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Copyright © 2015 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, Agile Mind, Inc.

Unit 7 Exponential relationships

In this unit, you will build your understanding of exponential functions by comparing and contrasting them with linear functions. You will learn how to analyze data to determine whether the data have a linear, exponential, or other nonlinear relationship. You will also learn how repeated multiplication leads to exponential functions, and compare that to the constant addition of linear functions.

OUTLINE

Topic 19: Exponents and exponential models In this topic, you will explore exponents. When working with exponents, you will identify some consistent patterns that are connected to basic arithmetic operations. These patterns make working with exponential expressions easier.

This topic will also cover exponential functions. In earlier topics, you investigated linear functions. However, in real life, the relationship between two quantities is not always linear. In these situations, exponential functions may be used to represent the relationship.

In this topic, you will:

• Develop and understand the laws of exponents • Simplify numerical and variable expressions involving exponents • Learn how exponential growth and decay are represented in situations, tables, and graphs • Determine whether a relationship represented by a table, rule, graph or statement can be modeled by an exponential

function

Topic 20: Reasoning with quantities Solving problems that involve exponential relationships often involves reasoning with especially small and large quantities. In this topic, you will practice using scientific notation to represent and perform operations on these kinds of numbers. In this topic, you will:

• Analyze data and represent situations involving exponential relationships • Reason about the relative magnitude of quantities • Use scientific notation to represent quantities and solve problems

Topic 21: Problem solving with exponential functions Earlier in the unit, you learned about non-‐linear relationships that grow exponentially. In this topic, you will more closely investigate the behavior of the family of functions known as exponential functions. You will study the nature of exponential growth and decay using different representations, including tables, graphs, and equations. In this topic, you will:

• Analyze graphs to identify exponential functions • Learn how exponential sequences build from previous terms • Learn how to model exponential relationships as functions of the for y = abx • Learn how changes in the parameters a and b for y = abx affect the graph of an exponential function • Describe the behavior of exponential functions

240 Unit 7 – Exponential relationships


Topic 19: Exponents and exponential models 241


EXPONENTS AND EXPONENTIAL MODELS Lesson 19.1 Models for exponents

19.1 OPENER Cell division is a basic property of living cells. In cell division, each “parent cell” produces two or more “daughter cells.” Consider cell division in which each parent cell produces two daughter cells.

1. Fill in the table to show the number of cells after each division.

Number of divisions

Number of cells

0 1 1 2 3 4

2. Graph the data from your table on the graph provided.

19.1 CORE ACTIVITY 1. Look at the data from the cell division scenario. Is the data linear? _________

a. How can you tell by looking at the table?

b. How can you tell by looking at the graph?

2. What pattern did you notice when watching the animation about cell division?

3. Use the table and graph you created in the opener to answer the following questions.

a. After how many divisions will the number of cells be 16? How could you use the table to figure this out? How could you use the graph to figure this out?

b. How many cells will be present after 5 divisions? How did you figure this out?

c. After how many divisions will the number of cells be 128? Explain.

4. Some cells divide into three cells at a time. Fill out the table to show the number of cells after each division. Then make



a graph of the data.

Number of divisions

Number of cells

0 1 1 3 2 3

5. Use the table and graph you competed in question 4 to answer the following question.

a. After how many divisions will the number of cells be 27?

b. How could you use the table to figure this out?

c. How could you use the graph to figure this out?

6. What pattern do you notice about the way the cells divide?

7. How is this table similar to the table from the Opener? How is it different?

8. How is this graph similar to the graph from the Opener? How is it different?



19.1 CONSOLIDATION ACTIVITY In Smalltown, a snowstorm has caused the power to go out across the community. The school principal has decided to use a phone tree to contact all of the students of Smalltown High School to let them know that school has been canceled. During the first round of phone calls, the principal calls the first two students on the phone tree. During the second round of phone calls, those two students each call two other students, and so on until all of the students have been notified.

1. Complete the table below that shows how many students are called during each round of phone calls.

Round Number of students called on this round of phone calls

1 2

2

3

2. How many students will be called during the fourth round of phone calls?

3. How did you figure out how many students would be called during the fourth round of phone calls?

4. Describe the pattern you notice in the number of students called in each round of phone calls.

5. Not all phone trees are set up the same way. In some phone trees, each person is only responsible for calling one other person. In other phone trees, each person is responsible for calling three people, or four people, or more! Work with your partner to investigate another type of phone tree. Your teacher will assign you a specific type of tree to investigate. You will then make a poster to present the results of your investigation to the class. Type of phone tree: ___________________________________ Your poster should include the following:

a. A table

b. A graph

c. A diagram

d. How many people will be called on the fourth round of phone calls? How did you figure this out?



