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Square Circles

  • Lesson
6-8
1
Measurement
Kristen Chandler
Location: unknown

This lesson allows students to use a variety of units when measuring the side length and perimeter of squares and the diameter and circumference of circles. From these measurements, students will discover the constant ratio of 1:4 for all squares and the ratio of approximately 1:3.14 for all circles.

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.

pdficon  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 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.

1110 overhead  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.)

    • Pennies
    • Paper clips
    • M&Ms
    • Lined paper (use the distance between lines as 1 unit)
    • Beads (identical size and shape)
    • Index finger (use the width of a student’s finger as 1 unit)
    • Pencil (use the width as 1 unit)
    • String (can mark inches or cm with a pencil on the string)
 

Assessments 

  1. Ask students to write a paragraph expressing the relationship of a circle’s diameter to its circumference. The title of the first activity sheet was What Changes, What Stays the Same? Ask students to explain the possible reason this title was used.
  2. Ask students to explain the difference between a variable and a constant.
  3. Allow students to consider the question on the last page of the What Changes, What Stays the Same overheads. Note that students should answer this question without the use of any measuring tools. Ask students to share their answers.

    Have students answer the following questions:

    • If the perimeter of a square is equal to the length of 32 football fields, what is the length of one side of the square? Use the ratio of P/s to set up an equation and solve.
    • If the circumference of a circle is 112 miles, what is the diameter of the circle? Use the ratio C/d to set up an equation and solve. (This is a good time to show students the pi key on a calculator and how to use it.)
Extensions 
  1. Require students to draw several circles on centimeter grid paper. Then, have them determine the radius and approximate area of each circle. By finding the ratio of Area ÷ Radius2, students will again see the appearance of the contstant pi.
  2. Students can create isosceles right triangles of different sizes and measure the lengths of one leg and the hypotenuse. Calculating the ratio of Hypotenuse ÷ Leg for each triangle will lead students to the discovery of the constant relating these two pieces, namely √2. 

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).]

Teacher Reflection 

  • 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?

Learning Objectives

Students will:

  • Identify various units of measure based on their appropriateness for each shape and size
  • Draw conclusions about the relationship of side/perimeter in squares and diameter/circumference in circles based on collected data
  • Through physical representations, develop the idea of a constant that relates a circle’s diameter and circumference, namely pi

Common Core State Standards – Mathematics

Grade 7, Geometry

  • CCSS.Math.Content.7.G.B.4
    Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.