X-Puzzles and Area ModelsFor Integers and Beyond…

Elizabeth Karrow

Diane Jacobs

Jennifer Smith

Marci Soto

Fitz Intermediate School - Garden Grove USD

Agenda

• X-Puzzles Diane Jacobs

• Area Model Liz Karrow

• Reverse Area Model Marci Soto

• X-Box Factoring Jennifer Smith

X-Puzzles

• Introduced in Pre-algebra (7th grade).

• Simple pattern, that is discovered, not taught.

X-Puzzles

Using the pattern in puzzles A and B, complete puzzles C, D and E.

Did you get?

10

7 1410

21 24

You discovered the pattern!

A. B. C. D. E.

5

6

3

2

7

12

4

3

5

2

3

7

12

2

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

F. G. H. I. J.

5

4

10

2

10

2

7

4

3

2

Try these!

X-PuzzlesStrengthen the skills by working backward.

K. L. M. N. O.

12

6

1

1

5

3

15

8

36

0

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

P. Q. R. S. T.

1

2

1

2

3

4

1

2

1

3

3

8

2

5

1

6

11

5

23

4

U. V. W. X. Y.

1

2

3

8

1

6

7

12

2

3

1

4

1

4

1

3

8

7

4

X-Puzzles

AA. AB. AC. AD. AE.

3x

2x

4 x 2

3x

5 x 3

x

3 x 2

7x 2

5 x

2

In Algebra X-Puzzles are used to reinforce the differences between combining like terms and multiplying exponents.

Working backwards reinforces the skills further. AF. AG. AH. AI. AJ.

2 x

4 x 2

10 x 2

2

2x 3

12 x 2

4 x

x 3

3x 2 x

3x 3

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

AK. AL. AM. AN. AO.

x 6

1

5x 2

3 x 3 2 x 2

2 x

4 x 3

x 6

x 2

x 2 49

x 2

AP. AQ. AR. AS. AT.

3

4 3

8

5 2

6 3

15

6

12

2 3

2 5

AU. AV. AW. AX. AY.

2 5

10

6

2 3

6 15

2 3

11

150

8 10

2 11

Area Model

What mistakes would your students make?

3(2x 5)Simplify:

Area Model

3(2x 5)Simplify:

Distributive Property teaches…Students often forget the second term: 6x 5

Students often forget the negative: 6x 15As well as other issues our students seem to encounter.

Area Model

3(2x 5)Simplify:

2x

3 5

6x 15

3(2x 5) 6x 15

The area model helps students avoid some of the most common mistakes.

Area Model

4(3x 3)Simplify:

3x

4 3

12x 12

4(3x 3) 12x 12

Area Model

2(x 4) 6(3x 5)Simplify:

x

2 4

2x 8

2(x 4) 6(3x 5) 2x 8 18x 30

What are the common mistakes your students would make simplifying this problem?

6 3x 5

18x 30

Area Model

3x(2x2 3x 1)Simplify:

2x2

3x 3x

6x3 9x2

3x(2x2 3x 1) 6x3 9x2 3x

What would the area model look like to simplify this problem? 1

3x

Area Model

(2x 3)(x 4)Simplify:

3 2x

x 2x2 3x

(2x 3)(x 4) 2x2 5x 12

What would the area model look like to simplify this problem?

4 8x 12

Area Model

(x 5)(3x2 3x 7)Simplify:

5

x 3x2

7

3x3

(x 5)(3x2 3x 7) 3x3 12x2 22x 35

What would the area model look like to simplify this problem?

3x

3x2 7x

15x2 15x 35

Reverse Area Model

Greatest Common Factor

• Students need to review the greatest common factor first before they can be successful at reverse area model.

Reverse Area Model• We are doing the area

model, but backwards.• We will give you this:

• You need to tell us this:

5x 10

x 105

x 2

2x2 xx

2x 1

x(2x 1)

5x2 15x35x

x 3x2

5x(x 3x2 )

26x 2x2x3x 1

(2 3)(3 1)x x

9x 33

x3 x2x2x 1

(x2 3)(x 1)

3x 33

4x3 2x22x

2x2 x

2x(2x2 x 1)

2x

1

10a4 6a32a2 5a2 3a

2a2 ( 5a2 3a 7)

14a2

7

5x3 2x2x25x 2

x2 (5x 2)9x3 2x2x29x 2

x2 (9x 2)

Common Factor?

x2

26x 2x2x

3x 1

(2 3)(3 1)x x

9x 336x2 2x2x

3x 1

21x 77

(2x 7)(3x 1)

Common Factor

(3x 1)

Reverse Area Model and Completing the Square

y 2x2 12x 14

Standard Form

y 2(x 3)2 4

In 6 easy steps!

to Vertex Form

Step 1:Identify a, b and c

a = 2 b = -12 c = 14 y 2x2 12x 14

y 2x2 12x 14

Step 2:Move c to left side.

14 14

y 14 2x2 12x

Step 3:Factor “a” out of right side using

the reverse area model.

y 14 2x2 12x

y 14 2(x2 6x )Leave room to complete the square!

2x2

12x

2

x2

6x

Step 4:Complete the square using the

reverse area model.

y 14 2(x2 6x 9 ) y 14 2(x2 6x )

18

y 4 2(x2 6x 9)

x2

3x

3x

x

x

3

3

9

Step 5:Write as a binomial squared.

y 4 2(x2 6x 9)

y 4 2(x 3)(x 3)

y 4 2(x 3)2

3x

3x

x

x

3

3

9

x2

Step 6:Move “k” to the left side.

y 4 2(x 3)2

4 4

y 2(x 3)2 4

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Graph:

y 2x2 12x 14

y 2(x 3)2 4

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Vertex: (3, -4)

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!

Using the x-box method

Given: 3x2 - 13x +12

36x2

-9x

-13x

-4x3x2

12-9x

-4xx

-3

3x -4

Answer: (x - 3)(3x - 4)

You try one!

• Given: 12x2 + 5x - 2

-24x2

5x

8x -3x

8x

-3x12x2

-2

3x

2

4x -1

Answer: (4x - 1)(3x + 2)

One more…

• Given: x2 - 10x - 24

Answer: (x + 2)(x - 12)

-24x2

-10x

-12x 2x

-12x

2xx2

-24

x

-12

2x

Same numbers that were in the x-puzzle!

It even works with the difference of 2 squares (“b” term is missing)!

• Given: 4x2-9

-36x2

0

-6x 6x

-6x

6x4x2

-9

2x

-3

2x 3

Answer: (2x - 3)(2x + 3)

It also works if the “c” term is missing!

• Given: 4x2-8x

0

-8x

0 -8x

-8x

04x2

0

x

-2

4x 0

Answer: 4x(x - 2)