If it was still not clicking, I might use the black square in the bottom left hand corner and ask for the side length and area to give them some momentum and information to use to find the area for each color. Once those measurements were better understood, the students would hopefully see how the squares relate and be able to transfer that onto the other squares in the image. I thought I could ask them to figure out the area, or even just side length, of the squares in the top right and bottom right corners. I also planned some questions I could ask them to help keep their struggle productive. I planned to give students time to struggle on their own. These figures require relating the smallest squares to ones slightly larger and then repeating the steps again. The challenge will come when they are asked to figure out the larger square’s areas. I think that the students will have a pretty easy time understanding the right side of this diagram because it’s six 1 foot lines and two 2 foot lines.
The black 4×4 square would also be a good starting point because it has individual blocks to the right and on the bottom, giving it a clear 4×4 layout. I think that my students will try to use the right hand side most frequently because it allows them to count 6 individual blocks and then figure out the 2×2 on the top and bottom. They may also be more comfortable with the concept of area and start with the black 4×4 square. Here’s what I was thinking… Students could use horizontal or vertical lines throughout the picture to establish a baseline for individual sizes of squares.
Students can use the totals of numbers that they come up with to then create ratios of the colors.īefore I tried it with my class, I did some brainstorming around how my students might try to make sense of the problem. However you establish that each small square is 1 square foot, once you do the area of each individual square can be solved with a simple area (l x w) problem. If the 3 yellow square and 3 black squares are six feet in length, then the remaining 4 feet have to be divided up between the lengths of the slightly larger blue and red squares. That might help you to establish a baseline for the different sizes of each square. There are 3 yellow squares, 3 black squares, a slightly bigger blue square and slightly bigger black square that all need to add up to 10 feet. Another approach is to look at the right side of the picture. Since it is a square, that would make it’s area 25 square feet. If you use the center of the picture (the 3 yellow squares, 2 white squares, and 1 big blue square), you might see that the yellow and white squares are five feet in length, which would mean the large blue square also has to be 5 feet in length.
After that, there are lots of ways to approach the problem and different ways to start. It’s impossible in this problem to get any of the answers by only looking at the relationships of the different sized squares.įor me, I used the 100 square feet total to figure out that each side of the figure is 10 feet. Also, using ratio can help to strengthen the idea that geometry and algebra are based on relationships and how those different quantities relate to each other. By giving one square foot for each of the small blocks, it helps to reinforce that idea. I want my students to have a better understanding of area as a concept of something measured in square units. This image came from an email sent out by Stanford mathematics educator Jo Boaler as a conversation starter – “What kind of math questions can we generate from an image like this?” It was also easy to pull in the concept of area as sort of an anchor to bring the other aspects of the question together. The ideas of ratio and relationship are important here because it’s impossible to get the answer to this question without comparing the different sizes of squares of different colors.
I think the visual clues will help students better understand the concept of area. I chose this problem because I love the visual nature and the puzzle-like feel to it.
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