LESSON PLAN. Unit: Problem Solving Day 1(Lesson 1) Date: Summer 2008

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1 LESSON PLAN Grade Level/Course: Grade 6 Mathematics Teacher(s): Ms. Green & Mr. Nielsen Unit: Problem Solving Day 1(Lesson 1) Date: Summer 2008 Topic(s): Poyla s Model: Steps to Problem Solving Resources: Transparencies: Worksheets: Problem Solving Strategies from George Polya (Reardon Solving Gifts, Inc.) Technology: Smart Board & Projector Manipulatives: Other Supplies: List Graphic Organizer Chart Assessments: Classwork: Produce four steps used for solving problems Assignment Options: Project/Performance Task: Quiz/Test: Objective Items Short Answer with Work Shown Extended Response Individual or Group Presentation: Share group results with class Notebook/Portfolio: Journal/Exit Slips: Observation(s): 2007 MN Math Standards and Benchmarks Number & Operations Data Analysis and Probability Algebra Geometry & Measurement X Problem Solving Instructional Strategies Launch: Questioning Brainstorming Demonstration Inquiry Setting Objectives/Goals Reinforcing Effort X Accessing Prior Knowledge Recording Information Graphic Organizers & Other Non-LRs Explore: Questioning X Cooperative Learning Instructional Technology X Presentations/Sharing Problem Solving Compare and Contrast Guided Practice X Problem-based Learning Simulations and Modeling Reinforcing Effort X Graphic Organizers & Other Non-LRs Summarize: Questioning Cooperative Learning Reinforcing Effort X Written or Oral Summaries X Compare and Contrast X Graphic Organizers & Other Non-LRs Providing Feedback X Discuss X Analyze Apply: Questioning Research Problem Solving Presentations and Exhibitions Project Design Connections to Other Disciplines

2 The Plan: LAUNCH (Introducing) Access student s prior knowledge about problem solving by posing several questions. How many of you have faced a problem? How did you go about solving the problem? Notes/Reflections/Vocabulary: 10 minutes Vocabulary *Metacognition Objective: Develop a list of steps that you use to solve problems. EXPLORE (Investigation(s))(Making, Investigating, Finding...) Students will work in small group to solve and record the steps that they use to solve a problem. The teacher will facilitate small group learning by posing questions. Problem: Brian has fourteen pets. All of his pets are cats and dogs. Brian has four more cats than dogs. How many dogs does he have? SUMMARIZE (Wrapping the Lesson) (Discussing, Writing...) Bring the group together to share list graphic organizer of problem solving steps. As a whole group, compare and contrast similarities and differences among the charts. Compile a class graphic organizer showing the four agreed upon steps to problem solving. Have students expand the four steps by asking them questions. What does understand the problem mean? What does it look like? What questions should you ask yourself? Etc. 25 minutes George Polya s Steps 1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back 15 minutes Homework Students will continue expanding upon the four problem solving steps. What does it mean? What does it look like?

3 Problem Solving Steps Step 1 Step 2 Step 3 Step 4

4 LESSON PLAN Grade Level/Course: Grade 6 Mathematics Teacher(s): Ms. Green & Mr. Nielsen Unit: Problem Solving Day 2 (Lesson 2) Date: Summer 2008 Topic(s): Understanding the Steps of Problem Solving Resources: Transparencies: Worksheets: Problem Solving Strategies from George Polya (Reardon Solving Gifts, Inc.) Technology: Smart Board & Projector Manipulatives: Other Supplies: List Graphic Organizer Chart Assessments: Classwork: Produce four steps used for solving problems Assignment Options: Project/Performance Task: Quiz/Test: Objective Items Short Answer with Work Shown Extended Response Individual or Group Presentation: Share group results with class Notebook/Portfolio: Journal/Exit Slips: Observation(s): 2007 MN Math Standards and Benchmarks Number & Operations Data Analysis and Probability Algebra Geometry & Measurement X Problem Solving Instructional Strategies Launch: Questioning Brainstorming Demonstration Inquiry Setting Objectives/Goals Reinforcing Effort X Accessing Prior Knowledge Recording Information Graphic Organizers & Other Non-LRs Explore: Questioning X Cooperative Learning Instructional Technology X Presentations/Sharing Problem Solving Compare and Contrast Guided Practice X Problem-based Learning Simulations and Modeling Reinforcing Effort X Graphic Organizers & Other Non-LRs Summarize: Questioning Cooperative Learning Reinforcing Effort X Written or Oral Summaries X Compare and Contrast X Graphic Organizers & Other Non-LRs Providing Feedback X Discuss X Analyze Apply: Questioning Research X Problem Solving Presentations and Exhibitions Project Design Connections to Other Disciplines

