Download Teacher Notes: Factors and Greatest Common Factors

Teacher Notes: Factoring Differences of Two Perfect Squares
Using algebra tiles model what a perfect square number looks like.
3
2
1
1
2
3
(1 tile)
(4 tiles)
(9 tiles)
Have students conduct a pair rally. Using one set of
student notes have students list as many perfect square
numbers as they can in 20 seconds. One student will start
with 1 give the paper to the other student to write 4 back to
write 9 etc. as many as they can do in 20 seconds. Tell
students this is without talking because other teams could
steal their answers. After the 20 seconds are up have all
students stand up. As you call out (or write on the board) a
perfect square number they don’t have down, students sit
down with their partner. It is interesting to find out how
many students remember the perfect square numbers.
(Awards could be given to winners, the last standing.) The
more perfect square numbers a student can recognize the
easier many areas of math will be.
Perfect Square Numbers: {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225}
Ask the following questions every time:
 Can anything be factored out?
 Is this a difference of two perfect squares?
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime.
x2 – 81
1.
2.
x
-9
x
x2
-9x
+9
+9x
-81
Sum: 0x
25x2 – 16
(1x)(81x)
(3x)(27x)
(+9x)(-9x)
3.
(5x + 4)(5x – 4)
5.
8.
2x2 – 98
Work on students understanding and using
the shortcut method.
Product: - 81x2
Or
100a2 – 1
(x + 9)(x – 9)
4.
(10a + 1)(10a – 1)
6.
17 – 68a2
16x2 – 36y2
(4x + 6y)(4x – 6y)
7.
-49x2 + 64y2
2(x2 – 49)
17(1 – 4a2)
64y2 – 49x2
2(x + 7)(x – 7)
17(1 + 2a)(1 – 2a)
(8y + 7x)(8y – 8y)
36x2 + 49
prime
9.
4
16a2 – 9
2
2
(4a + )(4a + )
3
3
10.
(x + 2)2 – 9
[(x+2) + 3][(x+2) – 3]
(x + 5)(x – 1)
Teacher Notes: Factoring Perfect Square Trinomials
1.
(a + b)2  (a + b)(a + b)
a
+b
a
a2
+ab
+b
+ab
+ b2
(a2 + 2ab +b2)
2.
Ask students if they can see the
pattern. Remember as they work
some students will pick up the pattern
quickly and will be able to use it, while
other students will never see the
pattern. The will continue to foil or box
to fine the product.
Perfect square trinomials:
(a – b)2  (a – b)(a – b)
a
-b
a
a2
-ab
-b
-ab
+ b2
(a2 – 2ab +b2)
(a + b)2 = a2 + 2ab +b2
(a – b)2 = a2 – 2ab +b2
When testing a trinomial ask the following questions:
 Is there a GCF for all three terms?
 Is the first term a positive perfect square number?
 Is the last term a positive perfect square number?
 Is the middle term the product of the square root of the first and last term?
 Is the second term positive or negative?
Determine whether each trinomial is a perfect squarer trinomial. If so, factor it.
3.
x2 – 8x + 16
4.
yes; (x – 4)2
5.
x2 +18x + 81
yes; (7x – 2)2
6.
16x2 – 56xy + 49y2
8.
yes; (4x – 7y)2
9.
64x2 – 72x + 81
no
4x2 – 28x + 49
yes; (2x – 7)2
yes; (x + 9)2
7.
x2 + 50x + 225
2x2 – 10x + 25
no
10.
1 2
x + 3x + 9
4
yes; (
1
x + 3)2
2