1 U n t er r ich t splan Multiplication - Expand ed and Stand ard Algorithms Altersgruppe: Ye ar 4 The Australian Curriculum: ACM N A07 5 NSW Mathematics K-10 Syllabus: M A2-6N A, M A2-6N A. Victorian Curriculum: VCM N A15 5, VCM N A15 6 NSW Numeracy Continuum K-10: 4.A5.1 Online-Ressourcen: M ul t i pl i c at i o n Al go ri t hms Opening Teacher present s Students play Ext ension Closing 8 1 0 1 2 1 2 5 ZIELE: E xpe ri e nc e an application of the Distributive Property P rac t i c e multiplication facts Learn multiple methods for multiplying De vel o p algebra skills
2 Ope ni ng 8 Let s practice our multiplication facts. Verbally quiz the students on their one-digit by one-digit multiplication facts. Try to ask each student at least one question. Emphasize that they need to memorize their times tables. Display the following rectangle: How can we find the area of this rectangle? We can count all the little squares inside it, or we can multiply 4 by 7 to get an area of 28 square units. Today we re going to use what we know about area of rectangles to help us multiply. T e ac he r prese nt s M ul t i pl i c at i o n Al go ri t hms 10 Present Matific s episode M ul t i pl i c at i o n Al go ri t hms - M ul t i pl i c at i o n Al go ri t hms to the class, using the projector. The goal of the episode is to present a visual model for the expanded algorithm for multiplication and then to connect the ideas to the standard algorithm. E xampl e :
3 What multiplication problem are we solving? Students can respond based on the episode. How does the blue rectangle represent the multiplication problem? The two rectangles together form one large rectangle. The length of that rectangle is one of the f ac t o rs in the multiplication problem, and the width is the other factor. What is the episode asking? The episode is asking how many of the grid squares are blue. How is this connected to the multiplication problem? The number of blue grid squares is the product of the multiplication problem. How is the large rectangle divided into two rectangles? The large rectangle has been divided into two smaller rectangles. The rectangle on the left has a width of 10. Then, whatever remains of the larger rectangle is part of the rectangle on the right. What is the area of each of the small rectangles? Enter the values that the students suggest by clicking on each. If the answer is correct, a copy of the answer will flow to the multiplication problem.
4 If the answer is incorrect, the answer will be colored brown. Ask the students to add the areas of both rectangles to find the total area (which is the final product of the multiplication problem). Enter this area by clicking on the. If the answer is correct, the episode will proceed to the next question. If the answer is incorrect, the question will wiggle. For the second and third multiplication problem, the rectangle is not drawn. Drag the mouse to create the rectangle. The fourth, fifth, and sixth questions will present multiplication problems using the standard algorithm. St ude nt s pl ay M ul t i pl i c at i o n Al go ri t hms 12 Have the students play M ul t i pl i c at i o n Al go ri t hms - M ul t i pl i c at i o n Al go ri t hms on their personal devices. Circulate, answering questions as necessary. E xt e nsi o n 12 How could we draw a rectangle to represent the problem 8 x 14? We could draw a rectangle with length 8 and width 14. How would the episode break this up into two rectangles? The episode would draw a vertical line, breaking up the 14 into 10 and 4. So there would be two rectangles: one with dimensions 8 by 10 and one with dimensions 8 by 4. What are the areas of the resulting rectangles?
5 The 8 by 10 rectangle has area 80, and the 8 by 4 rectangle has area 32. What is the area of the 8 by 14 rectangle? How do we find it? The area is 112 square units. We find it by adding 80 and 32, the areas of the two smaller rectangles. Let s write the equations for what we did on the board. Write: 8 x 14 = 8 x (10 + 4) 8 x 10 + 8 x 4 80 + 32 112 This is called the Distributive Property. It will work regardless of how we break up our rectangle. Let s look at the rectangle again. Display the following rectangle: Let s break this rectangle up differently. Let s break the 14 into 6 and 8. Display the following:
6 What are the areas of the two resulting rectangles? The rectangles have areas 48 and 64 square units. What do we do with those two areas to find the area of the 8 by 14 rectangle? We add them to find the area of the entire rectangle. When we add 48 and 64, we get 112 square units. Why does this make sense? We already found that the area of the 8 by 14 rectangle is 112 square units. This confirms that fact. Let s write down the equations for what we did on the board. Write: 8 x 14 = 8 x (6 + 8) 8 x 6 + 8 x 8 48 + 64 112 Can you generalize what we are doing? When multiplying, we can separate one of the factors into a sum. Then we can multiply the first factor by both adde nds. Then we add the results to get the final answer. Display the following rectangles. Ask the students to break the rectangles into two smaller rectangles, find the area of each, and then add to find the total area. Then the students should write down the equations that they used, demonstrating the Distributive Property.
7 Review solutions. Answer any questions the students may have. Discuss that while everyone should have the same final answer, the way that they divided the rectangles may differ. Cl o si ng 5 Display the following: 58 x 4 232
8 Let s consider the standard algorithm for multiplication. Display the following: 3 58 x 4 232 Look at the 3 above the 58 in the standard algorithm. Where does this 3 show up in the area of the rectangles? It is the 3 in 32, in the area of the smaller rectangle. In the standard algorithm, why do we need to write the 3 above the 5 in 58? The 3 stands for 30, or 3 tens. So we place it above the tens place. In the standard algorithm, why do we add the 3 to the product of 4 and 5? We add because we have 3 tens in addition to the 20 tens we get when we multiply 4 by 5 tens. It is the same as adding the two rectangles together to get the total area.