In the previous post, I described a workshop I was leading with seventh grade teachers recently. We were working on a problem in which the Wumps are given hats. The first hat is described by points in the coordinate plane. The other hats are transformations of the original hat and the problem has students investigate the effects on the image of multiplying and of adding and subtracting.
I had 6 small groups of teachers and there are 5 image hats. I had each small group use gridded chart paper and markers to draw the original hat and one assigned image hat on the same coordinate axes, as below.
I asked the sixth group to make up their own rule and to draw the original hat and the image, but not to show the rule on their chart paper.
We put the drawings on the classroom wall and gathered to talk about them. I asked the teachers to match the hats with the known rules, which went well. The group came to consensus about which was which and observed informally that:
- adding and subtracting moves the original figure around, and
- multiplying changes the size of the figure
I then asked teachers individually to think about what rule might have created the sixth image (see below).
Based on the rules above, I expected that most of the teachers would create a rule such as (2×-1, y+3). This is based on the two observations above, but applied in the opposite order. The most obvious change from the original to the image is that the image is twice as wide, hence the “2x”. The other change is that the hat moved. The left corner of the image is 1 unit to the left of the original’s corner, and the image is 3 units higher. This accounts for the “-1″ and the “+3″.
I then asked the group that made the hat to reveal their rule. Much to my teachers’ surprise, the rule was (2×-2,y+3).
2x-2, not 2x-1
This is what my friend Jim had been asking me about over the years. The reason 2×-2 is correct is that observation number 2 above is not complete. Multiplying changes the size of the original, and by necessity it moves the points around. Consider this: if two points define the left and right bottom corners of the Wump hat, and if the Wump hat doubles in size, then those endpoints cannot possibly both be in the same place. For then they would be the same distance apart and the hat could not have changed size.
So multiplying changes the size, but it also moves the points around. The hat moves to the right as a result of the “2x” part of the rule. So we need to subtract 2 (not 1) in order to shift the hat to its final resting place.
I felt that I had created a wonderful moment where teachers were ready to learn some mathematics. They had made a prediction and they had been presented with evidence that these predictions were incorrect. In resolving such conflicts, we have opportunities to learn.
A close look at the hat in question will reveal that there is something unexpected about the diagram. The scales on the two axes are not equal. This became an important point in the ensuing conversation and I’ll examine that in the next post.