Week 12 Day 1
Kristy Chimera
Casie Croff
Nichole Oswald

Title: Factoring Using the Distributive Property. Grades 9-12 Math A review

Materials:
	Whiteboard/Chalk board
	Whiteboard markers/chalk
	Worksheets and calculators

Lesson Overview:
	Students will recall the distributive property to help factor. Also, students will use grouping and the additive inverse property to factor. By doing what?

Lesson Objectives:
	Students will be able to apply the distributive property to help factor.
	Students will be able to combine terms with common factors
	Students will be able to generate the completely factored form of an algebraic 			expression.

NYS Standards:
	Key Idea 2: Number and Numeration
		2C. Apply the properties of real numbers to various subsets of numbers.
	Key Idea 3: Operations
		3A. Use addition, subtraction, multiplication, division, and exponentiation 
			with real numbers and algebraic expressions.
		3D. Use field properties to justify mathematical procedures.

Anticipatory Set: (5 minutes)
	Put the following problems on the board.
		1. 2a(6a + 8)	Did you want to use 4a?	Answer: 12a2 + 16a
		2. t(t - 4)		Answer: t2 - 4t
		3. 4v(2v + 4)		Answer: 8v2 + 16v
	Have three different students come up to the board to simplify each expression 
	using the distributive property. Tell students that we can do these same problems 	backwards, and that it is called factoring. So 12a2 + 16a = 2a(6a + 8). Have all 	students write these down in their notebooks on a sheet of paper titled “Factoring 	with distributive property”.

Developmental Activity/Guided Practice: (31 minutes)
	1. Hand out the worksheet titled “The Distributive Property Rocks!”
	2. Go over the guided notes section of the worksheet, filling in blanks and 
		explaining and answering questions along the way. (3 minutes)
	3. Go through section 1 problem a explaining and answering questions. Try to get 			as much student involvement as possible. (8 minutes)
	4. Have students work in pairs to complete section 1 problem b together the same 			way that problem 1a was done. While students are working, walk around 			the room and help any pairs that are having difficulty. Try to guide them in 		the right direction without giving them the answer (4 minutes).
	5. Go over answers, answering any questions students might have. (2 minutes)
	6. Go through sections 2 and 3 with the students explaining as you go. Have 			students answer questions along the way (what is FOIL (don’t use this method) and which represents the additive inverse property? These questions are on the 			worksheet). (14 minutes)

Closure: (3 minutes)
	Review what F.O.I.L (remove this) is, what GCF is, and what additive inverse is. If there is time, have them start the homework.

Independent Evaluation: (1 minute)
	Hand out the Bonus Homework. We did not have time to complete these in class, which is why it is a bonus homework. There are hints for the students, so they should be able to do most of the problem. Tell the students that this is a great way to get some bonus points, which a lot of them need. There is no reason why they should not at least TRY to complete the homework and hand it in tomorrow.

Name:_____________________________________________	Date:_____________

The Distributive Property Rocks!

Guided Notes:
	_____________ a polynomial (12a2 + 16a) means to find its _______________ factored form. 12a2 + 16a is not completely factored. We can go further find a common factor.  What common factor can we pull out factor from both terms? _______. After pulling factoring out 2a (isn’t there a larger value, say 4a?), we have 3a + 4 left. So the completely factored form of 12a2 + 16a is 2a(3a + 4) (distributing this results in 6a2 + 8a not 12a2 + 16a!).

Section 1:
	A) Distributive Property
To use the reverse of the distributive property to factor, we must first find the GCF (____________  _____________  ___________) of 12a2 and 16a.

12a2 = 2 * 2 * 3 * a * a	Factor Find the prime factorization of each number. Then circle the COMMON prime 
16a = 2 * 2 * 2 * 2 * a		factors. 

Recall: What is a prime number? ____________________________________________.
So the GCF is _________ = ________

Now we write each term as the product (multiplication) of the GCF and the remaining factors. So we have:
4a(3*a) + 4a(2*2) = 4a(3a) + 4a(4)
Now use the reverse of the distributive property to factor out the GCF which is 4a. So we are left with
4a(3a + 4), which is the completely factored form of 12a2 + 16a. 

	B) Use the reverse of the distributive property to find the completely factored form of 
18cd2 + 12c2d + 9cd

18cd2 =
12c2d =
9cd =
GCF is: ________ = ________
Write the completely factored form below:

Section 2: Use Grouping 
	C) Factor 4ab + 8b + 3a + 6 by grouping Rearrange these 4 so that the common factors are next to each other.

