Exponential Unit
Big Ideas:
-Developing the meaning of function and their graphs
-Compare linear (arithmetic growth) to exponential (geometric growth)
-y = 2x versus y = 2x….use tables here
-y = bx if b > 1 and 0 < b < 1
-y = abx + c vs. y = ax + b to develop meaning of a, c, and b
-Asymptote
Lesson 1 – Arithmetic and Geometric Sequences
(To the teacher: The goal of this first lesson is to compare arithmetic and geometric growth. You might want to spend a couple days on this lesson and adding more examples so that the students can develop an understanding between arithmetic and geometric sequences.)
Start with patterns and building on the patterns
1, 3, 5, 7, 9,…
What is the 200th number in that pattern? How would you get it?
(To the teacher: Move it back to table…make tables for this. This sequenced can be indexed both ways. The first term can be 0 or it could be 1. Then could both answers be correct? This idea can transfer the subsequent patterns.)
30, 25, 20, 15,…
Write a formula for the nth term and find the 50th term?
(To the teacher: Move it back to table…make tables for this)
Now give them the pattern
3, 9, 27, 81,…
What would be the 8th term in this sequence?
(To the teacher: Move it back to table…make tables for this)
How would you write a formula for the nth term?
16, 8, 4, 2,…
What would be the 6th term in this sequence?
(To the teacher: Move it back to table…make tables for this)
How would you write a formula for the nth term?
Lesson 2 – Laws of exponents
(To the teacher: The main goal here is to develop x0 = 1 and x-ais 1/xaas well as to connect exponentials to geometric growth.We have included three different lessons. Depending on where your students are you may scale these lessons down or use them in their entirety.)
Teacher Guide
Arithmetic Powers and the Generalizer (the Variable)
The goal of the next few lessons is to answer the following questions:
How can arithmetic powers be generalized in algebra?
And how do the operations behave with these generalizations?
How do we manipulate variables when we multiply and ivied?
What is the difference between x + x and x times x?
Opening Activity: Laws of Exponents
(To the Teacher: This lesson is an opportunity for your students to utilize their ability to recognize patterns and begin to create generalizations from those patterns. In the case of this lesson the generalizations will help lead toward developing an understanding of the exponent laws. One note on the opening activity is that it is very important for the students to write the results of the negative exponents as fractions and not decimals. We strongly encourage you to do this activity, as well as all other activities, for yourself before using it with your students. When examining the patterns be sure to look not only at the patterns in the columns, but also at possible patterns going across the columns. One can develop many of the exponential laws from this activity by simply introducing multiplication and division into their exploration and then generalizing their observations.
Technology note: If you have graphing calculators in the TI-84 Plus family the OS has recently been updated so that there is a built in fraction mode that enables the calculator to automatically display answers as fractions. Information about this can be found in TI Knowledge Base Solution 30891, the web address for this page is:
Additionally, if you are unfamiliar with the Wolfram Demonstrations Project, it has some free resources that may interest you for this topic as well as others. Here’s a link to the interactive online version:
Printing student activity note: You may want to print the two pages of this activity on separate pages so the students don’t have to flip the page back and forth while working on the writing prompts on the second page.)
Laws of Exponents:
Look at the following sets of powers. Find the solutions to each, changing all decimals to fractions. (You can use a calculator)
25 / 55 / 10524 / 54 / 104
23 / 53 / 103
22 / 52 / 102
21 / 51 / 101
20 / 50 / 100
2-1 / 5-1 / 10-1
2-2 / 5-2 / 10-2
2-3 / 5-3 / 10-3
2-4 / 5-4 / 10-4
2-5 / 5-5 / 10-5
(To the teacher: After groups have had some time to think about the following questions have a class discussion. You may want to document the observations, patterns, questions, and conjectures on chart paper so that there is a public record of their thinking for you as well as for them. The students’ questions and conjectures as well as extensions of their patterns become the centerpiece for moving the lesson forward. One approach to doing this is to have students self select what questions, conjectures, or patterns they want to continue to investigate and report back to the class on. Some questions that could be helpful as you think through how to move this lesson forward:
- How could you use a variable to generalize that pattern you found?
