Circles

Strand: Shape and Space (Measurement)
Outcome: 1

Step 1: Identify Outcomes to Address

Guiding Questions

What do I want my students to learn?
What can my students currently understand and do?
What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?

See Sequence of Outcomes from the Program of Studies

Strand: Shape and Space (Measurement)

Grade 6

Grade 7

Grade 8

Specific Outcomes
1.	Demonstrate an understanding of angles by: identifying examples of angles in the environment classifying angles according to their measure estimating the measure of angles, using 45°, 90° and 180° as reference angles determining angle measures in degrees drawing and labelling angles when the measure is specified.
2.	Demonstrate that the sum of interior angles is: 180° in a triangle 360° in a quadrilateral.
3.	Develop and apply a formula for determining the: perimeter of polygons area of rectangles volume of right rectangular prisms.

Specific Outcomes

Demonstrate an understanding of circles by:

describing the relationships among radius, diameter and circumference
relating circumference to pi
determining the sum of the central angles
constructing circles with a given radius or diameter
solving problems involving the radii, diameters and circumferences of circles.

Specific Outcomes

Determine the surface area of:

right rectangular prisms
right triangular prisms
right cylinders
to solve problems

Develop and apply formulas for determining the volume of right rectangular prisms, right triangular prisms and right cylinders.

Big Ideas

A circle is "a plane figure that has all its points the same distance from a fixed point called the center of the circle� (Cathcart 1997, p. 185). The radius is the distance from the centre of the circle to the edge of that circle while the diameter is a line segment passing through the centre of the circle with both endpoints on the circle. The circumference of a circle is the distance around or the perimeter of a circle.

In using any type of measurement, such as length or angle, it is important to discuss the similarities in developing understanding of the different measures: first identify the attribute to be measured, then choose an appropriate unit and finally compare that unit to the object being measured (NCTM 2000, p. 171).

When measuring the circumference, radius and diameter of circles, the attribute of length is being measured and appropriate units to measure length include millimetres, centimetres and metres. When finding the sum of the central angles of a circle, the attribute of angle measure is being used and the appropriate unit to measure angles is the degree.

The circumference of any circle divided by the diameter of that circle is a constant called pi and is written as π. "Pi (rather than some other Greek letter like alpha or omega) was chosen as the letter to represent the number 3.141592 ... because the letter [π] in Greek, pronounced like our letter 'p,' stands for 'perimeter'" (The Math Forum@Drexel).

Pi is an irrational number, a nonrepeating, nonterminating decimal.

Pi = 3.141592653589793238462643383279502884197169399375105820974944...

The value of pi is often approximated as 3.14 or 22/7.

Since pi is irrational, it cannot be expressed as a whole number divided by another whole number. Only rational numbers can be expressed as a/b, where a and b are integers and b is not equal to zero. Since pi is the ratio of the circumference to the diameter of a given circle, then the measure of either the circumference or the diameter or both cannot be whole numbers.

All measurements are approximate. When measuring the circumference and diameter of a circle, we use an approximate measure for each one, recognizing that at least one of the measures is an approximation for an irrational number.

Irrational numbers have a place on the number line and are therefore real numbers; i.e., real measures. For example, the measure of the hypotenuse of a right isosceles triangle with the congruent sides each one metre in length is metres, a real measure, a real number that is irrational.