Learning factoring with visual aids: a strategy for developmental math

							Learning factoring with visual aids:
a strategy for developmental math

                Ying Lin
           ylin2@pima.edu
       Pima Community College
         Downtown Campus

                                       1
      Student characteristics in
           MAT092/122
• Visual learners
• Tactile learners
• Social learners
• Having previously been taught the
  material, yet low level of mastering
• Common difficulties: fractions, factoring
• Today’s focus: factoring trinomials

                                              2
The importance of factoring in
         Algebra I

 Rational                  Quadratic
Expressions                Equations

              Factoring
  Rational
                          Square Roots
 Equations




                                         3
        Learning Factoring:
        A Reading Problem?
• A left-to-right presentation may encourage
  students to treat factoring as a tricky
  reading task, which has to be done
  together with arithmetic


     5x2 + 7x + 2 = (?x + ?)(?x + ?)


                                               4
       Factoring Trinomials as a
              Card Game
• User-friendly presentation
• Variant of the “trial-and-check” method
• Showing connection with FOIL
• Identifying different levels of problems
• Easy to implement in classrooms: colored index
  cards or colored post-its
• Eventually transfers to a paper-based, fast
  strategy


                                                   5
                    Level 1
• Instruction: “write two numbers on the cards, so
  that their sum is b, and their product is c”
• Ask students: “which clue is more useful?”

                                ?1        ?2
           ?1 +      ?2 = 7

            ?1 * ?2 = 12

                                                     6
                    Level 2
• Ask students: What if the coefficient b is a
  negative number?
  – Briefly review properties of negative numbers
  – Add signs to both numbers



                               ?1        ?2


                                                    7
                   Level 3
• Ask students: what if the constant c is a
  negative number?




                 ?1 +      ?2 = -1

                  ?1 * ?2 = -12
                                              8
          Level 4:
Factoring as Visual Learning
              ?1    ?1   ?2   ?2

        ?1         ?1
   2                     3
        ?2         ?2
        ?1 ?2 + ?2 ?1 = 5

                                   9
                     Level 4
• Instruction: Add two red cards to the game –
  coefficients of the binomials
   1. When you multiply the two green numbers, you
      should get the constant c
   2. When you multiply the two red numbers, you should
      get the leading coefficient a
   3. When you cross-multiply the red and green
      numbers then add the products, you should get b
• Level 1-3 are simplified versions of the full
  puzzle!


                                                      10
Where is FOIL in this game?

              ?1   ?1   ?2   ?2

        2          3
   2                    3
        1          1

                              11
            Level 4 Practice
Solve the following problems with
your index cards and pencil




                                    12
                  Level 5
• What if c is a negative number?
  – Hint: the green numbers carry signs!




                                           13
            Level 5 Practice
• Ask students: what’s your favorite
  strategy?




                                       14
        Factoring Completely
• General tip: the level of the puzzle depends on
  the leading coefficient a. Hence we can make
  the problem simpler
  – By factoring out the greatest common factor:



  – By factoring out the negative sign:




                                                    15
              Group Activity
• Students form groups of 2 or 3:
  1. One person constructs a trinomial puzzle by
     using the index cards
  2. The other person tries to “break the code” /
     factor into binomials also by the index card
     method
  3. Compare the numbers each person wrote on
     the cards. If they are different, discuss why.
  4. Switch roles.

                                                  16
    Related factoring problems
• Difference of squares:
                            2         +3

                            2          -3
• Perfect square:
                            2         +3
                            2         +3
• Trinomials containing more than one variable



                                                 17
              Things to note
• Seems effective for students who
  – find it difficult to learn factoring by grouping
  – are visual/tactile learners
• Using two colors is crucial in the beginning
• Most students can use the strategy without
  visual aids after 1 class
• Please let me know how it works for you!


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