1. X-Puzzles and Area Models For Integers and Beyond…
Elizabeth Karrow
Diane Jacobs
Jennifer Smith
Marci Soto
Fitz Intermediate School - Garden Grove USD

2. Agenda X-Puzzles Diane Jacobs Area Model Liz Karrow
Reverse Area Model Marci Soto
X-Box Factoring Jennifer Smith

3. X-Puzzles Introduced in Pre-algebra (7th grade).
Simple pattern, that is discovered, not taught.

4. X-Puzzles Did you get? You discovered the pattern!
Using the pattern in puzzles A and B, complete puzzles C, D and E.
Did you get? 10 21 24 7 10 14
You discovered the pattern!

5. X-Puzzles
After students have learned to add, subtract, multiply and divide integers, X-Puzzles are used for basic practice in daily warm-ups and homework.
Try these!

6. X-Puzzles
Strengthen the skills by working backward.

7. X-Puzzles
Fractions are an ongoing weakness. Regular practice with X-Puzzles increases skill and illuminates the difference between adding/subtracting fractions and multiplying/dividing fractions.

8. X-Puzzles
In Algebra X-Puzzles are used to reinforce the differences between combining like terms and multiplying exponents.
Working backwards reinforces the skills further.

9. X-Puzzles
X-Puzzles can also be used for polynomials and radicals.

10. Area Model
Simplify:
What mistakes would your students make?

11. Area Model Distributive Property teaches… Simplify:
Students often forget the second term:
Students often forget the negative:
As well as other issues our students seem to encounter.

12. Area Model
Simplify:
The area model helps students avoid some of the most common mistakes.

13. Area Model
Simplify:

14. Area Model
Simplify:
What are the common mistakes your students would make simplifying this problem?

15. Area Model
Simplify:
What would the area model look like to simplify this problem?

16. Area Model
Simplify:
What would the area model look like to simplify this problem?

17. Area Model
Simplify:
What would the area model look like to simplify this problem?

18. Reverse Area Model

19. Greatest Common Factor
Students need to review the greatest common factor first before they can be successful at reverse area model.

20. Reverse Area Model We are doing the area model, but backwards.
We will give you this:
You need to tell us this:

27. Common Factor?

28. Common Factor

29. Reverse Area Model and Completing the Square
Standard Form
to Vertex Form
In 6 easy steps!

30. Step 1:
Identify a, b and c
c = 14
a = 2
b = -12

31. Step 2:
Move c to left side.

32. Factor “a” out of right side using the reverse area model.
Step 3:
Factor “a” out of right side using the reverse area model.
Leave room to complete the square!

33. Complete the square using the reverse area model.
Step 4:
Complete the square using the reverse area model.

34. Write as a binomial squared.
Step 5:
Write as a binomial squared.

35. Move “k” to the left side.
Step 6:
Move “k” to the left side.

36. Graph:
Vertex: (3, -4)

37. X - Box Method for Factoring
Prior Knowledge
X - puzzle
Area model
Standard form of a quadratic equation
Benefits
Builds on prior knowledge
No more guessing involved
Organization
Fun!

38. Using the x-box method Given: 3x2 - 13x +12 3x -4 36x2 x 3x2 -4x -9x-3
-9x 12
-13x
Answer: (x - 3)(3x - 4)

39. You try one! Given: 12x2 + 5x - 2 4x -1 -24x2 3x 12x2 -3x 8x -3x 2 8x
Answer: (4x - 1)(3x + 2)

40. One more… Given: x2 - 10x - 24 x 2 -24x2 x2 2x x -12x 2x -12 -12x -24
Same numbers that
were in the x-puzzle!
Answer: (x + 2)(x - 12)

41. It even works with the difference of 2 squares (“b” term is missing)!
Given: 4x2-9 2x 3
-36x2 2x
4x2 6x
-6x 6x -3
-6x -9
Answer: (2x - 3)(2x + 3)

42. It also works if the “c” term is missing!
Given: 4x2-8x 4x x
4x2
-8x -2
-8x
-8x
Answer: 4x(x - 2)