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.

Standards & Objectives

Academic standards
CCSS.Math.Content.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge...
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...
CCSS.Math.Content.7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,...
CCSS.Math.Content.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
GLE 0606.4.3
Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes.
GLE 0606.4.4
Develop and use formulas for surface area and volume of 3-dimensional figures.
SPI 0606.4.4
Calculate with circumferences and areas of circles.
SPI 0606.4.5
Determine the surface area and volume of prisms, pyramids and cylinders.
SPI 0606.4.6
Given the volume of a cone/pyramid, find the volume of the related cylinder/prism or vice versa.
TSS.Math.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and...
TSS.Math.7.G.B.3
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...
TSS.Math.7.G.B.5
Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles,...
TSS.Math.8.G.C.7
Know and understand the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.
 
Alignment of this item to academic standards is based on recommendations from content creators, resource curators, and visitors to this website. It is the responsibility of each educator to verify that the materials are appropriate for your content area, aligned to current academic standards, and will be beneficial to your specific students.
 
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: