Patterns in Mutliplication and Division
Lesson 23 of 24
Objective: SWBAT multiply and divide integers by competing with one another in a game.
This lesson follows up on a game students learned in an earlier lesson (Integer Product Game). It gives them more practice with integer multiplication and division and helps them internalize the "rules" for multiplying and dividing. This is important for them to develop automaticity and not have to think about and make sense of each and every problem. The lesson is also designed, however, to help them to relate to the context of hot and cold cubes when they are unsure of or can't remember the rules. While it is important for students to develop automaticity with basic calculations it is also important for them to rely on understanding rather than memory. Understanding why and how the mathematics works is much more reliable than memory.
- Which two ways can you claim the -8 spot?
- Which two ways can you claim the +9 spot?
- If paperclips are on the factors -3 and -4, which one would you move and where to claim the +24 spot?
I also place paperclips on the factors -3 and -4 to prepare them for problem three. Students begin the warm up as I circulate to check in homework.
Before we go over the warm up we go over last night's homework and I ask them how many of the 16 problems they think are division problems. Answers may vary, but I want them to see that all of them could be considerred division problems because in each we are looking for a factor not a product. I would ask students to explain why they think a problem in a certain section is a division problem homework multiplying and dividing notes. The section with division signs is obvious, so we only have to explain the two sections without division signs.
The first two warm up problems ask what two ways they can claim the -8 spot and then the +9 spot. The only two ways possible on the board are -4x2 & -2x4 and -3x-3 & 3x3. I ask these questions for two reasons:
- One is to get them looking for factors instead of products, because that is when they are dividing.
- The other reason is because in their group discussion when they are working on the warm up I want them to start discussing the patterns in the signs of the factors that produce positive and negative products.
As students are discussing I might hear:
- "the factor has to be positive because..."
- "one of the factors has to be negative because..."
- "if one factor is negative the other has to be ____ because..."
Not only does this help them generalize the patterns into rules, but it also helps them engage in argumentation(mp3).
Problem three asks them which paperclip they move and to where if they are on the factors -3 & -4 and they want to claim the +24 spot. (move the -3 to -6) When we go over this one I ask what the division problem was that they did when they were looking for the factor -6. Someone may say -4 x ____ = +24 or +24 / -4 = ? If no one suggests the later I would ask what else the division problem might look like.
I tell students we are going to play the Integer Product Game again today. I tell them that they might forget the rules for multiplying especially since they just started and that we are going to demonstrate how they can use the hot and cold cube context to help them remember whenever they forget.
On the board I write:
- x + = (a negative times a positive is...)
+ x - = (a positive times a negative is...)
- x - = (a negative times a negative is...)
+ x + = (a positive times a positive is...)
As I go over each one at a time I relate it to the context of hot and cold cubes. For example, with the first one I point to the negative sign and say that since the first factor is negative that means we are taking out bags of cubes and since the second factor is positive that means "hot cubes", "so we are taking out hot cubes, which is..." (negative since it lowers temperature). This goes really quickly because I speak in partial sentences and have them to complete my sentences for the next ones: "if the first factor is positive that means....", etc. In order to shift the level of abstraction I go back over them and use the terms negative and positive in place of hot and cold cubes. So I would say "putting in negatives leaves negatives" and "taking out negatives leaves positives", etc.
Next I want them to notice the patterns so I circle the two with factors of opposite signs and then the two with factors of the same sign and ask them to look for a more general pattern. I point to the top two and say that "both of these products are...." (negative) "so, if one factor is positive and one is negative the product will be..." (negative) Similarly, for the next section I say "if both factors have the same sign the product will be..." (positive)
Then I leave the rules on the board while they play the game.
For the remainder of class the students play the Integer Product Game Board. As they play I circulate and highlight good blocking and share realizations when someone makes them. One thing that often comes up after they have played for a while is that you can't always make the product you want immediately, because you can only move one of the paper clips so, at least one of them has to be on a factor that will result in the desired product. If neither is on one of the necessary factors you can't make the product you want right away. What this means for them is that they want to block early.
Another great thing about using this game to help them learn and practice integer multiplication and division is that they correct each other. If they put their token on the wrong product they lose that turn, so they have an interest in catching each other in a mistake. When they do catch each other in a mistake they have to explain what the mistake was. This is one way to engage students in argumentation and critiquing the work of others (mp3).