# 26 Squares – Sum of Squares #MFM2P

I’ve started my #MFM2P course again this year with a set of activities using the 26 Squares thought up by Al Overwijk & Bruce McLaurin (there’s a write-up about how Al uses them here). The 26 squares are a set of squares cut out of grid paper; a 1×1, 2×2, 3×3, … all the way to a 26×26 square. Each group gets one full set.

We started with the perimeter investigation that I blogged about last year.

Our second activity was the area investigation that I blogged about already last year.

Our third activity involved creating right-angled triangles with our 26 squares, starting with a 3-4-5 triangle:

Students were asked “What do you notice?” about this and other right-angled triangles made from our squares. Several commented that the area of the two smaller squares add together to make the area of the largest square (well, it took some prompting to get them to express themselves w/ the proper mathematical terminology!). And they remembered from past Math classes that this is the Pythagorean Theorem with the equation a2 + b2 = c2.

I gave students the measurements of 3 sides of a triangle and asked them to verify if it is right-angled or not. I forgot to take photos, but they all had the idea of using the P.T. equation to check that the two side are equal.

Finally I gave them this problem:
A right triangle has two smaller sides measuring 28cm and 45cm.
Determine the length of the longest side:

The homework was to practice Pythagorean Theorem on Khan Academy.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

# 26 Squares – Area #MPM2D #MFM2P

For my MFM2P group this followed the Perimeter activity I did with the 26 Squares manipulatives (partially pictured at right). For my MPM2D group, this was their first introduction to working with the 26 squares manipulatives. For both groups this was their first introduction to Quadratic relations and parabolas.

Predict: What is the relationship between side length and area of a square?

Create a table of values:

This was done in their groups at their boards.

I had to encourage groups to count the grid on their squares. Many were calculating the side length times 4, while others were trying to square the side length but doubling instead. For each of those groups, I redirected them to our physical squares cut out w/ grids [pictured at top of post] & asked them to count the area of a 2×2 square, then a 3×3 square, and so on.

Graph: Back in their seats students were given this handout & asked to graph by hand the data from their table.

Linear VS Quadratic: Students were asked to choose which type of relation they thought this was.

And why:We then discussed the shape of the graph being a curved line & the first differences being not equal (which only some students had pointed out).

First & second differences: Groups were sent back to their boards & their table of values with this prompt:

We discussed that second differences being equal means this is a Quadratic relation; a new key term for us. The black writing on the whiteboard above is my own addition during the class discussion.

Desmos & Quadratic regression: Back at their seats, individually students used Desmos to perform a quadratic regression on their table of values. They had this prompt on their handout from earlier:The 2P students had practiced performing a linear regression with Desmos the day before during the Perimeter investigation. The 2D students had mostly never seen Desmos before. I walked around helping students that got stuck or couldn’t find where they’d mistyped something & gotten an error. The result was:at which point I did some direct teaching about how to use the a, b, and c value determined by Desmos to write out an equation for the relationship between side length and area. I also introduced the word parabola to them while we looked at the graph from Desmos, zooming in & out.

In their groups at their desks they had 4 application questions to work on:and this became the homework for the MFM2P class as we ran out of time in class.

Key features of a quadratic graph:

With the 2D students I had time left to do some direct teaching about y-intercept, x-intercept / zeros, vertex, max/min, & axis of symmetry. Their homework was on Khan Academy to identify these key features given an equation that they could graph using Desmos.
For my 2P students this lesson came a few days later with class time to work on the Khan Academy exercise set.

My reflection: I wish I had asked at the end of the activity for students to restate in words the relationship between side length & area.

Folder w/ handout & Pear Deck interactive slideshow here.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

# 26 Squares – Perimeter #MFM2P

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.

Investigation Question

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”.

Handouts & Pear Deck interactive slideshow here
Desmos activity here
26 squares here

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)