Students pract ice. min. E xpe ri e nc e aligning polygons with a grid to determine area. P rac t i c e finding area of triangles and quadrilaterals

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1 1 U n t er r ich t splan Calculating Area of Polygons Altersgruppe: Grade 4, Grade 3 Virginia - Mathematics Standards of Learning (2009): 3.10b, 3.9d Virginia - Mathematics Standards of Learning (2016): 3.8.b, 4.7 Fairfax County Public Schools Program of Studies: 3.10.b.1, 3.9.d.1, 3.9.d.2 Online-Ressourcen: Shape s o n t he Gri d Opening Teacher present s Students pract ice Math pract ice Closing M at h Obj ect ives E xpe ri e nc e aligning polygons with a grid to detere area P rac t i c e finding area of triangles and quadrilaterals Learn to find the dimensions of a polygon when the area is given De vel o p spatial sense

2 2 Ope ni ng 5 Display the following: Ask the students which of the polygons have the same area. Ask them to write and explain their answers in their notebooks. When the students are done writing, share. Ask: Which polygons have the same area? How do you know? The triangle and parallelogram have the same area, 10 square units. The rectangle and trapezoid have the same area, 20 square units. Explanations may vary. Some students may use formulas. Others may notice that the trapezoid can be converted into the rectangle by drawing a perpendicular line segment from the top left vertex of the trapezoid to the base and moving the triangle that is formed to the opposite side of the trapezoid. Similarly, the parallelogram can be seen to have half the area of the rectangle. If we draw a perpendicular line segment from the top left vertex of the parallelogram to the base, we can move the triangle that is formed to the other side of the parallelogram to form a 2 by 5 rectangle. We can see that the 2 by 5 rectangle is half the size of the 4 by 5 rectangle. We can also see that the area of the right triangle is half that of the rectangle. Since both the right triangle and the parallelogram have half the area of the rectangle, then their areas must be equal.

3 3 T e ac he r prese nt s M at h game : Shape s o n t he Gri d - Area: P o l ygo ns 10 Present Matific s episode Shape s o n t he Gri d - Area: P o l ygo ns to the class, using the projector. The goal of the episode is to find the area of various types of polygons by using a grid under the polygon. Say: Please read the question. The question asks, What is the area of the triangle? Ask: How can we detere the area of this triangle? First, we should move and rotate it to align it with the grid. Then we can multiply the base times the height and divide by 2 to find its area. Move the triangle so that it is aligned with the grid. Ask: What is this triangle s area? Click on the to enter the students answer.

4 4 If the answer is correct, the episode will proceed to the next question. If the answer is incorrect, the question will wiggle. The episode will present a total of six questions. The questions will ask for the area of different types of triangles and quadrilaterals. St ude nt s prac t i c e M at h game : Shape s o n t he Gri d - Area: P o l ygo ns 12 Have the students play Shape s o n t he Gri d - Area: P o l ygo ns on their personal devices. Circulate, answering questions as necessary. M at h prac t i c e : Area o f P o l ygo ns Wo rkshe e t 16 Arrange the students in groups of 4. Distribute graph paper and rulers. Ask the students to draw as many different polygons as they can that have area 12 square units. When the students are done working, share. Ask students to come to the board to draw some of the polygons that they drew with their group. Assug that the dimensions are whole numbers, here is a description of possible polygons. For any triangle, the base and height could be: 1 unit and 24 units, 2 and 12 units, 3 and 8 units, or 4 and 6 units. For any rectangle or parallelogram, the base and height could be: 1 and 12 units, 2 and 6 units, or 3 and 4 units. For any trapezoid, the height and the sum of the bases must multiply

5 5 to 24: the height could be 1 unit and the bases could add to 24, the height could be 2 units and the bases could add to 12, the height could be 3 units and the bases could add to 8, the height could be 4 units and the bases could add to 6, the height could be 6 units and the bases could add to 4, or the height could be 8 and the bases could add to 3. It is also possible that the students will create other polygons, such as: Say: Let s talk about the triangles we drew. What has to be true of all the triangles if they have an area of 12 square units? How do you know? The base and the height of all the triangles must multiply to 24. The area formula for a triangle is times base times height. So the base and height must multiply to 24 so that when we multiply by we get 12. The f ac t o r pai rs of 24 are 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Say: Let s talk about the rectangles we drew. What has to be true of all the rectangles if they are to have area 12 square units? How do you know? The base and the height of all the rectangles must multiply to 12, since the area formula for a rectangle is base times height. The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4. Ask: How is finding parallelograms with area 12 similar to finding rectangles with area 12? To find area of a parallelogram, we multiply base times height, just as we do for a rectangle. Thus, the base and height options are the same for a non-rectangular parallelogram as for a

6 6 rectangle. Ask: What has to be true of all the trapezoids that have area 12 square units? How do you know? The product of the height and the sum of the bases must be 24. The area formula for a trapezoid is. So the height and the sum of the bases must multiply to 24 so that when we multiply by we get 12. Assug we are using whole numbers, the only way to get a product of 24 is to use the factor pairs: 1 and 24, 2 and 12, 3 and 8, and 4 and 6. We use one of the factors as the height and the other as the sum of the bases. However, we cannot form a trapezoid with height 12, because then the bases would have to add to 2. The only way to have a sum of 2 with whole numbers is to add 1 and 1. If the bases are equal, then the polygon is not a trapezoid. Similarly, we cannot use a height of 24, because then the sum of the bases would be 1, and we cannot get a sum of 1 using whole numbers (except for 0 and 1, which would form a triangle). If students use fractional lengths, then trapezoids with heights of 12 and 24 are possible. (For example, a trapezoid could have a height of 12, one base equal to and the other base equal to. Or we could have a trapezoid with a height of 24, with one base equal to and the other base equal to.)

7 7 Cl o si ng 4 Display the following: Hand out a small piece of paper. Ask the students to answer the following questions on the piece of paper: 1. The sum of the areas of which two shapes is 20? 2. How do you know? Collect the papers, to review later. The areas of the rectangle and the parallelogram add to 20. The rectangle has area 8 square units and the parallelogram has area 12 square units. (The triangle has area 10 square units, and the trapezoid has area 16 square units.)