Display the following polygon. Ask students to find its area. They should copy the shape and do all the calculations in their notebook.

Transcription

1 1 План урока The Distributive Property Возрастная группа: 3 rd Grade, 4 t h Grade Онлайн ресурсы: Spl i t and Co nq ue r Opening Teacher present s Students pract ice Math Pract ice Closing M at h Obj ect ives E xpe rii e nc e partitioning rectangles into smaller rectangles P rac t i c e calculating area of rectangles Learn to use the distributive property De vel o p algebra skills Ope ni ng 8 Display the following polygon. Ask students to find its area. They should copy the shape and do all the calculations in their notebook.

2 2 When the students are done working, share. Ask: How did you solve this problem? Responses may vary. Solicit as many methods as possible. Most students will cut the polygon into two rectangles, either horizontally or vertically. If they make a horizontal line, then they have two rectangles: one with dimensions 2 by 3 and one with dimensions 4 by 12. So the areas of those rectangles are 6 and 48, for a total of 54 square units. Other students may make a vertical line, creating a rectangle that is 4 by 9 and a rectangle that is 6 by 3. The areas of those rectangles are 36 and 18, for a total of 54 square units. Say: There are multiple ways to solve the problem. Sometimes it is easier to find the area of a shape by dividing it into smaller pieces. This can be true even of a rectangle. Sometimes, it is easier to find the area of a rectangle by dividing it into smaller pieces so that the multiplication is easier. Let s look at a rectangle that is 18 by 7. Display the following rectangle: Say: We could find the area by multiplying 7 by 18, but we might not know the answer to that problem in our heads. So instead, we can break the rectangle into two smaller rectangles. Display the following rectangles:

3 3 Ask: How do we know that we haven t changed the size of the rectangle? The dimensions are still 7 by 18. The 18 has been broken up into 10 and 8. Say: Now we can find the area of each rectangle and then add them together to find the total area. In doing so, we will have found the answer to the problem 7 times 18. What are the areas of each smaller rectangle and the total area of the entire rectangle? The smaller rectangles have areas 70 and 56 square units. We add those together to get the area of the entire rectangle, which is 126 square units. So now we know that the pro duc t of 18 and 7 is 126. Say: Today, we re going to break up a large rectangle into smaller rectangles as a way to find the area of the large rectangle. T e ac he r prese nt s M at h game : Spl i t and Co nq ue r - Di st ri but i ve Law 12 Present Matific s episode Spl i t and Co nq ue r - Di st ri but i ve Law to the class, using the projector. The goal of the episode is to practice the distributive property by partitioning a rectangle into two smaller rectangles and finding the sum of the areas of the smaller rectangles. Example :

4 4 Ask: What do the instructions ask us to do? We need to find the area of the blue rectangle. Say: I don t see a blue rectangle. I see two green rectangles. Where is the blue rectangle? The blue rectangle is covered by two yellow rectangles. The layering of colors makes the rectangles appear green. Ask: What is the area of each of the green rectangles? Click on the to enter the students suggestions for each area. If the answers are correct, the entry will remain white. If the answers are incorrect, the entry will turn brown. Ask: What is the sum of the two areas? What is the area of the blue rectangle? Click on the to enter the students suggestion.

5 5 If the answer is correct, the episode will present the equation: 5x13 = (5x10)+ (5x3) = 65. If the answer is incorrect, the instructions will wiggle. Ask: What does the equation show? The equation has three parts: on the left is the product of the dimensions of the blue rectangle, in the middle is the sum of the areas of the two green rectangles, and on the right is the total area. Say: This is an example of the Di st ri but i ve P ro pe rt y. y The Distributive Property says that the product of a number and a sum is equal to the sum of the products of the number and each of the adde nds. In the middle of the equation, where does the 10 come from? The blue rectangle is broken up into two rectangles. One of the rectangles has width 10. To find the area of that rectangle, we multiply 10 by its height. That is where the 10 comes from. Click on the to proceed to the next problem. The episode will present a total of five problems. Only the first problem has the smaller rectangles given. For the remainder of the problems, you need to create the smaller rectangles that overlap the large blue one. The idea is to divide the width of the blue rectangle into two parts, where one part has width 10. St ude nt s prac t i c e M at h game : Spl i t and Co nq ue r - Di st ri but i ve Law 12 Have the students play Spl i t and Co nq ue r - Di st ri but i ve Law on their personal devices. Circulate, answering questions as necessary.

6 6 M at h P rac t i c e : T he Di st ri but i ve P ro pe rt y Wo rkshe e t 12 Distribute graph paper and rulers. Ask the students to complete the following assignment: 1. Pick a multiplication problem where one of the factors is greater than 10 and one is smaller. 2. Write down the multiplication problem and draw a rectangle with the factors as its dimensions. 3. Divide the rectangle into two smaller rectangles. 4. Find the area of each of the smaller rectangles and the area of the large original rectangle. Show your work. 5. Explain in writing what the Distributive Property says and how the drawing illustrates the property = A possible response: The bottom rectangle is divided into two parts. The rectangle on the left has area 7 10 = 70 square units, and the rectangle on the right has area 7 6=42 square units. So the area of the original large rectangle is = 112

7 7 square units. This shows: 7 16 = 7 (10 + 6) = (7 10) + (7 6) = = 112 We can break 16 up into 10 and 6. Then we have two rectangles. The sum of the areas of the two rectangles must be equal to the area of the original rectangle because they cover the same amount of space. So we can multiply the height of each of the smaller rectangles (7), by the widths of each of the rectangles (10 and 6) to get the areas of the smaller rectangles and add them to get the area of the original rectangle. The Distributive Property says that multiplying a sum by a number is equal to multiplying each of the addends by that number and then adding the products. That is what we have done here, so we have illustrated the Distributive Property. Collect the students work to display later.

8 8 Cl o si ng 3 Ask: How could we use the distributive property to help us multiply 14 by 8? We could split 14 into 10 and 4. Then we multiply each of them by 8. We get 80 and 32. Then we add those two numbers to get the answer, 112. Fourteen times 8 is 112. Ask: How could we use the distributive property to help us multiply 17 by 9? We could split 17 into 10 and 7. Then we multiply each of them by 9. Ten times 9 is 90, and 7 times 9 is 63. Adding 90 and 63, we get 153. The product of 17 and 9 is 153. Say: So far, we have only multiplied two-digit numbers smaller than 20 by a one-digit number. But we can use the distributive property with larger numbers too. How could we use it to multiply 22 by 8? We could split 22 into 20 and 2. Then we multiply each of them by 8 and get 160 and 16. We add 160 and 16 to get 176, our answer. Say: Suppose we had split 22 into 11 and 11. How could we have used the distributive property then? If we split 22 into 11 and 11, we could multiply each 11 by 8. So we get 88 each time. We add the 88 and 88, and we get 176, our answer. Say: There is not one right way to split up a number. We can do it any way we want. The distributive property will allow us to arrive at the correct answer.