HOMEWORK 19.1 Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. The flu virus is known to spread quickly. A recent outbreak began with a single contaminated person, but the number of infected people is increasing rapidly! Each week, there are ten times as many people infected as the week before. Fill out the table to show the number of infected people after each week.

Number of weeks

Number of infected people

0 1 1 10 2 3 4

2. How many people will be infected after 5 weeks?

3. If this rate of infection were to continue, how many weeks would it take for 1 million people to be infected?

4. An outbreak of a less aggressive strain of the flu virus is occurring in another area of the country. This outbreak began with a single person, but the number of infected people only triples each week. Fill out the table to show the number of infected people after each week.

Number of weeks

Number of infected people

0 1 1 3 2 3 4

5. How many people will be infected after 5 weeks? After 6 weeks?

6. Compare the two flu viruses.

a. How are the data in the tables similar? How are they different?

b. How would the graphs of the two sets of data be similar? How would they be different? (It may help for you to sketch a graph of each set of virus data.)



STAYING SHARP 19.1

Review

ing pre-‐algebra ideas

1. What is the area of a rectangle that measures 4 cm by 5 cm? Provide a sketch to support your work.

2. If you doubled the dimensions of the rectangle in question 1, what would be the ratio of the area of the larger rectangle to the area of the smaller rectangle?

Practic

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cepts

3. Solve the equation 3x + 5 = 7x – 3. Show evidence of the method you used. Check that your solution is correct.

4. Solve the equation in question 3 using a different method. Show evidence of the method you used.

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5. If you are given $3 on day 1 and the amount you are given doubles every day, how much will you be given on day 4?

6. For the situation in question 5, you can use exponents to write expressions that give the amount of money on each day. The amount of money you are given each day is as follows:

Day 1: 3•20 Day 2: 3•21 Day 3: 3•22 Day 4: 3•23

What exponential expression would help you find how much money you will be given on day 20 if the money you get continues to double every day?



Lesson 19.2 Multiplication and division rules for exponents

19.2 OPENER Janice and Alejandro are working with the rules they developed yesterday for the phone tree situation. They are trying to figure out how many people would be called during the fourth round of phone calls. Take a look at their discussion:

1. Who is correct, Janice or Alejandro? Explain why you think he or she is correct.

2. How would you explain to the other student why he or she is not correct?

19.2 CORE ACTIVITY 1. Your teacher will assign you a table to fill out. For the table you are assigned, find the value of each exponential

expression.

Expression 23 � 28 27 � 24 25 � 26 211

Value

Expression 31 � 37 32 � 36 35 � 33 38

Value

Expression 44 � 41 42 � 43 43 � 42 45

Value

Expression 53 � 54 52 � 55 51 � 56 57

Value

2. What do you notice about the values in your table?

3. Is there a relationship among the exponents in each expression you investigated?

4. What do you notice about the values in the other tables?

5. Is there a relationship among the exponents in the expressions within each of the other tables?



6. Now, find the value of each exponential expression in the following two tables.

Expression 312

310 37

35 36

34 32

Value

Expression 213

210 27

24 26

23 23

Value

7. What do you notice about the values in each table? 8. Is there a relationship between the exponents in the expressions within each of the tables?

9. In your own words, state a general rule that could be used to multiply or divide expressions with like bases.

10. Rewrite each exponential expression below using only one exponent.

a. x � x b. 2x � 3x c. x5

x 2

11. Try to extend the patterns you noticed above to expressions containing variables by writing each expression below with a single variable.

a. 4x2 � 7x3 b. x31

x17

c. -‐3x4 � 5x6 d. 6a8

3a5

12. Complete the statement to write a general rule for multiplying expressions with exponents. Then convert the statement to an algebraic representation.

base exponent add quotient product sum difference

Verbal representation of the rule

When you multiply two exponential expressions with the same base, the ______________ is an exponential

expression with the same ______________ and an ______________ that is the ______________ of the two

exponents of the initial powers.

xm � n xn xm – n xm xm + n Algebraic representation of the rule For any number x except zero, the following property is true:

_______ � _______ = __________



13. Complete the statement to write a general rule for dividing expressions with exponents. Then convert the statement to an algebraic representation.

base exponent divide add quotient product sum difference

Verbal representation of the rule

When you divide two exponential expressions with the same base, the ______________ is an exponential expression

with the same ______________ and an ______________ that is the ______________ of the two exponents of the

initial expressions.

xm�n xmn

xm x

n xm

– n xm

+ n

Algebraic representation of the rule

_______ = __________

19.2 REVIEW END-OF-UNIT ASSESSMENT




1. Would you rather rewrite the expression n47 ·∙ n25 with a single exponent by writing out all the factors or by using the Law of Exponents for Multiplication? Explain why.

