I have very strong opinions about manipulatives. I like them. I really do, as long as they are natural, demonstrate a clear pattern, or give students an alternative way to work out a problem. To me this seems obvious. Why would you have students work with manipulatives if they do not clearly illustrate and support the objective of the lesson? Unfortunately in BC our curriculum is trying to shoehorn manipulatives into the curriculum. I get their effort to make things more concrete, but I feel as if manipulatives should be a supplement, not a requirement. Regardless of this opinion, I still have to find ways to make these manipulatives accessible, clear, and engaging, and this is my effort to do so.
The Battle of the Integers (Army Men kick inergers chips arses)
Ok, yes I am a loser. I just need to get that out of the way, but hey it works! How do I do this in class? Here goes.
Start the video: The opening sequence is enough time to hand out bags of army men. I hand them out one ziploc bag per pair, and worksheet/notes per student.
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Then I press pause when "Integer addition" shows up on the screen. This is the easy part. We talk about what happens in war. In integer war they always kill each other off in pairs (zero pairs to be specific). Kids catch on to this pretty quick. So we can begin with our first example. This is our basic addition example. Have students start with three positive army men, then add three more. Easy peasy! I play the video for a few seconds to let those three positive guys come in. Now we have six, so ask the students what if the 4 red guys came to the battle scene? The students recreate the battle scene, write down there answer, and then view it on the video! (This involves pausing and playing, and you really have to know where to stop it).
Then students work out their own examples, they can use the army men, and they plot their actions on a number line. I have tried to make the work sheet one that slowly removes steps (I think I saw Jason Buell do this, and I love it). By the end students get a pretty good grasp of integer addition. It is just practice (I pick integer addition war, or a pre-alg with pizazz that I snagged from Dan Meyer).
Where battle starts getting tough... (AKA Subtraction, and its difficulties...)
The next day we start with some small activity (probably war), and then we jump into subtraction. I remember when I was a kid, and learning about subtracting a negative number I felt a little like this...
Subtracting negative integers is not intuitive (to me at least) in the slightest. In fact it was not until my modern algebra class in third year university, that I actually understood why a negative times a negative is a positive. It is thus not a surprise that this was the hardest part for me to teach, but after a few tries I finally got it.
I start by asking the students to make a battle situation that will result in a zero, but uses ten men (in total). Students work through it, and with a little bit of help and guiding questions, they get 5 red and 5 blue. I then introduce them to the idea of retreat. Subtraction is like retreating, so I tell them,"from this battle, retreat (or subtract) +3 guys. What do we have left?"
Students show me and tell me, "Negative 3! You have -3 left!" I get them to write this down (I am working on a worksheet still, but for now blank paper will do). So I decided to pull out my infinite cloner from my Smart notebook, and get them to show me on the board.
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We reset to zero and build it up, make bigger battles, and try questions like 0-(+5), 0-(+10), it doesn't take long to get that subtracting a positive, is like adding a negative. Then I have them reset to zero, and this time, with out removing any men, they have to make the battle scene (I haven't thought of a better name than battle scene) equal to +2. In this case they should have 7 positive guys, and 5 negative guys. I ask them the same thing, "now remove 3 positive guys, (+2) - (+3)," students get really used to recognizing how it helps. The basic steps are as follows (you can do it for any number).
- Reset to zero
- For the first number of the equation, add those army men (start with zero, go to positive, then go to negative).
- Subtract the number (in battle terms, tell which side should retreat, start with positive, work to negative).
- Fight the battle, see what remains.
Go through the combinations, and keep asking students, "Do you notice patterns?" Follow along on the smartboard. Here is an example of what it looks like on a Smartboard.
Move on Soldier!
It took me at least three full lessons of falling flat on my face (having kids more confused than when we started), but this way finally worked. Students could see the patterns and the rules clearly. Kids who could memorize the rules did, those who didn't could make the battle with the army men. I was happy, but now I have to do multiplication. I have some ideas, but nothing that I feel inspired by, whereas I am proud of this. Any tips? I'd love to hear them...
A completely pointless epilogue
Just for fun, I created this, and put it up in my class using block posters