On the first day of school, students tend to expect the typical ice breaker activities, the lists of rules and procedures, and the “what did you do on your summer vacation” discussions. For better or worse, I like to get right into the math. I want my students to realize that every moment we have together is precious and that bell-to-bell mathematical exploration will be the norm; we will have plenty of time to get to know each other as we work together on challenging tasks as mathematicians.
So my students entered Day 1 and saw a circle of desks with this on each desk:
Once we were all seated, I asked the students if they had any questions or comments about anything they saw. Students shared that the circle arrangement made them feel “more welcome” in class, now that we could all see and listen to each other. I had arranged the desks in a circle as I felt creating an environment where everyone felt like active participants, where no one had their back to anyone else, where participants felt safe and included, to be a necessary component to truly increasing discourse. The students’ reactions told me we were off to a good start.
I then asked my students to look at the paper on their desks and to think about what was mathematically interesting to them about the figure. Many talked about intersecting lines, made assumptions about the shapes being squares, wondered if they were different sizes, contemplated what fraction of the area of the whole each smaller square represented. We cleared up some misconceptions about perimeter and area, squares and rectangles. And then Day 1 was over.
On Day 2, I gave them some information about the figure:
- The large figure is a rectangle.
- The rectangle is partitioned into 11 squares.
- None of the squares is the same size.
- The smallest square measures 9 cm by 9 cm.
Then we decided what questions we would answer:
- What are the sizes of the other squares?
- The large figure looks like it could be a square, is it?
Many students reached for their rulers and tried find the answers by measuring. This led to great discussions about models and representations, scale drawings, exact answers vs. estimates, precision. A lively confluence of struggle and perseverance filled the room as students accessed this challenging task. One group took a guess about the length of one of the lines and, in working through some of the other lengths, came to the contradictory result of two squares being the same size. They realized that they needed to revise their guess in order to comply with the known information.
On Day 3, one student made a discovery. The large figure can’t be a square, the student said, because the top side is composed of a side of the largest square and the side of another, and the left side is also the side of the largest square plus another, and the two smaller squares can’t be the same size. So the top and the left side must be different lengths.
At the end of each class period, the students were so involved with the problem that I had trouble moving them out of my room and on to their next class.
In the end, several groups solved the task with guess and check strategies and presented their work to the class. I plan to revisit the task once we have worked on solving equations and ask the students to generalize their reasoning by using a variable for the unknown length. But the best outcome was a group of students who were actively engaged in solving a challenging task, who used their time productively and collaborated when appropriate, who understood the norms and expectations of their math class without my stating anything explicitly, just by getting right into the math.
Note: I found the problem at the site of Phillips Exeter Academy, page 60, within an outstanding problem set that I plan to borrow from extensively this year. In the spirit of 3-act math, I decided to strip the information out for the initial presentation and have the students pose some questions themselves.