Dear Parents and Caregivers, We appreciate the support you give your child in learning mathematics. You are a vital partner in this learning. We would like to share some information to help you better understand Arizona’s College and Career Ready Standards. This is one in a series of letters intended to help you understand the work your child brings home. We will highlight some of the vocabulary and strategies we will use to build understanding and develop underlying mathematical ideas. This letter addresses circumference and area of a circle in seventh grade. End-­‐of-­‐year goals The goal of middle school mathematics is to extend the strong foundational knowledge developed in elementary school to new topics as students prepare for high school. Students build on their previous understanding of geometry and measurement to come to an understanding of the relationships between parts of a circle. By the end of seventh grade, students should have an understanding of pi as the ratio of the circumference of a circle to its diameter. They should be able to give an informal explanation of the relationship between the circumference and area of a circle. They should discover and know the formulas for the area and circumference of a circle and use them to solve real-­‐world and mathematical problems. Vocabulary •
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circumference: the distance around a circle diameter: the distance across a circle through its center; twice the length of the radius radius: the distance from the center of a circle to any point on its circumference; half the length of the diameter pi: the ratio of the circumference of a circle to its diameter represented by the Greek letter 𝜋; numerical value is approximately !Diagram!of!Circle
or 3.14 !diameter
!radius
Discovering and understanding pi (𝝅) Students in seventh grade have not previously studied pi. They should develop an understanding of pi as the ratio of the circumference of a circle to its diameter. To discover pi, students can measure around the circumference of a circle with a string, mark the distance one time around on the string, then lay the string flat and measure the distance with a ruler, recording the distance or length of the circumference on a piece of paper. Students should then use the string and/or the ruler to measure the diameter of the circle and record this length on the paper. To find the ratio, they divide the circumference length by the diameter length. If students measure several differently sized circles and find the ratios of circumference to diameter for each circle, then take the average of all the ratios, they will discover that the average result will approximate to 3.14, or pi. This process can also lead to the discovery of the circumference formula, which is C = 𝜋𝑑, where C is the circumference measurement and d is the diameter measurement. The circumference can also be represented as C = 2πr, where r is the measurement of the radius of the circle, since the diameter is twice the length of the radius. Students also will learn that pi is an approximation, since the ratio of circumference to diameter (
number, specifically a decimal that neither ends nor repeats. Mesa Public Schools/Grade 7/Circles/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).
) is an irrational The relationship between circumference and area of a circle Students can build on their knowledge of the area formula for a rectangle (A = base • height) to find the area formula for a circle. If a student divides a circle into wedges and cuts out the wedges, they can be laid out as shown below. !πr
When the circle is laid out as a rectangle, students can see that the base length is ½ of the circumference (2πr) and the height is the length of the radius, resulting in: Area = Base • Height Area = (2πr) • r Area = πr • r Area = πr2 !πr
Using circumference and area formulas to solve real-­‐world problems Students apply their knowledge of the circumference and area formulas to solve problems such as: The seventh grade class is building a mini-­‐golf game for the school carnival. The end of the putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? Students need to remember the formula for area of a circle (A = πr2), and that radius is half the diameter. To solve the problem, they could follow these steps: A = 3.14(5)2 A = 3.14(25) A = 78.5 ft2 After finding the area of the circle, students should discuss that carpet is not sold in circular pieces and what that means in this situation. If the students were to purchase a piece of carpet that measured 10 feet wide by 8 feet long, for an area of 80 square feet, would that be sufficient to cover the circle? Students should be led to realize that the 8-­‐foot measurement wouldn’t be long enough to cover the circle, since the diameter is 10 feet. How to help at home •
Help your child understand that pi is not an exact number and that 3.14 is an approximation. •
Give your child the opportunity to measure the circumference and diameter of several circular objects around the home (clock, trash can, door knob, etc.) and have her discover patterns in the relationships of these measurements. •
For a video illustrating the discovery of the relationship between circumference and area, visit the following website: http://learnzillion.com/lessons/819-­‐find-­‐the-­‐area-­‐of-­‐a-­‐circle •
For a video further illustrating the concepts in this letter, visit the following website: https://www.khanacademy.org/math/geometry/basic-­‐geometry/circum_area_circles/v/area-­‐of-­‐a-­‐circle •
Remember, making mistakes is a part of learning. Mesa Public Schools/Grade 7/Circles/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).