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3. Math Alliance – Teaching All Learners Summer 2011 Beth Schefelker Chris Guthrie Melissa Hedges Quantitative Analysis (Q.A): Surfacing Operation Sense in Problem Solving www.mmp.uwm.edu

4. Thinking about problem solving What comes to mind when you hear the words “problem solving”? What is the purpose of problem solving in mathematics? What do your student think of when they hear those words? What strategies do you use to help them?

5. What Advice Would You Give To Students? Louise has a bag full of marbles. On the way to the classroom she dropped the bag and 31 of the marbles went under the bookcase. Louise divided the leftover marbles into 4 piles with 16 marbles in each pile. How many marbles were in the bag when it was full?

6. Polya’s Four Steps to Problem Solving First: Understanding the Problem Second: Devising A Plan Third: Carrying Out The Plan Fourth: Looking Back

7. Learning Intention Success Criteria We Are Learning To… Develop an understanding of quantitative analysis with contextual situations. Success Criteria… Apply quantitative analysis process in order to reason through two-step word problems. Make connections to Standards for Mathematical Practice #2 Reason Abstractly and Quantitatively

8. NCTM says… To use algebra for solving a problem, the focus of attention is not on getting numerical answers to each step of the solution but on the operations used. Key Idea To Consider… It is important, therefore, that students get experience in identifying which operation they are using to solve problem.

9. Quantitative Analysis “…the process of coming to understand the quantities and relationships between those quantities in a word problem.” Quantity vs. Value A quantity is anything that can be measured or counted. The value of the quantity is its measure or the number of items that are counted. It involves a number and a unit. Clement, L. & Bernhard, J. (2005). A problem-solving alternative to using key words. Mathematics Teaching in the Middle School. 10(7) pp.360-365.

10. Quantity vs. Value Examples Hamburger costs $1.57 per pound Quantity: Cost of hamburger per pound Value: $1.57 per pound Melissa’s dog weighs 40 lbs Quantity: weight of Melissa’s dog Value: 40 lbs Leslie saved 365 nickels Quantity: # of nickels Leslie saved Value: 365 nickels

11. Thinking about the Dieter’s Problem Using a Think Aloud to clarify the problem: Read the problem. Flip your paper over. Retell the problem. Clarify vocabulary Use guiding questions to quantitatively analyze the problem. Thinking about where to start…

12. Quantitative Analysis of Dieter’s Problem Guiding Questions for Step # 4 What quantities are involved in this situation? For which quantities do we know the values? For which quantities do we not know the value? What quantities are we trying to find? Which quantities are critical to the problem?

13. QuantityValue Known - record Unknown Useful in solving the problem Quantitative Analysis Worksheet

14. Discuss these questions when your chart is complete Are any quantities related to other quantities in the situation? Could these relationships help us find any unknown values? Would drawing a diagram or acting out the situation help to answer any of the above questions?

15. Dieter’s Problem Two people who have been on diets are talking: Dieter A: “I lost 1/8 of my weight – I lost 19 pounds.” Dieter B: “I lost 1/6 of my weight, and now you weigh 2 pounds less than I do.” What was Dieter B’s original weight? Clement, L. & Bernhard, J. (2005). A problem-solving alternative to using key words. Mathematics Teaching in the Middle School. 10(7) pp.360-365.

16. Rethinking the process Talk to a shoulder partner and retell the process used to work through the dieter’s problem. What connections can be made to Polya’s Problem Solving? Polya’s Four Steps to Problem Solving First: Understanding the Problem Second: Devising A Plan Third: Carrying Out The Plan Fourth: Looking Back

17. Connecting to the Math Practice Standards Reason Abstractly and Quantitatively Read practice standard #2 silently to yourself Highlight 3 phrases in the standard that connect to the dieters task. Turn and talk with a neighbor about your finding.

18. Benefits of Quantitative Analysis Supports students as they make sense of mathematics an develops operation sense. Strengthens ability to reflect on their own thinking and make it explicit. Provides a tool to communicate understanding. Validates good problem solving.

19. Time to practice! Work with a table partner to… Pick one of the four problems on the table. Read the problem. Practice retelling the problem. Quantitatively analyze the problem. Use the guiding questions on the chart to help select the quantities and the values. Complete chart and discuss relationships between quantities. 1. Write an equation(s) using the quantities. No values please. 2. Explain why you selected the operation(s) you did.