- How would you test that conjecture? What new observations can we make if we use negative numbers? Do our generalizations still hold when we use negative numbers?
- What new patterns do we see when we multiply or divide within a column? Is there a way to generalize these patterns? – What’s meant by “multiply …within the column” by example is: 22 * 23 = 25
- What new patterns do we see when we multiply or divide across the columns? Is there a way to generalize these patterns? – What’s meant by “multiply across the columns” by example is: 25 * 55 = 105
- Does your generalization hold for decimals? For fractions? For radicals?
- What new conclusion can we make if we raised all the things we’ve already worked with to the 2nd power? Example: (21)2 or (53)2
Today’s lesson is about exploration and getting comfortable with the ideas of exponents and how the exponent can be represent as repeated addition. The next two lessons will explicitly address multiplication and division and then negative exponents and zero exponents. But these next two lessons are guides. You might approach the lessons differently based on the information you gather from this initial lesson. For example you might have groups work on different patterns discovered and work to show why they are always true and how we would work with them using variables. This might take longer but could lead to deep understanding)
What observations do you have of the results?
What patterns do you see?
What questions do you have?
Are there any conjectures you can you make?
What statement can you make that holds true all the time?
Activity 2: Further Investigation
What would you like to continue investigating?
What hypothesis do you have about the outcome of your investigation?
What questions might you ask yourself to help in your investigation? Think back on the questions you have been asked through the course of the activities so far this year for ideas.
(To the Teacher: Part of what we are wanting to encourage in our students at all times is their ability to problems solve, which often comes back to the questions we ask ourselves during the problem solving process and we want to foster this in our students as well. It’s always good to have some questions in your back pocket that can aid your students. Some of the questions given in the previous note would be good ones and the questions from the opening activity could also be useful. In addition to those here are some other more generic questions you could ask:
- How could we create a set of examples (data) to make observations?
- What patterns do we notice in our examples (data)?
- What conjectures do we have based on this pattern?
- Can we generalize this pattern with a variable?
- Is this generalization true all the time? How do we know? Can we prove it?
We want our students to engage in the process of problem solving and so we have to be thoughtful in how we are going to interact with them. We do not want short-circuit their process by immediately providing answers or next steps.)
Student Activity Sheet
Name______
Date______
Opening Activity: Laws of Exponents
Look at the following sets of powers. Find the solutions to each, changing all decimals to fractions. (You can use a calculator)
25 / 55 / 10524 / 54 / 104
23 / 53 / 103
22 / 52 / 102
21 / 51 / 101
20 / 50 / 100
2-1 / 5-1 / 10-1
2-2 / 5-2 / 10-2
2-3 / 5-3 / 10-3
2-4 / 5-4 / 10-4
2-5 / 5-5 / 10-5
What observations do you have of the results?
What patterns do you see?
What questions do you have?
Are there any conjectures you can you make?
What statement can you make that holds true all the time?
Activity 2: Further Investigation
What will are you going to continue investigating?
What hypothesis do you have about the outcome of your investigation?
What questions might you ask yourself to help in your investigation? Think back on the questions you have been asked through the course of the activities so far this year for ideas.
Teacher Guide
Exponents and the operations of multiplication and division
Aim: How does multiplication and division behave when working with exponents?
(To the Teacher: Today we want to see how the operations of multiplication and division evolve into working with exponents and create generalizations of the observed behavior. The first activity will build on yesterday’s observations. Then we build from there.)
Opening Activity:
Look back at the chart in yesterday’s lesson…
25 / 55 / 10524 / 54 / 104
23 / 53 / 103
22 / 52 / 102
21 / 51 / 101
Do you observe anything in this table that will help us to understand what 22 times 23 equals? How about 51times 53 equals?