Step 1: Group terms with common factors.
		4ab = 2 * 2 * a * b
		8b = 2 * 2 * 2 * 2 * b
		3a = 3 * a
		6 = 2 * 3
So we have (4ab + 8b) + (___+___  )

Step 2: Factor out the GCF from each grouping.
		4b(__+__) + 3(__+__)

Step 3: Use the distributive property.
		(a + 2)(4b + 3)

Step 4: Checking you work. Use F.O.I.L to check your work. What is F.O.I.L? Warning: I don’t approve of this method.  Use “double” distributing instead.

		        F        O          I          L
(a + 2)(4b + 3) = a(4b) + a(3) + 2(4b) + 2(3)
		= 4ab + 3a + 8b + 6, which is what we started with! 

Section 3: Additive Inverse Property

Recall: Which of the following is an example of the additive inverse property?
1) (5 - x) and (x - 5)	2) (x - 2) and (x + 2) 	3) 2x and -4x	4) 3 and -7

	D) Factor 35x - 5xy + 3y - 21 using the additive inverse property.

35x - 5xy + 3y - 21 = (35x - 5xy) + (3y - 21)		By Grouping
		        = 5x(7 - y) + 3(y - 7)		By factoring out the GCF from each
		        = 5x(-1)(y - 7) + 3(y - 7)		7 - y = -1(y - 7)
		        = -5x(y - 7) + 3(y - 7)		5x(-1) = -5x
		        = 					By reverse of the Distributive Property
This is the completely factored form of 35x - 5xy + 3y - 21.

Name:_______________________________________________	Date:_____________

Bonus Homework: Solving an equation in factored form

1) Solve (d - 5)(3d + 4) = 0 	CHECK your answer!!
This was not done in the notes.  Bonus or not, you want the students to Succeed at it.  Write another polynomial and have the students factor it.
HINT: Set both pairs equal to 0. Your 2 answers should be written as a solution set 
{   ,   }

2) Solve by factoring: x2 = 7x	HINT: Get 0 on one side, then solve.
See above statement.

Name:____________Answer Key_______________________	Date:_____________

The Distributive Property Rocks!

Guided Notes:
	Factoring____ a polynomial (12a2 + 16a) means to find its completely_____ factored form. 12a2 + 16a is not completely factored. We can go further. What can we pull out from both terms? __2a__Don’t you mean 4a?_. After pulling out 2a, we have 3a + 4 left. So the completely factored form of 12a2 + 16a is 2a(3a + 4).
See above for the correct wording.
Section 1:
	A) Distributive Property
To use the distributive property to factor, we must first find the GCF (__greatest__  common_______  __factor___) of 12a2 and 16a.

12a2 = 2 * 2 * 3 * a * a	Find the prime factorization of Factor each number. Then circle the COMMON prime 
16a = 2 * 2 * 2 * 2 * a		factors. 

Recall: What is a prime number? __a number that can only be divided by itself and 1. 
So the GCF is 2*2*a____ = __4a____

Now we write each term as the product (multiplication) of the GCF and the remaining factors. So we have:
4a(3*a) + 4a(2*2) = 4a(3a) + 4a(4)
Now use the distributive property to factor out the GCF which is 4a. So we are left with
4a(3a + 4), which is the completely factored form of 12a2 + 16a. 

	B) Use the reverse of the distributive property to find the completely factored form of 
18cd2 + 12c2d + 9cd

18cd2 = 2 * 3 * 3 * c * d * d
12c2d = 2 * 2 * 3 * c * c * d
9cd = 3 * 3 * c * d
GCF is: 3*c*d___ = __3cd___
Write the completely factored form below:

3cd(6d) + 3cd(4c) + 3cd(3)
3cd(6d + 4c + 3) This is completely factored.

Section 2: Use Grouping 
	C) Factor 4ab + 8b + 3a + 6 by grouping

Step 1: Group terms with common factors.
		4ab = 2 * 2 * a * b
		8b = 2 * 2 * 2 * 2 * b
		3a = 3 * a
		6 = 2 * 3
So we have (4ab + 8b) + (3a + 6)

Step 2: Factor out the GCF from each grouping.
		4b(a + 2) + 3(a + 2)

Step 3: Use the distributive property.
		(a + 2)(4b + 3)

Step 4: Checking you work. Use F.O.I.L to check your work. What is F.O.I.L? _You multiply the first plus the outside, plus the inside, plus that last.                          _______
See above statement.
		        F        O          I          L
(a + 2)(4b + 3) = a(4b) + a(3) + 2(4b) + 2(3)
		= 4ab + 3a + 8b + 6, which is what we started with! 

