Title: Graphing Systems of Inequalities

Grade: Algebra (grades 9-12)

Materials: 
	Pen/pencil
	Worksheet (I will have enough copies for everyone)
	Chalkboard/chalk
	Whiteboard/whiteboard markers
	Dr. Cushman, do we have the following materials or do I need to get them?
		Big paper
		Markers
		Rulers
		Masking tape
		Calculators-each student should have their own

Lesson Overview:
	Students will learn how to graph systems of inequalities using a graphing calculator. Students will interact and share ideas with each other during an activity. Students will also present their work to the class.

Learning Objectives:
	Students will be able to generate a graph of inequalities (synthesis level)
	Students will be able to interpret what the graphs mean (comprehension level)
	Students will compare and contrast different graphs (analysis level)

NYS Standards:
	Key Idea 1: Mathematical Reasoning
		1A. Construct valid arguments
		1B. Follow and judge the validity of arguments
	Key Idea 3: Operations
		3A. Use addition, subtraction, multiplication, division, and exponentiation 
			with real numbers and algebraic expressions.
	Key Idea 7: Patterns/Functions
		7A. Represent and analyze functions using verbal descriptions, tables, 
			equations, and graphs.
	
Anticipatory Set: (3 minutes)
	Review what an inequality looks like. What each means <, >, =  Also review on a number line on the board. If it is x < 5, how do we show that? What about     x  7? Have students come up to the board and show what this means (they should draw an open circle around 5 and shade to the left, and also draw a closed circle around 7 and shade to the right).
	
Developmental Activity: (12 minutes)
	1. Hand out worksheet and calculators to students
	2. Work through example with students
	3. Have students interpret the graph on the bottom of their paper
		Walk around and spot check students’ work while they’re writing
	4. Go over student responses to what the graph means. 

Guided Practice: (15 minutes)
	1. Break students into groups of two or three depending on the attendance in your 
		group.
	2. Hand out large paper and markers to each group
	3. Assign roles to each student. Hand out the “Role Responsibilities” sheet. Go 
		over the worksheet quickly. If you have groups of 2, then one is the 
		recorder and one is the reporter. If it is a group of 3, then one is the 			reporter, one is the recorder, and one is the task manager. Make sure you 			let the students know frequently how much time they have left for the 			activity since the clock doesn’t really work in the room and we can’t all 			see it anyway!
	3. Give each group a different problem to do. Tell the students that they can work 
		on the floor if they prefer for more space. 
	4. Walk around the room and help groups if needed

Closure: (9minutes)
	Have each group reporter come up to the board, tape their paper up, and explain 
		what they did and how to interpret their graph. How are the graphs from 			each group different?

Independent Practice: (1 minute)
	Hand out the entrance ticket for tomorrow. Explain that to get into class 
		tomorrow, they must present this sheet and it WILL be graded. 

Name:_______________________________________________	Date:___________

Graphing Systems of Inequalities

To solve a system of inequalities, you need to find the _________ pairs that satisfy all the inequalities involved. One way to do this is to _______ the inequalities on the same coordinate plane. The solution set is represented by the ________________, or overlap, of the graphs.

Example: Solve the system of inequalities by graphing 

y < -x + 1
y  2x + 3

What do we put into Y1? _____________________
What do we put into Y2?_____________________

What are the boundaries of the solution set? ____________________________________

Is y = -x + 1 included in the graph of y < -x + 1? ________ Why or why not? ________ ________________________________________________________________________

Is y = 2x + 3 included in the graph of y  2x + 3? ________ Why or why not? ________
________________________________________________________________________

What does this graph mean?_________________________________________________
________________________________________________________________________

Name:_______________________________________________	Date:___________

Graphing Systems of Inequalities

To solve a system of inequalities, you need to find the _ordered_ pairs that satisfy all the inequalities involved. One way to do this is to _graph_ the inequalities on the same coordinate plane. The solution set is represented by the __intersection__, or overlap, of the graphs.

