Sometimes CAMI meetings have a mind of their own. This one followed a direction we didn’t expect!
We started with a question about this image:
The catch? We wanted to spark our creativity by exploring the world of wrong answers.
What is the Area? WRONG ANSWERS ONLY
Here’s what we came up with:
- 25 because it’s 5 across and 5 down looking at it sideways, but there are squares in between those rows.
- One side is a mirror image of a pyramid, but there is no pyramid there. Something is separating the pyramid from its reflection. There is a column of 9 squares separating the pyramid from its reflection.
- 0 because if the area was zero, there wouldn’t be a shape at all.
- Base x height is area, it looks like 9 across and 9 tall, so the area would be 81 square units, but that would fill in all the corners.
- 15 because I can see a 3×3=9 inside and I see a 3×2 next to it. So 15 is in it but not all of it. It’s the area of some of the figure.
- 101 because it looks three-dimensional, so the area must be a lot larger than what we actually see.
- The height is 9 squares and 3 squares make the width, so that would equal 27. I made it into a big rectangle. (I think this is a wrong answer.)
- 25 because there’s a 5×5 square inside it but that isn’t all of it.
- 81 – 25, but we don’t remember why.
- 4 because there are 4 points and there’s more than just the points. Also not 8 and not 12 and not 16 or 17 because that’s just traveling along the axes towards the center.
- It seems like an 8-sided star – the points are not all the same size.
Next, we dug a little deeper into this image, comparing it to another that appeared related:
Here are our thoughts:
Same
- Both blue
- Both made out of a grid
- Both have a single square in the middle
- They are the same, but one is tilted.
- Both would be difficult to describe the pattern as it grows.
- Both make me think of weaving – easter baskets and chair seats.
- Both are built of the same units. They could each be stacked into a pile of tiles.
- I see a plus or an X in both of them.
Different
- The area is different – one is smaller and one is bigger. The one on the right is 6 across and the other is 5 across when you tilt your head.
- One is a diamond and one is a square.
- The one on the left is named Marie and the one on the right is named Bob. (The left feels female and the right feels male to me.) [The one on the right is taking up too much space.]
- The one on the right is like a lattice or a chain link fence.
- One is a diamond made of squares and the other is a square made of diamonds.
- The left is flat on the top and bottom and the right is pointy, so I think the one on the left is taller.
- You can walk up the white steps or the blue steps on the left, but you couldn’t walk up the steps on the one on the right. Negative space is more evident in the one on the left. A rock climber might like the one on the right.
These discussions really opened up our creativity and our desire to look at beautiful mathematical objects. Our shared space quickly started to fill with connected images:
Inspired by some of these objects, some of us explored drawing lines inside the image to find triangles, stars, and other shapes.
The group exploring this path especially got into investigating 8-pointed stars and the numbers and symmetries we saw in them. Below are some of the images that grew from that conversation.
What do you see in them? What ideas do they inspire?
Another way of exploring the shape came out of this question:
Starting with one of the original figures, what happens if we fill in the rest of the “square”?
For example:
Looking at just the green, we rearranged the pieces to look similar to the original figure. The figure felt similar, but like an even version (8 + 8 + 6 + 6 + 4 + 4 + 2 + 2 = 40)
We decided to create similar figures so we would have more to explore to look for patterns.
The blue figure below looks like our original figure, but it is 11 squares high and 11 squares wide (as opposed to 9).
The area of the blue is 11 + 9 + 9 + 7 + 7 + 5 + 5 + 3 + 3 + 1 + 1 = 61 squares
The area of the orange is 10 + 10 + 8 + 8 + 6 + 6 + 4 + 4 + 2 + 2 = 60 squares
Again, the area of the blue is a little more than half the area of the entire square, and the area of the orange is a little less.
What happens if we make the height and width 13 squares?
Another pattern emerged when we started playing around with creating a grid in the center of each square unit.
For example:
The area of the blue figure here is 61 square units.
If we look at the grid by itself, we saw it had an area of 40 square units.
We also noticed that it looked a lot like the figure we got we we added corners to our original figure:
There were several patterns that we wanted to investigate about the oddness and the evenness of all these different, yet related, figures.
Here is a figure that is similar to our original figure, except this one has a height and width of 13 squares. We called it Figure 13.
We wondered, if we created a grid in the center of each square unit for Figure 13, would that have the same area as the figure we can build by completing the square for Figure 11, removing the center, and pushing the corners together?
Here’s what we found:
As with many CAMI meeting, this investigation left us with even more questions than we started with, including:
- So far, we only explored odd figures (Figure 9, Figure 11, and Figure 13). What about even figures? What if the height and width are an even number, like Figure 12?
- Why is the area of the figure we made from the corners always 1 less than the area of Figure 9, Figure 11, Figure 13, etc.?
- What is going on with all the oddness and the evenness?
- Why is there a relationship between the areas of the figures we made with grids and the figures we made by pushing together the corners?
- Why do the even figures we explored have a double row in the middle of the height and the width?
Where does your creativity take you when you begin with these figures?
