Multiplicative Reasoning and Word Problems
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1 Multiplicative Reasoning and Word Problems Dr. Roger Fischer EMAT Project Facilitator Montana State University December 2, 2016
2 OVERVIEW Sample Analogous Tasks Algebraic Techniques and Visual Models Challenges
3 ANALOGOUS PROBLEM SAMPLE FROM THE GROUP 1. Some oranges and apples were counted into a basket. Their ratio was 32 oranges to 18 apples. If there are 384 oranges in the basket, how many apples are there? 2. The two polygons are similar. Find x and y.
4 ONE SOLUTION STRATEGY ORANGES PROBLEM 18x apples 32 oranges 384 oranges = 18 apples x apples ( ) 32 oranges 18 apples = 18 x apples 32 x oranges = oranges 32 x oranges 32 oranges = oranges 32 oranges ( ) 384 oranges x apples x = x = x = 216
5 ONE SOLUTION STRATEGY TRIANGLES PROBLEM = x 13 ( ) 32 ( x = ) = x = x 26 = x
6 DISCUSSION Are these proportion problems analogous? Why or why not?
7 DISCUSSION Are these proportion problems analogous? Why or why not? 1. What is the value of the cross-multiplication approach?
8 DISCUSSION Are these proportion problems analogous? Why or why not? 1. What is the value of the cross-multiplication approach? 2. What are its limitations?
9 DISCUSSION Are these proportion problems analogous? Why or why not? 1. What is the value of the cross-multiplication approach? 2. What are its limitations? 3. Consider how a student would approach this problem if they had been trained to solve all proportions with cross-multiplication: x 2 x 1 = 1 x 1
10 DISCUSSION Are these proportion problems analogous? Why or why not? 1. What is the value of the cross-multiplication approach? 2. What are its limitations? 3. Consider how a student would approach this problem if they had been trained to solve all proportions with cross-multiplication: x 2 x 1 = 1 x 1 4. Multiplying both sides by the LCD reveals the analogy between the two problems.
11 DISCUSSION Are these proportion problems analogous? Why or why not? 1. What is the value of the cross-multiplication approach? 2. What are its limitations? 3. Consider how a student would approach this problem if they had been trained to solve all proportions with cross-multiplication: x 2 x 1 = 1 x 1 4. Multiplying both sides by the LCD reveals the analogy between the two problems. 5. Cross multiplication is equivalent, but can be perceived as magical.
12 SAMPLE ANALOGOUS PROBLEM Can you solve them? 1. In a dice game, a winner gets 5 points and a loser gets 2 points. A student played this game 20 times. His total score was 67 points. How many times did he win, and how many times did he lose? 2. In a dice game, a winner gets 5 points and a loser gets 2 points. A student played the game several times. His total was 27 more points than what he would have earned had he lost all the games, and it was 33 fewer points than what he would have earned had he won all the games. How many games did the student play, and what was the score? These problems originally came from Watanabe (2015). Whitney Peterson (woot woot!) modified the original story context.
13 ALGEBRAIC TECHNIQUE 1. Let x be the number of wins and let y be the number of losses. Then 5 points 1 win x wins = 5x points and 2 points 1 loss y losses = 2y points.
14 ALGEBRAIC TECHNIQUE 1. Let x be the number of wins and let y be the number of losses. Then 5 points 1 win x wins = 5x points and 2 points 1 loss y losses = 2y points. 2. A system of equations that represents this situation is x + y = 20 (1) 5x + 2y = 67 (2)
15 ALGEBRAIC TECHNIQUE 1. Let x be the number of wins and let y be the number of losses. Then 5 points 1 win x wins = 5x points and 2 points 1 loss y losses = 2y points. 2. A system of equations that represents this situation is x + y = 20 (1) 5x + 2y = 67 (2) 3. Solving (1) for x gives and substituting into (2) gives 5(20 y) + 2y = y + 2y = y = = 3y 33 3 = y = 11
16 AREA MODEL APPROACH
17 ALGEBRAIC TECHNIQUE REVISITED 1. Let x be the number of wins and let y be the number of losses. Then 5 points 1 win x wins = 5x points and 2 points 1 loss y losses = 2y points.
18 ALGEBRAIC TECHNIQUE REVISITED 1. Let x be the number of wins and let y be the number of losses. Then 5 points 1 win x wins = 5x points and 2 points 1 loss y losses = 2y points. 2. A system of equations that represents this situation is x + y = 20 (3) 5x + 2y = 67 (4)
19 ALGEBRAIC TECHNIQUE REVISITED 1. Let x be the number of wins and let y be the number of losses. Then 5 points 1 win x wins = 5x points and 2 points 1 loss y losses = 2y points. 2. A system of equations that represents this situation is x + y = 20 (3) 5x + 2y = 67 (4) 3. Solving (3) for y and substituting into (2) gives 5x + 2(20 x) = 67 5x x = 67 3x = 67 3x = = 27 x = 27 3 = 9
20 AREA MODEL APPROACH REVISITED
21 MORE AREA MODELS Is it possible to solve this problem with an area model? In a dice game, a winner gets 5 points and a loser gets 2 points. A student played the game several times. His total was 27 more points than what he would have earned had he lost all the games, and it was 33 fewer points than what he would have earned had he won all the games. How many games did the student play, and what was the score?
22 MORE AREA MODELS Why yes... yes it is. Thanks Whitney!
23 MORE COWBELL... I MEAN COMPLEXITY! Is it possible to solve this problem with an area model? 1. The sisters baked cookies last weekend and want to split them between their relatives. One sister said, If we give 6 cookies to each, there will be 21 extra cookies left, but if we give 8 cookies to everyone, we are 9 cookies short. How many relatives are there, and how many cookies did the sisters bake?
24 AREA MODEL COOKIE PROBLEM
25 CHALLENGES 1. How do we get students to consider area models as a way to solve these problems?
26 CHALLENGES 1. How do we get students to consider area models as a way to solve these problems? 2. How do we help students who are proficient with these visual reasoning tools learn the power and usefulness of more advanced mathematical tools like linear equations? (Watanabe (2015, p. 157))
27 THOUGHTS FROM THE AUTHOR Teachers could give students a hypothetical student s solution to a problem that uses a visual representation tool and ask them to explain how the visual model represents the problem situation. Then, they could follow up by using visual representations to solve other problems (p. 157).
28 THOUGHTS FROM THE AUTHOR These visual representations are adequate when the focus is on solving particular problems... If the classroom discussion focuses only on the correctness of the answers, [visual and algebraic] solution strategies would be equally valid. On the other hand, class discussions that extend beyond the correctness of answers or build on student solutions... may be enriched by these visual representations (p. 157).
29 THANKS FOR BEING A PART OF OUR PROGRAM! Dr. Roger Fischer Mathematics Instructor Gallatin College Montana State University 201 Hamilton Hall Bozeman, MT roger.fischer@montana.edu
30 REFERENCES I Watanabe, T. (2015). Visual reasoning tools in action. Mathematics Teaching in the Middle School, 21(3),
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