Standard: 8.EE.1: Know and apply properties of integer exponents to generate equivalent numerical expressions.

Learning Goal: Students will be able to develop and use the rules for multiplying expressions with integer exponents to simplify expressions.

Overview:

Develop: Ponzi Pyramid Scheme, Multiplying Exponents Table

Solidify: Kinesthetic Finding Rules of Exponents Activity, Laws of Exponents Activity Sheet, Exponents Pamphlet

Practice: Spinner Game, Power Bingo, Simplifying Expressions Puzzle

Name________________________ Period______ “Ponzi” Pyramid Schemes Max has received this email. It describes a scheme for making money.

1. If that process goes as planned, how much money would be sent to Max? Show your calculations. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 2. What could possibly go wrong? Explain your answer clearly. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 3. Why do they make Ponzi schemes like this illegal? ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ Copyright © 2011 by Mathematics Assessment Page 2 Ponzi Pyramid Schemes Resource Service. All rights reserved.

From: A Crook Date: Thursday 15th January 2009 To: B Careful Subject: Get rich quick! Dear friend, Do you want to get rich quick? Just follow the instructions carefully below and you may never need to work again: 1. At the bottom of this email there are 8 names and addresses. Send $5 to the name at the top of this list. 2. Delete that name and add your own name and address at the bottom of the list. 3. Send this email to 5 new friends.  

“Ponzi” Pyramid Schemes Rubric Points

Section points

1. Gives correct answer: $59 or $1953125 (accept 2 million dollars) Partial credit: $58 or $390625 5 to any power greater than 2

3 (2) (1)

3

2. Allow up to 4 points for any sensible statements such as: People might not send any emails. They might not send any money. The email may go into the Spam folder.

4 x 1

4

3. Allow up to 3 points for any sensible statements such as: It is a scam because: It gives people an unrealistic expectation of what they will receive. It takes people’s money and they may not receive any.

3 x 1

3

Total Points 10

Adapted from http://map.mathshell.org.uk/materials/tasks.php?taskid=278&subpage=expert

Name________________________ Period______ Multiplying Exponents Table

Complete the table using exponents.

Repeated Multiplication Exponent

2 * 2 * 2

62

4 * 4 * 4 * 5 * 5

25 * 34 * 63

(-3) * (-3) * (-3) * (-3)

(-8)2

(-2) * (-2) * 5 * 5 * 5 * x * x * x * y * y * z

(-1)3 * 4 * x5 * y * z3

Make a prediction of what would happen if we multiplied two expressions with the same base.

Complete the table using exponents.

Expression Repeated Multiplication Exponent Product

(example) 23 * 24 (2 * 2 * 2) * (2 * 2 * 2 * 2) 27

32 * 32* 34

4 * 43 * 45

23 * 26 * 32 * 35

Make up your own

Was your prediction correct?

What patterns do you observe from the table above?

What “short-cut” or rule would summarize your findings?

FINDING RULES OF EXPONENTS ACTIVITY Teacher Guide: Source: 8th-grade-math.barrow.wikispaces.net

Materials: pencil/paper, chart or butcher paper, index cards or paper, markers, tape Question: How can you use patterns to discover rules for multiplying integers?

• Prepare the tables ahead of time on chart paper and post in the room. Prepare the index cards with the following numbers:

1. 11 cards with a 2 on them 2. 8 cards with a 3 on them 3. 13 cards with a 5 on them 4. 1 card with a 6 on it 5. 1 card with a 1 on it 6. 1 card with a 7 on it

• Give out the cards to students so that every student has at least one card. Students will model the problem as you facilitate a discussion of it. For example, in 28*23, 8 students will stand side by side holding up their cards to represent 28, and 3 students will stand next to them to represent multiplying by 23. Now they see that there are 11 students with 2’s, so the answer is 211.

• After completion of each problem, students record the data in their own tables. • Questions to ask the class during/after presentations:

1. In the Products problems, how are the exponents in the factors related to the exponent in the product? • The exponents in the factors are added to get the exponent in the product.

2. What is the product of 107*104 as a power? • 1011

3. When you multiply powers with the same base, what do you do with the exponents? • You add the exponents.

4. What is the product of 72*77 as a power? How did you find the answer? • 79; you add the exponents.

5. What do you notice about the bases of the numbers being multiplied? • The bases are the same number.

6. Will the rules you developed for products work for 23*43? Why or why not? • No, the rules will not work because the bases are not the same number.

Answers to tables:

Products Expression Expression written as repeated

multiplication Number of factors Product as a power

24*23 (2*2*2*2)*(2*2*2) 7 27

31*34 3*(3*3*3*3) 5 35

52*54 (5*5)*(5*5*5*5) 6 56

THE NEXT PAGE CAN BE PROJECTED/PRINTED FOR THE STUDENTS.

Name______________________ Period________

FINDING RULES OF EXPONENTS ACTIVITY

Student Record Sheet

Question: How can you use patterns to discover rules for multiplying and dividing integers?

Products

Expression Expression written as repeated multiplication

Number of factors

Product as a power

24*23

31*34

52*54

Name______________________ Period________

FINDING RULES OF EXPONENTS ACTIVITY

Student Record Sheet

Question: How can you use patterns to discover rules for multiplying and dividing integers?

Products

Expression Expression written as repeated multiplication

Number of factors

Product as a power

24*23

31*34

52*54

Name_______________________ Period_________ Laws of Exponents Activity Sheet 1. Write a4b7 x a2b3 without using exponents. DO NOT multiply them out.

a. How many a’s are being multiplied? How many b’s are being multiplied?

b. Rewrite your expression using exponents. 2. Write m4n3 x m3n without using exponents. DO NOT multiply them out.

a. How many m’s are being multiplied? How many n’s are being multiplied? b. Rewrite your expression using exponents.

Look at Questions 1 and 2 above. What patterns do you notice? What rule can you generalize about multiplying expressions with like bases? Try your new rule: 3. (x2y4z3 )(xy5z7 ) 4. (r6s4t3 )(r2s3t2 )(rs7t5 )

Complete the Exponents Pamphlet for the Product of Powers Rule. Students should keep the pamphlet throughout the whole unit to solidify their understanding of the exponent rules.

Spinner Game:

Directions:

1. Toss a number cube to determine how many times you spin the spinner. 2. Spin the spinner to get your first variable with an exponent. Record it. 3. Continue spinning and recording the number of times shown on the number cube. 4. Write the multiplication problem and answer using the Multiplication Rule of

Exponents. 5. Repeat 10 times.

Power Bingo Power Bingo uses game sheets of 16 squares containing monomials that are the products of any three of the following monomials.

3x 3x 3x 3x 3y 3y 3y 3y x2 x2 x2 x2 y2 y2 y2 y2

Three tiles are drawn from the sack, and the three monomials picked are announced to the class and displayed on the overhead. If a square on your game sheet contains the product of the three monomials announced, you may cover or mark that square. The first player to cover four squares in a row vertically, horizontally, or diagonally wins the game. 1. One possible monomial is 9x3y , the product of 3x, 3y, and x2 . How many possible products do you think there are? How did you arrive at your answer? 2. Find all possible monomials that are products of any three of the 16 monomials listed above. Write your products below. 3. How does the number of monomials in part 2 compare with your answer in part 1? 4. You are almost ready to play Power Bingo. Before you can play, you need to fill your game sheet with different possible products of three monomials that might be drawn from the sack. When you are done, you are ready to play. 5. Once you have played Power Bingo, think about changes you may make in filling out your game sheet? Why?

Power Bingo Game Sheet 1

Power Bingo Game Sheet 2