UNIT TITLE: Problem Solving

Author: S. M. Stellfox  
E-Mail Address:  [email protected]
School: Bloom Trail : Chicago Heights, IL
School Home Page URL: BloomDistrict206.org

THEME : Problem Solving & Critical Thinking

BROAD CONCEPT: The ability to interpret problems, arrive at a solution, and evaluate the solution requires persistence.  Many problems do not have easy solutions that are quickly solved.  True problems usually require more than one attempt at a solution.    The student that is a problem solver will need to persevere to arrive at a correct solution.  The problem solver will be flexible in his/her problem solving approach.

GRADES:9 - 12

INTEGRATED SUBJECTS: Language Arts, and (depending on problems) Science, Social Studies

UNIT GOALS AND PURPOSE: Students will become problem solvers.  They will learn to read a problem critically and apply appropriate strategies to solve the problem.  Once they have arrived at a solution, they will evaluate their solution to determine if the solution is correct.  Students will gain the ability to develop and critique various strategies.  Students will be evaluated by their participation in discussion, their representations of different problem solving strategies, their ability to solve problems, and their explanation of their problem solving method and evaluation of their solution. 
 

TIME FRAME:   8 to 10 days.
 
 

OBJECTIVES:

In this lesson, students will:

• The student will learn how to interpret problems.
•  The student will learn different problem solving strategies.
•  The student will apply those strategies to various problems. 
•  The student will learn to evaluate the results of those strategies.
•  The student will communicate the problem, a strategy, and a solution to classmates.

MATERIALS:

• General materials and equipment Paper, pencils, log books, rulers, graph paper

• Technology tools : Computer, Calculator, Digital Camera

• Software Used in this Unit

_x_Desktop Publishing- ( Print Artist)         _x_ Brainstorming (Inspiration)

_x_Word Processing ( MS Word)               _x_Spreadsheet (Excel)

__CD-ROM Encyclopedia ( Encarta)    _x_Power Point
 
 

STATE STANDARDS ADDRESSED BY THIS UNIT:

            3 A. Use correct grammar, spelling, punctuation, capitalization and structure.

            3B. Compose well-organized and coherent writing for specific purposes and audiences.

            3C. Communicate ideas in writing to accomplish a variety of purposes.

            5A. Locate, organize, and use information from various sources to answer questions, solve problems and communicate ideas.

            7C. Select and use appropriate technology, instruments and formulas to solve problems, interpret results and communicate findings.

            8D. Use algebraic concepts and procedures to represent and solve problems.

           

N.C.T.M. STANDARDS ADDRESSED BY THIS UNIT:

To meet new challenges in work, school, and life, students will have to adapt and extend whatever mathematics they know. Doing so effectively lies at the heart of problem solving. A problem-solving disposition includes the confidence and willingness to take on new and difficult tasks. Successful problem solvers are resourceful, seeking out information to help solve problems and making effective use of what they know. Their knowledge of strategies gives them options. If the first approach to a problem fails, they can consider a second or a third. If those approaches fail, they know how to reconsider the problem, break it down, and look at it from different perspectives—all of which can help them understand the problem better or make progress toward its solution. Part of being a good problem solver is being a good planner, but good problem solvers do not adhere blindly to plans. Instead, they monitor progress and consider and make adjustments when things are not going as well as they should (Schoenfeld 1985).

In high school, students' repertoires of problem-solving strategies expand significantly because students are capable of employing more-complex methods and their abilities to reflect on their knowledge and act accordingly have grown. Thus, students should emerge from high school with the disposition, knowledge, and strategies to deal with the new challenges they will encounter.

 

STUDENT ACTIVITIES WHICH WILL PROMOTE ENGAGED LEARNING:

In lesson 1, students, working in groups, will brainstorm on various problem solving strategies.  Students will research problem solving strategies on the World Wide Web.  They should discuss the strengths and weaknesses of each.  Students will use Inspiration to organize their problem solving strategies.  It should be emphasized that there is no one correct way to solve a problem.  In fact, there may be several strategies to arrive at a solution.  Each group must keep a log of their discussions and activities.  These lessons may be adapted to meet special learning needs of the individual through the groupings used, the timeline for the lessons, and the level of problems to be solved.

