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 …
June 2, 2014 at 11:40 am
>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.
Did anyone rotate the inner square 45º? By “scaffolding” the solution in terms of r, you shoehorn students into one contrived solution pathway and more elegant primary school solutions are easily missed.
June 2, 2014 at 12:05 pm
No one did. Thanks for the reminder that providing the scaffolded questions even if later in the task can change what students see and do. I just read Dan’s post “You can always add. You can’t always subtract.” http://blog.mrmeyer.com/2014/you-can-always-add-you-cant-subtract/ You both always give me something more to think about to make my students’ experiences more authentic.
June 6, 2014 at 9:33 am
I really like opening up this task to student noticings and wonderings. A similar technique I’ve been using for some problems is to give students a figure that has a number of angles/side lengths/etc that can be determined, and see how many they can find. Sometimes I have a specific goal I want them to get to, other times I just want to see how much they can fill in. I really like having students start these on their own, then share what they’ve found with a partner — I’ve found it creates excellent math discussions, and gives students who struggle a chance to answer a few simple questions while working toward a bigger one.
A few examples:
https://app.box.com/s/bcbezbibzlteh1kxvcum from http://mathymcmatherson.wordpress.com/things-ive-made/
The tough part for me is figuring out the best way to facilitate these problems as a class after students work on them — I’ve had students share out one piece of information at a time so that as many people as possible can participate, but I also want to embrace different methods of getting to the answer.
Thanks for your ideas!
June 6, 2014 at 4:24 pm
Hi, Dylan. Thanks for your comment. I just found your blog & look forward to learning from you!
Facilitating class work and discussion around tasks is definitely challenging. What is productive struggle for one student might not be for another, and so while we want all students to have the opportunity to make sense of problems and persevere in solving them, we also want them to learn some math by the time the bell rings.
Have you read Smith & Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions? Thinking through their practices for teachers has made a difference in my classroom.
Also, I like your idea of students sharing out one piece of information at a time. I did something like that with an Illustrative Math task on our last day of class and will blog about it soon.
June 7, 2014 at 5:18 pm
I haven’t read the book, though I’ve read summaries and I really like the way they break down facilitating discussions into 5 key components. Sequencing especially is one I’ve been thinking a lot about. Is the book worth a read beyond the 5 big ideas?
June 7, 2014 at 5:26 pm
I do think it is worth it. I picked up on ways to think about selecting and sequencing that I wouldn’t have thought of without reading through their examples. I’ve thought a lot about sequencing since I read it. It makes me feel like I’m conducting a symphony when it happens during class … which I’m sure comes from their using the word “orchestrate”! But it’s like a symphony that you and your students get to compose yourselves instead of playing something someone has written.