Transferring Knowledge of Multiplicative Structures
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1 Transferring Knowledge of Multiplicative Structures Dr. Roger Fischer EMAT Project Facilitator Montana State University November 11, 2016
2 OVERVIEW Sample Analogous Tasks Definition of Multiplication Transferring the definition of multiplication
3 ANALOGOUS PROBLEM SAMPLE FROM THE GROUP 1. Jill had 72 candies. After giving the same number of candies to each of her three brothers, she had 48 candies left. How many candies did she give to each brother? 2. James has 122 new songs on his iphone. After listening to the same number of songs each day for a week, he had 66 songs that he had not heard. How many songs did he listen to each day?
4 ANALOGOUS PROBLEM SAMPLE FROM THE GROUP Jill had 72 candies. After giving the same number of candies to each of her three brothers, she had 48 candies left. How many candies did she give to each brother? Solution. Let x represent the number of candies each brother gets. Then Jill gave away 3 brothers x candies = 3x candies. 1 brother Thus, 72 candies 3x candies = 48 candies (72 48) candies = 3x candies 24 candies 3 candies = x 8 = x
5 ANALOGOUS PROBLEM SAMPLE FROM THE GROUP James has 122 new songs on his iphone. After listening to the same number of songs each day for a week, he had 66 songs that he had not heard. How many songs did he listen to each day? Solution. Let x represent the number of songs listened to per day. Then James listened to 7 days x songs = 7x songs over 1 day the week. Thus, 122 songs 7x songs = 66 songs (122 66) songs = 7x songs 56 songs 7 songs = x 8 = x
6 RECALL OUR DEFINITION OF MULTIPLICATION If A and B are nonnegative numbers, then A B or A B which we read as A times B, means the total number of objects in A groups if there are B objects in each group. Beckmann (2014, p. 140) All the examples featured in today s discussion were drawn from Chapter 4 of Beckmann (2014).
7 EXAMPLE: COMMUTATIVITY OF MULTIPLICATION For all numbers A and B, A B = B A. Why?
8 EXAMPLE: COMMUTATIVITY OF MULTIPLICATION 3 rows 5 marbles 1 row = 15 marbles
9 EXAMPLE: COMMUTATIVITY OF MULTIPLICATION 3 rows 5 marbles 1 row = 15 marbles 5 columns 3 marbles 1 column = 15 marbles
10 EXAMPLE: AREA OF A RECTANGLE The area of a rectangle of length l units and width w units is given by A = l w. Why?
11 EXAMPLE: AREA OF A RECTANGLE 2 rows 3 squares 1 row = 6 squares
12 EXAMPLE: AREA OF A RECTANGLE 2 rows 3 squares 1 row = 6 squares 3 columns 2 squares 1 column = 6 squares
13 LIFE IS NOT AN INTEGER 3 rows 4 squares 1 row = 12 squares 4 columns 3 squares = 12 squares 1 column 1 2 inch 1 3 inch = 1 6 square inches 1 6 square inches 12 squares = 2 square inches 1 square
14 EXAMPLE: VOLUME OF A RECTANGULAR PRISM The volume of a rectangular prism that has a base of length l units, a width of w units and a height of h units has volume Why? V = l w h.
15 EXAMPLE: VOLUME OF A RECTANGULAR PRISM 2 rows 1 layer 3 cubes 1 row = 6 cubes 1 layer
16 EXAMPLE: VOLUME OF A RECTANGULAR PRISM 2 rows 1 layer 3 cubes 1 row = 6 cubes 1 layer 4 layers 6 cubes 1 layer = 24 cubes
17 EXAMPLE: ASSOCIATIVITY OF MULTIPLICATION For all numbers A, B, and C, A (B C) = (A B) C. Why?
18 EXAMPLE: ASSOCIATIVITY OF MULTIPLICATION We found that 4 layers 6 cubes 1 layer = 24 cubes
19 EXAMPLE: ASSOCIATIVITY OF MULTIPLICATION We found that 4 layers 6 cubes = 24 cubes 1 layer We can express this as ( 2 rows 4 layers 1 layer 3 cubes ) = 4 (2 3) cubes 1 row
20 EXAMPLE: ASSOCIATIVITY OF MULTIPLICATION ( ) 4 columns 1 group 2 groups 3 cubes = (4 2) 3 cubes 1 column
21 AN INVITATION For our next meeting: 1. Find two mathematical objects that have similar mathematical structures. 2. For each mathematical object, construct a word problem whose solution would involve the use of said mathematical object. 3. Send me a digital copy of your tasks within two weeks. 4. I will typeset the problems and include them in future sessions for discussion and feedback. 5. All solution authors will remain anonymous.
22 THANKS FOR COMING! Dr. Roger Fischer Mathematics Instructor Gallatin College Montana State University 201 Hamilton Hall Bozeman, MT
23 REFERENCES I Beckmann, S. (2014). Mathematics for elementary teachers with activities, 4/e (4th ed.). Boston, MA: Addison-Wesley.
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