Download x + 2 - Pacent Learning Solutions

Diamond Problems
throughout Middle School
Bob Battinich
Pacent Learning Solutions
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Outcomes


Introduce Diamond Problems
Look at the different types and uses of diamond
problems from whole numbers to factoring
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Diamond Problems
Product
a●b
a
b
Sum
a+b
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Diamond Problems: Basic
Given: Both Numbers
24
6
4
10
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Diamond Problems: Moderate
Given: Product and 1 Number
36
4
9
13
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Diamond Problems: Moderate
Given: Sum and 1 Number
56
7
8
15
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Basic Operations (+, –, ×, ÷)
Addition
Subtraction Multiplication
Division
?
a
b
a
?
a
b
b
a
?
b
a+b
b–a
a●b
b÷a
?–a=b
?+a=b
?÷a=b
a●?=b
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Diamond Problems: Advanced
Given: Product and Sum
60
4
15
19
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Fractions
5
6
5
18
51712
6
66 6
1
3
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Decimals
1.93
6.755
3.5
3.5
1.93
3.50
5.43
1.93
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Integers
Common Mistakes
Sign Errors
Most common mistake in Algebra
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Integers
36
-4
-9
-13
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Integers
48
-6
-8
-14
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Integers
-60
4
-15
-11
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Integers
-36
-3
12
9
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Integers
24
-4
-6
-10
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Integers
-6
-6
1
-5
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Integers
-36
-6
6
0
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Integers
5
-5
-1
-6
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Monomials
Common Mistakes
?
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Monomials
x2
x
x
2x
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Monomials
-10x2
-2x
5x
3x
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Monomials
12x4
-4x2
-3x2
2
-7x
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Monomials
-12x
-3x
4
-3x + 4
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Monomials
-6x3
2x2
-3x
2
2x -
3x
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Monomials
-18x5
-6x3
3x2
-6x3 + 3x2
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Polynomials
2x2 – 11x + 12
2x – 3
x–4
3x – 7
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Polynomials
-6x2 + 29x – 35
2x – 5
-3x + 7
-x + 2
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Factoring Quadratics

4 methods for factoring Quadratics
–

Simple, Complex, Difference of Two Squares, and
Trinomial Squares
Guess and Check using Factors
–
–
Created Frustration
Students shut down
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Ex.
2
x
–4
4x – 12
(x – 1)(x + 12)
(x – 12)(x + 1)
(x – 2)(x + 6)
(x – 6)(x + 2)
(x – 3)(x + 4)
(x – 4)(x + 3)
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Factoring Simple Quadratics
a 2+b
ax
bx + c where a = 1
●
#1
(x +
#2
)(x +
)
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Ex.
2
x
–4
4x – 12
-6
2
(x – 6)(x + 2)
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Ex.
2
x
-12
– 15
15x + 36
-3
(x – 12)(x – 3)
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x2 –
+ 81
0x – 81
0
Ex.
-9
9
(x – 9)(x + 9)
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x2 –
+ 49
0x – 49
0
Ex.
-7
7
(x – 7)(x + 7)
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Ex.
2
x
-4
8 + 16
– 8x
-4
(x –(x4)(x
– 4)–2 4)
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Ex.
2
x
8
16 + 64
+ 16x
8
(x +(x8)(x
+ 8)+2 8)
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Ex. 12x2 + 7x – 10
(x – 1)(12x + 10)(2x – 1)(6x + 10)(3x – 1)(4x + 10)
(x + 1)(12x – 10)(2x + 1)(6x – 10)(3x + 1)(4x – 10)
(x – 10)(12x + 1)(2x – 10)(6x + 1)(3x – 10)(4x + 1)
(x + 10)(12x – 1)(2x + 10)(6x – 1)(3x + 10)(4x – 1)
(x – 2)(12x + 5) (2x – 2)(6x + 5) (3x – 2)(4x + 5)
(x + 2)(12x – 5) (2x + 2)(6x – 5) (3x + 2)(4x – 5)
(x – 5)(12x + 2) (2x – 5)(6x + 2) (3x – 5)(4x + 2)
(x + 5)(12x – 2) (2x + 5)(6x – 2) (3x + 5)(4x – 2)
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The Generic Rectangle
23 ● 47
20
40
+
7
+
3
800
●
●
120
●
140
21
●
800 + 120
+ 140 + 21
1081
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The Generic Rectangle
(x + 4)(2x – 5)
2x
-5
x
2x
●2
●
-5x
+4
4
●
8x
-20
●
2 –
2 +
2x2x
5x3x+ -8x
20– 20
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Factoring Complex Quadratics
ax2 + bx + c where a ≠ 1
GCF
●
#1
#2
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Ex. 6x2–13x – 8
● 2
-48x
3x
-16x
3x
-8
2x
+1
(3x – 8)(2x + 1)
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Ex. 4x2 + 4x –15
● 2
-60x
10x
2x
-6x
-3
2x
+5
(2x – 3)(2x + 5)
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Ex. 6x2 + 5x – 4
● 2
-24x
-3x
3x
8x
4
2x
-1
(3x + 4)(2x – 1)
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Ex. 9x2 –30x +25
● 2
225x
-15x
-15x
3x
-5
3x
-5
– – 5)
(3x (3x
– 5)(3x
2
5)
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Ex. 12x2 –23x +10
● 2
120x
-8x
-15x
4x
-5
3x
-2
(4x – 5)(3x – 2)
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Ex. 6x2–11x + 4
●2
24x
-3x
3x
-8x
-4
2x
-1
(2x – 1)(3x – 4)
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Contacts

Bob Battinich
–
–

Tom Bjorkman
–
–

(916) 296-3958
[email protected]
(530) 219-2533
[email protected]
Keith Smith
–
–
(530) 672-6148
[email protected]
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