5 The Plan: LAUNCH (Introducing) The students will recall the four steps of problem solving by referring back to previous lesson. George Polya s Steps 1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back Objective: Understand what the four steps of problem solving mean. EXPLORE (Investigation(s))(Making, Investigating, Finding...) As a whole group, continue to expand what the four steps of problem solving mean. Using the Smart Board, create a class graphic organizer of the shared information. First, focus on steps 1, 3, and 4. Second, focus on step 2 and have students generate problem solving strategies that they use to solve math problems. Ask students, to recall strategies that they used on Day 1. Notes/Reflections/Vocabulary: 5 minutes Vocabulary *Strategies 40 minutes Problem Solving Strategies Guess and check Make an organized list Draw a picture or diagram Look for a pattern Make a table Use a variable Solve a simpler problem Experiment Act it out Work backwards Use deduction Change your point of view SUMMARIZE (Wrapping the Lesson) (Discussing, Writing...) Ask students reflective questions. What was the main concept/skill we learned? Can someone restate our problem solving model? What is the difference between a problem solving model and a problem solving strategy? What is the best problem solving strategy and why? Facilitate the discussion so that students realize that there are many different paths to get to a solution and some strategies are more effective than other depending on the problem. 5 minutes

6 LESSON PLAN Grade Level/Course: Grade 6 Mathematics Teacher(s): Ms. Green & Mr. Nielsen Unit: Problem Solving Day 3 (Lesson 3) Date: Summer 2008 Topic(s): Exploring Problem Solving Strategies Resources: Transparencies: Worksheets: Circle Problem Technology: Smart Board & Projector Manipulatives: Other Supplies: Assessments: Classwork: Solve a problem by applying the Guess & Check Strategy Assignment Options: Project/Performance Task: Quiz/Test: Objective Items Short Answer with Work Shown Extended Response Individual or Group Presentation: Share group results with class Notebook/Portfolio: Journal/Exit Slips: Observation(s): 2007 MN Math Standards and Benchmarks Number & Operations Data Analysis and Probability Algebra Geometry & Measurement X Problem Solving Instructional Strategies Launch: Questioning Brainstorming Demonstration Inquiry Setting Objectives/Goals Reinforcing Effort X Accessing Prior Knowledge Recording Information Graphic Organizers & Other Non-LRs Explore: Questioning X Cooperative Learning Instructional Technology X Presentations/Sharing Problem Solving Compare and Contrast Guided Practice X Problem-based Learning Simulations and Modeling Reinforcing Effort X Graphic Organizers & Other Non-LRs Summarize: Questioning Cooperative Learning Reinforcing Effort X Written or Oral Summaries X Compare and Contrast X Graphic Organizers & Other Non-LRs Providing Feedback X Discuss X Analyze Apply: Questioning Research X Problem Solving Presentations and Exhibitions Project Design Connections to Other Disciplines

7 The Plan: LAUNCH (Introducing) Have students revisit the list of problem solving strategies created during day 2. Let the students know that they will be focusing on using the Guess and Check strategy to solve a problem. Discuss what Guess and Check means. Objective: Use the Problem Solving Model to apply the Guess and Check problem solving strategies. EXPLORE (Investigation(s))(Making, Investigating, Finding...) In groups of 4 or 5, have the students complete the Circle Problem. Make sure that groups are following the Problem Solving Model and discussing key questions for each step. Step1: What are you asked to find out or show? Step 2: What strategy do you plan to use? Step 3: Why is it important to keep a list of you guesses and checks? Step 4: Does your answer make sense? Are there multiple solutions for this problem? Could you have done this problem another way maybe even an easier way? Problem: Place the digits 1, 2, 3, 4, and 5 in these circles so that the sums across (horizontally) and down (Vertically) are the same. Is there more than one solution? Notes/Reflections/Vocabulary: 10 minutes Vocabulary *estimate *reasonable *horizontal *vertical 30 minutes Problem Solving Strategies Guess and check Make an organized list Draw a picture or diagram Look for a pattern Make a table Use a variable Solve a simpler problem Experiment Act it out Work backwards Use deduction Change your point of view SUMMARIZE (Wrapping the Lesson) (Discussing, Writing...) Bring the group together to share their solutions. Compare and contrast similarities and differences among the groups. Ask students reflective questions. Why did the Guess and Check strategy work with this problem? Why is it important to keep a list of you guesses and checks? Can you solve this problem using a different strategy? 10 minutes Extra Practice Problem: Jamie went out to her grandfather's farm. Her grandfather has pigs and chickens on his farm. She noticed that there were a total of 26 heads and 68 feet among them. How many chickens and how many pigs did her grandfather have?

