Learning factoring with visual aids: a strategy for developmental math

1. Learning factoring with visual aids: a strategy for developmental math Ying Lin ylin2@pima.edu Pima Community College Downtown Campus

2. 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

3. The importance of factoring in Algebra I Rational Expressions QuadraticEquations Factoring Rational Equations Square Roots

4. 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 + ?)

5. 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

6. 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 + = 7 ?1 ?2 * = 12 ?1 ?2

7. 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

8. Level 3 • Ask students: what if the constant c is a negative number? + = -1 ?1 ?2 * = -12 ?1 ?2

9. Level 4: Factoring as Visual Learning ?1 ?1 ?2 ?2 ?1 ?1 2 3 ?2 ?2 ?1 ?2 + = 5 ?2 ?1

10. Level 4 • Instruction: Add two red cards to the game – coefficients of the binomials • When you multiply the two green numbers, you should get the constant c • When you multiply the two red numbers, you should get the leading coefficient a • 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!

11. Where is FOIL in this game? ?1 ?1 ?2 ?2 2 3 2 3 1 1

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

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

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

15. 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:

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

17. Related factoring problems • Difference of squares: • Perfect square: • Trinomials containing more than one variable 2 2 +3 +3 2 2 -3 +3

18. 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!