2. Use a rule of exponents to rewrite each expression with a single exponent. If it is not possible to do so, explain why not.

a. d3 ·∙ d11 b. k8 ·∙ y4

c. c7 ·∙ c

d. n11 ·∙ n23

3. Joni said that she could rewrite the expression d3 ·∙ d11 as d33. What mistake did Joni make? How would you explain to Joni the correct way to rewrite this expression?

4. Write two different exponential expressions involving multiplication that are equivalent to z11.

5. Rewrite each expression using a single exponent.

a. (3x4) ·∙ (5x2) b. (2s7) ·∙ (3s5)

6. Rewrite each expression using only two exponents.

a. (p4 x2) ·∙ (p5 x3) b. (5y3z11) ·∙ (7y12z4)

7. Jackson was rewriting some exponential expressions. He claims that 34 � 23 is equivalent to 67 since “all you have to do is multiply the bases and add the exponents”. Is Jackson’s reasoning correct? If not, how would you to explain to Jackson that his reasoning is incorrect?



8. Use the laws of exponents to rewrite each expression. If it is not possible to do so, explain why not.

a. 11

4

pp

b. n 8

n 5

c. 6

3

dz

d. r 7s 2

r 4s 2

9. Consider the expression 5 2

3 3

x yx y

.

a. Without simplifying, evaluate the expression for x = 5 and y = 7. (You may get some big numbers.)

b. Rewrite the expression using the laws of exponents.

c. Evaluate the expression you wrote in part b with the same values: x = 5 and y = 7.

d. Did you get same value for parts a and c? Why?

10. You have developed the Law of Exponents for Division and the Law of Exponents for Multiplication.

Compare and contrast the Law of Exponents for Division and the Law of Exponents for Multiplication. Specifically mention what you know about the base and the exponents in each case.

Compare (Ways the two laws are similar) Contrast (Ways the two laws are different)

11. Think about your performance on the Unit 6 end-‐of-‐unit assessment.

a. Are you pleased with your performance on the end-‐of-‐unit assessment? Circle one: Yes / No

b. Does your performance reflect your understanding of the topics in Unit 6? Circle one: Yes / No

If you answered “No”, why do you think this?

c. Based on your answers to parts a and b, do you need to revise the goal you wrote at the end of the last unit? If so, write your new goal below along with any enabling goals that will help you reach your goal.



STAYING SHARP 19.2 Re

view


1. What is the area of a right triangle whose legs are 4 cm and 5 cm? Provide a sketch to support your work.

2. If you tripled the dimensions of the triangle in question 1, what would be the ratio of the area of the larger triangle to the smaller triangle?

Practic


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3. Solve the equation 8 -‐ 2x = 3x – 7. Show evidence of the method you used. Check that your solution is correct.

4. Solve the equation in question 3 using a different method. Show evidence of the method you used.

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5. In a particular sequence, each term is 3 more than the last. The sequence begins with 2, 5, 8. Write the next four terms of the sequence.

6. In a particular sequence, each term is 3 times the last. The sequence begins with 2, 6, 18. Write the next four terms of the sequence.



Lesson 19.3 Other rules for exponents

19.3 OPENER 1. Find the values of the following expressions.

a. b.

c.

2. How do you know the values you found are correct?

3. Rewrite the following expressions using a single exponent: a. b.

4. How do you know that the expressions you found are correct?

5. Compare each original expression to its rewritten expression. Do you notice a pattern in the relationship between the exponents in the original expression and the exponent in the rewritten expression?

19.3 CORE ACTIVITY

1. Maggie created the table on the right.

a. What pattern do you notice in the two columns of the table?

b. Based on this pattern, what do you think 20 will be? Explain.

2. Use the pattern you found in question 1 to complete the table. Find the value of 2-‐3.

3. How is the value of 2-‐3 related to 23?

4. What is the value of 2-‐5? Justify your answer in at least two ways.

5. Think about the expression . Use the Law of Exponents for Division to rewrite using a single exponent. What do you notice?

