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.
- Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge...
- 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...
- Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,...
- 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.
- 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...
- 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...
- Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles,...
- Know and understand the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.
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- 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.
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?
- 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.
- What Changes, What Stays the Same? Activity Sheet
- What Changes, What Stays the Same Overheads
- Alternate units of measure, such as:
- Paper clips
- 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)