SHOW 113 PROGRAM SYNOPSIS Segment 1 (1:20) OOPS! PERIMETER A careless draftsman mistakenly calculates the perimeter of a rectangle by adding its length and width. He realizes too late that the perimeter is the length of all four edges, so he should have doubled his first result. All Shapes and Sizes: Measuring Perimeter and Area Segment 2 (1:09) AREA OF IRREGULAR SHAPES This animation illustrates that shapes can have the same area but different perimeters. Segment 3 (2:11) MATH MIMES Two not-so-silent mimes use a large rope loop to show that rectangles can have the same perimeter but different areas. When we talk about the area of a rectangle, we mean the area of the region inside the rectangle. The rectangle is just the four line segments. INTRODUCTION Measurement of perimeter and area is a topic traditionally included in upper elementary school mathematics one that has caused confusion for countless students. This tape illustrates in a lighthearted way some of the relationships between area and perimeter, suggesting that (1) if the perimeter is kept constant, the area can vary; and (2) if the area is kept constant, the perimeter can vary. BEFORE VIEWING Segment 4 (5:28) GENERAL MATHPITAL: DOUBLING THE AREA The incompetent doctors of General Mathpital are at it again, this time trying to double the area of a rectangle. They learn that doubling both the length and the width creates a rectangle that is too large in fact, the new rectangle has four times the area, not twice the area. You may want to review the terms perimeter (the total length of the edges of a shape) and area (the size of the region inside a shape). DURING VIEWING OOPS! PERIMETER STOP THE TAPE right after the draftsman declares that the perimeter is the sum of the length and width to ask if that s correct. (No; as he soon realizes, that s only half the perimeter.) GENERAL MATHPITAL STOP THE TAPE when doubling the length and width is first suggested, to see if students are able to predict that the resulting rectangle will be too large. 81
MathTalk AFTER VIEWING A follow-up to the MATH MIMES segment can be done with a piece of string that is a little more than 12 feet long, with the ends tied together to make a loop of exactly 12 feet. First, ask two volunteers to use the loop of string to create a rectangle that they think is twice as long as it is wide, with input from the rest of the class. Once they have decided on a rectangle, you can measure it with a yardstick. The lengths of the sides should be 2 feet and 4 feet. Ask: If this is a rectangle, with length 4 feet and height 2 feet, what is its perimeter? (12 feet) What is the area inside? (8 square feet) Then ask two other students to use the same loop of string to create a rectangle with a larger area. (The largest rectangle that can be made is a square, 3 feet on a side. Its area is 9 square feet.) When the area is changed, does the perimeter change too? (no) Next ask another pair of students to create a rectangle with an area that is smaller than 8 square feet. (This can be done by making one side shorter than 2 feet.) There are shapes other than rectangles with a perimeter of 12 feet and an area greater than 9 square feet. A circle, for instance Demonstrate why you can use the following shortcut: measure only the shorter side. (You can figure out the length of the longer side by subtracting the length of the shorter side from 6 feet.) Compute the area. It may be easier to calculate in inches rather than feet. For example, if students create a rectangle one side of which is 1 inch, then the other side must be 5 feet 11 inches long. It s easier to figure the area of a shape like that in square inches (71" x 1", or 71 square inches). Can you make the area even less? (Yes. For example, you can make it in. x71 in., which has an area of 35.75 in. less than square foot.) Can it be even smaller? (yes) Can the area of a rectangle be zero? (No, because it would no longer be a rectangle!) To express the area in square feet, divide by 144 the number of square inches in one square foot. In this case, the area is just a little less than square foot. 82
MEASUREMENT activity GAME: ROLLING FOR RECTANGLES The purpose of this activity is to explore how changes in the lengths and widths of rectangles affect their areas and shapes. 1. Pass out copies of the reproducible page and a number cube to each group. Draw students attention to the small rectangle on the grid. Notice that it is like the rectangle on the tape, except much smaller. The width is 2 units and the length is 5 units. What is its area? (10 square units) MATERIALS copies of reproducible page 86 one number cube numbered 1 6 per small group pencils rulers 2. Each student rolls the number cube twice, recording the results on the lines labeled first number and second number. (If a student gets two ones, he or she rolls again; otherwise the result is pretty dull.) 3. Then students draw a new rectangle and find its area. The new rectangle s sides are determined by multiplying the length of the original rectangle by the first rolled number and its width by the second rolled number. For example, a student who rolls a 3 and a 6 draws a rectangle that is 6 units by 30 units, with an area of 180 units. 4. Encourage students within each group to compare and discuss their results. How could they find the area of the new rectangle easily? (Since the area of the original rectangle is 10, they multiply 10 times the product of the two numbers they rolled.) 1st number x 2 2nd number x 5 5. Discuss the results as a class. Here you can explore the shapes of the new rectangles, asking questions like these: Did anyone get a rectangle that was a square? If not, what numbers would you have to roll to get a square? (The only way is to get a 5 first and a 2 second, giving a square that is 10 units on a side.) Did anyone get a rectangle that was similar to the original one? in other words, exactly the same shape, but a different size? (To get a rectangle that has the same proportions, each side must be multiplied by the same number by rolling two 3s or two 5s, for instance.) If your rectangle isn t similar to the original, is it wider (taller) or longer? (Relate this to which of the rolled numbers is larger. If the second number is larger than the first, then the new rectangle is not similar because it is too long.) EXTENSION Ask students to construct two scale models of the original rectangle, making them 2 inches by 5 inches. Ask them to cut each one into the eight pieces shown here. Can they rearrange the 16 pieces into a single rectangle that has the same proportions but twice the area of each of the original ones? (Yes; like this) 83