Example: Solve the system of inequalities by graphing 
y < -x + 1
y  2x + 3

What do we put into Y1? ____-x + 1___________
What do we put into Y2?___2x + 3____________

What are the boundaries of the solution set? _y = -x + 1 and y = 2x + 3 ______________

Is y = -x + 1 included in the graph of y < -x + 1? __No____ Why or why not? _Because it is y is less than -x _+ 1, not y equals -x + 1_____________________________________

Is y = 2x + 3 included in the graph of y  2x + 3? Yes_____ Why or why not? _Because it is y is less than or equal to 2x + 3___________________________________________

What does this graph mean? The solution set is the area above y = -x + 1 and below and including the line y = 2x + 3._______________________________________________ 
_______________________________________________________________________

Role Responsibilities

Recorder

	Your job is to record the work on paper. Make sure your words are clear and that you write neatly so the whole class can read it easily. Be sure to write down all steps that were taken in the process to solve the problem. You are responsible for knowing your groups’ work and all steps taken in the problem solving process!

Reporter

	Your job is to report your work to the class when we come back together at the end of the period. Make sure you know and understand what steps your group has taken to solve the problem. You are responsible for knowing your groups’ work and all steps taken in the problem solving process!

Task Manager

	Your job is to keep everyone on track with the assignment. If someone is starting get off track just remind them politely that you only have so many minutes to finish the assignment. You are responsible for knowing your groups’ work and all steps taken in the problem solving process!

***Even though you are assigned specific roles, you are ALL responsible for your groups’ work and the steps that you took to solve the problem. As always, remember to treat everyone in your group and in the class with R-E-S-P-E-C-T at ALL times. This assignment is meant to be fun!

Solve by graphing: Think about what you need to put into Y1 and Y2!!
Will you include graph paper that can be used and then attached to the chart paper?  It would be a great idea.

x - y < -1
x - y < 3

Solve by graphing:

y > 3
y > -x + 4

Solve by graphing:

y  -x + 3
y  x + 3

Solve by graphing: Think about what each inequality means!

y > 0
x  0

Name:___________________________________________	Date:_____________

Entrance Ticket

Directions: Write down the steps that need to be taken to solve the following system of inequalities on a graphing calculator. If you have a calculator at home, you may work the problem out. If you do not have a calculator at home, just write down the steps. That is, go as far as you can without the calculator.

3x - 4y < 1
x + 2y  7

Name:___________________________________________	Date:_____________

Entrance Ticket

Directions: Write down the steps that need to be taken to solve the following system of inequalities on a graphing calculator. If you have a calculator at home, you may work the problem out. If you do not have a calculator at home, just write down the steps. That is, go as far as you can without the calculator.

3x - 4y < 1
x + 2y  7

Multiplying Monomials
Week 7 - Day 4
Casie Croff
Kristy Chimera
Nichole Oswald

Title:  Multiplying Monomials
Grade Level:  Algebra 1
Topics:  Identifying, Multiplying, and Simplifying Monomials
	
Materials:
Teacher:   Multiplying Monomials Worksheets (I will provide copies for all groups.)
	     Chalk and Chalkboard or Markers and Whiteboard

Students:  Multiplying Monomials Worksheets
	     Calculators
	     Pen / Pencil

Lesson Overview: 
Students will take part in a discussion to determine if it is possible to multiply monomials and how they think you could go about multiplying monomials.  Students will multiply monomials and simplify expressions involving powers of monomials.

Lesson Objectives: 
Students will compare and contrast expressions to classify monomials.
Students will examine products of various monomials for relationships between exponents. 
Students will propose and describe rules for multiplying monomials.
 
NYS Standards:    	     
A.A.12:  Multiply and divide monomial expressions with a common base, using the properties of exponents.
A.A.13:  Add, subtract, and multiply monomials and polynomials.