Section 3: Additive Inverse Property

Recall: Which of the following is an example of the additive inverse property?
1) (5 - x) and (x - 5)	2) (x - 2) and (x + 2) 	3) 2x and -4x	4) 3 and -7

	D) Factor 35x - 5xy + 3y - 21 using the additive inverse property.

35x - 5xy + 3y - 21 = (35x - 5xy) + (3y - 21)		By Grouping
		        = 5x(7 - y) + 3(y - 7)		By factoring out the GCF from each
		        = 5x(-1)(y - 7) + 3(y - 7)		7 - y = -1(y - 7)
		        = -5x(y - 7) + 3(y - 7)		5x(-1) = -5x
		        = (y - 7)(-5x + 3)			By Distributive Property
This is the completely factored form of 35x - 5xy + 3y - 21.

Name:_______Answer Key______________________________	Date:_____________

Bonus Homework: Solving an equation in factored form

1) Solve (d - 5)(3d + 4) = 0 	CHECK your answer!!
HINT: Set both pairs equal to 0. Your 2 answers should be written as a solution set 
{   ,   }

This all needs to be changed to something the students can do without HINTS.
d - 5 = 0	or 	3d + 4 = 0		Check: Substitute 5 and -4/3 for x into the 
d      = 5		3d        = -4		original equation.
			d          = -4/3	      (d - 5)(3d + 4) = 0	                  (d - 5)(3d + 4) = 0
The solution set is {5, -4/3}		    (5 - 5)(3(5) + 4) = 0      ((-4/3) - 5)(3(-4/3) + 4 = 0
					         0  *   4          = 0		    (-19/3)(0)           = 0
						  0              = 0			0                = 0

2) Solve by factoring: x2 = 7x	HINT: Get 0 on one side, then solve.

x2 - 7x = 0					Check: Substitute 0 and 7 for x into the 
x(x - 7) = 0					original equation.
x = 0		or 	x-7 = 0			x2 = 7x			            x2 = 7x
			x    = 7			02 = 7(0)			72 = 7(7)
Solution set is {0, 7}				0   = 0				49 = 49

Week 12 Day 2
Kristy Chimera
Casie Croff
Nichole Oswald

Title: Factoring Trinomials grades 9-12 Math A review

Materials:
	Whiteboard/chalkboard
	Whiteboard markers/chalk
	Worksheets and calculators

Lesson Overview:
	Students will learn how to factor a trinomial. Students will also learn how to check their solutions using both the F.O.I.L (rename “double” distribute)  method, and the calculator.

Lesson Objectives:
	Students will be able to analyze graphs.
	Students will be able to explain why the solution works graphically.
	Students will be able to generate the factors for a trinomial.

NYS Standards:
	Key Idea 2: Number and Numeration
		2C. Apply the properties of real numbers to various subsets of numbers.
	Key Idea 3: Operations
		3A. Use addition, subtraction, multiplication, division, and exponentiation 
			with real numbers and algebraic expressions.
		3D. Use field properties to justify mathematical procedures.

Anticipatory Set: (4 minutes)
	Put the following on the board. Have the students do the problem in their notes on 
	a sheet of paper. Then have one volunteer come up to the board and complete and 	explain the problem. Use F.O.I.L (“double” distribute or even the Box Method – see me to learn this one) to evaluate (x + 2)(x + 3)
	Answer: (x + 2)(x + 3) = (x2 + 3x + 2x + (2*3)) Use x2 instead of x2.
			 = x2 + 3x + 2x + 6
			 = x2 + (3 + 2)x + 6
			 = x2 + 5x + 6
	Now ask what the prefix “tri” means? Answers could include: tricycle, triangle, 	triathlon. Tell the students that today we’re going to factor trinomials!

Developmental Activity:
	1. Hand out worksheet titled: x2 + bx + c
	2. Go through the notes section of the worksheet. Fill in the blanks as you get to 
		them. Try to have the students come up with the answers. (2 minutes)
	3. Go through cases 1 and 2 with the students. Go slowly and answer any 				questions that come up. You may need to spend a little time on finding the 		factors of a number, they may not remember this. (10 minutes)
	4. Make sure the students understand how to use the calculator to check their 			answer when you get to the check in case 2. 