25

2534

3473

73

2 2 2 23 3 3 3⋅ ⋅ ⋅ x 5 ⋅x 5 ⋅x 5

42

42

42

42

24 16

23 8

22 4

21 2

20 ?



6. Complete the table to further investigate zero and negative exponents. Fill in the middle column with the process that gives you the final, simplified answer in the third column.

32 3⋅3 9

31 3 3

30 1

3-‐1

3-‐2

3-‐3

7. What is the value of 3-‐5?

8. Demonstrate what you have learned about zero and negative exponents by stating the value of each expression.

Expression Value Expression Value

a. 6-‐2 d. –(50)

b. 100 e. 7-‐1

c. –(62) f. (–5)0



9. Use what you know about the Law of Exponents for Division to show that any number other than zero raised to the zero power will be equal to 1. Fill in the blanks.

0 x1 x0 -n xn 0 – n n – n 1

10. Use the fact that any number, besides zero, raised to the zero power will be 1, and the Law of Exponents for Division to

show what the expression is equivalent to. Fill in the blanks.

0 x1 x0 -n xn 0 – n n – n 1

11. Fill in the blanks to rewrite the expression (132)3 using repeated multiplication. Then write the answer using a single

exponent using the Law of Exponents for Multiplication.

( ________ ) ⋅ ( ________ ) ⋅ ( ________ ) = ________

12. Use repeated multiplication to rewrite the following expressions and express the final answer using a single exponent.

a. (293)4 = b. (x6)8 =

c. (y4)5 = d. (57)3 =

13. In each problem in question 12, how are the exponents in the original expression related to the exponent in the answer?

1xn



14. Fill in the blanks to rewrite the expression (5�3)4 using repeated multiplication. Then write your answer without parentheses. Hint: Do not simplify 5�3 to 15.

(5�3)4 = (________) ⋅ (________) ⋅ (________) ⋅ (________) = ________

15. How can you use repeated multiplication and the associative and commutative properties to simplify the following expressions? Rewrite each expression without parentheses. Do you see any patterns across all of the examples?

a. (3x)4 = b. (2t)3 c. (7ab)-‐6

19.3 CONSOLIDATION ACTIVITY In this activity, you will work with your partner as you continue to investigate the properties of integer exponents through a card sort activity. You will also develop two more rules for exponents that you investigated earlier in the lesson.

1. Write down the pairs of cards that show a correct simplification in the card sort activity.

2. When a power is raised to a power, the algebraic rule is (xm)n = ________ .

3. When a product is raised to a power, the algebraic rule is (xy)m = ________.




1. Find the value of each expression a. 100 = _________ b. 10-‐1 = _________ c. 34 = _________ 4-‐3 = _________

2. Marcus worked the following problem out in his notebook. Is his answer correct? Did he do the work correctly? If not, explain to Marcus what he did right and also what he needs to do to correct his work.

3. Justify why the following statement is true.

4. In the following two examples, students have used the laws of exponents to solve problems. Each example contains a mistake. First, identify the mistake. Then, solve the problem correctly, using the same method that was used in the original problem.

a. Judy is trying to rewrite the expression . She says, “The expressions being divided have the same base, so I can use the Law of Exponents for Division. The difference of the exponents is 7 – 2 = 5, so it must come out to w5.”

b. Tim is trying to evaluate the expression 2-‐3. He says, “I know 23 = 2 ·∙ 2 ·∙ 2 = 8. Since the exponent is -‐3 instead of 3, the value of 2-‐3 = -‐8.

!!

�

13−2 =1

−132 =1169

!!

�

54 =15−4

2

7

ww



5. Find the value of each expression in two different ways, as shown in the example.

Expression Use the Law of Exponents for Multiplication, then evaluate.

Evaluate each term of the expression, then multiply.

Example: (24)(2-‐2) (24)(2-‐2) = 24 ·∙ 2-‐2

= 24+(-‐2)

= 22 = 4

(24)(2-‐2) = (24)( )

= 16 ·∙

= 4

a. (36)(3-‐1)

b. (5-‐3)(57)

c. (42)(4-‐5)

6. For each of the following expressions, use the Law of Exponents for Multiplication and/or the Law of Exponents for Division to rewrite the expression using a single exponent. Then, use the properties of integer exponents to evaluate the expression.