Anticipatory Set:  (7 minutes)
~Pass out “Who am I” worksheet to students.
~Read riddle aloud for students and help break down key elements of the problem (see answer key for suggested questions to ask).
~Students will have about 5 minutes to work on problem.
~Go over problem with students.

Developmental Activity:  (27 minutes)
~Pass out Multiplying Monomials worksheets to students.
~Go over definition of monomials and example 1 asking for students suggestions during discussion.  (Be sure to have students give a reason for why the given expression is a monomial or not.  The reason should include key points from the definition of monomial.)
~Students work in groups of 3 or 4 to complete examples 2 and 3.
~Go over answers with the class asking for student volunteers to share answers based on their group work.
~Students complete example 4 independently.
~Ask for student volunteer to work out or explain the problems for the class.
~Students work in same groups to complete examples 5 and 7.
~Go over answers with the class asking for student volunteers to share answers based on their group work.
~Students complete examples 6 and 8 independently.
~Ask for student volunteer to work out or explain the problems for the class.
~Briefly discuss how properties can be combined to simplify more complex monomial expressions.
~Work through example 9 with the class asking for students’ suggestions on how to simplify.
This could take longer – especially when trying to discuss math with teenagers!  What is optional?  What is essential?  Maybe have discussion for the first few and then not?
Closure:  (3 minutes)
~Summarize the rules for multiplying monomials.  Students should fill in the blanks and complete the chart in their notes.

Assessment:  (3 minutes)
~Pass out exit ticket to students.
~Students should turn in before leaving class or finish for homework if time runs out.

Resources:  http://www.tcnj.edu/~billman2/Unit%20Plan/Lesson%20Plans.htm

Name:  ______________________			Date:  ____________
Multiplying Monomials and Simplifying Powers of Monomials

What is a Monomial?
A monomial is a _______________, a _______________, or a _______________ of a number and one or more variables.  An expression involving the division of variables is not a monomial.
Example 1:  Identifying Monomials
Expression			Monomial?
		a.	         -5				
		b.	       p + q
		c.	          x
		d.	       
		e.	      

Example 2  
         is the product of two 5's.  .
                      is the product of four 5's.   .
Using the above as an example, describe a2 and a3.  Remember to explain your reasoning and to show each step.

Example 3:  Product of Powers  
       
Using this as an example, show the product of a2 and a3.  Do not leave out any steps.

Do you see a relation between the exponents in the example and in your work?  Do you think this can be made into a rule?  If so explain the rule and explain your reasoning.

Example 4: 
Simplify each expression:  
a. (5 x7 )( x6 )					b. (4ab6 )(-7a2b3 )   

 

Example 5:  Simplifying Powers of Monomials 
          is three's.  .  We just discovered a rule to help us multiply monomials.  This rule tells us.  
Using this example, explain what (this was missing) is equal to.  Write a full explanation of your thoughts and reasoning.  

Do you see a relation between the exponents in the example and in your work?  Do you think this can be made into a rule?  If so explain the rule and explain your reasoning.

 

Example 6:  
Simplify 

Example 7:  Power of a Product
Using example 2 as an example, describe (xy)4 .  Remember to show each step.

Do you see a relation between the exponents in the example and in your work?  Do you think this can be made into a rule?  If so explain the rule and explain your reasoning.

Example 8:    
Simplify   (6ab)3   

NOTE:  Sometimes the rules for the Power of a Power and the Power of a Product are combined to simplify more complex expressions involving exponents.
To simplify an expression involving monomials, write an equivalent expression in which:
~each base appears exactly once
~there are no powers of powers
~all fractions are in simplest form

Example 9:  
Simplify    ( 1 xy4)2 [(-6y) 2]3              Can you use Equation Editor to make fractions?                                                  
	        3

Summing it all up
Property	Rule	Symbolically	
Product of Powers	You may multiply powers with the same base by _______________ exponents.	 	
Power of a Power	You can find a power of a power by _______________ exponents.	 	
Power of a Product	A power of a product is the product of the powers.	 	
Power of a Monomial	The power of a power property and the power of a product property can be combined into the power of a monomial property.	 	

