Learning factoring with visual aids: a strategy for developmental math

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```							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

Expressions                Equations

Factoring
Rational
Square Roots
Equations

3
Learning Factoring:
• A left-to-right presentation may encourage
students to treat factoring as a tricky
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
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

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

13
Level 5 Practice
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!

18

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