Expression Rewrite using a single exponent Evaluate

7. Complete the math journal.

State the rule in your own words Give an example that shows this rule

Rule for rewriting an exponential expression that has been raised to an exponent

Rule for rewriting a product that has been raised to an exponent

�

122

�

14

75 ⋅7−7

1126

1125

127 ⋅12−5

122



8. Find the value of each expression in two different ways, as shown in the example. (You may end up with some large numbers!)

Expression Use the Law of Exponents for a Power, then evaluate.

Evaluate, then raise to a power.

Example: (23)-‐2 (23)-‐2 = 23·∙(-‐2)

= 2-‐6

= =

(23)-‐2 = (23)-‐2

= (8)-‐2

= =

a. (36)-‐1

b. (5-‐3)4

9. Ray confused the rules for rewriting an exponential expression that has been raised to an exponent and rewriting a product that has been raised to an exponent. For each problem below, explain to Ray which rule he should have used and why. Then rewrite each expression correctly.

a. (5x)3 = 5x • 3 = 15x

b. (x2) 7 = x 7 • 27 = 128 x 7

126

164

182

164




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1. A regular nonagon has nine sides that are all the same length. If each side of a regular nonagon measures 4 cm, what is its perimeter?

2. If you doubled the dimensions of the nonagon in question 1, what would be the ratio of the perimeter of the larger nonagon to the perimeter of the smaller nonagon?

Practic


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3. Consider the following systems of equations problem.

An office has two different kinds of imagers: copiers and scanners. Each copier uses an average of 400 watts of power and each scanner uses an average of 35 watts of power. There are 15 imagers in all, and their total energy usage is 3080 watts.

Identify what you think this problem is looking for.

4. Consider the following systems of equations problem.

Aunt Betty uses oranges and pineapples to make 12 pounds of fruit salad. If oranges weigh 0.5 pounds each and pineapples weigh 2 pounds each, and she uses 21 fruits in all, find the number of oranges and the number of pineapples she uses.

Define variables and write two equations to model the situation described in the problem.

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5. Janis got the answer to this problem in less than one second. How did she do that?

6. Find the value of each y-‐coordinate and sketch a graph of these points.

(1,20), (2,21), (3,22), (4,23)

!!

�

(34 ⋅85 ⋅1612 )0



Lesson 19.4 Rational exponents

19.4 OPENER 1. Using the example of n = 8 as a guide, fill in the table.

2. Rewrite 625 as the product of four like factors.

3. Using your answer to question 2, rewrite 6254 as a single integer. Use the table you completed in question 1 as a model.

19.4 CORE ACTIVITY 1. Complete this table to show whole number powers of 3.

30 31 32 33 34 35

2. According to the table in question 1, what is the principal square root of 9? Why?

3. What symbol is used to indicate the principal square root?

4. Provide three numerical examples that show how taking the square root of a number “undoes” the action of squaring a number.

5. What is another name for the small number inside the notch of the radical sign?

6. You now know that “9 to the power” means the same thing as “the square root of 9.” Use fractional exponential expressions to complete the last three rows in the table.

�

12



7. How is the index on the radical sign related to the fractional exponent?

8. Use what you know about exponents to show that 2713 = 3 . (You may want to refer back to the table you constructed

in question 6.)

9. Isabel was given the following sketch of a square on a test. When asked to find the perimeter, Isabel incorrectly answered that the perimeter of this square is 16 centimeters. Explain to her where she might have made a mistake and how she could arrive at the correct answer of 32 centimeters for the perimeter of the square.

10. A square has a perimeter of 40 centimeters. Explain how to find the area of the square.

11. Bart has been asked to find the square root of 16. He is not sure which key sequence to use in his calculator, so he tries the two different sequences shown here. Explain, using order of operations, how he got 8 for an answer for one sequence of keys and 4 as an answer for the other. Which answer is the correct square root?

A = 64 cm2



19.4 CONSOLIDATION ACTIVITY 1. Amelia took a quiz in her Algebra class. It only took her one minute to answer the following five questions. Find the

solutions to the problems.

a. (53)13 = ______ b. 132 = ______

c. ( 7 )2 = ______

d. (102415 )5 =

______ e. 344 =

______ Explain how Amelia could have finished the quiz so quickly.

2. Write the following in two ways using symbols. Find the value, and state a reason for that value. Use the example as a guide.

