This lesson follows an exploration of negative numbers using a context provided by a story about a chef that uses hot and cold cubes to adjust cooking temperatures. (Cooking with Mathmaster Chef series) Students have learned that adding cold cubes or removing hot cubes will decrease the temperature and vica versa. This model helps them make sense of what is happening in integer addition and subtraction. At this point students have used the number line to model problems that are given verbally, have written mathematical models for the verbal models. Today they will be given the mathematical model and be asked to make sense of it using a verbal model and a number line. Students are aware that adding the opposite is equivalent to subtraction, but have not used it in numeric expressions. Today they will translate the numeric expression into a verbal model using the context of the hot and cold cubes, and use a numberline to solve.
I tell a story that there were once three little bears, mama bear, papa bear, and baby bear. Papa bear liked his chocolate really hot, mama bear liked her chocolate frozen, and baby bear liked his chocolate somewhere in between. The warm up gives integer problems that result in the three temperatures and students need to determine which is which. I purposely use numbers that are larger than they are used to to force them to consider what is happening with hot and cold cubes. They have had some experience with integer addition and subtraction, but they are usually taught short cuts and don't understand why they work, so they remember them incorrectly. Sadly they are very resistant to letting go of these shortcuts and they continue trying to use them incorrectly. This lesson sets them up to predict incorrectly and then be surprised.
This warmup tells students:
Papa Bear likes his cocoa hot, Mama Bear likes her cocoa frozen, and Baby Bear likes his at exactly zero degrees. Who gets which cocoa?
The choices are:
I may need to scaffold by asking them if they think the hot chocolate will have a negative temperature or a positive temperature? Really positive or just a little positive? Similarly for the iced shake.
I suspect many will think that +50 + (-50) will be the most positive and think this is the hot chocolate, because it is addition and because they will mistakenly think that 50 - (-50) is zero. I ask them to explain their reasoning using the context of hot and cold cubes:
As I circulate I ask if they are coming to some conclusions about who gets which cup of cocoa. I suspect there may be some dissagreement and some surprise as their results don't match their predictions. I want to address the surprise and ask them what happened that they didn't expect and what made them change their minds. These are the moments in which students are confronting misconceptions like "adding always makes the numbers bigger" and "subtracting always makes the numbers smaller."
In this activity students work in pairs or trios to sort cards 3 bears that result in either a positive, negative, or zero. Students are told that each numeric or variable expression makes cocoa for either Papa Bear, Mama Bear, or Baby Bear and their job is to sort out who's cocoa is who's. I circulate to check in on their progress. I am looking for partners who look like they are having trouble making the categories or who look like they are finishing.
For those who are struggling I ask what temperature each bear likes to help them remember what they are looking for. I might then focus their attention on a single card (probably one that equals zero) and ask
Partners that look like they have all or most of their cards in categories I ask if there are any they are unsure about. If there are I ask some probing questions to see what makes them unsure, I may ask them to think about hot and cold cubes and set up a number line to help them decide. Then I walk away to give them time to work it out before I come back. If students are sure about all of the cards I will point out any mistakes by saying "there are two cards wrong in this category" for example. If the cards are all correctly placed I tell students I am going to take some away and they need to try to explain how they know the remaining cards are in the correct category. If partners still have any wrong after given a chance to correct them I make sure they are among the cards they have to explain.
This is when I want students to share their thoughts about the cards I asked them to explain. I expect students to model their understanding with verbal explanation, with examples (for the variable expressions), and with number lines. I ask them to show us on their white board, the big white board or projector, or I model for them as they explain.
I expect to hear things like "we know that 40 - (-20) will be positive because subtracting negatives is like taking out cold cubes, which increases the temperature, so we move to the right on the number line". They may use the number line as they explain or I might model it for them as they explain. I would ask questions like "why does it make sense that there is a negative number in this problem, but we never went to the left on the numberline?" It may take input from several students.
I want to make sure students explain addition problems as well because integer addition is also counterintuitive for students since it seems to contradict what they already know about addition. They are used to adding numbers in addition problems and they are used to getting a sum that is bigger than both the addends. I ask them to explain how they know -40 + 10 will be negative and -10 + 20 will be positive. I also ask them why it makes sense that when adding a positive and a negative number together we didn't end up with a bigger number? Hopefully someone will mention that it is like subtracting, which I ask them to explain. I would ask what got taken away? It would be the +10 in -40 + 10 and the -10 in -10 + 20. If no one mentions that it is like subtracting, I would ask why we don't add the 40 and the 10 in -40 + 10. As part of the explanation I would ask how we could represent these on a number line and then highlight that part the cancels out.
Another expression I want to discuss is x + (-x). With this one I ultimately want to hear that any number when you add it to it's opposite will cancel out or add up to zero. I might ask here "why does it make sense that we can add two numbers together and get zero?"
Students often want to call x and -x the "same" number, but I make sure to draw a number line and point out they are on opposite sides of zero, so we call them opposites. Because it is strange to think of the "same" number as opposite I want to explore this "sameness" with opposites. My students do not have a "let's try it and see" attitude when they get stuck, so I want to encourage them in this manner. I ask them if the problem would be easier to think about or explain if the variables were numbers and suggest we try that then. I want them to have the freedom to work on a simpler problem when they get stuck to help them persevere.