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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

ProblemSecond: Devising A PlanThird: Carrying Out The PlanFourth: 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 ExamplesHamburger costs $1.57 per pound

Quantity: 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?

Quantity ValueKnown - recordUnknown

Useful in solving the problem

Quantitative Analysis Worksheet

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.