Name:  ________________________________		Date:  ________________

Who am I?
I am a certain odd, single digit number.  When I am multiplied by myself 4 times, the sum of the digits in this product is equal to me.  What number am I?

Name:  ANSWER KEY							Date:  ________________

Who am I?
I am a certain odd, single digit number.  When I am multiplied by myself 4 times, the sum of the digits in this product is equal to me.  What number am I?
Answer:  7
7*7*7*7 = 2401
2 + 4 + 0 + 1 = 7

Notes:
You may give the students some helpful hints by helping them interpret the question or parts of it as you read it aloud.
For example:
odd:  What numbers are odd?
single digit:  What does that mean?

Name:  ANSWER KEY					Date:  ____________
Multiplying Monomials and Simplifying Powers of Monomials

What is a Monomial?
A monomial is a number, a variable, or a product of a number and one or more variables.  An expression involving the division of variables is not a monomial.

Example 2:    
         is the product of two 5's.  .
                      is the product of four 5's.   .
Using the above as an example, describe a2 and a3.  Remember to explain your reasoning and to show each step.
a2  is the product of two a's.
a2 = a*a
           a3 is the product of three a's.
         a3  = a*a*a

Example 3:  Product of Powers
       
Using this as an example, show the product of a2 and a3.  Do not leave out any steps and explain your reasoning.
a2 * a3 = (a*a) * (a*a*a) = a*a*a*a*a = a5                 

Do you see a relation between the exponents in the example and in your work?  Do you think this can be made into a rule?  If so explain the rule and explain your reasoning.
The exponent in the answer (product) = the sum of the exponents of the monomials being multiplied.

Example 4:  

Example 5:  Simplifying Powers of Monomials 
          is three's.  .  We just discovered a rule to help us multiply monomials.  This rule tells us.  
Using this example, explain what is equal to.  Write a full explanation of your thoughts and reasoning.  
(x2)3 is three (x2)’s
(x2)3 =  (x2) (x2) (x2) =  x2+2+2   = x6
Do you see a relation between the exponents in the example and in your work?  Do you think this can be made into a rule?  If so explain the rule and explain your reasoning.
The exponent on the answer (product) = the product of the exponents on the monomial being raised to a power.

Example 6:  

Example 7:  Power of a Product
Using example 2 as an example, describe (xy)4 .  Remember to show each step.

Do you see a relation between the exponents in the example and in your work?  Do you think this can be made into a rule?  If so explain the rule and explain your reasoning.
The exponent is being distributed to each variable in the parenthesis.

Example 8:    
Simplify (6ab)3

NOTE:  Sometimes the rules for the Power of a Power and the Power of a Product are combined to simplify more complex expressions involving exponents.
To simplify an expression involving monomials, write an equivalent expression in which:
~each base appears exactly once
~there are no powers of powers
~all fractions are in simplest form

Example 9:  
Simplify    ( 1 xy4)2 [(-6y) 2]3                                                                      
	        3

Summing it all up
Property	Rule	Symbolically	
Product of Powers	You may multiply powers with the same base by adding exponents.		
Power of a Power	You can find a power of a power by multiplting exponents.		
Power of a Product	A power of a product is the product of the powers.		
Power of a Monomial	The power of a power property and the power of a product property can be combined into the power of a monomial property.		

Name:  _____________________________			Date:  _______________

Exit Ticket

Write 64 in three different ways using exponents.
Example:  64 = (22)3

Name:  ANSWER KEY			Date:  _______________

Exit Ticket

Write 64 in three different ways using exponents.
Example:  64 = (22)3

Answers will vary.
Sample answers:   26
(23)2
26 * 24
2 * 25
23 * 23