8 Guess & Check Name Date Circle Problem Place the digits 1, 2, 3, 4, and 5 in these circles so that the sums across (horizontally) and down (Vertically) are the same. Is there more than one solution? Guess & Check 1 Guess & Check 2 Guess & Check 3 Guess & Check 4 Guess & Check 5 Guess & Check 6 Guess & Check 7 Guess & Check 8 Guess & Check 9

9 LESSON PLAN Grade Level/Course: Grade 6 Mathematics Teacher(s): Ms. Green & Mr. Nielsen Unit: Problem Solving Day 4 (Lesson 4) Date: Summer 2008 Topic(s): Exploring Problem Solving Strategies Resources: Transparencies: Worksheets: Dart Problem Technology: Smart Board & Projector Manipulatives: Other Supplies: Assessments: Classwork: Solve a problem by applying the Make an Organized List Strategy Assignment Options: Project/Performance Task: Quiz/Test: Objective Items Short Answer with Work Shown Extended Response Individual or Group Presentation: Share group results with class Notebook/Portfolio: Journal/Exit Slips: Observation(s): 2007 MN Math Standards and Benchmarks Number & Operations Data Analysis and Probability Algebra Geometry & Measurement X Problem Solving Instructional Strategies Launch: Questioning Brainstorming Demonstration Inquiry Setting Objectives/Goals Reinforcing Effort X Accessing Prior Knowledge Recording Information Graphic Organizers & Other Non-LRs Explore: Questioning X Cooperative Learning Instructional Technology X Presentations/Sharing Problem Solving Compare and Contrast Guided Practice X Problem-based Learning Simulations and Modeling Reinforcing Effort X Graphic Organizers & Other Non-LRs Summarize: Questioning Cooperative Learning Reinforcing Effort X Written or Oral Summaries X Compare and Contrast X Graphic Organizers & Other Non-LRs Providing Feedback X Discuss X Analyze Apply: Questioning Research X Problem Solving Presentations and Exhibitions Project Design Connections to Other Disciplines

10 The Plan: LAUNCH (Introducing) Have students play a dart simulated game by toss a ball at target on the floor. Further, activate students prior knowledge by asking them what they know about darts. Let students know that they will use a different problem solving strategy to solve a problem about darts. Have students revisit the list of problem solving strategies created during day 2. Have students make predictions about what strategy they might use to solve a problem about darts. Let the students know that they will be focusing on Making an Organized List as a strategy to solve the dart problem. Discuss what an organized list my look like and how might it be helpful to solve a problem. Objective: Use the Problem Solving Model to apply the Make an Organized List problem solving strategies. EXPLORE (Investigation(s))(Making, Investigating, Finding...) In groups of 3 or 4, have the students complete the Dart Problem. Make sure that groups are following the Problem Solving Model and discussing key questions for each step. Step1: What are you asked to find out or show? How many darts do you have? Step 2: What strategy do you plan to use? How are you going to keep you list organized? Step 3: What patterns or trends do you see in your organized list? Step 4: Does your answer make sense? Are there other ways to organize you list? Could you have done this problem another way maybe even an easier way? Notes/Reflections/Vocabulary: 10 minutes Vocabulary *Patterns *Trends 30 minutes Problem Solving Strategies Guess and check Make an organized list Draw a picture or diagram Look for a pattern Make a table Use a variable Solve a simpler problem Experiment Act it out Work backwards Use deduction Change your point of view Problem: Three darts hit this dart board and each scores a 1, 5, or 10. The total score is the sum of the scores for the three darts. There could be three 1 s, two 1 s and one 5 and two 10 s, and so on. How many different possible total scores could a person get with three darts? SUMMARIZE (Wrapping the Lesson) (Discussing, Writing...) Bring the group together to share their solutions. Compare and contrast similarities and differences among the groups. Ask students reflective questions. Why did the Make an Organize List strategy work with this problem? Why is it important to organize the list? Can you solve this problem using a different strategy? Homework Use the Make an Organized List Strategy to List the 4-digit numbers that can be written using each of 1, 3, 5, and 7 once and only once. Extension Have students play ball toss darts and record their scores and combinations. 10 minutes