In lesson 2 students and teachers will discuss understanding the problem.  Problem solving requires the solver to understand the problem.  Working in groups, students will receive nonstandard problems to present to the class.  They are not to solve the problem but are to explain their problem for their classmates.  Students should be able to determine what the problem is asking them to discover and  present the problem to the class.  To present their problem, students may use desktop publishing to create posters or handouts.  They may use Power Point to present their problem in a slide show.  The students are not concerned with a solution at this point.  The goal is to make sure they understand what the problem is asking them to discover.  Each group will document their problem and its explanation in their log.

In lesson 3, the student groups will determine an appropriate strategy for their problem.  In their log, they must offer justification for their choice of strategy.  For example: guess and check would be a good strategy for their particular problem because . . .  Students may choose more than one strategy.   The strategy or strategies and justification must be documented in the log.

In lesson 4, students will attempt to apply their chosen strategy. The lesson will vary depending on problems.  A group using a guess and check strategy may use a spreadsheet to generate random solutions or a spreadsheet may be used to graph results.  Groups needing to research their problem may use Encyclopedia Britannica Online, Google or another appropriate source.  This lesson may last a few days depending on topics and students.

In lesson 5, students must demonstrate that their solution is correct, or determine that their solution is not correct.  Using diagrams, graphs, charts (created in Excel),  equations, or explanations the group must offer proof that their solution is correct.  The outline for this proof must be documented in the log.  If the solution is determined to be incorrect, the group should examine their strategy and its application.  Was the strategy not appropriate, was a mistake made in its application?  Depending on time the teacher may have the group reattempt to solve their problem.

In lesson 6, students will develop a slide show presentation of their strategy and solution for their classmates.  It will include an explanation of their problem, the strategy they used to solve it, and a validation of their solution (or explanation of why the solution was not correct).

In lesson 7, students will develop a guideline for selecting an appropriate strategy.  They should include at least three different types of problem solving strategies.  Each strategy should have an explanation of why to choose that strategy and how to implement it.  Students may record this information in a variety of ways.  They could use Inspiration to map their ideas, they could use posters diagrams, or they could use word processing to explain their strategies.

In lesson 8, students groups will develop a nonstandard problem for their classmates to solve.  Students will need time to research their problem.  The World Wide Web and current events can provide an excellent resource.   Students should remember that a nonstandard problem is one that the solver has not seen before.  The teacher should emphasize that problems are not to be too easy, but they are not to be impossible.  Students do not have to have a solution but they should have an understanding of the problem and a possible solution.

In lesson 9, student groups will exchange problems, determine an appropriate strategy, and attempt to solve the problems.  Students will discuss problems, strategies, and solutions as a group.

In lesson 10, students will write a reflection paper on the problem solving process.  They should include an explanation of problem solving, its strategies, what makes a good problem, and when and how they will use problem solving in their life.  Students may choose to write their drafts with a word processor or by hand but final copy will be typed.

 

SHARE YOUR IDEAS:

The culminating project for this project will be a slide show demonstrating the understanding and possible solution of an authentic problem.  Students will research a topic that interests them and write a proposal for a problem that exists.  (For example: The student parking lot is overcrowded.  What are possible solutions and how much would each cost?  Could the parking lot be expanded, if it is, what would be the cost, and should students pay a parking fee?  Should there be a limit on the number of parking permits issued?  How would that limit be determined?)  The students will research their problem, look at various strategies to determine a solution, and present  their solutions in an appropriate medium (such as Excel or Inspiration).  The slide show should represent their problem solving process.  Students may take digital pictures to illustrate their show.  Students should include a title slide and end slide with appropriate credits.  The slide show should have at least 10 slides.  These slides should document the problem, the strategy or strategies to arrive at a solution, and a feasible solution.

