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X-Puzzles and Area Models
For Integers and Beyond…
Elizabeth Karrow Jennifer Smith
Diane Jacobs
Marci Soto
Fitz Intermediate School - Garden Grove USD
Agenda
•
•
•
•
X-Puzzles
Area Model
Reverse Area Model
X-Box Factoring
Diane Jacobs
Liz Karrow
Marci Soto
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?
A.
3

 

6
5
B.
2
4
12
7
C.
3
10
5
D.
2
7


 
 
You
discovered

21
3
7
10



the
pattern!
E.
12
24
2
14
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!
F.
G.
5



4

H.
10

2
I.
10

4
7
2

J.


3

2
X-Puzzles
Strengthen the skills by working backward.
K.
12
L.
1
6




M.
1

N.
5
3


15
8


O.
36
0
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.
P.
Q.
1
2


U.


R.
V.





3
1 8
2
1
6



W.
X.
2
3
7
12


2
5
1
3
1
2
3
4
1
2
S.


1
4

T.
3
8
1
6

1
4
1

Y.
1



1
5
2
3
8
7
4
3
4
X-Puzzles
In Algebra X-Puzzles are used to reinforce the
differences between combining like terms and
multiplying exponents.
AA.
AB.
3x

4x2
3x
5 x 3
AD.
x
AE.
2
3 x 2 7 x
5x
2
Working
backwards
reinforcesthe skills further.







AF.

2x
AC.
4 x
2x

2
AG.
10 x


AH.
2
2
AI.
2x3
x3



3x 3


3x 2  x


AJ. 12
4x
x2
X-Puzzles
X-Puzzles can also be used for polynomials and radicals.
AK.
x2
AL.
x6
x6
1

 AQ.
AP.

5 2
8

AV.
10
 6

AW. 6 15
6
2 5





4x  3
2 5

AT.


12
15
2 3

AX.
AY.
150
2 11
2 3
2
x  2
2x
AS.

6 3


AO.

AR.

4 3

AU.
3 x 3  2 x 2

3

5x2



AN. x 2  49
AM.
 11
3





8 10
Area Model
Simplify:
3(2 x  5)
What mistakes would your students make?
Area Model
Simplify:
3(2 x  5)
Distributive Property teaches…
Students often forget the second term:
6x  5
Students often forget the negative:
6x  15
As well as other issues our students seem to encounter.
Area Model
3(2 x  5)
2 x students
5
The area model helps
avoid some of3the6most
x 15
Simplify:
common mistakes.
3(2 x  5)  6x  15
Area Model
Simplify:
4(3x  3)
3x 3
4 12x 12
4(3x  3)  12 x  12
Area Model
Simplify:
2( x  4)  6(3x  5)
x
3
x
5
4
What are the common mistakes
your
make
8 would
2 xstudents
x 30
6 18
2
simplifying this problem?
2( x  4)  6(3x  5)  2x  8  18x  30
Area Model
Simplify:
3x (2x  3x  1)
2
What would the 2area model look
1
3
x
2
x
like to simplify this problem?
3x 6 x 9 x
3
2
3x
3x (2x  3x  1)  6 x  9 x  3x
2
3
2
Area Model
(2x  3)( x  4)
2x 3
What would the area model look
2 3x
x
like to simplify x
this2problem?
Simplify:
4 8x 12
(2 x  3)( x  4)  2x  5x  12
2
Simplify:
Area Model
2
(x  5)(3x  3x  7)
3x area3model
What would the
look
x 7
like to simplify
this
problem?
3
2
x
7x
2
3x
3x
5 15x 15x
2
35
(x  5)(3x  3x  7)  3x 3  12x 2  22x  35
2
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:
5x
• You need to tell us
this:
5
10
x
2
x
10
2x
x
2x
2
1
x
x(2x  1)
x
3x
2
5x 5x 15x
2
3
5x(x  3x )
2
1
3x
2x
6x
2
2x
3
9x
3
(2 x  3)(3 x  1)
x
x
2
3
x
1
3
3x
x
2
3
(x  3)(x  1)
2
2x
2x
2
4x
3
x
2x
1
2
2x
2x(2x  x  1)
2
5a
2a
2
2
3a
7
10a 6a 14a
4
3
2
2a (5a  3a  7)
2
2
5x
x
2
5x
3
2
2x
9x
2
x (5x  2)
2
x
2
2
3
9x 2x
2
x (9x  2)
2
Common Factor?
x
2
1
3x
6x
2x
3
2
3x
2x
9x
3
2x 6x
(2 x  3)(3 x  1)
2
7 21x
1
2x
7
(2x  7)(3x  1)
Common Factor
(3x  1)
Reverse Area Model and
Completing the Square
Standard Form
y  2x  12x  14
2
to Vertex Form
y  2(x  3)  4
2
In 6 easy steps!
Step 1:
Identify a, b and c
y  2x  12x  14
2
a = 2 b = -12 c = 14
Step 2:
Move c to left side.
y  2x  12x  14
2
14
14
2
y  14  2x  12x
Step 3:
Factor “a” out of right side using
the reverse area model.
y  14  2x  12x
2
x 6x
2
2 2x 12x
y  14  2(x  
6x 
2
Leave room to complete
the
square!



2
)
Step 4:
Complete the square using the
reverse area model.
x
3
x x 2 3x
3 3x 9

 14

y

 2(x  6x  9 ))
2
18
2
y  4  2(x  6x  9)
Step 5:
Write as a binomial squared.
x
3
x x 2 3x
3 3x 9



y  4  2(x  6x  9)
2
y  4  2( x  3)( x  3)
y  4  2(x  3)
2
Step 6:
Move “k” to the left side.
y  4  2(x  3)
2
4
4
2
y  2(x  3)  4
Graph:
y  2x  12x  14
2
y  2(x  3)  4
2
Vertex: (3, -4)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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
3x
-4
x
3x2
-4x
-3
-9x
12
-4x
-13x
Answer: (x - 3)(3x - 4)
You try one!
• Given: 12x2 + 5x - 2
4x
-1
3x
12x2
-3x
2
8x
-2
-24x2
8x
-3x
5x
Answer: (4x - 1)(3x + 2)
One more…
• Given: x2 - 10x - 24
x
2
x2
2x
-12x
-24
-24x2
x
-12x
2x
-10x
-12
Answer: (x + 2)(x - 12)
Same numbers that
were in the x-puzzle!
It even works with the difference of
2 squares (“b” term is missing)!
• Given: 4x2-9
2x
3
-36x2
2x
-6x
4x2
6x
-6x
-9
6x
0
-3
Answer: (2x - 3)(2x + 3)
It also works if the “c” term is
missing!
• Given: 4x2-8x
0
4x
0
x
4x2
0
-2
-8x
0
-8x
0
-8x
Answer: 4x(x - 2)