11 Make An Organized List Name Date Darts Organized List # of 1 s # of 5 s # of 10 s Score

12 LESSON PLAN Grade Level/Course: Grade 6 Mathematics Teacher(s): Ms. Green & Mr. Nielsen Unit: Problem Solving Day 5 (Lesson 5) Date: Summer 2008 Topic(s): Exploring Problem Solving Strategies Resources: Transparencies: Worksheets: Stock Car Race Problem Technology: Smart Board & Projector Manipulatives: Other Supplies: Assessments: Classwork: Solve a problem by applying the Draw a Diagram Strategy Assignment Options: Project/Performance Task: Quiz/Test: Objective Items Short Answer with Work Shown Extended Response Individual or Group Presentation: Share group results with class Notebook/Portfolio: Journal/Exit Slips: Observation(s): 2007 MN Math Standards and Benchmarks Number & Operations Data Analysis and Probability Algebra Geometry & Measurement X Problem Solving Instructional Strategies Launch: Questioning Brainstorming Demonstration Inquiry Setting Objectives/Goals Reinforcing Effort X Accessing Prior Knowledge Recording Information Graphic Organizers & Other Non-LRs Explore: Questioning X Cooperative Learning Instructional Technology X Presentations/Sharing Problem Solving Compare and Contrast Guided Practice X Problem-based Learning Simulations and Modeling Reinforcing Effort X Graphic Organizers & Other Non-LRs Summarize: Questioning Cooperative Learning Reinforcing Effort X Written or Oral Summaries X Compare and Contrast X Graphic Organizers & Other Non-LRs Providing Feedback X Discuss X Analyze Apply: Questioning Research X Problem Solving Presentations and Exhibitions Project Design Connections to Other Disciplines

13 The Plan: LAUNCH (Introducing) Have students revisit the list of problem solving strategies created during day 2. Let the students know that they will be focusing on using the Draw a Diagram strategy to solve a problem. Discuss what drawing a diagram means and how it is helpful. Notes/Reflections/Vocabulary: 10 minutes Vocabulary Objective: Use the Problem Solving Model to apply the Draw a Diagram problem solving strategies. EXPLORE (Investigation(s))(Making, Investigating, Finding...) In groups of 4, or 5, have the students complete the Stock Car Problem. Make sure that groups are following the Problem Solving Model and discussing key questions for each step. Step1: What are you asked to find out or show? What are the five types of cars? What does each line tell you? Step 2: What strategy do you plan to use? Step 3: How does a diagram or picture help you? Step 4: Does your answer make sense? Could you have done this problem another way maybe even an easier way? Problem: In a stock car race, the first five finishers in some order were a Ford, a Pontiac, a Chevrolet, a Buick, and a Dodge. The Ford finished seven seconds before the Chevrolet. The Pontiac finished six seconds after the Buick. The Dodge finished eight seconds after the Buick. The Chevrolet finished two seconds before the Pontiac. In what order did the cars finish the race? SUMMARIZE (Wrapping the Lesson) (Discussing, Writing...) Bring the group together to share their solutions. Compare and contrast similarities and differences among the diagrams. Ask students reflective questions. Why did the Draw a Diagram strategy work with this problem? Can you solve this problem using a different strategy? 30 minutes Problem Solving Strategies Guess and check Make an organized list Draw a picture or diagram Look for a pattern Make a table Use a variable Solve a simpler problem Experiment Act it out Work backwards Use deduction Change your point of view 10 minutes Homework Problem: Four friends ran a race: Matt finished seven seconds ahead of Ziggy. Bailey finished three seconds behind Sam. Ziggy finished five seconds behind Bailey. In what order did the friends finish the race?

14 Draw a Diagram Name Date Stock Car Race In a stock car race, the first five finishers in some order were a Ford, a Pontiac, a Chevrolet, a Buick, and a Dodge. The Ford finished seven seconds before the Chevrolet. The Pontiac finished six seconds after the Buick. The Dodge finished eight seconds after the Buick. The Chevrolet finished two seconds before the Pontiac. In what order did the cars finish the race? What strategy did you use?