keep thinking MathTalk PERIMETER PLAY The animation showed three arrangements of 21 squares. Each one has an area of 21 square units, but the perimeter varies from 44 to 36 to 20 units. Draw the figures on the board or overhead projector. Students can use graph paper or square tiles to explore the next questions. MATERIALS centimeter graph paper (page 132) OR 21 small square tiles for each student 1. Describe what makes the perimeters of these shapes different. (Students will make a variety of observations here. In general, the shapes that are more compact, or solid, have the smaller perimeters. The ones that are more strung out have larger perimeters.) 2. How many ways of arranging the squares to get a perimeter of 44 can students find? There are many; the simplest is probably a long rectangle, like this: After students have explored this question for a while, ask if anyone can find a way to get a figure with a perimeter larger than 44. (It s not possible.) Why not? (You can think of building up a shape by starting with a single square and successively sticking on additional squares. Each square you stick on will add, at most, 2 to the total perimeter, because it covers up 1 existing unit of perimeter while it has, at most, three exposed sides. The first square has a perimeter of 4; to that you add 20 squares that contribute 2 units each. That s a total of 4 + (20 x 2), or 44.) 3. Is there a way to make the perimeter smaller than 20? (No, not if the 21 squares remain intact.) 4. In this module, we suggest that of all rectangles with the same perimeter, the one with the largest area is always a square. Students may wonder why that s true. Here s one way of proving it. Suppose you have a rectangle that isn t a square, and a square with the same perimeter as the rectangle. Put one on top of the other so that one corner lines up, like this: The square is made up of A and B, and the rectangle is made up of B and C. Since the square and the rectangle have the same perimeter, the two short line segments indicated by the arrows have the same length. 84 So one side of A is as long as one side of C. But the other side of C is shorter than the other side of A. So the area of C is less than the area of A. And that means the area of the rectangle we started with is less than the area of the square we started with. Imagine the math mimes transforming the square into the rectangle. A B C
MEASUREMENT FOR THE PORTFOLIO 1. Are there any other ways (other than the 3x7 rectangle) of arranging the 21 squares so that the perimeter is 20? Surprisingly, there are! Here are a few examples. Can your students find more? 2. Invite students to explore these questions: You know that you can arrange 21 squares to make a figure with a perimeter as large as 44 or as small as 20. Can you make figures with every whole-number perimeter between 20 and 44? It s clear that you can t have an odd number for a perimeter (assuming, of course, that the squares have to touch each other along full edges). Why not? (Look at the vertical sides of the figure: For each unit length on the right side of the figure there s a unit length on the left side. Similarly, the unit lengths in the top and bottom sides of the figure can be paired off with each other. So the perimeter must be an even number.) What about having every even number between 20 and 44 as a perimeter is that possible? (Yes.) How could you convince someone that this is possible without drawing a large number of individual pictures? (One way to think about this is to start with this arrangement, which has a perimeter of 44: Now move a block from the right end up to the top row, like this; This has a perimeter of 42. Keep moving blocks from the bottom to the top row. Each time you move a block you reduce the perimeter by 2. That will get you down to a perimeter of 26; you ll have to modify this approach a little to get perimeters of 24 and 22.) CURRICULUM CONNECTIONS The ideas of area and perimeter can be clarified for students by asking questions like the ones suggested in this module whenever you discuss measurement of plane figures. For instance: If we doubled the length of this figure, what would happen to the area? Can you think of a shape that has the same area but a longer perimeter? A shorter perimeter and a larger area? And so on. By focusing on the relationships between the two concepts, both can become more solidly understood. Math Talk MEASUREMENT CONNECTIONS Measured Steps: Measuring Length Scoping Out the Area: Measuring Area Sizing Things Up: Exploring Scale and Ratio Drawing to scale means drawing shapes that are similar but of another size 85
NAME MathTalk ROLLING FOR RECTANGLES 1. Roll the number cube twice, recording what you get. (If you get 1 both times, roll until the second number is something different.) 1st number 2nd number 2. Now draw a rectangle that is FIRST NUMBER times as tall and SECOND NUMBER times as long as the rectangle below. What is the area of the rectangle you drew? On the back of this sheet, tell how you found the area just by knowing the numbers you rolled with the number cubes. square units. 86 1995 Children s Television Workshop