26 squares is an introductory investigation I use in MFM2P. It comes from Al Overwijk & Bruce McLaurin. The idea is that you can use the same set of manipulatives – 26 squares of varying sizes with an overlaid grid – to run investigations/activities to introduce each of the 3 strands in the course; linear relations, quadratic relations, and measurement & trigonometry (similar triangles, Pythagorean Theorem, trig). Today’s first investigation introduced linear relations.
What is the relationship between the side length & the perimeter of a square?
Students were asked to predict the relationship. A sample of responses:
Table of Values: Groups were sent to their VNPS station to create a table of values of side length & perimeter using their squares to collect data.
Some groups correctly counted the perimeter using the grid. At least one group was squaring side length, so I went over and we talked about counting perimeter using the grid & they changed their table of values. One group (red marker) decided to measure the lengths with a ruler instead of counting w/ the grid.
Graph: Back at their desk students graphed their data by hand on this handout (forgot to take photos of student work here). I then had them all decide whether or not this was a linear relation & why. This led to a class discussion of the graph being a straight line as well as the pattern in the perimeters. At this point, groups were sent back up to their boards to determine the first differences for their table & we discussed their findings (again, I forgot photos here).
Equation: Back at their desks once again students worked their way through this short Desmos activity I created asking them to create a graph & perform a linear regression to find the line of best fit. A summary of the student work from Desmos:Students then completed 4 practice problems on the earlier handout to solve for either perimeter or side length given the other. This all took 2 days and they had time at the end to start the homework which was a Khan Academy exercise set titled “Slope Intuition”.
Update: I added a 3rd day to wrap-up this activity and talk about representations. Students completed this handout:They had to name the 3 different representations & explain how they are all related to each other. After 5 minutes of working on it themselves, I had them get up & walk around the room to read each others’ sheets in a gallery walk type style. Then they returned to their seats & could add, change or erase anything from their own notes. I then led a class discussion about the connections of slope & y-intercept between the 3 different representations.
Reflection: I wish I’d included a “word” representation such as “Perimeter is equal to 4 times the side length”.
– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)