15 NCTM Problem Solving Standard for Grades 6 8 Instructional programs from prekindergarten through grade 12 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. Resources Reardon Problem Solving Gifts, Inc. Problem solving document Grandfather s Farm Problem Problem Solving Website

16 TEACHING PROBLEM SOLVING STRATEGIES IN THE 5 12 CURRICULUM (Thank you George Polya) GOAL The students will learn several Problem Solving Strategies and how use them to solve non-traditional and traditional type problems. The main focus is to get students to THIMK! (I know it s supposed to be THINK, but I just wanted to get your attention. I did. ) OBJECTIVES Upon completion of this unit, each student should: Know George Polya s four principles of Problem Solving Have an arsenal of Problem Solving Strategies Approach Problem Solving more creatively Attack the solution to problems using various strategies Acquire more confidence in using mathematics meaningfully PREREQUISITES The prerequisites for the students will vary. The teacher will need to read the examples and exercises to decide which problems are appropriate for your students and the level of mathematics that they understand. Most of these problems were originally written for elementary and middle school mathematics students. However, many of these problems are excellent for high school students also. MATERIALS SOURCES This document Calculators are encouraged (graphing or scientific is adequate) Option: Creative Problem Solving in School Mathematics by George Lenchner, 1983 How To Solve It, George Polya, 1945 Creative Problem Solving in School Mathematics, George Lenchner, 1983 NCTM Principles and Standards, 2000 Mathematical Reasoning for Elementary Teachers, Calvin T. Long and Duane W. DeTemple, 1996 Intermediate Algebra and Geometry, Tom Reardon, 2001 Problems Sets from Dr. G. Bradley Seager, Jr., Duquesne University, 2000 Where ever else I can find good problems! C 2001 Reardon Problem Solving Gifts, Inc. Page 1

17 TEACHER BACKGROUND INFORMATION There is a poetry and beauty in mathematics and every student deserves to be taught by a person that shares that point of view. Long and DeTemple Problem Solving is one of the five Process Standards of NCTM s Principles and Standards for School Mathematics The following is taken from pages 52 through 55 of that document. Problem Solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Solving problems is not only a goal of learning mathematics but also a major means of doing so. Students should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort and then be encouraged to reflect on their thinking. By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages. Problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program. Problem solving in mathematics should involve all five content areas: Number and Operations, Algebra, Geometry, Measurement, and Data Analysis & Probability. Problem Solving Standard Instructional programs from prekindergarten through grade 12 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 The teacher s role in choosing worthwhile problems and mathematical tasks is crucial. By analyzing and adapting a problem, anticipating the mathematical ideas that can be brought out by working on the problem, and anticipating students questions, teachers can decide if particular problems will help to further their mathematical goals for the class. There are many, many problems that are interesting and fun but that may not lead to the development of the mathematical ideas that are important for a class at a particular time. Choosing problems wisely, and using and adapting problems from instructional materials, is a difficult part of teaching mathematics. C 2001 Reardon Problem Solving Gifts, Inc. Page 2

18 INTRODUCTION PROBLEM SOLVING STRATEGIES FROM GEORGE POLYA George Polya ( ) was one of the most famous mathematics educators of the 20 th century (so famous that you probably never even heard of him). Dr. Polya strongly believed that the skill of problem solving could and should be taught it is not something that you are born with. He identifies four principles that form the basis for any serious attempt at problem solving: 1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back (reflect) 1. Understand the problem What are you asked to find out or show? Can you draw a picture or diagram to help you understand the problem? Can you restate the problem in your own words? Can you work out some numerical examples that would help make the problem more clear? 2. Devise a plan A partial list of Problem Solving Strategies include: Guess and check Solve a simpler problem Make an organized list Experiment Draw a picture or diagram Act it out Look for a pattern Work backwards Make a table Use deduction Use a variable Change your point of view 3. Carry out the plan Carrying out the plan is usually easier than devising the plan Be patient most problems are not solved quickly nor on the first attempt If a plan does not work immediately, be persistent Do not let yourself get discouraged If one strategy isn t working, try a different one 4. Look back (reflect) Does your answer make sense? Did you answer all of the questions? What did you learn by doing this? Could you have done this problem another way maybe even an easier way? C 2001 Reardon Problem Solving Gifts, Inc. Page 3