Guided Practice:
	1. Have the students work together (groups of 2 or 3) to work through cases 3 and 
		4. Walk around the room and help any students who are stuck. They may 			have a hard time coming up with all of the factors. Try to give some hints 			without giving away the answers. (15 minutes)
	2. Come back together as a class and go over what the answers are for cases 3 and 
		4. Answer any questions that come up along the way. (4 minutes)

Independent Practice: (5 minutes)
	Hand out the exit ticket. The students should be able to figure out how to do this. 			There is a hint on there for them as well. Collect the exit ticket before the 			students leave the room.

Name:_____________________________________________	Date: _____________

x2 + bx + c

Notes:
	Notice, in the previous example we had x2 + (3 + 2)x + (2*3) where m = 3, and 
n = 2. What?  This is confusing – and I know what you are trying to say.  Use words – “the factors of 6 add up to 5, what are they?”  So bx = m + n, and c = n*m. This is true for all trinomials of the form x2 + bx + c. The coefficient (number in front of the variable) of the middle term is the ______ (addition) of m and n and the last term is the __________ (multiplication) of m and n. We will use this pattern to factor quadratic trinomials of the form x2 + bx + c. Let’s look at 4 cases.

Case 1:  b and c are both positive.

Factor x2 + 6x + 8		Use “Factors of ___ add up to ___” b = _________	c = _________
We need two numbers whose sum is 6 and whose product is 8. Let’s look at the factors of 8: what can we multiply to get 8? Let’s look at a table:

Factors of 8	Sum of the factors	
      1, 8	              9	
      2, 4	              6	

Since we want our sum to be 6 and our product to be 8, we will use 4 and 2. 
So x2 + 6x + 8 = (x + m)(x + n) which is the pattern
		= (x + 2) (x + 4)  so m = 2 and n = 4

Let’s check using F.O.I.L (rename “double” distributing or use the Box Method)
	(x + 2)(x + 4)
F       O     I	L
x2 + 4x + 2x + 8 = x2 + 6x + 8 which is what we started with.

Case 2: b is negative and c is positive

Factor x2 - 10x + 16		Use “Factors of ___ add up to ___” b = _________	c = _________
Now we need two numbers whose sum is -10 and whose product is 16, so we need to find the factors of 16. We need to look at negative numbers so when we add them, we will get a negative 10.

Factors of 16	Sum of the factors	
    -1, -16	          -17	
    -2, -8	          -10	
    -4, -4	           -8	

It looks like -2 and -8 will work. Let’s see what we get.
x2 - 10x + 16 = (x + m)(x + n)
	          = (x - 2)(x - 8)    so m = -2 and n = -8

Check: (x - 2)(x - 8) = x2 -8x -2x +16
                                 = x2 -10x + 16

Alternate way to check: Go to y = on your calculator. In Y1, type x2 - 10x + 16. In 
Y2 type (x - 2)(x - 8). Hit graph. Why does only one graph appear? _________________
_______________________________________________________________________

Case 3:  b is positive and c is negative

Factor x2 + x - 12		Use “Factors of ___ add up to ___” b = _________	c = _________
This time we need a sum of 1 and a product of -12. Let’s look at the factors of -12.

Factors of -12	Sum of the factors	

      	           	

		

The correct factors are ____ and _____. 
So x2 + x -12 = (x + m)(x + n)
                      = (x      )(x      )   	so m = _____ and n = ______

Check: (x    )(x    ) = 
		       = 

Case 4:  b is negative and c is negative

Factor x2 - 7x - 18			Use “Factors of ___ add up to ___” b = _________	c = _________
Now we need a sum of -7 and a product of -18 so let’s look at the factors of -18.

Factors of -18	Sum of the factors	

     	           	

The factors we should use are _____ and _____.

x2 - 7x - 18 = (x + m)(x + n)
                    = (x    )(x    )            so m = _____ and n = _____

Check: use the calculator to check. Enter x2 - 7x - 18 into Y1 and (x    )(x    ) into Y2.
Is this the solution? _______

Name: ______________________________________________	Date: _____________

Exit Ticket: Solve an Equation by Factoring

Solve x2 + 5x = 6  Then check your solutions.	This was not taught in the notes.  Change this to “Factor”.  No HINTS allowed!

HINT: Get all terms on one side of the equation, factor, then set parts equal to 0.