                                                10 Slides                                                          15 points

                                                Title Slide                                                         5 points

                                                End Slide                                                          5 Points

                                                Explanation of problem                                     25 points

Strategy                                                            30 points

Solution                                                            20 points

            Total                                                    100 points

                                               
 
 

STUDENT AND TEACHER ROLES:  The student is actively developing a collection of problem solving strategies.  The problems should be chosen to be difficult, but not frustrating.  The student will have the opportunity to choose problems that may be of personal interest.  The assessment is not based on acquiring a single “correct” answer but is ongoing through the unit.  The assessment is meant to be instructional and guide student work.  The teacher monitors discussion, guides students in their strategy selection by providing options not answers. Students are expected to explore different strategies.  Students may fail to arrive at a correct solution without penalty as long as they demonstrate understanding of the problem, and why their solution is incorrect.
 
 

ADDITIONAL REFERENCE PRINT RESOURCES STUDENTS MAY USE :

World Book Encyclopedia, newspapers
 
 

THE FOLLOWING BOOKS CONTAIN INFORMATION ABOUT THIS TOPIC:

How to Solve It: A New Aspect of Mathematical Method by G. Polya

The New Sourcebook for Teaching Reasoning and Problem Solving in Junior and Senior High School by S. Krulik and J. Rudnick
 
 
 

FOR ADDITIONAL ONLINE INFORMATION AND MEDIA PERTAINING TO THIS UNIT, TEACHERS MAY WANT TO USE THE FOLLOWING INTERNET ARTICLES:  (minimum 6 sites)

Title1: MERLOT

URL1: merlot.org

Description1: Multimedia Educational Resource for Learning and Online Teaching provides links to online learning materials and encourages the exchange of ideas, assignments, and feedback among faculty and students.
 
 

Title2: Finite Math

URL2: www.math.ncsu.edu/ma114/modules/c3d4.html

Description2: How to solve nonstandard problems
 
 

Title3: Problem Solving Strategies
 URL3: mathcounts.org/Problems/strategies.html

Description3: Problem solving strategy of considering a simpler case of a problem is used with other strategies.  This site provides discussion and examples of this strategy.
 
 
 

Title4: Key Curriculum Press | Problem Solving Strategies
 URL4: www.keypress.com/catalog/products/ textbooks/Prod_ProbSolv.html  
 
Description4: Crossing the River with Dogs and other Mathematical Adventures,  source of problems. 
 

Title5: Ten Math Strategies for Problem Solving
 URL5: www.fcps.k12.va.us/DeerParkES/ kids/diane/Math/tenstrat.htm

Description5: List of strategies for problem solving.  Strategies are organized in an easy to read table form.
 

Title6: Fifty Problem Solving Strategies
 

URL6: www.une.edu.au/psychology/staff/malouff/problem.htm

Description6: Fifty problem solving strategies explained and listed by type. Also included strategies to help you understand the problem.
 
 
 

RELEVANT INFORMATION, STUDENTS WILL VISIT THE FOLLOWING WEB SITES: (minimum 6 sites)

Title1: Problem Corner

URL1: math.smsu.edu/~les/POTW.html 

Purpose of Use1: Source of Problems

Description1: This site offers mathematics problems for high school students.
 
 

Title2: Aunty Math

URL2: dupagechildrenmuseum.org/aunty/index.html

Purpose of Use2: Students will submit answers and solution strategies to solve Mathematical problems

Description2: Interactive Web site tht encourages children to submit solutions and strategies to solve math challenges.
 
 

Title3: Problem Solving Strategies

URL3: www.mav.vic.edu.au/PSTC/general/strategy.htm

Purpose of Use3: This site can provide student with step-by-step list of how to solve a problem.  Students may use this site when working on their culminating project.

Description3: . If you are having some trouble solving a math problem, these ideas might get you going.

 
 
 

Title4: Kids@work: Problem Solving Strategies

URL4: www.cyberjourneys.net/books/cybermath2/ student/strategies.html

Purpose of Use4: Students will use this site when developing problem solving process.

Description4: Site walks students through problem solving. Explains importance of recording steps taken to solve problem.
 
 
 

Title5: Problem Solving
URL5: mathcounts.org/Problems/problems.html

Purpose of Use5: Students may use this page to get ideas when they are creating problems to exchange with classmates.

Description5: Examples of different problems with different strategies.
 
 
 

Title6: Word Problems For Kids

 URL6: www.stfx.ca/special/mathproblems/welcome

Purpose of Use6: Students can select their own problem to solve.

Description6: Set of selected problems designed to improve thinking skills.