Students pract ice. min. E xpe ri e nc e aligning polygons with a grid to determine area

Transcription

1 1 U n t er r ich t splan Calculating Area of Parallelograms Altersgruppe: Grade 4, Grade 3 Virginia - Mathematics Standards of Learning (2009): 3.10b, 3.9d Virginia - Mathematics Standards of Learning (2016): 3.8.b 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 Class discussion Closing 8 min 1 2 min 1 2 min 1 2 min 3 min 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 rectangles Learn to find the area of a parallelogram De vel o p the concept that area is preserved when parts of a polygon are rearranged Ope ni ng 8 min

2 2 Display the following questions. Ask the students to write their answers in their notebooks. What is a re a? What is the area of this rectangle? When the students are done working, share. Ask: What does area mean? Area is the number of square units inside a flat object. Ask: What is the area of the displayed rectangle? How do you know? The rectangle has area 8 square units. We can count the small squares within the rectangle. Since each small square is 1 square unit, then the entire rectangle has area 8 square units. Ask: Why do we use square units with area when we only use units when we describe pe ri me t e r? Area is the number of square units inside a flat object. Area describes a 2-dimensional space. To cover a 2-dimensional space, we use small squares. Those small squares are themselves 2-dimensional, so we talk about square units. With perimeter, we are looking at a 1-dimensional unit, the length around a shape. Ask: Is this the only rectangle that has area 8 square units? Or could we draw a different rectangle that also has area 8 square units? This is not the only rectangle with area 8 square units. We can also draw this rectangle:

3 3 Say: In today s episode, we are going to explore area of paral l e l o grams. What is a parallelogram? A parallelogram is a q uadri l at e ral with two pairs of paral l e l sides. T e ac he r prese nt s M at h game : Shape s o n t he Gri d - Area: P aral l e l o grams 12 min Using Preset Mode, present Matific s episode Shape s o n t he Gri d - Area: P aral l e l o grams to the class, using the projector. The goal of the episode is to find the area of parallelograms by using the grid under the polygon.

4 4 Mat hespiel - Area: Parallelograms - Exercício de para 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 tell it is a square? It is a quadrilateral with 4 ri ght angl e s where all 4 sides are the same length. Ask: How can we find its area?

5 5 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. The second question will ask for the area of a rectangle, and the third question will ask for the area of a parallelogram. Rotate the parallelogram and align it with the grid. Say: Because the parallelogram does not have 4 right angles, it is more difficult to count the squares on the grid. How can we find the area of this parallelogram? Responses will vary. A possible response: We can imagine cutting through the parallelogram vertically from an upper corner of the parallelogram to the base of the parallelogram. We will have removed a triangle. We can then place this triangle on the opposite side of the parallelogram to form a rectangle that is 4 units by 6 units. The area of the parallelogram is the same as the area of the rectangle that we formed.

6 6 Ask: So what is the area of the parallelogram? Click on the to enter the students answer. St ude nt s prac t i c e M at h game : Shape s o n t he Gri d - Area: P aral l e l o grams 12 min Have the students play Shape s o n t he Gri d - Area: P aral l e l o grams on their personal devices. Circulate, answering questions as necessary. Cl ass di sc ussi o n 12 min Display the following rectangle: Ask: What is the area of this rectangle? How do you know? The area of this rectangle is 12 square units. The rectangle contains 12 small squares. We can find the area by counting the small squares. Ask: Is there a faster way to determine the area besides counting the squares? We can multiply the number of squares along one side of the rectangle by the number of squares on the other side of the rectangle. Say: Yes. Along one side of the rectangle, there are 3 squares.

7 7 Along the other side of the rectangle, there are 4 squares. We can multiply 3 by 4 to get 12 square units, the area. Write on the board: Display the following rectangle: Say: Let s multiply to find the area of this rectangle. State its length and width. This rectangle is 6 units by 3 units. Ask: What is this rectangle s area? How do you know? This rectangle has area 18 square units. We can multiply its length by its width, 6 by 3, to get its area, 18 square units. Say: Suppose we have a rectangle that is 5 units long and 4 units wide. What is its area? How do you know? It has area 20 square units. We can find its area by multiplying 5 by 4, its length by its width. Say: Suppose we have a rectangle that is 8 units long and 2 units wide. What is its area? How do you know? It has area 16 square units. We can find its area by multiplying 8 by 2, its length by its width. Ask: Are there other rectangles that have area 16 square units? How do you know?

8 8 Yes, there are other rectangles that have area 16. Since 16 is a c o mpo si t e number, it has more than 2 f ac t o rs. We can use any of its factors to form different rectangles. Therefore, we could have a rectangle that was 1 by 16 or 4 by 4. Both rectangles also have area 16 square units. Display the following parallelogram: Say: In order to determine the area of this parallelogram, let s set it on top of a grid. Display the following: Ask: How do we find the area of this parallelogram? We can make a vertical line from the top left corner down to the base. Then we can move the triangle that we have created from the right side of the parallelogram to the left side. When we do that, we have created a rectangle that is 5 units by 4 units. We can see that the area of that rectangle is 20 square units. Therefore, the area of the original parallelogram must also be 20 square units because we have not changed the area. The amount

9 9 of space that the rectangle covers is identical to the amount of space that the parallelogram covers. Ask: Can we generalize? What is the formula for the area of a parallelogram? A parallelogram can always be rearranged into a rectangle. Since we know that the area of a rectangle is length times width, we know that the area of a parallelogram is the distance from the top left corner down to the base multiplied by the base. Say: Yes, to find the area of a parallelogram, we multiply the base times the height. Write on the board: Display the following:

10 10 Ask: What is the area of this parallelogram? How do you know? The area of this parallelogram is 12 square units. The base is 4 units long. The height is 3 units. We multiply the base by the height to get 12 square units. Display the following: Ask: What is the area of this parallelogram? How do you know? The area of this parallelogram is 18 square units. The base is 3 units long. The height is 6 units. We multiply the base by the height to get 18 square units. Ask: What is the area of a parallelogram with base 5 units and height 6 units? How do you know? The area is 30 square units. We multiply 5 by 6 to get 30. Say: Please draw and label a parallelogram that has area 24 square

11 11 units. Responses will vary. A possible response: Cl o si ng 3 min Say: Suppose your friend is absent today. How would you explain to him or her how to find the area of a parallelogram? To find the area of a parallelogram, we multiply the base times the height. Display the following: Say: Label the base and the height. Ask: Why does it make sense that multiplying the base times the height gives the area of a parallelogram? We can rearrange the parts of the parallelogram by removing the triangle to the left of the height. Then if we attach this triangle to the right side of the parallelogram, we have formed a rectangle. The base remains the same length. The height of the

12 12 parallelogram is now the width of the rectangle. We can see that multiplying the base times the height will give the area of the rectangle. Since we have not changed the amount of space that the polygon covers, the area of the parallelogram is equal to the area of the rectangle.