Example: The seventh root of 2187 = , because 37 = 2187.

a. The square root of 121 b. The cube root of 216

c. The sixth root of 64

3. Fill in the table and then graph the relationship on the grid provided. (Hint: You can use your calculator if you need to!)

Expression Value

161

1612

160

16−12

16-‐1

4. Use the graph you created to estimate the value of 1614 .

5. Explain how you used the graph to estimate the value of

1614 .

6. What is the exact value of 1614 ? How do you know? How

does this value compare to the value you estimated using the graph?

HOMEWORK 19.4

! 21877 = 218717 = 3

-1 -0.5 0 0.5 1

-5

5

10

15



Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. Create a table for powers of 4, as we did for powers of 3.

Powers of 4 4 as a root 4 as a root

2. Describe at least two patterns you see in the “Powers of 4” table.

3. How are the exponents of the base 4 (first column) related to the fractional exponent when 4 is a root (second column)?

4. In the table below are the answers that Khanh submitted during an online assessment. Identify which three answers are incorrect, describe the errors Khanh made, and provide the correct answers.

Expression Value Correct or Incorrect? If incorrect, what is the correct value?

50 0

9-‐1 -‐9

49 7

16

�

14 4

5. Carrie claims that the following statement is true: 25 = 102412 Samantha, her classmate, says “There’s no way these two

expressions have the same value.” Who is correct? Why?

6. Sketch a square with an area of 25 square centimeters. How long is each side? In complete sentences, explain how you got the answer, using the vocabulary word square root.

7. Martha is asked to find the value of 814 . She uses her calculator as shown here. Is she correct? Why or why not?

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12



STAYING SHARP 19.4

Review


1. What is the volume of a rectangular solid measuring 3 cm wide by 4 cm deep by 5 cm tall? Provide a sketch to support your work.

2. If you doubled the dimensions of the rectangular solid in question 1, what would be the ratio of the volume of the larger rectangular solid to the volume of the smaller rectangular solid?

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3. Is a = 3, b = 4 a solution to this system of equations?

5a + 2b = 23 7a – 6b = 3

Show evidence for your answer.

4. Will the lines represented by the equations y = 2x + 3 and 2y – 4x = 12 ever intersect? (You may use the grid below.)

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5. Which do you think is bigger, 210 or 102? a. Guess: b. Compute the value of each expression. (Were

you right?) c. Find the difference between the values of the

expressions.

6. Which table shows a linear relationship? Justify your answer.

Table A Table B x y x y 1 4 1 1 2 7 2 4 3 10 3 9 4 13 4 16 5 16 5 25



Lesson 19.5 Comparing linear and exponential growth

19.5 OPENER Jen and Ben are stacking checkers on a grid. They each have a grid that has six squares as shown in the figure.

Jen’s pattern: Start with 5 checkers on the first square of the grid. Each of the remaining squares will have 5 more checkers than the previous square. Ben’s pattern: Start with 1 checker on the first square of the grid. Each of the remaining squares will have twice as many checkers as the previous square. 1. Who will have more checkers on the fourth square on

the grid? How many more?

2. Who will have more checkers on the sixth square on the grid? How many more?

3. Suppose there were ten squares on the grid instead of six, and Jen and Ben continued stacking checkers according to

the patterns described above. Who would have more checkers on the tenth square? Explain how you can be sure of the answer without actually extending the pattern.

19.5 CORE ACTIVITY 1. How many insects did Barry and Red each start off with?

2. Record and graph the fruit fly and fire ant data for the four-‐week observation period.

Week

Fruit flies

Fire ants

0

1

2

3

4



3. How does the total number of fruit flies change each

week?

4. How does the total number of fire ants change each week?

5. What type of function models the fruit fly data? 6. Does this same type of function model the fire ant data? Explain.

7. Study the data in the table to determine the growth pattern for the populations of fruit flies and fire ants. Then use the appropriate operations and numbers to complete the puzzle.

+2 +1 +20 �2 -‐20 �1

8. The populations of the two species of insects have different growth patterns.

a. How is the pattern of change in the fruit fly data related to the shape of the graph?

b. How is the pattern of change in the fire ant data related to the shape of the graph?

19.5 ONLINE ASSESSMENT Today you will take an online assessment.



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1. Johnny has an account at a local bank. The following table shows the amount of money he has in his account after x months. Graph the data.

Time (in months) Amount (in dollars) 0 1 1 3 2 9 3 27 4 81

Is the relationship linear or exponential? How do you know?

