Reasoning Word Problems and Algebra

Critical Thinking is important in all areas of life.  Math is not an exception.  I believe that it is important to teach students to think “outside the box” in mathematics. 

In this series of lessons:
Students will learn to observe and note details of sequential patterns and then a math problem.
Students will begin to think critically as they approach a math word problem. 

Prior Knowledge:
Students should have a knowledge of basic mathematics:  fractions through percents.
Students should be familiar with basic algebraic equations.

            A.  Introduction to Critical Thinking
                        1.  Observing Patterns
                        2.  Critical Thinking Warm Up
            B.  Approaching a Word Problem
                        1.  Poly’s Process for Solving Word Problems
                        2.  Create a Word Problem Graphic Organizer
            C.  Applying Knowledge
                        Critical Thinking Lesson - Pay Check Problem
            D.  Reviewing and Reflecting
                        1.  Writing about Math – Solving Equations 
                        2.  Learning from Mistakes
            C.  Expanding the Concepts
                        Approach several Algebraic Word Problems.  Use the problems from
                                    a.   Math Memos
                                    b.  problems similar to the HiSET/GED/TASC practice test problems.

College & Career Readiness Standards for Adult Education: Mathematics 7.EE.2; 7 .EE.3; 7.EE.4

Common Core State Standards for Math:  Algebra  A –CED.1, A-REI.1, A-REI.2, A-REI.3


A.  Introduction to Critical Thinking Lessons.

Use these as lessons to introduce Critical Thinking.  Then, use these and items like them to “warm up” students before beginning a Critical Thinking Lesson.

            1.  Pascal’s Triangle

            Show Pascal's Triangle to students.  Talk with students to help them find patterns in the triangle.  Move from observing to figuring how rows have changed.  Finally, ask students to duplicate the pattern and create new rows.                 

            Pascal's Triangle Resources

                Resources For teacher ideas. These are not in public domain.    
               Pascal's Triangle Worksheet      


            2.  What is Next? 

            Students will observe patterns as they change.  Rows might be added/subtracted, etc.       They will observe the changes and predict how to add rows to the configuration.  State the change predictions mathematically.

              Find the Pattern Lesson:      

Download: Find_the_Patterns_ui1pmVQ.pptx

            3.  The Arch Problem
            Students will observe patterns as they change.  They will observe the changes and predict how to add rows to the configuration.       The Arch Problem     

            4. A+ Click Math Problem

            Choose an A+ Click Math problem and allow students to work in Groups of 2 -4 to solve problems.

            Come together as a group and discuss the problem.  How did they solve it? What worked?  What did not work?

        A+ Click Math Problem       


B.  How to Approach a Word Problem:

            Use the following word Problem:  You want to hire a band for a dance. The live band charges $75 per hour. Each attendee pays $5.00 admission. If the band is to play for 3  hours, how many tickets do you need to sell to break even?  Discuss how to work word problems.            

            1.  Share Poly’s Process for Solving Word Problems

           Poly's Process

Download: OneStepAtATime-AS-ProblemSolver_L3EqunM.pdf

            Ask the following questions:

            1. Do you understand all the words used in the problem?
            2.  Find the facts in the problem.
            3.  What is the question?
            4.  Restate the problem in your own words.
            5.  Can you think of a picture or diagram that might help you understand the problem?
            6.  Is there enough information to enable you to find a solution?              

            Devise a Plan:

            1.  What words might give a clue as to how you can solve the problem?
                 (Key Words)
            2.  Can you draw a picture or chart to help you understand the problem?
            3.  Will a formula help you to solve this problem?
            4.  How will you solve this problem? 

            Carry Out the Plan:
                1.  Write out a mathematical sentence that will help you to solve the problem.
                2.  Work the problem.
                3.  If what you tried does not work, re-evaluate.

            Look Back

                1. Check your work.
                2.  Think about how you solved the problem.
                3.  What worked/did not work?
                4.  How can you use what you learned with future problems?           

            2.  Students should devise a graphic organizer similar to the one below.               

Download: Word_Problem_Graphic_Organizer.pdf


            Other Resources:



            CRITICAL THINKING LESSON - Pay Check Problem

Download: The_Pay_Check_Problem.pptx

            The Paycheck Problem will teach critical thinking in order to create an equation that would solve a real-life problem.  Use the simple version of the Pay Check Problem.  Students will better understand concepts.  You might try the more difficult version after students become adept at solving problems critically. 

                Resource – 
                     Paycheck Problem pdf 

         Prior Knowledge:  Students should understand how to work with simple linear equations.   

         How to use this:  Present this lesson using the dialogue in the PowerPoint slides. Begin with a critical thinking warm-up (patterning). Then present the pay chart.  Ask the students to figure out how the pay was figured. Guide them as they work.  As students give answers, write them on a whiteboard.             After the lesson:  Take a few moments to talk about the problem.  How could the students use the methods in this lesson to work other problems?  Can they give specific examples?


            1.  Writing about Math – Solving Equations          

            Take a few moments after you teach a lesson to have the students write about the problem.  This resource combines language arts with math.  It allows the student to think and reason out a problem.  When a student can explain how to do a problem, s/ he probably understands the concept.

                More specifically: 

            Variation:  Place students in groups of two.  Have them explain how to work a problem similar to the ones taught in the lesson to another student. 

            Variation:  Group activity.
            After teaching and modeling few problems, have students help you to solve them. As students give suggestions, write them on a whiteboard.  Then discuss the problem and the solutions. One student may say, “Subtract 7 from each side.”  Another might add, “Divide the variable by 6,” etc.  Validate every suggestion – students are brainstorming.
            2.  Learning from Mistakes
            Take a few moments every day for this activity.
            Put a problem on the board.  Give each student a 3 x 5 card.  Have them work the problem and hand it to you. 
            Choose a card with the WRONG process or answer.  Have students discuss why this is wrong. How can we fix this? 
            Thank the (anonymous) student for submitting this problem.  Note that we can learn from our mistakes.  We build our math thinking capabilities by looking at these kinds of examples.                

Link:  Learning From Mistakes
Direct route: 


E.  Expanding: 
            Approach several Algebra Word Problems – allowing students to work together reasoning them out.
            1.  Use the problems from
                More specifically:   Choose Math Memos
            2.  Work with problems similar to the HiSET/GED/TASC practice test problems.


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