Activity 2: Multiplication and Exponents
Of the representations below rewrite into groups the ones that are equal. (You may use a calculator.)
35 / 31 * 32 / 3434 * 32 / (3*3)*(3*3*3) / (3*3*3*3)*(3*3)
(3*3)*(3*3) / 3*3*3*3*3*3 / 32 * 31
32 * 34 / 32 * 32 / 3*3*3*3*3
33 / 32 * 33 / (3)*(3*3)
(To the Teacher: There are many observations that can be made of the different groupings of equal representations. Many of which will help students see the connection between exponents and repeated multiplication, as well as that order doesn’t matter when we multiply with exponents, commutativity holds. One way this could be differentiated is to provide some groups with the groups already established. You want to support students to get to xatimes xb = xa+b)
What observations do you have of the groups?
What patterns do you see?
What questions do you have?
Are there any conjectures you can you make?
What statement can you make that holds true all the time?
How could this be generalized using a variable?
What would x3 * x5 equal? Why?
Activity 3
Earlier in this unit we learned about combining like terms in addition and subtraction. Now we are learning to multiply. In a few sentences describe the difference between
x + x and x * x
(To the Teacher: students often confuse these two ideas so it important for them to be able to talk conceptually about the difference between the two of them.)
Activity 3: Division and Exponents
Of the representations below rewrite into groups the ones that are equal. (You may use a calculator.)
/ 41 /43 / /
/ 46 /
/ / 44
42 / /
What observations do you have of the groups?
What patterns do you see?
What questions do you have?
Are there any conjectures you can you make?
What statement can you make that holds true all the time?
How could this be generalized using a variable?
What would x5 / x2equal? Why?
(To the Teacher: Its recommended that you alter these home works to best fit where your class. This may mean adding a few more exercises for practice, but be thoughtful with how you use such exercises as doing exercises ad nauseam will do more harm than good.)
Homework Assignment
- What was the most interesting discovery from today’s investigation for you? What made this discovery interesting to you?
- How would you explain why 25 * 23 = 28?
- A big idea in mathematics is doing and undoing. Why would multiplication and division with variables and exponents be described as doing and undoing?
Simplify all of the following:
A) 32 * 35B) 64 * 67 C) x3 * x5 D) m7/m2
Now based on your understanding multiplication and division try to answer the following and be ready to talk about why you think your answer is correct
1) m3n4 * mn2
2) x6y3
x2y
3) (x3z5)2
Student Activity Sheet
Name______
Date______
Opening Activity:
Look back at the chart in yesterday’s lesson…
25 / 55 / 10524 / 54 / 104
23 / 53 / 103
22 / 52 / 102
21 / 51 / 101
Do you observe anything in this table that will help us to understand what 22 times 23 equals? How about 51times 53 equals?
Activity 2:
Multiplication and Exponents
Of the representations below rewrite into groups the ones that are equal. (You may use a calculator.)
35 / 31 * 32 / 3434 * 32 / (3*3)*(3*3*3) / (3*3*3*3)*(3*3)
(3*3)*(3*3) / 3*3*3*3*3*3 / 32 * 31
32 * 34 / 32 * 32 / 3*3*3*3*3
33 / 32 * 33 / (3)*(3*3)
What observations do you have of the groups?
What patterns do you see?
What questions do you have?
Are there any conjectures you can you make?
What statement can you make that holds true all the time?
How could this be generalized using a variable?
What would x3 * x5 equal? Why?
Activity 3
Earlier in this unit we learned about combining like terms in addition and subtraction. Now we are learning to multiply. In a few sentences describe the difference between
x + x and x * x
Activity 4: Division and Exponents
Of the representations below rewrite into groups the ones that are equal. (You may use a calculator.)
/ 41 /43 / /
/ 46 /
/ / 42
What observations do you have of the groups?
What patterns do you see?
What questions do you have?
