Lesson 10

This warm-up gets students ready to think about making estimates of quotients. Estimation will be central in how students learn about the circumference and area of a circle.

Instruct students to find a method of estimation other than performing long division. 

Ask students to share their strategies for each problem. Record and display their explanations for all to see. Ask students if or how the factors in the problem impacted the strategy choice. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone solve the problem the same way but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

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 in a future lesson.

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 is a proportional relationship between these quantities. This makes sense because the squares can all be viewed as scale copies of one square and both the side length and the diagonal should change by the same scale factor.

Because of measurement error, the points do not lie exactly on a line. Whenever analyzing proportional relationships through experimentation, small errors can be expected.

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, even though 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 and poll the class on the measurements they got for each square. If desired, give students 1–2 minutes to plot more points on their graph for the values they added to their tables.

Use questions like the following to lead students to the idea that the length of the diagonal and the perimeter are proportional to each other. Even though the data does not lie perfectly on a line, the inconsistencies are caused by measurement error.

• “Would it makes sense to have \((0,0)\) as a possible point?” (Yes, since a square with diagonal 0 cm would have a perimeter of 0 cm.)
• “The data is not perfectly lined up. Do you think it should be? What could be causing the inconsistencies?” (Measurement error.)
• “If you had a square with diagonal 1 cm, what would the perimeter be?” (About 2.8 cm.)

Consider adding a column to the table with the quotient of the perimeter of the square divided by the length of the diagonal to help students identify the constant of proportionality for the relationship. Sample data is included in the table below. While the level of precision of their measurements does not justify including this many decimals places for the quotients, they are shown in this sample table because 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.

When we collect data through measurement, we usually will introduce small errors into the data. Even though the data will look a little bit “bumpy,” it will often show the underlying relationships between two quantities. In this case, we were able to use the plotted data to estimate the constant of proportionality. Once we have an estimate for the constant of proportionality, we can find other diagonal lengths given the perimeter or other perimeters given the diagonal length. 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. The table not only helps to confirm this but also gives the approximate constant of proportionality.

This activity builds on the previous one as students plot a different relationship, comparing the length of each square diagonal to its area. Students use the length measurements from the previous activity and use them to calculate the area of the squares. Unlike in the previous activity, this time both the table and the graph show that the relationship between diagonal length and area is not close to being proportional. As a result, the measurement error does not play a role in this activity. Rather, the focus is on identifying that a relationship is not proportional: this is an example of MP7 because students will identify that there is no constant of proportionality in the table (not even allowing for measurement error) and will correspondingly notice that the plotted points are not close to lying on a line.

Keep students in the same groups. Tell them to calculate the area of the same 3 squares they measured in the previous activity. Give students 4 minutes of group work time. Display this blank grid and have students plots points for their measurements.

Give students 2–3 more minutes of group work time followed by whole-class discussion.

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.

Invite students to share their observations about the graph. Important points 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 \((0,0)\).
• The area of the square looks to grow at a faster rate for larger diagonals than for the short diagonals.

To reinforce that the relationship is not proportional, instruct students to calculate the quotient of the area divided by the length of the diagonal for their square. Consider displaying a table of their measurements and adding a third column to record their quotients. Students should realize that the differences between these quotients are too large to be accounted for by just measurement error.

If time permits, consider using this picture to reinforce that length scales linearly while area does not:

• “If the side length of the small square is \(s\), what is the side length of the large square?” (\(2s\))
• “If the diagonal of the small square is \(d\), what is the diagonal of the large square?” (\(2d\))
• “If the area of the small square is \(A\), what is the area of the large square?” (\(4A\))
• “Is the relationship between the diagonal and the area proportional?” (No, when the diagonal doubles, the area quadruples; if the relationship were proportional, both the diagonal and the area would double.)