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 volume of cylinders, cones, and spheres in eighth 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 the understanding of circles and volume from seventh grade to find the volume of cylinders, cones, and spheres. By the end of eighth grade, students should understand the relationship between the volumes of cylinders and cones, and the volumes of cylinders and spheres to the corresponding formulas. Students use this understanding to solve real-­‐world volume problems. Vocabulary
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volume: a measure of the amount of space occupying a three-­‐dimensional shape, expressed in cubic units diameter: a line segment that passes through the center of a circle and divides it in half radius: the distance from the center point of a circle to any point on its circumference, represented by the variable r, and is equal to half of the diameter pi: the ratio of the circumference of a circle to its diameter, represented by the symbol π and is approximately equal to 3.14 Volume A common misconception among middle grade students is that volume is a “number” that results from figuring a formula. These students do not recognize that volume is the measure of the amount of space occupied. This can be addressed by providing students with opportunities for hands-­‐on experiences filling three-­‐dimensional objects to see how much will “fill up” the space inside the object. Volume of a cylinder Students can make the connection that the volume of a right rectangular prism can be thought of as the area of a base (l • w) times the height of the prism; therefore, the area of the base of a cylinder (the area of a circle is πr2) times the height of the cylinder should lead to the volume of the cylinder. Mesa Public Schools/Grade 8/Volume of Cylinders, Cones, Spheres/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).
Once students make this connection, they can begin to solve real-­‐world problems involving the volume of a cylinder, such as: James wants to plant pansies in his new planter. He wonders how much potting soil he should buy to fill the planter. Use the measurements in the diagram below to determine the planter’s volume. Volume (V) = πr2h V = 3.14 • 202 • 100 V = 125,600 cm3
The radius (20 cm) is one half of the diameter (40 cm). Volume of a cone Students begin to understand the volume of a cone in relation to the volume of a cylinder by using a cone and cylinder with the same base and height measurements. Students will discover that it takes 3 full cones to fill the cylinder. 1
Therefore, the volume of a cone is 1/3 the volume of a cylinder or πr2. 3
Volume of a sphere Students can explore the volume of a sphere and its relationship to the volume of a cylinder. For example, they might estimate the diameter and radius of a whiffle ball and use these measurements to make a cylinder out of paper. The height and diameter of the cylinder must equal the diameter of the ball. Cover all the holes in the ball except one and fill the ball with rice. Pour the rice into the cylinder and decide what fraction of the cylinder is filled. You can find a video demonstrating a similar exploration at http://www.youtube.com/watch?v=aLyQddyY8ik r3
How to help at home • Check out the following videos that explain some of the math in this newsletter: http://www.youtube.com/watch?v=aLyQddyY8ik http://www.youtube.com/watch?v=xuPl_8o_j7k
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Visit http://www.wikihow.com/Calculate-­‐the-­‐Volume-­‐of-­‐a-­‐Sphere with your child for a step-­‐by-­‐step explanation to help find the volume of a sphere. •
Remember, making mistakes is a part of learning. Mesa Public Schools/Grade 8/Volume of Cylinders, Cones, Spheres/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).