Bonnie Vondracek Susan Pittman

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Bonnie Vondracek Susan Pittman
August 2224, 2006 Washington, DC
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• Through problem solving, students can
  experience the power and utility of mathematics.
  Problem solving is central to inquiry and
  application and should be interwoven throughout
  the mathematics curriculum to provide a context
  for learning and applying mathematical ideas.
• NCTM 2000, p. 256

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NCTM Problem-Solving Standard

• Instructional programs . . . should enable all
• students to
• Build new mathematical knowledge through problem
  solving
• Solve problems that arise in mathematics and in
  other contexts
• Apply and adapt a variety of appropriate
  strategies to solve problems
• Monitor and reflect on the process of
  mathematical problem solving
• Principles and Standards for School Mathematics
  (NCTM 2000)

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Welcome!

• Lets begin with a
• Brainteaser
• Math Starter
• Mathematic Motivator
• Math Bender

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Math Starter

• The following letters represent the digits of a
  simple addition
• S E N D
• M O R E
• M O N E Y
• Find the digits that represent the letters to
  make this addition correct. Each letter
  represents a unique digit and M is not equal to
  D.

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Math Starter

• The following letters represent the digits of a
  simple addition
• 9 5 6 7
• 1 0 8 5
• 1 0 6 5 2
• Find the digits that represent the letters to
  make this addition correct. Each letter
  represents a unique digit and M is not equal to
  D.

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GED Mathematics Test Overview

• Four Content Areas
• Number Operations and Number Sense
• Measurement and Geometry
• Data, Statistics, and Probability
• Algebra, Functions, and Patterns

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GED Mathematics Test Overview

• Three Question Types
• Procedural
• Conceptual
• Application

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GED Mathematics Test Overview

• Procedural questions require students to
• Select and apply correct operations or procedures
• Modify procedures when needed
• Read and interpret graphs, charts, and tables
• Round, estimate, and order numbers
• Use formulas

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GED Mathematics Test Overview

• Sample Procedural Test Question
• A companys shipping department is receiving
• a shipment of 3,144 printers that were
• packed in boxes of 12 printers each. How
• many boxes should the department receive?
• PLEASE DO NOT WRITE IN THIS TEST BOOKLET.
• Mark your answer in the circles in the grid on
• your answer sheet.

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GED Mathematics Test Overview

• Conceptual questions require students to
• Recognize basic mathematical concepts
• Identify and apply concepts and principles of
  mathematics
• Compare, contrast, and integrate concepts and
  principles
• Interpret and apply signs, symbols, and
  mathematical terms
• Demonstrate understanding of relationships among
  numbers, concepts, and principles

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GED Mathematics Test Overview

• Sample Conceptual Test Question
• A salesperson earns a weekly salary of 225 plus
  3 for every pair of shoes she sells. If she
  earns a total of 336 in one week, in which of
  the following equations does n represent the
  number of shoes she sold that week?
• (1) 3n 225 336
• (2) 3n 225 3 336
• (3) n 225 336
• (4) 3n 336
• (5) 3n 3 336

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GED Mathematics Test Overview

• Application/Modeling/Problem Solving questions
  require students to
• Identify the type of problem represented
• Decide whether there is sufficient information
• Select only pertinent information
• Apply the appropriate problem-solving strategy
• Adapt strategies or procedures
• Determine whether an answer is reasonable

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GED Mathematics Test Overview

• Sample Application/Modeling/Problem Solving Test
  Question
• Jane, who works at Marine Engineering, can make
  electronic widgets at the rate of 27 per hour.
  She begins her day at 930 a.m. and takes a 45
  minute lunch break at 1200 noon. At what time
  will Jane have made 135 electronic widgets?
• 145 p.m.
• 215 p.m.
• 230 p.m.
• 315 p.m.
• 515 p.m.

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What Does Math Involve?

• Memory
• Language
• Sequencing
• Spatial ordering
• Critical thinking
• Good problem-solving strategies
• Number sense
• Reasoning
• Making connections

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Thinking With Numbers

• Are your students ready to tackle a math problem
  with confidence?
• Do they have a briefcase filled with
  problem-solving strategies that help them when
  they encounter a new problem?
• Do they get confused about how to solve problems?
• Do they have fun posing problems with math?

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An Effective Problem Solver

• Reads the problem carefully
• Defines the type of answer that is required
• Identifies key words
• Accesses background knowledge regarding a similar
  situation
• Eliminates extraneous information
• Uses a graphic organizer
• Sets up the problem correctly
• Uses mental math and estimation
• Checks the answer for reasonableness

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What is Problem Solving?

• According to Michael E. Martinez
• There is no formula for problem solving
• How people solve problems varies
• Mistakes are inevitable
• Problem solvers need to be aware of the total
  process
• Flexibility is essential
• Error and uncertainty should be expected
• Uncertainty should be embraced at least
  temporarily

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Improving Problem-Solving Skills

• Solve problems out loud
• Explain your thinking process
• Allow students to explain their thinking process
• Use the language of math and require students to
  do so as well
• Model strategy selection
• Make time for discussion of strategies
• Build time for communication
• Ask open-ended questions
• Create lessons that actively engage learners
• Jennifer Cromley, Learning to Think, Learning to
  Learn

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Generating Questions

• At the ABC Adult Center, thirty-three students
  from Mr. James class took and passed the GED
  Mathematics Test with a 420 or above. Twenty-five
  percent of the class did not pass the test. How
  many students took the test?

