The reason I teach multiplying this way is so students have some understanding of why the "rules" for multiplying work, so that they have a way to remember them when they forget. If they can relate the idea of multiplying a negative by a negative with "taking out cold cubes" then it makes sense that the product would be positive, because taking out cold cubes raises the temperature, which is a positive change. They may even start thinking more abstractly by thinking of it as "taking out negatives leaves positives". Being able to make some sense out of the abstract mathematical concepts helps student rely on understanding rather than memory. (mp1) These students struggle to remember their multiplication facts, let alone the rules for multiplying integers. They need some way to make sense of the new ideas.
Before going over the warm up multiplying integers I go over the homework premultiplying from last night, because I want the students to notice how I count the addends in order to connect the repeated addition to multiplication. I also want them to see the way I use the "open" number line to reinforce the idea of multiplication. I tell them ahead of time to look for what I do on each problem when I figure out the answers. I tell them it's subtle and they might miss it, but if they look carefully they might catch it. I go over it under the document camera so they can see me counting with my pen. Some kids may say I'm adding, but I ask if they think I am saying "5, 10, 15, 20" in my head. Most of them say they think I'm saying "1, 2, 3, 4", so we clarify that I am counting. I ask how that helps me find the answer and someone will likely say it tells me how many times to add the number or what number to multiply by. Some students did each "jump" individually on the number line, so I also want them to see the benefit of the open number line homework premultiplying completed so they can use this as a simpler more useful tool.
I demonstrate the warm up in the same way that I went over the homework, first by counting to find the answer and then showing the jumps on the open number line. I ask what multiplication problem I might have done and I might ask a student to come up and demonstrate the number line. The last question on the warm up says that Mathmaster Chef put in two handfuls of cold cubes and lowered the temperature 30 degrees. It asks how many cubes could have been in each handful. I would write down any combination that equals a sum of 30. However, because of the problems from the homework and the warm up I expect them to come up with 15 & 15 pretty easily. This sets the stage for the powerpoint exploration on multiplying integers.
This powerpoint Mathmaster Chef Multiplies follows another that I used to introduce adding and subtracting integer in earlier lessons (Mathmaster series). On slide two I tell the story that sometimes Mathmaster Chef needs to make big temperature changes all at once, like increasing the temperature 100 degrees or decreasing 50 degrees. He doesn't want to put in or take out one cube at a time, because it would take far too long. Instead he has his assistants package cubes in bags like the ones in the slide and he can add them in bunches.
Slide 3 gives an example of raising the temperature 20 degrees. It shows 4 bags containing 5 hot cubes each being added to the pot. I ask students to explain what is being added to the pot and ask how they know it will increase the temperature 20 degrees. I expect them to say that adding hot cubes will raise the temperature and that 4 times 5 is 20. If they don't say both I would prompt by asking "how do we know it will raise the temperature?" or "How do we know it will be 20 degrees?".
Slide 4 gives students a chance to work together to come up with some ways to increase the temperature 40 degrees by putting in cubes. Answers may vary and I encourage groups to come up with as many as they can. I write their responses on the board and ask them how they know it will raise the temperature and how they know it will be 40 degrees. I want them to relate adding hot cubes to multiplying a positive number by a positive number. When they tell me the multiplication to support their answer I write that on the board and when they give me the equivalent method, for example "adding 10 bags of 4 hot cubes" and "adding 4 bags of 10 hot cubes", I want them to remember the commutative property. When I ask them to prove the second method they will say 4x10 and I point out that it is the same reason the first one works out. For subsequent pairs now I expect some of them to point out that it is the same reason as it's equivalent.
Slides 5 - 10 go through a similar process with adding cold cubes and taking out hot and cold cubes. A student may start refering to cold cubes as negative, for example "putting in 6 bags of -10". If a student does this I would ask the kids why this makes sense and encourage them to start representing the ideas mathematically.
I make big flash cards out of recycled manila folders cut in half. Today they would practice adding and subtracting integers. I have students stand behind their desks for flash cards. They do not raise their hands or call out the answer. I choose who to call on and I expect their answer in one second or less. This theoretically forces everyone to do the problem and get the answer ready in their head. Once a student gets the answer right they sit down and are done (most keep doing the problems in their head anyway). If a student doesn't know the answer or gets it wrong I ask another student. Once a student gets it right I go back to the last one and explain it or have another student explain it. I like to leave the students who struggle standing a little longer, because they get more practice and they can check themselves on each problem another student does. This is a great way to review past or preview prior knowledge for future lessons.