Name: _________Answer Key___________________________	Date: _____________

Exit Ticket: Solve an Equation by Factoring

Solve x2 + 5x = 6  Then check your solutions.	
Change this.
HINT: Get all terms on one side of the equation, factor, then set parts equal to 0.

x2 + 5x = 6						Check: Substitute 1 and -6 for x 
x2 + 5x - 6 = 0						into the original equation.
(x - 1)(x + 6) = 0				     x2 + 5x = 6		        x2 + 5x = 6
x - 1 = 0	or	x + 6 = 0		   12 + 5*1 = 6		(-6)2 + 5(-6) = 6
x       = 1		x        = 6		    1 + 5     = 6		        36 - 30 = 6
Solution set is {1, -6}				       6        = 6		             6     = 6

Name:_______Answer Key____________________________	Date: _____________

x2 + bx + c

Notes:
	Notice, in the previous example we had x2 + (3 + 2)x + (2*3) where m = 3, and 
n = 2. So bx = m + n, and c = n*m. See above for wording. This is true for all trinomials. The coefficient (number in front of the variable) of the middle term is the _Sum__ (addition) of m and n and the last term is the Product___ (multiplication) of m and n. We will use this pattern to factor quadratic trinomials of the form x2 + bx + c. Let’s look at 4 cases.

Case 1:  b and c are both positive.

Factor x2 + 6x + 8		See above. b = ___6_____	c = ____8____
We need two numbers whose sum is 6 and whose product is 8. Let’s look at the factors of 8: what can we multiply to get 8? Let’s look at a table:

Factors of 8	Sum of the factors	
      1, 8	              9	
      2, 4	              6	

Since we want our sum to be 6 and our product to be 8, we will use 4 and 2. 
So x2 + 6x + 8 = (x + m)(x + n) which is the pattern
		= (x + 2) (x + 4)  so m = 2 and n = 4

Let’s check using F.O.I.L  See above.
	(x + 2)(x + 4)
F       O     I	L
x2 + 4x + 2x + 8 = x2 + 6x + 8 which is what we started with.

Case 2: b is negative and c is positive

Factor x2 - 10x + 16		See above. b = _-10_____	c = ___16____
Now we need two numbers whose sum is -10 and whose product is 16, so we need to find the factors of 16. We need to look at negative numbers so when we add them, we will get a negative 10.

Factors of 16	Sum of the factors	
    -1, -16	          -17	
    -2, -8	          -10	
    -4, -4	           -8	

It looks like -2 and -8 will work. Let’s see what we get.
x2 - 10x + 16 = (x + m)(x + n)
	          = (x - 2)(x - 8)    so m = -2 and n = -8

Check: (x - 2)(x - 8) = x2 -8x -2x +16
                                 = x2 -10x + 16

Alternate way to check: Go to y = on your calculator. In Y1, type x2 - 10x + 16. In 
Y2 type (x - 2)(x - 8). Hit graph. Why does only one graph appear? Because the factors of the trinomial form the same line as the trinomial, so the graphs are on top of each other. 

Case 3:  b is positive and c is negative

Factor x2 + x - 12		See above. b = ___1_____	c = ___-12___
This time we need a sum of 1 and a product of -12. Let’s look at the factors of -12.

Factors of -12	Sum of the factors	
      1, -12	          -11	
     -1, 12	           11	
      2, -6	           -4	
     -2, 6	            4	
      3, -4	           -1	
     -3, 4	            1	

The correct factors are _-3_ and __4__. 
So x2 + x -12 = (x + m)(x + n)
                      = (x - 3)(x + 4)   	so m = _-3__ and n = __4___

Check: (x - 3)(x + 4) = x2 + 4x - 3x - 12
		          = x2 + x - 12

Case 4:  b is negative and c is negative

Factor x2 - 7x - 18			See above. b = _-7______	c = __-18____
Now we need a sum of -7 and a product of -18 so let’s look at the factors of -18.

Factors of -18	Sum of the factors	
      1, -18	          -17	
     -1, 18	           17	
       2, -9	           -7	
      -2, 9	            7	

The factors we should use are 2 and -9.

x2 - 7x - 18 = (x + m)(x + n)
                    = (x + 2)(x - 9)            so m = 2 and n = -9

Check: use the calculator to check. Enter x2 - 7x - 18 into Y1 and (x + 2)(x - 9) into Y2.
Is this the solution? _Yes___

Title/Topic:  Factoring the Difference of Perfect Squares
Grade Level:  Algebra 1

Materials:  
Teacher:  Graph paper for students 
	    Scissors for students
Worksheets (Let’s Review Perfect Squares, Difference of Squares, Frayer Model and Exit Slip)
Whiteboard
Markers
Students:  Graph paper
	    Scissors
Worksheets (Let’s Review Perfect Squares, Difference of Squares, and Exit      Slip)
Pen/Pencil
(Note:  I will provide graph paper, scissors, and worksheets.)

