In this lesson students play a game called the Integer Product Game. It involves multiplying factors to claim spaces on a game board. As they impove their strategies from claiming random spaces to intentionally claiming specific spaces to get six in a row or to block their opponent they begin to divide rather than multiply. The purpose of this lesson is for students to realize the connection between multiplication and division of integers so they understand that the rules for division are the same as for multiplication. Many of my students struggle with basic math facts and are intimidated by calculation. The game format takes the sting out of the calculation by putting it in the background and making it simply a tool for them to use in fun. It also motivates them to take an interest in learning the rules for multiplication, because if they get it wrong they lose their turn. After they have been playing the game and having fun with it for a little while I show them that as they have been blocking or seeking a specific product by looking for a factor, they have already been dividing and we discuss the connection between multiplying and dividing. Students seem to gain some confidence and often ask if they can continue playing the game. Before they play the game I tell them they will learn how to divide integers, but first we will have a little fun. In order to play students need a game board, which I have laminated, two paperclips, and a cup of two color chips as tokens.
As I circulate I do a couple of things, I highlight strategies like blocking that I see in each game and I redirect them to the hot and cold cube context. As they have just started working with and "discovering" the rules for integer multiplication they are still unsure. As I circulate they ask me if, for example -2 times -6 is positive. This is when I ask them to think about the hot and cold cube context by asking them what the problem is telling us to do with hot and cold cubes (taking out 2 bags of 6 cold cubes). I ask what this does to the temperature (increases) and whether that is negative or positive. Asking them to think about the context forces them to make sense of the math on their own and it will help them figure out the rules in the future whenever they forget them. (mp1)
The Warm up multiplication patterns asks students to think about patterns in integer multiplication to help them generalize the rules. The first question tells them that Chole has multiplied two number together and gotten a negative answer. It asks them what this tells us about the two numbers. If students are not used to open ended questions they may need to be prompted to try out some specific cases and see what kinds of numbers will work and not work to make a negative product. I would expect students to figure out that one factor would be negative and one positive. I would also ask them to explain how they know. If they are not sure how to explain I would suggest they think about it in the context of hot and cold cubes. Explanations might be along the lines of "putting in cold cubes (negatives) lowers the temperature (negative product) and "taking out hot cubes (positives) lowers the temperature as well". Numbers are purposely left out of this problem so that students can see that any positive times any negative should have a negative product. Briefly I point out that when one factor is positive and one negative the product will be negative. I may do this simply be starting the sentence and letting them finish it. "So, if one number is positive and one negative the product will be...."
The second problem tells them that Alexis multiplied two numbers together and got positive 24. Students are asked what two numbers he might have multiplied. Answers may vary for this one. Write down all the suggestions and have students explain how they know some of them are true. Briefly I point out that when both factors have the same sign the product is positive. I may do this simply be starting the sentence and letting them finish it. "So, if both factors are the same the product will be...."
If students have not had much experience with more general open ended questions you might want to switch the two problems. Problem two more clearly suggests that students need to come up with examples than problem one.
In this exploration students will be introduced to the Integer Product Game. I pair students up and tell them they will be competing with each other in a game similar to tic-tac-toe and bingo. The game board consists of a grid of spaces with positive and negative numbers in it and a list of factors at the bottom. I show them the Integer Product Game Board under the document camera and tell them they win by claiming 6 spaces in a row vertically, horizontally, or diagonally.
Next I go over how they claim a space. When two paperclips are placed on factors from the list at the bottom of the board their product corresponds to one of the spaces and the player gets to claim that space by placing one of their tokens on it. When they correctly claim 6 in a row they win. If they are incorrect and their opponent catches them they lose their turn and no one claims a space.
Lastly, I show them how to begin and how play continues. The first player only gets to place one paperclip and so, doesn't get to claim a space. The second player then takes his/her turn and places the second paperclip on a factor. He/she multiplies the factors and claims the space with that product by placing his/her token on that space. Each player takes turns, but can only move ONE paperclip during their turn. I demonstrate 3 or four moves to reinforce this rule of moving only one.
I distribute game boards, paperclips, and tokens to each pair of students. I also leave displayed under the document camera the answers to problems 1-4 from last night's homework multiplying integers in which they "discovered" the multiplication rules.
As students play the game I circulate and look to make sure students are getting started and are understanding the rules. I expect them to ask some questions like "can we place paperclips on the same factor?" and "what if the space is already claimed by my opponent?" If they ask about the same factor I know they are trying for one of the spaces with a 9 or 16, so they are already starting to divide. In answer to their question I just tell them that it is the only way to claim some of the spaces. If they ask the second question it most likely indicates that they are NOT starting to divide, because they are not thinking ahead for which space the product. I simply tell them they don't get to claim a space if it is already taken. This may be enough motivation to get them to start thinking ahead.
As I circulate I am also looking to see if players are blocking each other or not. If it looks like someone has blocked their opponent I bring that to the attention of the class so they take the hint to start trying that strategy. I will also look for evidence that they are not blocking, for example, if one player has 3 or 4 in a row. I will suggest that their opponent should try to block them. I may even help by asking what product they should try to make in order to block them and then ask them to think about what factors they need to do that. Every once in a while I may ask my "blockers" to raise their hands to encourage them.
With about 5 minutes left I have them pack up and return the supplies and then ask them to raise their hands if they were trying to block their opponent. This is when I inform them that they have already been dividing. They will not understand how this is true and some may even disbelieve me, because I did not directly teach them how. So, I tell them I will prove it to them. I show them a game board with three or four tokens in a row and ask what they should do to keep this player from winning. They will tell me they need to claim the remaining spaces in the row before their opponent can. I ask them how they intend to do this and they go through the process of explaining how they find the factor(s) that will produce the desired products. For example, if they are trying to claim the space -24 and the clips are on factors -4 and -3, they will tell me to move the one on -3 to the factor +6. I ask how they decided on +6 and expect them to tell me that -4 x +6 = -24. I model this on the board by writing -4 x ____ = -24 and tell them that this is the math they did in their heads as they asked themselves "-4 times what equals -24". As they agree I tell them that when they are looking for a factor instead of the product, they are dividing and that the math they were actually doing was -24 divided by -4.
I write a couple more problems on the board and ask them to figure out in their math family groups what division we are actually doing when we look for the factors:
-2 x ____ = +6 5 x ____ = -20
Their homework multiplying and dividing has four division problems. The rest are multiplication like the two problems above, so, really all are division problems.