In this activity, students examine the relationship between the length of the diagonal and the perimeter for squares of different sizes. This prepares students for examining the relationship between the diameter and circumference of circles.

Students graph their measurements and find that the points look like they are close to lying on a line through the origin, suggesting that there could be a proportional relationship between these quantities. This makes sense because the squares can all be viewed as scale copies, so both the side length and the diagonal should change by the same scale factor. As students measure multiple squares and notice patterns in their measurements, they express regularity in repeated reasoning (MP8).

Whenever analyzing proportional relationships through experimentation, small errors can be expected. As students make the simplifying assumptions that the variability is likely due to measurement error and that the underlying relationship is proportional, they are modeling with mathematics (MP4).

In the digital version of the activity, students use an applet to plot points on the coordinate plane. The applet allows students to add, remove, adjust, and label points. The digital version may help students graph quickly and accurately so they can focus more on the mathematical analysis.

The goal of this discussion is for students to see that there is a proportional relationship between the length of the diagonal and the perimeter of a square. However, it is difficult to measure accurately enough to get an exact constant of proportionality.

First, ask students to share what they noticed about their graphs. Next, display a table like the one in students’ books or devices, ask students to share the measurements they got for each square, and record their answers. If desired, give students 1–2 minutes to plot more points on their graph for the values that they added to their tables.

To lead students to the idea that the length of the diagonal and the perimeter are proportional to each other, ask questions like:

• “The data is not perfectly lined up. Do you think it should be? What could be causing the inconsistencies?” (Yes. When we collect data through measurement, we usually will introduce small errors into the data.)
• “Would it make sense to have as a possible point?” (Yes, since a square with diagonal 0 cm would have a perimeter of 0 cm.)
• “If we added a point to the graph to represent a square with diagonal 1 cm, what would the perimeter be?” (about 2.8 cm)

To help students identify the constant of proportionality, consider adding a column to the table with the quotient of the perimeter divided by the diagonal. For example:

These sample quotients include more decimal places than is justified by the level of precision of the measurements. It is important that students notice the variability in the numbers and come to their own realization that these quotients are all approximately 2.8.

Even though the data does not lie perfectly on a line, the small inconsistencies can be attributed to measurement error. In a situation like this, analyzing the graph is a powerful method to check visually to see if a relationship looks like it may be proportional. Even though the data will look a little bit “bumpy,” it will often show the underlying relationships between two quantities.

In this activity, students examine the relationship between the length of the diagonal and the area for squares of different sizes. Students see that the relationship between diagonal length and area is not close to being proportional. With the table, students identify that there is no constant of proportionality (even after allowing for measurement error). With the graph, students notice that the plotted points are not close to lying on a line.

Some students may struggle to calculate the area of the squares, or may use the length of the diagonal as if it were the side length. Prompt them with questions like “How can you calculate the area of a rectangle?” “What is the length and width of your square?”

Some students may measure the side length of each square again, instead of dividing the perimeter from the previous activity by 4. This strategy is allowable, although you can also prompt them to consider if they no longer had access to a ruler, is there a way they could use the information they already recorded to find this measurement.

Some students may struggle to organize the information. Prompt them to add a column to the table in the previous activity to record the side length of the squares.

The goal of this discussion is for students to recognize that the relationship between the length of the diagonal and the area of squares of different sizes is not a proportional relationship.

First, invite students to share what they noticed about the graph. Key observations include:

• As the length of the diagonal increases, the area of the square increases.
• The points do not look like they lie on a line and definitely do not lie on a line that goes through .
• The area of the square looks to grow at a faster rate for the larger diagonals than for the shorter diagonals.

To reinforce that the relationship is not proportional, ask students to calculate the quotient of the area divided by the length of the diagonal for their squares. Consider displaying a table of their measurements and adding a third column to record their quotients. For example:

Ask questions like:

• “Can the differences between these quotients be explained by measurement error? Why or why not?” (No, the quotients are not close enough to the same value.)
• “Is the relationship between the length of the diagonal and the area of the square a proportional relationship? How do you know?” (No, the table does not have a constant of proportionality. The graph is not a straight line through the origin.)

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the question, “How is this relationship, between the diagonal and the area, different from the relationship we saw earlier, between the diagonal and the perimeter?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback: 

• “_____ makes sense, but what do you mean when you say. . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

Here is an example of a second draft: “The first relationship was proportional, because the quotients of the measurements were approximately the same, and the graph was close to a straight line. The differences could be due to measurement error. This relationship is not proportional, because the quotients are too different to be explained by measurement error, and the graph is not close to a straight line.”

As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.