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.
What Changes, What Stays the Same Activity Sheet
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 What Changes, What Stays the Same Overhead.
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.
What Changes, What Stays the Same Overheads
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.)
Questions for Students
1. How can we change the formula P = 4s into an equation with P and s on the same side of the equals sign?
[Dividing both sides by s gives P/s=4, which is a relationship that holds for any square.]
2. Though we may already know P = 4s for squares, why are some of our ratios P ÷ s not coming out to exactly 4?
[Errors in measurement may cause slight errors in the ratio of perimeter to side length. However, all ratios should be close to 4.]
3. There is a constant that relates a square’s side to its perimeter, and there is a constant that relates a circle’s diameter to its circumference. Is there a similar constant for a rectangle? Why or why not?
[There is not a constant relationship between a rectangle’s perimeter and its width, nor is there a constant relationship between a rectangle’s perimeter and its length. However, there is a constant relationship between a rectangle’s perimeter and the sum of its width and length. That is, for all rectangles. More commonly, this is written as P = 2(w + l).]
- What were some of the problems students encountered when using the different units of measure?
- Are there better items that can be used next time? Which ones worked particularly well?
- Were students focused and on task throughout the lesson? If
not, what improvements could be made the next time this lesson is used?
- How did students demonstrate that they were actively learning?
- Were the students led too much in the lesson? Did the students need more guidance?
- Did you find it necessary to make any adjustments during the lesson?
- Did the materials that the students were using affect classroom behavior or management?