Are there any conjectures you can you make?
What statement can you make that holds true all the time?
How could this be generalized using a variable?
What would x5 / x2equal? Why?
Homework
Name______
Date______
What was the most interesting discovery from today’s investigation for you? What made this discovery interesting to you?
How would you explain why 25 * 23 = 28?
A big idea in mathematics is doing and undoing. Why would multiplication and division with variables and exponents be described as doing and undoing?
Simplify all of the following:
A) 32 * 35B) 64 * 67 C) x3 * x5 D) m7/m2
Now based on your understanding multiplication and division try to answer the following and be ready to talk about why you think your answer is correct
1) m3n4 * mn2
2) x6y3
x2y
3) (x3z5)2
Teacher Guide
Working with zero and negative exponents
(To the Teacher: Since students were given these for homework which required them to think in new ways it is important to go over these and have students talk about the mathematical reason for the procedures they have chosen to use. Take your time. It is important to learn how students transfer their conceptual understanding to a new problem.)
You are going be given the same problems from the homework to work on for the first five minutes. Be ready to explain how you would work with these problems and the mathematical reason for doing what you did?
Opening Activity
1) m3n4 * mn2
2) x6y3
x2y
3) (x3z5)2
Activity 2: Zero Exponents
Relook at the Table from two days ago. What did you observe about zero exponents?
Why do you think that the same answer always came up?
Think about the following:
1) What is true about any number divided by itself?
2) So based on the previous question what is 23 equal to?
23
3) Now what did we say you did in division and exponents when you divided a number by a number with the same base?
4) So based on the previous question what would another answer to 23 ?
23
5) How can you use the your answers to the previous for questions to say that 20 = 1?
(To the Teacher: In this exercise we are trying to get the students to think about the fact if two different answers are equal to the same expression then they must be equal to each other. We are in the field of proof and would be important for students to think about this idea. If a= b and a =c then a must be equal to c.)
6) How would you generalize this idea?
Some examples to ponder: Simplify the following:
- 70
- (7x)0
- 7x0
- 7x0y
- 7x0 * 5
Activity 3: Negative Exponents
Relook at the Table from two days ago. What did you observe about negative exponents?
Why do you think that the same answer always came up?
Think about the following:
1) Again, what did you learn about division and working with exponents when you divided a number by a number with the same base?
2) So what would 23 equal?
25
3) If we wrote 23 in expanded form it would look like 2*2*2 ,
25 2*2*2*2*2
4) If we simplified the expression above by cancelling what would be left in the numerator and what would be left in the denominator?
5) How can we use our answers from the previous two questions to say that 2-2 = ½?
(Again we are in the field of proof using the transitive property. We want students to think about the conceptual reason for negative exponents and not just the rule.)
6) Generalize the rule for negative. Begin with x-a =.
Some examples to ponder: Simplify the following. Be ready to talk about the mathematical reason for what you did.
2-3
y-5
32
34
x2y3
x4y2
Homework
Name______
Date______
Simplify all of the following. Show the work you did to get the answer.
6-2
x-3
8-3 * 82
(m3)2
(m-3)2
b3 * b-3
b3 * x3
7x * 7y
How does 2 * ½ = 1 help us make some sense of why 20 equals 1?
Is 78 * 7-3 equal to ? Provide evidence and an explanation for your response.
Student Activity Sheet
Working with zero and negative exponents
Opening Activity
You are going be given the same problems from the homework to work on for the first five minutes. Be ready to explain how you would work with these problems and the mathematical reason for doing what you did?
1) m3n4 * mn2
2) x6y3
x2y
3) (x3z5)2
Activity 2: Zero Exponents
Relook at the Table from two days ago. What did you observe about zero exponents?
Why do you think that the same answer always came up?
Think about the following:
1) What is true about any number divided by itself?
2) So based on the previous question what is 23 equal to?
23
3) Now what did we say you did in division and exponents when you divided a number by a number with the same base?