Square Circles

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 is a hands on activity to formulate what pi is and the relationship between a circle and a square.

Weblink address (URL): 
Square Circles

Standards & Objectives

Learning objectives: 

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.
Essential and guiding questions: 

Questions for Students:

  • How can we change the formula P = 4s into an equation with P and s on the same side of the equals sign?
  • Though we may already know P = 4s for squares, why are some of our ratios P ÷ s not coming out to exactly 4?
  • 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?

Lesson Variations

Blooms taxonomy level: 
Understanding
Extension suggestions: 

Extensions:

  • 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 constant pi.
  • 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. 

Helpful Hints

Materials:

  • What Changes, What Stays the Same? Activity Sheet 
  • What Changes, What Stays the Same Overheads 
  • Rulers
  • Calculators
  • Alternate units of measure, such as:
  • 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)

References

Contributors: 
Roxann Johnson
Originator: 
Ayers Institute for Learning & Innovation