Lesson Overview:
Students will find the formula for factoring the difference of perfect squares using paper squares.  Students will factor the difference of perfect squares using the formula found ().

Lesson Objectives:
Students will manipulate paper squares to determine how the areas and side lengths relate to factoring the difference of squares.
Students will break down components of similar squares and subdivide the sides and edges. 
Students will combine and reassemble the parts of squares in order to relate their areas and sides to factoring the difference of perfect squares. 

NYS Standards:
Key Idea 3 - Operations
3A. Use addition, subtraction, multiplication, division, and exponentiation with real numbers and algebraic expressions.
3B. Use integral exponents on integers and algebraic expressions.
Key Idea 4 – Modeling/Multiple Representation
4A. Represent problem situations symbolically by using algebraic expressions, sequences, tree diagrams, geometric figures, and graphs.
Key Idea 5 - Measurement
5I. Use geometric relationships in relevant measurement problems involving geometric concepts.

Anticipatory Set:  (5 minutes)
-	Pass out Let’s Review Perfect Squares Worksheet to all students.
-	Students complete worksheet.
-	Go over answers as a class asking for student suggestions and answers.
Developmental Activity:  (25 minutes)
- Pass out the Difference of Squares Worksheet to all students.
- Students work in pairs to complete the activity and worksheet.  The teacher should be circling around the room as the students are working to answer any questions and monitor completion of the activity.
- Go over answers to worksheet as a class asking for student suggestions and answers.  Be sure to discuss the formula for factoring perfect squares which they just found.
- Complete examples 1-3 as a class.  Allow students to tell you how to work through each example.

Closure:  (5 minutes)
-	Pass out Frayer Model Worksheet to all students.
-	Review formula for finding the difference of perfect squares and characteristics of such problems.
-	Ask students for some examples and nonexamples of the difference of perfect squares.  The teacher may offer some additional examples and nonexamples.
-	Students fill in the Frayer Model throughout the discussion.

Assessment:  (5 minutes)
-	Pass out Exit Slip to all students.
-	Students complete exit slip and turn in before leaving class.

Name:  _____________________________				Date:  ____________

Let’s Review Perfect Squares

Directions:  Circle any number and/or expression that you think is a perfect square.  Next to those that you circle, write the factors that make it a perfect square.

2			49						25

			18			9			

			12						16

14						4			8

Name:  ________________________				Date:  ____________

Difference of Squares

Step 1:  Use a straightedge to draw two squares similar to those shown below (I don’t see any squares shown below, I am missing a page?) on graph paper.  The larger square has side length a; and the smaller square has side length b.

1.  Write an expression representing the area of the larger square.

2.  Write an expression representing the area of the smaller square.

3.	Write an expression representing the area of the both squares.

Step 2:  Cut the smaller square out of the larger square.  Label the lengths of each of the sides/edges.  This might take a while – note this in the Developmental Activity to spend the needed time explaining how to find the expressions representing the areas.

4.  Write an expression representing the area of the remaining region.

Step 3:  Cut the remaining region into 2 congruent pieces as shown below (Again, I see nothing “below”).

Step 4:  Rearrange the two congruent regions to form a rectangle.

	5.  What is the length of the rectangle?

	6.  What is the width of the rectangle?

	7.  Write an expression representing the area of the rectangle.

8.	Explain why -  = (a + b)(a – b)

9.  Explain why 

We can use this pattern to factor binomials that can be written in form - .

Factoring the difference of squares

Formula:  ____________________________________

Examples:  Factor each binomial.

a. - 25					

b.   - 12a

c.  

Name: _________________________			Date: __________

Factoring the Difference of Perfect Squares

Definition/Rule	Characteristics	
Examples	Nonexamples	

Name:  ___________________________				Date:  ____________

Exit Slip
Factor 			Not nice!  You want the students to Succeed.  Find another problem that is aligned with the notes.

(Hint:  You will need to apply several different factoring techniques.)

Name:  ___________________________				Date:  ____________

Exit Slip
Factor 

(Hint:  You will need to apply several different factoring techniques.)

Name:  ANSWER KEY				Date:  ____________

Let’s Review Perfect Squares

Directions:  Circle any number and/or expression that you think is a perfect square.  Next to those that you circle, write the factors that make it a perfect square.

2			49 = 						25 = 

			18			9 = 			

 and x2x2x2x2	12		    		16 = 

14					4 = 			8

Name:  ANSWER KEY				Date:  ____________

Difference of Squares

Step 1:  Use a straightedge to draw two squares similar to those shown below (What?).  The larger square has side length a and the smaller square has side length b.

