Students pract ice. мин. E xpe ri e nc e aligning polygons with a grid to determine area. multiple strategies for finding area of a trapezoid

Transcription

1 1 План урока Calculating Area of Trapezoid s Возрастная группа: 3 rd Grade, 4 t h Grade 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 Онлайн ресурсы: Shape s o n t he Gri d Opening Teacher present s Students pract ice Class discussion Closing 8 мин 1 0 мин 1 2 мин 1 4 мин 3 мин M at h Obj ect ives 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 Learn multiple strategies for finding area of a trapezoid De vel o p the area formula for a trapezoid Ope ni ng 8 мин Display the following po l ygo ns. Ask students to find their areas in their notebooks.

2 2 When the students are done working, share. Ask: What is the area of the triangle? How did you find it? We can find area of a triangle by multiplying the base times the height and dividing by 2. If we multiply 2 by 4 and divide by 2, we get 4 square units. The area of the triangle is 4 square units. Ask: How can we find the area of the pe nt ago n? Responses may vary. A possible response: We can think of the pentagon as a square with a triangle on top. We can find the area of the square and the triangle and then add them. Ask: What is the area of this pentagon? How do you know? The pentagon has area 20 square units. We multiply 4 by 4 to get the area of the square, 16. We already know that the area of the triangle is 4. The area of the pentagon is the sum of these areas. Sixteen plus 4 is 20. Say: Today we are going to look at area of a t rape zo i d. There are multiple ways to find the area of a trapezoid. One way is to divide the trapezoid into parts, and then add the areas of each of the parts. Before we begin today s episode, let s talk about trapezoids for a minute. Say: When we talk about trapezoids, many of us picture something like this.

3 3 But this is a special type of trapezoid known as an i so sc e l e s t rape zo i d. We could also have a trapezoid that looks like this. Ask: What makes the first trapezoid different from the second? In the first trapezoid, the sides that are not paral l e l are equal in length. Say: Correct. In an isosceles trapezoid, the non-parallel sides are equal in length. T e ac he r prese nt s M at h game : Shape s o n t he Gri d - Area: T rape zo i ds 10 мин Using Preset Mode, present Matific s episode Shape s o n t he Gri d - Area: T rape zo i ds to the class, using the projector. The goal of the episode is to find the area of trapezoids by using a grid under the polygon.

4 4 Say: Please read the question. The question asks, What is the area of the square? Ask: How can we determine the area of the square? We can rotate the square and move it to align it with the grid behind it. Move the square so that it is aligned with the grid. Ask: How can we find its area? We can count the small squares of the grid to find its area. Ask: What is the area of this square? Click on the to enter the students answer. If the answer is correct, the episode will proceed to the next question. If the answer is incorrect, the question will wiggle.

5 5 The episode will present a total of three questions. The second question will ask for the area of a rectangle, and the third will ask for the area of a trapezoid. Encourage the students to find the area of the trapezoid in more than one way. St ude nt s prac t i c e M at h game : Shape s o n t he Gri d - Area: T rape zo i ds 12 мин Have the students play Shape s o n t he Gri d - Area: T rape zo i ds on their personal devices. Circulate, answering questions as necessary. Cl ass di sc ussi o n 14 мин Ask the students to turn to a partner to brainstorm all the different methods that could be used to find the area of this trapezoid. When they are done discussing their methods, share. Ask different students for their methods until all the different strategies have been presented. Responses will vary. Possible methods include: 1. Divide the trapezoid into a 3 by 4 rectangle and two 1 by 4 triangles. Find the area of each and add. 2. Draw a line from the top left corner to the base. Remove the triangle formed on the left. Reflect and rotate it and attach it to the right side of the figure. Find the area of the resulting 4 by 4 square. 3. Draw a 4 by 5 rectangle around the trapezoid. Find the area of the rectangle.

