Unit title*  Problem Solving
Subject area* Mathematics
Secondary subject 
Grade*5th grade  
Duration/time* 37-40 fifty minute lessons
Overview* Through solving a series of difficult mathematic problems by applying and adapting appropriate strategies, students will increase their mathematical understanding and their willingness and ability to solve problems.  Groups of students will summarize their new problem solving abilities by preparing a Hyperstudio presentation about problem solving.
   The four steps in problem solving are: 1. Understand the problem, 2. Devise a plan, 3. Carry out the plan, and 4. Look back.
   The nine strategies for problem solving are: 1. draw a picture/diagram, 2. guess and check, 3. use logical reasoning, 4. choose an operaton, 5. act out the problem/use objects, 6. make a table or chart, 7.  work backwards, 8. make an organized list, and 9. use a pattern.
Technology overview*

click on level for description

 
Level 1: Students will access information. 
Students use lecture information to learn problem solving steps and strategies.
Level 2: Students will collect data for research.
As students solve different types of problems with different strategies, they develop categories of problems and approaches.  Students use calculators to solve problems; as they do so, they are collecting information about which strategies will solve which types of problems.
Level 3: Students will analyze information.
Students use their sets of problems to find examples of different types, then write their own, in groups.
Level 4: Students will synthesize and communicate information.
Students use Power Point to present their six problems that use six of the nine strategies.  Each card must present the problem and show the solution process.
Focus questions*What is problem solving?  
Why is it important to have good problem solving skills?
What are the four steps and the major strategies used in problem solving?
Content standards and benchmarks*

(Mastery)

 Michigan Mathematic Content Standards and Benchmarks:
MA. I.1.2  Represent and record patterns and relationships in a variety of ways including tables, charts, and pictures.
 MA.I.1.5 Apply their experiences with patterns to help solve problems and explore new content.
 MA.IV.1.4 Apply their understanding of number systems to model and solve problems.
 MA.IV.2.5 Select and appropriate numbers and representations in order to solve problems.
 MA.IV.3.5 Apply their understanding of number relationships in solving problems.
 MA.V.1.4 Apply operations efficiently and accurately in solving problems. MA.V.2.4  Use analytic thinking to describe situations and solve problems.
Content standards and benchmarks*

(Secondary)

  
Technology standards and benchmarks*

(Mastery)

3-5.1.1  Use keyboards and other common input and output devices efficiently and effectively.
3-5.4.5  Use technology tools for individual and collaborative writing, communication, and publishing activities to create knowledge products for audiences inside and outside the classroom.
3-5.6.8  Use technology resources for problem solving, self-directed learning, and extended learning activities. 
Technology standards and benchmarks*

(Secondary)

  
Resources* Most of the problems and cooperative learning strategies from this unit came from these sources:
  • Cooperative Learning by Spencer Kagan
  • Math Strategies: Problem Solving and Performance-based Assessment by Options, Inc.
  • Problem Solving Tips for Teachers by the National Council of Teachers of Mathematics
  • Scott-Foresman/Addison Wesley Math (fifth grade)
  • The New Sourcebook for Teaching Reasoning and Problem Solving in Elementary School by Stephen Krublik and Jesse A. Rudnick
  • Historical Connections in Mathematics by AIMS Education Foundation
    •
  • Hyperstudio software

    I posted these problem solving posters in the room during the unit.
Lesson Plan(s)*Throughout this unit, students work in groups to solve problems.  Students learn to apply steps and strategies, to determine what data/information is necessary to solve a problem, and to determine when there is not enough information to solve a problem.  Each student must be prepared to answer and to explain to the class how their team arrived at the solution.  A spinner is used to determine who is called upon, so all students need to be prepared to answer.  
Lesson 1:  Define/discuss problem solving and the four steps to problem solving.  Discuss the use of estimation in the fourth step.  Review calculator usage.   Introduce the "draw a diagram or picture" strategy and the "guess and check" strategy.
Lesson 2:  Play the "Game of Nim."  Discuss the "use logical reasoning" strategy.  Play the Venn Diagram Game.
Lesson 3:  Discuss the "choose an operation" strategy.
Lesson 4:  Review previously discussed concepts.  Use the "Showdown" cooperative learning structure to solve problems in teams.
Lesson 5:  Introduce "act out the problem/use objects" and "make a table or chart" strategies.
Lesson 6:  Review and practice previously learned concepts and strategies, using the "Numbered Heads Together" structure.
Review:  "Roundtable" review of concepts in teams and a review of strategies.
Assessment:  Problem solving quiz
Lesson 7:  Introduce the "work backwards" strategy.
Lesson 8:  Introduce the "make an organized list" strategy.
Lesson 9:  Introduce the "use a pattern" strategy.
Lesson 10:  Practice the "use a pattern" strategy.  (Page 73; Historical Connections in Mathematics by AIMS Education Foundation)
Lesson 11-12:  Practice all strategies.
Lesson 13:  Roundtable review: discuss problems with too little or too much information.
Lesson 14:  In their teams, students choose from four problems to solve.  After completion of the problem, they are to answer questions individually regarding the process/strategies the group used to solve the problem.  (Pages 32-35; Math Strategies: Problem Solving and Performance-based Assessment by Options, Inc.)
Lesson 15: Students do a similar activity to previous lesson, but do so independently.  (Pages 28-31; Math Strategies: Problem Solving and Performance-based Assessment by Options, Inc.)
Lesson 16:  Practice skills and strategies.
Assessment:  Problem solving test--Day 1.
Assessment:  Problem solving test--Day 2.
Lesson 17:  Introduce "problems" without questions.
Lessons 18 (10 days):  In teams of four, students will create a Power Point presentation to demonstrate their understanding of problem solving, the four steps, and the nine strategies studied.  They will create problems for the strategies and share the stack with another 5th grade class.
Example student materialsTina will find one or two examples from students.  Student reflection pieces could go here, too.
Assessment*Assessments included:  a daily problem solving assignment, a problem solving quiz,  individual problem solving and reflection/response, done through a structured interview and  scored with a rubric, a Multi-Step Math Task, pages 36-39 from Math Strategies: Problem Solving and Performance-based Assessment, scored with a rubric, and individual reflection on group problem solving.  The culminating activity for the unit is the creation, in groups of four, of a six slide Power Point presentation.  The six student created problems will utilize six of the nine strategies, and students will conduct the problem solving portion with another fifth grade class.  This assessment plan details some of the criteria for the assessments.
ReflectionTo choose and apply the four problem solving steps and nine strategies, students must be able to form generalizations about which strategies solve which types of problems.  Some of the problems have extraneous information and some lack a question; students learn to analyze which information is critical.
Home activities   
Credits Tina Marenger, Soo Hill Elementary School, 5219 18th Road, Escanaba, MI  49829     tmarenger@dsisd.k12.mi.us