1.  Write an expression representing the area of the larger square.

2.  Write an expression representing the area of the smaller square.

9.	Write an expression representing the area of the both squares.
What is expected here?
Step 2:  Cut the smaller square out of the larger square.  Label the lengths of each of the sides/edges.   What is expected here?

4.  Write an expression representing the area of the remaining region.

Step 3:  Cut the remaining region into 2 congruent pieces as shown below (What?).

Step 4:  Rearrange the two congruent regions to form a rectangle.

	5.  What is the length of the rectangle?	a + b	

	6.  What is the width of the rectangle?	a - b

	7.  Write an expression representing the area of the rectangle.          (a + b)	(a – b)

	8.  Explain why -  = (a + b)(a – b)      Answers will vary.  The area of the remaining region is -  which can be rearranged to form a rectangle with area          (a + b)(a – b).  Since the area is the same for both shapes, they are equal.  Hence 
-  = (a + b)(a – b).

9.  Explain why      Answers will vary.  The area of the both squares was which is not equal to the area of the final rectangle (a + b)(a – b) because the smaller square with area was removed from it.

Note:  We can use this pattern to factor binomials that can be written in form - .

Factoring the difference of squares

Formula:  -  = (a + b)(a – b) or (a - b)(a + b)

Examples:  Factor each binomial.

a.  - 25 							
     - 25  = 				Write in the form - .			                                                                                                 	      = 			Factor the difference of squares.

b.   - 12a					
      - 12a = 		The GCF of  and -12a is 12a.
                        = 		and 
	            =  		Factor the difference of squares.

c.  
      = 			The GCF of and -162 is 2.
	           = 		 and 
	           = 		Factor the difference of squares.
	           = 		 and 
	           = 	Factor the difference of squares.

Name: _________________________			Date: __________

Factoring the Difference of Perfect Squares

Definition/Rule

-  = (a + b)(a – b)

OR

-  = (a - b)(a + b)	Characteristics

Subtraction of 2 perfect squares

Can be constants or variables

Answers will vary.	
Examples

Answers will vary

	Nonexamples

Answers will vary	

Name:  ANSWER KEY				Date:  ____________

Exit Slip
Factor 
See above.
(Hint:  You will need to apply several different factoring techniques.)

			Original polynomial.
       = 		Factor out the GCF.
       = 		Group terms with common factors.
       = 		Factor each grouping.
       = 			 is a common factor.
       = 	Factor the difference of squares,,  into (x+1)(x-1).

Topic: Graphing Quadratic Functions
Grade: 9th
Section: 10-1

Materials: calculators, worksheets, chalkboard/whiteboard, chalk/markers
**ALL WORKSHEETS WILL BE PROVIDED

Lesson Overview: Students will learn how to graph quadratic functions, both by hand and by using the calculator.  They will also find the equation of the axis of symmetry and the coordinates of the vertex of a parabola.  Students will work through a series of questions first as a class and then independently. This is too much for 40 minutes.  Maybe two days worth here – or more.

Lesson Objectives:
•	Students will be able to identify the equation of a parabola, and then create graphs of parabolas by plotting ordered pairs. [Application]
•	Students will analyze the graphs of parabolas in order to find the maximum/minimum point (vertex), as well as the axis of symmetry. [Analysis]
•	Given an equation for a parabola the students will be able to write the equation of the axis of symmetry, find the coordinates of the vertex, identify whether it is a maximum or minimum, and then graph the function. [Evaluation]

NYS Standards:
•	Key Idea 7: Patterns/Functions- Students apply linear, and quadratic functions in the solution of problems.
•	Key Idea 7: Patterns/Functions-Students represent and analyze functions using verbal descriptions, tables, equations and graphic forms of functions.

Anticipatory Set: (3 mins)
Ask the students what it looks like when you bounce a ball to another person (Hmm, I don’t think of a parabola when I visualize this situation – more of a V.  Try toss a basketball into a hoop or toss paper into a trash can – even have a student do it).  Draw on the board a ball bouncing (this should look like a parabola).  Then ask the students what it looks like when someone dives into a pool, touches the bottom (try launching a rocket) and then swims back up to the top.  Draw this on the board (again this should look like a parabola).

