As we finished up our unit on Geometric Measure and Dimension, we used a task from the Mathematics Assessment Project called Circles and Squares.
This task is no different from many other tasks in scaffolding the work that students will likely need to do in order to answer the final question – what is the ratio of the areas of the two squares. Instead of going straight to the questions, I just showed students a picture of the diagram and asked them to complete the prompt “I wonder …”.
if the right triangle inside the smaller square is half of the larger right triangle. 1
what is the side length of either square or t he diameter\\radius of the circle 1
if the area of the sectors in the circle have the same area as the curved triangle thing 1
what the similarity ratio is of the small square to the large square 1
if there is a proportional relation between the area of the big square and the area of the little square, regardless of the radius of the circle 1
if circles and squares can keep going into each other infinitely 1
what are the dimensions of the two squares? 1
is the small square similar to the large square 1
what the ratio of small to big square is 1
if the area of the circle not encompassed by the smaller square is the same as the bigger squar 1
how the radii of all three shapes are related 1
if the squares are simular. 1
if the area of all of the shapes are related 1
is the area between the smaller square and circle equivalent to the area between the larger square and circle 1
if the smaller square is half the larger square 1
what is the radius of the smaller square compared to the larger square? 1
why there isnt another circle 1
what we could possibly do with all of the possible calculations 1
if the shapes are similar anc how they relate to each other 1
Is the square scale facator 1.5? 1
if the smaller square is proportional to the larger one. 1
why the smaller square is significant. 1
how the areas of the shapes relate to each other 1
if we can figre ot the areaof the space between the shapes 1
if the the smaller square has ((1)/(2))the area of the largest? 1
why there is a circle represented in between 2 squares 1
do all these have the same center 1
I’m not sure what I expected, but I am still always surprised how often what students wonder is tied to our learning goals for the lesson. Next I asked them to estimate the ratio of the area of the smaller square to the larger square.
And then I gave them the handout, which had the scaffolding questions from the Mathematics Assessment Project. As students worked, I sent Quick Polls to assess their work. Students kept working with their teams after submitting their responses. I only stopped them for a whole class discussion when I felt like their responses needed that.
After the first question, I went to the table of the three who didn’t answer correctly to find out their thinking.
After the second question, I stopped them for a moment to ask why the “Quick Poll grader” had marked both of the first two responses correct.
After the third question, I deselected “Show Correct Answer” and asked the teams to decide which expressions we should mark as correct, equivalent to 1:2.
This task provided students a good opportunity to both reason abstractly and quantitatively and look for and make use of structure.
We neglected to go back to the class estimates to discuss how they had done. There’s always next year, right?
And so the journey continues …