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Time Out for a Problem

• A bee, starting in cell A of the honeycomb design
  wishes to stroll to cell G via a path of
  connected cells. Each step in the journey must
  take the bee to a neighboring cell with a letter
  higher in the alphabet. (For example, A-B-D-E-G
  is a valid path.) How many different routes are
  there from A to G?

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Math Graphic Organizers

• Common graphic organizers
• Hierarchical diagrams
• Sequence charts
• Compare/contrast charts

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Math Graphic Organizers

• Polynomials

Polyas 4-Step Problem Solving Method
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Math Graphic Organizers

• A New Kind of Graphic Organizer
• Builds comprehension skills
• Requires analysis of the problem
• Encourages the use of a variety of strategies
• Incorporates estimation
• Shifts focus to the process of problem solving

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Math Graphic Organizer

• Asks for the main idea
• What is happening in the problem?
• Asks the question
• What is the problem asking you to do?
• Lays out the facts
• What is pertinent, what is irrelevant?
• Checks to see if the answer relates to the
  question asked

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Using a Graphic Organizer
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Four Steps to Problem Solving

• Understand the problem
• Devise a plan
• Carry out the plan
• Examine the solution obtained

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Four Steps to Problem Solving

• Understand the problem
• What is the unknown?
• What are the data and conditions?
• Can you satisfy the condition?
• Is there sufficient information to determine the
  unknown?
• Can you draw a figure?
• Can you write down the different parts?

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Lets Try a Simple Problem

• Four boys work together painting houses. For
  each house they paint, they get 256.00. Each
  house will be painted a different color. If the
  boys work for 4 months and their expenses are
  152.00 per month, how many houses must they
  paint for each of them to have 1,000.00?

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Four Steps to Problem Solving

• Find out using the Graphic Organizer

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Four Steps to Problem Solving

• Devise a plan
• Is there a connection between the data and the
  unknown? What is it?
• Have you see a similar problem?
• Could you restate the problem?
• What strategy can you use to solve this problem?

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Whats Your Strategy?

• Compute or simplify
• Use a formula
• Guess, check, and revise
• Consider a simpler case

• Eliminate
• Make a table, chart, or list
• Look for patterns
• Work backwards
• Make a model or diagram

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Four Steps to Problem Solving

• Choose a strategy using a Graphic Organizer

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Sample Relationship Sentence

• Divide the total amount that the boys want to
  earn in the given time period and the total
  amount of expenses in the given time period by
  the amount earned per house.

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You Try It!

• Julia Child was roasting a turkey. It has been
  out of the oven for 20 minutes. The turkey was
  roasting for 4 hours and 15 minutes. The oven was
  preheated for 10 minutes. If it is now 500 p.m.,
  then what time did Julia put the turkey in the
  oven?

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Sample Relationship Sentence

• From the current time, take the amount of time
  that the turkey was roasting in the oven and the
  amount of time it has been out of the oven. This
  will give you the time Julia put the turkey into
  the oven.

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Four Steps to Problem Solving

• Carry out the plan
• Use the selected strategy to solve the problem
• Follow the plan in sequence
• Complete the computations to obtain the answer
• Show all work
• Can you see clearly that each step is correct?

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Four Steps to Problem Solving

• Solve it using a Graphic Organizer

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Four Steps to Problem Solving

• Examine the solutions obtained
• Did you answer the question asked?
• Did you check your results?
• Is your answer in the correct units?
• Does your answer seem reasonable?
• Could you solve the problem differently?

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Four Steps to Problem Solving

• Look back using a Graphic Organizer

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Four Steps to Problem Solving

• A side of square B is four times the length of a
  side of square A. How many times greater is the
  area of square B than the area of square A?
• Square A Square B

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Four Steps to Problem Solving

• Byron purchased a 5,000 certificate of deposit
  (CD) at his local bank. The CD will pay him 7
  percent simple interest at the end of two years.
  In dollars, how much INTEREST will Byron have
  earned from his CD at the end of the two-year
  period?

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Reflection Questions

• What are some important things to consider as you
  select rich mathematics problems for your
  students to solve?
• If your students have little background with
  problem-solving strategies, how could you help
  them develop and use such strategies in your
  classroom?
• Why is communication a critical element of the
  problem-solving standard?

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Reflection Questions

• In what ways is this lesson a rich topic for
  exploration?
• How does the problem provide a basis for
  mathematical discussion among the students?
• What is the role of the teacher in setting the
  classroom environment for effective problem
  solving? Be specific. What can you do in your
  classroom to help students learn by exploring new
  concepts in a problem-solving situation?
• Students might work on problems in groups or
  individually. What are the advantages and
  disadvantages to each? How do each of these kinds
  of working environments, or the two combined,
  elicit problem solving?
• How do a teachers questions help students
  solidify their understanding of the mathematical
  concepts developed in a problem?
• What techniques can teachers use to help students
  get started on solving rich problems?