than negatives"
- "more hot cubes are added than cold cubes"
- "we are going further in the positive direction than the negative"
- "the positive number is farther from zero"
- "since we are going farther in the positive direction, when we go back in negative direction we are still on the positive side"
- "not enough to get all the way to the negative side"
If students say the positive number is larger or bigger I like to point out that their thinking is correct, but the word bigger is misleading because a positive number is always "bigger" than a negative.
I ask how students know that -15 + (+10) and 7 +(-30) will both be negative. I expect the explanations to be similar in form to the last question just in reverse.
I ask how students know that -8 + (-8) and -3 + (-200) will both be negative. Students may say that
- "adding cold cubes and adding more cold cubes just makes it colder"
- "we go to the left for the first negative number and keep going left"
- "we are adding more negatives to negatives so nothing cancels out"
- "we never get into the positive side of the number line"
Asking "how do you know" is the key question that gets students to develop reasoning practices and develop number sense!
Now that students can explain how they know a sum will be positive or negative I want them to notice the pattern addition pattern explanations that tells them whether to add the numbers or find the difference.
For example, we know that -10 + (+2) will be negative, but what do we do to figure out exactly how many negatives? I want them to notice that we find the difference when we are adding a positive to a negative number. It might help to circle just the 10 and the 2 and ask what we do with these two numbers to find the sum of -10 and +2. I emphasize that we already know it will be negative so we can ignore the signs. Once they figure out that the sum is -8 they should decide that we subtract the 10 and the 2. If you think they won't figure that out you can just give the the sum of -8 before asking what to do with 10 and 2.
I ask then what we do with the 7 and 5 to find out how many negatives we end up with in -7 + (-5). They should say they add the 7 and the 5 to find -12.
It is critical to now ask them why we would subtract for one addition problem, but add for the other. I tell them this seems really weird that we do two different things for addition problems and its different from any other addition they have ever done. I ask them to have a discussion in their math family groups. I remind them to make sure they use number lines or symbols or hot & cold cubes to help them decide. You have to allow for a slow evolution of ideas in group discussions. To help level the playing field I like to circulate and call attention to an idea that comes up in a group. It might be something like "this family is talking about how in one problem some of the positives and negatives get canceled out", "Isabella is showing on a number line what happens to our original steps when we go back in the opposite direction", "Juan is showing how when we are going further in the same direction no canceling happens", "this family is talking about what happens when we add hot cubes and then add cold cubes".
In the end I want a class discussion to bring out the ideas that when we add a positive to a negative they cancel or take away from each other, which I would model on a number line and with plus and minus symbols. I would ask what got canceled or taken away? I might ask what part shows zero. I also want them to share that when we add integers with the same sign, either both positive or both negative, that no canceling happens, we are traveling further in the same direction, so getting more and more negative.
Normally I don't allow this long for flash cards, but this is a big transition point when they are able to start doing integer addition mentally and I want to have two complete rounds. I ask students to pack up and stand behind their desks. I hold up a flash card. Students do not raise hands or shout out. I call on a student of my choice and expect an answer in 1 second or less, which means theoretically that everyone needs to do the problem in order to get the answer ready because I don't give any think time after I call their name. I like to leave struggling students standing a while longer for two reasons. One is that they get to see and hear the right answers and possibly explanations first and they also get more practice before sitting down. Once a student gets the right answer they sit down and are done for this round. If a student gets the answer wrong or says they don't know I move on to another student. After someone has gotten it right I go back and explain it or have a student explain it.
The cards today have integer addition problems adding two numbers.