2. At the same time, Allison decides to start an account at the same bank. She deposits $1 to start the account and then adds $20 to the account each month after that. Complete this table that shows how much money Allison has in her account after x months. Then, graph the data.

Time (in months) Amount (in dollars) 0 1 1 2 3 4

Is the relationship linear or exponential? How do you know?

3. Compare Johnny’s and Allison’s bank accounts. Which account has an additive pattern of growth and which one has a multiplicative pattern of growth?

4. How long will it take for Johnny to have more money in his account than Allison? Why do you think this happens?




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1. Sketch a regular parallelogram with twice the perimeter of the one shown.

2. In question 1, what is the ratio of the area of the larger parallelogram to the smaller parallelogram?

Practic


cepts

3. Solve this system of two linear equations using the substitution method.

2x + y = 8

y = x – 10

4. Solve this system of two linear equations using the linear combination method.

2x – 5y = 18

y – x = -‐10

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5. Describe any patterns you see in this sequence of ordered pairs: (1,1), (2,4), (3,9), (4,16)

6. If you graph the four ordered pairs in question 5, will the graph be linear? Justify your answer.



Lesson 19.6 Exponential decay

19.6 OPENER Five friends are having lunch together at a restaurant. As they are leaving the restaurant they notice a candy dish by the door. Each friend takes exactly half of the candy that is in the bowl as he or she walks by.

1. Fill out the table to show how much candy each friend will get if there were originally 32 pieces of candy in the dish. The first entry has been filled in for you.

Friend Pieces of candy

1 16

2

3

4

5

2. Graph the data from the table.

3. How is this graph similar to other graphs that you have seen in previous lessons in this topic? How is it different?

19.6 CORE ACTIVITY 1. Write a definition for the term half-‐life.

2. Complete the table below. What is happening to the amount of chromium-‐51 during each month?

Months Amount of chromium-‐51

0 1000 grams

1 500 grams

2 250 grams

3

4

3. Do you think the pattern of chromium-‐51 is an example of exponential growth or exponential decay? Justify your answer.

4. What is a common multiplier and what is the common multiplier in these data?

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5. Extend the table to 8 months.

Months Amount of chromium-‐51

4

5

6

7

8

6. Sketch a graph of the data from the table relating the amount of chromium-‐51 to the number of months.

Amou

nt of chrom

ium-‐51 (grams)

Months

7. What will happen to the amount of chromium-‐51 as the number of months gets very large?

8. Carbon-‐14, the isotope scientists use to perform carbon dating, has a half-‐life of about 6,000 years. Suppose an object (such as a fossil) contains 800 grams of carbon-‐14 today. How many grams of carbon-‐14 will remain in the fossil 30,000 years from now? (You can use the table to help you determine the answer!)

Years from now (in thousands)

Amount of carbon-‐14 (in grams)

0

6

12

18

24

30

-2 -1 0 1 2 3 4 5 6 7 8

-200

200

400

600

800

1000



9. Use the table you created in question 8 to make a graph of the relationship between the amount of carbon-‐14 and time.

Amou

nt of carbo

n-‐14

(grams)

Years from now (in thousands)

10. Will there ever be zero grams of carbon-‐14 left? How do you know?

-6 0 6 12 18 24 30

-200

-100

100

200

300

400

500

600

700

800



19.6 REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions.

Problems we did well on: Skills and/or concepts that are addressed in these problems:

Problems we did not do well on: Skills and/or concepts that are addressed in these problems:

Addressing areas of incomplete understanding

Use this page and notebook paper to take notes and re-‐work particular online assessment problems that your class identifies.

Problem #_____ Work for problem:





HOMEWORK 19.6

Before you leave class, make sure you have found the activities in your Student Activity Book you need to review and re-‐work to prepare for the mid-‐unit assessment. Then record any notes or additional instructions based on whole-‐class discussion of the homework assignment:

Next class period, you will take a mid-‐unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts by reviewing related course materials.

Important skills and ideas you have learned so far in the unit Exponential relationships:

• Simplify expressions involving exponents

• Understand and use the laws of exponents

• Recognize the difference between linear growth and exponential growth

• Represent exponential growth using tables, graphs, and algebraic rules

Homework Assignment

Part I: Study for the mid-‐unit assessment by reviewing the key ideas listed above.

Part II: Complete the online More practice in the topic Exponents and exponential models. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study.

Part III: Complete Staying Sharp 19.6.




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1. Sketch a rectangular prism with dimensions that are three times the one shown.

