Which Way Do We Go?
Lesson 5 of 24
Objective: SWBAT use a number line to solve integer addition and subtraction.
This is a lesson that requires students to make sense of the problem before solving it. Often with integer operations students rush to begin before they have developed a negative number sense. Thinking about what the problem could mean in terms of hot and cold cubes (Cooking with Mathmaster Chef series) and temperature helps students figure out which direction to go on the number line. Students need experience with this in order to internalize the pattern that adding one type (negative or positive) of integer is the same as subtracting the other.
The warm up has students focus on interpreting the numeric expression in the context of hot and cold cubes. The notes section has them focus on what the hot and cold cubes do to the temperature, which determines the direction to go on the number line. It is important to slow students down here, which is why I give them sentence frames and number lines in their notes. This forces them to think about the problem and make sense of it before trying to solve it.
In this warmup which way on a number line students are given three integer addition and subtraction problems and are asked what they might be telling us to do with hot and cold cubes (Cooking with Mathmaster Chef series).
The first problem, +5 + (-3) tells us to ""put in 5 hot cubes and put in 3 cold cubes". It might be necessary to go over each symbol by asking what part of the expression represents putting in, hot cubes, cold cubes, etc.
The second problem, -2 - (+4) could be interpreted in two ways because the negative on the 2 could represent "taking away" or "cold cubes". The fact that it can only be one OR the other needs to be clarified and pointed out that there would be two "negative" signs (one for taking away and one for cold cubes) if it were indicating "taking out cold cubes". This could be interpreted as "putting in 2 cold cubes and taking out 4 hot cubes" OR "taking out 2 hot cubes and taking out 4 hot cubes". Emphasize that the two options would be equivalent, by relating it to what happens to temperature. This will be the focus in our notes section.
The third problem, 3 - (-2) can only be interpreted as "putting in 3 hot cubes and taking out 2 cold cubes". Point out the two "negative" signs here to reinforce the point made in the second problem. Sometimes students may get confused when there is no positive or negative sign as with the 3. Remind them that with no sign, it can't possibly be "take out" or "cold cubes" and that no sign means positive.
Provide students with the guided notes White board notes integers. Students are first asked to predict the answers to the four problems before we complete the sentence frames and the number line. I expect them to predict incorrectly and be surprised by the first two. When students see subtraction they often automatically find the difference between the numbers which does not always work with negative numbers.
Then we go over each problem one at a time together. I ask what the numeric expression could be telling us with hot and cold cubes and ask them to explain where that is represented in the expression. Next I ask them what effect the hot and cold cubes have on the temperature and what direction that tells us to go on the number line.
The first problem, -6 - (-3) can be interpreted in two ways. The -6 could be "putting in cold cubes" or "taking out hot cubes", the - (-3) can only mean "taking out cold cubes". Ask for alternate interpretations so that both possibilities come out in order to reinforce their equivalence. Have students explain why it makes sense that they are equivalent. Make sure to ask what "taking out cold cubes", for example, does to the temperature in order to determine what direction to go on the number line.
When moving in opposite directions on the number line I ask my students what happens to the original "steps" we took. This helps them notice that moving in opposite directions on the number line "cancels out" the original moves. Ask who was surprised by the answer and got something different than they predicted. I expect some students may have gotten +3 because they are incorrectly remembering rules (two negatives make a positive). After working out the number line ask students to put their final answer on a white board and hold it up on the count of three.
Go through the same process for the next three problems. The second one is probably the one most students will be surprised by because it seems to go against what they already know about subtraction. They expect to find the difference when subtracting, but this problem added the two numbers. Explore this for a while in the hot and cold cube context and on the number line. Reinforce that "taking out hot cubes" decreases the temperature so that we move in the same direction on the number line as when we "put in cold cubes".
I want students to work on the homework integers sentence frames number lines for a few minutes in class, because I know that once they get home a lot of them just rush through and may skip the sentence frames or not use the sentence frames to think about what it does to the temperature. I tell them to start on number 5 for two reasons. One is that it is on the backside of the page and they often forget to turn over their homework and see that there is a second side. These are also the subtraction problems, which, if they rush through them, they will likely get wrong and never gain any negative number sense.
I circulate and make sure they are interpreting the expressions correctly. More importantly I want to see that they are taking the time to use the hot and cold cube context to help them make sense of the math. If they are going the wrong direction on the number line I know they are not thinking about what the hot and cold cubes are doing to the temperature. The answers I expect to see if they are making these mistakes are -6-(-4)=-10, 3-(-4)=-1 or 1, -2-(-6) =8 or -8 or -4, and 3-5=2. If I see these answers I go back and see if the sentence frames are right. If they are then I know they are not taking the time to think about what "taking out hot cubes", for instance, does to temperature. As soon as I ask this question they usually see that they've gone the wrong direction on the number line. If not, I ask which direction that tells them to go. I make my questioning fairly succinct ("what does that do to the temperature?" "so, what direction is that on the number line?") so that students see that this "extra step" doesn't take long.
After instructing students to pack up I ask them to stand behind their desks. I use recycled manila folders cut in half as flash cards which I hold up for the whole class. Students do not shout out and do not raise their hand. When I call on a student I expect their answer in 1 second or less. Theoretically, this forces all of them to do the problem in order to get the answer ready in their heads. If a student answers incorrectly or doesn't know the answer I call on another student. I may ask that student or another student to explain it, so the first student understands. Once a student gets the answer right they sit down and they are done. I like to leave struggling students standing for a while so they get more exposure. Usually, even after they sit down, students will continue doing the problems, but for those that I think are a little more apathetic I would leave them standing longer as well, because once they get one right they are more likely to stop participating when given the chance.
Today's flash cards would have an integer on it and students would be asked to say what it could be telling us to do in terms on hot and cold cubes. For example -5 could be "putting in cold cubes" or "taking out hot cubes". If they get it right I would ask them what that does to the temperature before having them sit down.
I want to reinforce that this is the important process of the homework and also to show them how little time it takes. This is the part that requires thinking and helps them make sense of the math that otherwise seems contrary to their prior experience.