This lesson is designed for students working in pairs. When students arrive to class, they should have the
What Changes, What Stays the Same activity sheet and rulers at their desks.
To begin, ask students to measure the side length and perimeter of each square on the activity sheet using the ruler. They should record the measurements in either inches or centimeters on the activity sheet. When students have done this once for each square, ask them to measure the diameter and circumference of the circles on the activity sheet. This will probably spur a discussion of how to measure the circumference with a ruler.
After brainstorming ways that this can be done, hand out alternative units of measure to be used (M&Ms, paper clips, pennies, identical beads, etc.). It is best if these are already divided up so that they can be handed out quickly. Also, be sure that there are enough for each pair of students to measure the perimeters and circumferences. You may wish to discuss how each unit of measure can be used, or you may prefer that the students discover this on their own. It might be helpful to use pennies on the overhead projector to demonstrate how the students can use them to measure. The teacher can also discuss with students how they may have to estimate portions of a unit if the measure is not exactly an integer. Allow students to find a second measure of the squares and both measures of the circles using at least one non‑traditional unit of measure.
Once students have filled in the activity sheet, the class can use this data to complete the first page of the overhead What Changes, What Stays the Same. The teacher can record the findings of the student pairs. Students may be given copies of this, too, if the teacher wishes for each student to have a copy of the table. At this point, the teacher should lead the students into identifying a relationship between a square’s perimeter and its side. Many students will know that Perimeter = 4 × Side, or P = 4s, but try to get students to think of the 4 as a constant that is equal to P ÷ s. Label the third column of the overhead Perimeter ÷ Side, and calculate the constant for each square. It is important for students to see that this relationship is the same regardless of the square’s size or unit of measure, which makes it a "constant." A discussion of constant versus variable may be necessary here.
Now gather the data for the circle in a similar manner and record results on the second page of the overhead. The relationship for the side and perimeter of a square is somewhat obvious since the sides combine to form the perimeter. However, for circles, it is not clear that the diameter contributes directly to the circumference other than a longer diameter results in a larger circumference. Ask the students to go along with your investigation of whether there might be a similar constant for this situation and label the third column of this Overhead Circumference ÷ Diameter. Ask the students to calculate this ratio for each set of data. Once students are convinced that all values are similar and there might be a constant, the name of this constant (π) and a better approximation can be given. Students can also measure the diameter and circumference of other circles in the room if there is still a question about the existence of this constant.
It is very important that students see that pi is a constant rather than a variable. Its value does not change regardless of the size of the circle or units of measure, and pi always represents the same number. (Students often think of pi as a variable similar to x and y, rather than a constant value like √5.)