2. In question 1, what is the ratio of the volume of the larger rectangular prism to the volume of the smaller rectangular prism?

Practic


cepts

3. Solve this system of two linear equations using substitution.

y = 3x – 5

3y – x = 1

4. Solve this system of two linear equations using a method of your choice.

4x + y = 7

2x + y = 1

Was your method a good choice? Explain why or why not.

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5. This table represents a linear function, but is only partly filled in. Find the rate of change and complete the table.

x y

1 5

2

3 1

4

5

Rate of change =

6. Here is a table of exponential expressions. Complete the table.

x 12

⎛

⎝⎜

⎞

⎠⎟

x Value of

12

⎛

⎝⎜

⎞

⎠⎟

x

0

1

2

3

4

Does this table show a constant rate of change? Justify your answer.



Lesson 19.7* More rational exponents

19.7 OPENER 1. Write each of the following fractions as a product of a unit fraction (a positive fraction with a 1 in the numerator) and

an integer. An example has been done for you.

Ex: 56= 16⋅5

a. 34

b. 23

c. 57

d. 35

e. 43

f. −23

2. Write each of the following expressions using a single exponent.

a. (52)4 b. (73)−5 c. (x 9)0 d. (x −6)−2

19.7 CORE ACTIVITY

1. Evaluate the expression 823 . For each step in the process, provide a justification (for example, the property used).

823

Justification:

= 813⋅2

= (813 )2

= 83⎛⎝⎜

⎞⎠⎟2

= (2)2

= 4

Use the graph of y = 8x to answer the following questions.

2. You have shown algebraically that 823 = 4 . Does the graph

support this? Explain.

3. Use the graph to estimate the value of each of the following expressions:

a. 843 c. 8

−13

b. 8−23 d. 8

13

y = 8x



4. Find the values of the following expressions algebraically.

a. 843 b. 8

−23

5. Are the values you found algebraically the same as the values you estimated from the graph? Why or why not?

6. Between the values you found using the graph and the values you computed algebraically, which are most accurate? Explain.

19.7 CONSOLIDATION ACTIVITY 1. For each expression, use the graph to estimate the solution.

a. 974

b. 912

c. 934

d. 932

e. 9−12

f. 954

g. 9−14

2. Explain how you used the graph to estimate the value of each expression.

y = 9x

-2 -1 0 1 2

-10

10

20

30

40

50

60

70

80



3. Find the values of the following expressions algebraically.

a. 912 b. 9

32 c. 9

−12

4. How do your answers for question 3 compare to the estimates you made in question 1? Explain.

5. Bronte used his calculator to estimate the values of several expressions from question 1. Two of his answers are correct and two are not. Identify which two are correct and describe how the graph from question 1 supports his estimate. For the two estimates that are not correct, tell Bronte how you can tell by looking at his graph that his answer is incorrect.

Bronte’s estimate Correct or Incorrect

Use the graph to support your decision about whether Bronte is correct or incorrect

a. 974

Correct

Incorrect

b. 934

Correct

Incorrect

c. 954

Correct

Incorrect

d. 9−14

Correct

Incorrect




1. Jude computed the values for two exponential expressions. His work is shown below. Did Jude do the computations correctly? If so, justify each step he took using the properties of exponents and the properties of real numbers. If not, explain what he did wrong and what he should do to correct his work.

a. 223 = (22)

13

= 413

= 143

= 164

b. 27

43 = (27

13 )4

= 34

= 81

2. For each expression below, find the value using algebra.

a. 3245

b. 32−15

c. 3225

d. 320.6

3. On the right is a graph of the function y = 32x. Does the graph support the values you computed in the question above? Explain how you could use the graph to check your solutions.

y = 32x

-1 0 1

-8

8

16

24

32



STAYING SHARP 19.7

Review


1. What is the volume of a cube with a side length of 2 inches?

2. If you double the side length of the cube in question 1, what would be the ratio of the volume of the larger cube to the volume of the smaller cube?

Practic


cepts

3. Solve the system of equations using any method.

y = 2x + 9 2x + y = 1

How did you choose what method to use to solve the system of equations?

4. Solve the system of equations using any method.

2x – y = 2 x + y = 7

How did you choose what method to use to solve the system of equations?

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5. Describe any patterns you see in this sequence of ordered pairs: (1,3), (2,9), (3,27), (4,81)

6. If you graph the four ordered pairs in question 5, will the graph be linear? Justify your answer.

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