Developmental Activity: (27 mins)
1.)	Tell the students that the function describing the ball bouncing or diving in the pool is an example of a quadratic function.
2.)	Pass out “Graphing Quadratic Functions”.  
3.)	After introducing the standard form of a quadratic function- point out to the students that this polynomial is of degree 2 and the exponents are positive.
4.)	For example 1- Ask the students what they notice about the y-values. (they should notice the pattern in which they repeat)
5.)	For example 2- Ask the students why the graph opens downward.
6.)	Ask the students what it means for something to be symmetrical. (a figure can be folded in half so that each half matches the other exactly)
7.)	Show the students how to graph both examples on the calculator.  
8.)	Point out how to find the vertex on the calculator.

Closure: (5 mins)
Discuss what they learned about what the axis of symmetry is.  When does a parabola have a maximum? A minimum?  What is a vertex?  These are just a few suggestions.

Assessment: (5 mins)
Have the students complete the Exit Ticket and hand it in before they leave. This will be graded.

Name:	Answer Key								Date:

“Graphing Quadratic Functions”

A QUADRATIC FUNCTION can be written in the form , where .
This form of the quadratic function is called STANDARD FORM.  The graph of a quadratic function is called a PARABOLA.

A parabola can either open UPWARD or DOWNWARD.

1.) Graph Opens Upward
Use a table of values to graph .
x	y	
 -2	11	
-1	1	
0	-5	
1	-7	
2	-5	
3	1	
4	11	

Consider the standard form of a quadratic equation for this example.  The value of the coefficient, “a” is POSITIVE and the curve opens UPWARD.

The lowest point on the graph is called the MINIMUM.

What is the minimum of this graph?  (1, -7) How was this found?  Did you describe toggling left and right?  Did you use the CALC function?  Did you Trace?

2.) Graph Opens Downward
Use the table of values to graph .
x	y	
-1	-6	
0	-1	
1	2	
2	3	
3	2	
4	-1	
5	-6	

The value of the coefficient, “a” is NEGATIVE, and the curve opens DOWNWARD.
Therefore the graph has a highest point, called the MAXIMUM.
What is the maximum for this graph? (2,3)  See notes above.
**The maximum and minimum point of a parabola is called the VERTEX.

Parabolas are symmetrical figures.  They have an axis of symmetry.  The VERTEX is the only point on the parabola that is on the axis of symmetry.  The axis of symmetry is written as x = the x-value of the vertex. This is a very hard concept for students and might take more than one sentence!

What is the axis of symmetry for the parabola in example 1? X =1
What is the axis of symmetry for the parabola in example 2? X =2
Hmm, I see a pattern that has nothing to do with the sentence above.  I see that example 3 will be X=3.  You should think about changing the formatting.

Name:											Date:

“Graphing Quadratic Functions”

A _______________________ can be written in the form , where .
This form of the quadratic function is called ________________________.  The graph of a quadratic function is called a _________________.

A parabola can either open ________________or _________________.

1.) Graph Opens Upward
Use a table of values to graph .
x	y	
 -2	11	
-1	1	
0	-5	
1	-7	
2	-5	
3	1	
4	11	

Consider the standard form of a quadratic equation for this example.  The value of the coefficient, “a” is ________________ and the curve opens ______________.

The lowest point on the graph is called the _________________.

What is the minimum of this graph?  

2.) Graph Opens Downward
Use the table of values to graph .
x	y	
-1	-6	
0	-1	
1	2	
2	3	
3	2	
4	-1	
5	-6	

The value of the coefficient, “a” is _________________, and the curve opens ___________________.
Therefore the graph has a highest point, called the ___________________.
What is the maximum for this graph? 
**The maximum and minimum point of a parabola is called the ____________.

Parabolas are symmetrical figures.  They have an axis of symmetry.  The ____________ is the only point on the parabola that is on the axis of symmetry.  The axis of symmetry is written as x = the x-value of the vertex.

What is the axis of symmetry for the parabola in example 1? 
What is the axis of symmetry for the parabola in example 2? 

Name:	Answer Key							Date:

Exit Ticket

Graph the equation , using the table.
X	Y	
-2	-5	
-1	0	
0	3	
1	4	
2	3	
3	0	
4	-5	

What are the coordinates of the vertex? (1,4)

Write the equation of the axis of symmetry?  X=1

Is the vertex a maximum or minimum?  Maximum

Name:										Date:

Exit Ticket

Graph the equation , using the table.
X	Y	
-2	-5	
-1	0	
0	3	
1	4	
2	3	
3	0	
4	-5	

What are the coordinates of the vertex? 

Write the equation of the axis of symmetry?

Is the vertex a maximum or minimum?