This lesson is intended to help students "discover" the rules for integer multiplication. It provides a template that shows the "anatomy" of a multiplication problem that helps students understand what the problem means in terms of hot and cold cubes. This context helps students understand why, for example, a negative times a negative turns out positive and helps them remember the rules when they forget. The rules for integer multiplication are confusing and when students just try to memorize them without being given a context that helps them make sense of it they remember them incorrectly or incompletely. (mp1) They are often unsure about the rules and mix them up with addition as well.
The first question in the Warm up multiplication 2 asks student how we could lower the temperature 30 degrees by first putting in bunches of cubes and then by taking out bunches of cubes. Answers may vary and I take the first couple to review how they know that this will "lower" the temperature and how they know it will lower it by the 30 degrees. For example, they may say that we could "put in 2 bags or 15 cold cubes" or "put in 3 bags of 10 cold cubes". I would underline "put in" and "cold" as they explain why it will lower the temperature and then underline the numbers as they explain that multiplying shows how much the temperature will decrease. This helps them understand the "template" for discovering the multiplication rules in the exploration section of this lesson.
The second problem in this warm up helps them relate the numerical expression to the context of hot and cold cubes. It asks what they think the recipe +4 x (-20) is telling us to do. I remind them that when I am asking them what they think that I don't expect them to absolutely sure. This helps them feel safe sharing their ideas. I write down what ever suggestion they start out with. I expect the numbers will be right, but they may get the type of cubes or whether they are being put in or taken out wrong. A typical answer might be "it is telling us to take out 4 bags of 20 cold cubes". I ask students for thumbs up or down and I expect it to be split, because no one is sure. I underline the 4 and the 20 and ask if we agree on this part, which I expect they do. Pointing out first what is right about the response helps students feel more comfortable sharing, because they are not completely wrong. I would also ask what might have given the student the idea that there are 20 cold cubes. Students would identify the negative sign in front of the 20. I then circle the positive sign in front of the 4 and say that this part must then be telling us whether we are putting in or taking out the cubes. By now most students have figured out that the cubes are being put in, not taken out. This also helps students feel more comfortable making mistakes. Instead of the teacher pointing out and correcting the mistake the students themselves figure out the correction. Even the student who made the mistake in the first place has figured it out on their own, so he/she doesn't feel the sting of making the mistake, but more the joy of learning.
Now I project the Number of bunches template or "anatomy of a multplication problem" on the white board, which should remind them of the problem we just "disected". It shows that the first factor tells us whether we are putting in or taking out bunches of cubes as well as the number of bunches being removed or added. If the first number is positive bunches are being added, if it is negative they are being removed. The second number tells us whether they are bunches of hot or cold cubes as well as the number of cubes in each bunch. This template is also printed in their homework to help them "discover" the rules for multiplying.
I write 4 problems, one at a time on the board under the projected template and ask students to use the template to figure out the product. For each problem I write down what it is telling us to do in order to figure out the produce. For example:
+ 4 x (+2) says "put in 4 bunches of 2 hot cubes", which is +8 because it raises the temperature 8 degrees. Students will need to be walked through how to use the template to "translate" the problem with prompting questions like "if the first number is positive that tells us what?", etc. When figuring out the product I find it helpful to think about it in two parts, first the numeric part then the sign. I prompt with "4 times 2 is what?" and then "putting in hot does what?"
Other problems that I do this will are: +3 x (-5), -2 x (+5), -3 x (-2) because it goes through all the permutations of factors.
I leave the template and the responses to the four problems on the white board while they are figuring out problems on their white boards. I give them one problem at a time to work on with their math family groups and they raise their boards to show me all at the count of three. That way they don't all copy off of the first few people who raise their boards and no one can opt out. I have them work on problems like the ones we worked out together using the template:
-4 x (+2) -3 x (-5) 2 x (-5) -3 x (-3)
They use the template in the homework multiplying integers to "discover" the rules for multiplying integers in the first section and then practicing with them in the last section.