6 6 Subtract the area of the two triangles that are within the rectangle but outside of the trapezoid. Say: Use one of these methods to find the area of the trapezoid. What is its area? The trapezoid has area 16 square units. Say: The methods that we can use to find the area of an isosceles trapezoid don t all apply when we are looking at a trapezoid where the l e gs are not equal. What method that we used before are we unable to use with this trapezoid? We cannot easily turn this trapezoid into a rectangle. If we draw a line from the top left corner down to the base, the triangle that is formed on the left does not fit on the right side of the figure because the legs are different lengths. Say: We can divide this trapezoid into a rectangle with triangles on either side, or we can draw a rectangle around the trapezoid, find the area of the rectangle, and then subtract the area that is not included.

7 7 Say: Let s look at the diagram on the left. What is the area of each piece, and what is the area of the trapezoid? The triangle on the left has area 2 square units. The rectangle in the middle has area 12 square units. The triangle on the right has area 4 square units. Altogether, that is 18 square units. Say: Let s look at the diagram on the right. What is the area of the rectangle and the triangles within the rectangle? What is the area of the trapezoid? How do you know? The rectangle is 4 by 6, so it has area 24 square units. The area of the triangle on the left is 2 square units. The area of the triangle on the right is 4 square units. To find the area of the trapezoid, we subtract the area of both triangles from the area of the rectangle. So we subtract 2 and 4 from 24. Subtracting 6 from 24 gives 18. The area of the trapezoid is 18 square units. Say: We are able to use our knowledge of area of rectangles and triangles to find the area of trapezoids. What are the formulas for area of rectangles and area of triangles? To find area of a rectangle, we multiply length by width. To find area of a triangle, we multiply times the base times the height. Say: Yes. Now we will turn our attention to finding a formula for the area of trapezoids. Distribute two pieces of paper and scissors to each student. Display the following instructions. Ask the students to complete them. 1. Cut one piece of paper into two equal pieces.

8 8 2. Cut a trapezoid out of one of the pieces. 3. Trace the trapezoid on the other piece of paper and then cut it out. You should now have 2 identical trapezoids. 4. On a second piece of paper, place the two trapezoids side by side so that the edges line up. Trace them. How many different shapes can you make? When the students are done, ask a student to come to the front of the room to share the finished shapes. While the trapezoids may look different, there are 4 possible configurations: Say: So we ve formed two he xago ns and two q uadri l at e ral s. How will this help us determine a formula for the area of a trapezoid? We can use the quadrilaterals. Both quadrilaterals that we formed are parallelograms. We know that the area of a parallelogram is base times height. So the area of the trapezoid is going to be half that. Ask: How do the dimensions of the parallelogram relate to the dimensions of the trapezoid?

9 9 The height of the parallelogram is equal to the height of the trapezoid. The base of the parallelogram is equal to the sum of both bases of the trapezoid. Say: Yes. Now we have the formula for the area of a trapezoid. Say: Let s return to the trapezoid that we already looked at. Say: We already figured out that this trapezoid has area 18 square units. Now let s apply the formula to make sure we get the same answer. What is the height of this trapezoid? What is the length of the bases? The height is 4 units, and the bases are 6 units and 3 units. Say: Explain how to use the formula to calculate the area. The formula for area of a trapezoid is times the sum of the bases times the height. First, let s add the bases. If we add 6 and 3, we get 9. Then we multiply times 9 times 4. That is 18 square units. Say: So we have verified again that the area of this trapezoid is 18 square units. Let s use the trapezoid area formula to find the area of another

10 10 trapezoid. Ask: What is the area of this trapezoid? Explain how you found your answer. To find the area of the trapezoid, first we add the bases. When we add 6 and 12, we get 18. Then we multiply 18 by and by the height, 4. Half of 18 is 9. When we multiply 9 by 4, we get 36 square units, the area of this trapezoid. Cl o si ng 3 мин Distribute a small piece of paper to each student. Ask the students to use two different methods to find the area of the displayed trapezoid. Collect papers, to review later. A possible response:

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