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Math Alliance – Teaching All LearnersSummer 2011
Beth SchefelkerChris Guthrie
Melissa Hedges
Quantitative Analysis (Q.A): Surfacing Operation Sense in
Problem Solving
www.mmp.uwm.edu
Thinking about problem solvingWhat 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?
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?
Polya’s Four Steps to Problem SolvingFirst: Understanding the
Problem
Second: Devising A Plan
Third: Carrying Out The Plan
Fourth: Looking Back
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
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.
Quantitative Analysis“…the process of coming to understand the quantities and relationships between those quantities in a word problem.”
Quantity vs. ValueA 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.
Quantity vs. Value Examples
Hamburger costs $1.57 per poundQuantity: Cost of hamburger per poundValue: $1.57 per pound
Melissa’s dog weighs 40 lbsQuantity: weight of Melissa’s dogValue: 40 lbs
Leslie saved 365 nickelsQuantity: # of nickels Leslie savedValue: 365 nickels
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…
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?
Discuss these questions when your chart is completeAre 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?
Dieter’s ProblemTwo 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.
Rethinking the processTalk 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 SolvingFirst: Understanding the ProblemSecond: Devising A PlanThird: Carrying Out The PlanFourth: Looking Back
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.
Benefits of Quantitative AnalysisSupports 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.
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 thequantities and the values. Complete chart and discuss
relationships between quantities. Write an equation(s) using the quantities. No values
please. Explain why you selected the operation(s) you did.