LESSON 7: Properties of Exponents [OBJECTIVEntnmath.kemsmath.com/Level H Lesson Notes/Grade 8-...

Mathematics Success – Grade 8 T161

LESSON 7: Properties of Exponents

[OBJECTIVE]The student will know and apply the properties of integer exponents to generate equivalent numerical expressions.

[PREREQUISITE SKILLS]Order of operations with whole number exponents

[MATERIALS]Student pages S71 – S89Pull a Power Cards Pages 1 and 2 (1 copy of each per student pair) T187 – T188Pull a Power Number Tiles (1 copy per student pair) T189CalculatorScissors

[ESSENTIAL QUESTIONS]1. Explain how to simplify an expression using the product of powers.2. How can you simplify an expression using the quotient of powers? Justify your

thinking.3. Explain the process of simplifying an expression by raising a power to a power.

[WORDS FOR WORD WALL]base, exponent, laws of exponents

[GROUPING]Cooperative Pairs (CP), Whole Group (WG), Individual (I)*For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson.

[LEVELS OF TEACHER SUPPORT]Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer

[WARM-UP] (IP, WG) S71 (ANSWERS ON T175.)Have students turn to S71 in their books to begin the Warm-Up. Students will use knowledge of order of operations with exponents to simplify expressions. Monitor students to see if any of them need help during the Warm-Up. Have students complete the warm-up and then review the solutions as a whole group. {Graphic Organizer}

[HOMEWORK]Take time to go over the homework from the previous night.

[LESSON] [1 – 2 Days (1 day – 80 minutes) – M, GP, WG, CP, IP]

Mathematics Success – Grade 8T162

MODELING

Introduction to Exponents

Step 1: Direct students’ attention to the top of S72. • Have students look at the top left box. • Partner A, explain the numerical expression that is written in the box.

(Possible answers: 23, a number raised to a power, exponents) • Partner B, what do we call the small “3” that is raised to the right of

the “2”? (an exponent) • Partner B, what do we call the “2” when it has an exponent? (base) • Partner A, explain what the expression means. (The base multiplied

three times: two times two times two) • Partner B, what is the value of 23? (8)

Step 2: Direct students’ attention to the graphic organizer on S73. • Partner A, identify the exponential expression in Row 1. (24) • Partner B, explain the meaning of this expression. (2•2•2•2 = 16) • Partner A, identify the second exponential expression in Row 2. (23) • Partner B, explain the meaning of this expression. (2•2•2 = 8)

Step 3: Have student pairs discuss what they notice about the changes in the expression from Row 1 to Row 2 looking specifically at the change in the exponent and base.

• Partner A, as you look at the two expressions, 24 and 23, identify and describe the bases. (The base is a 2 and stays the same.) Record.

• Partner B, as you look at the two expressions, 24 and 23, identify and describe the exponents. (The exponent in the first expression is a 4 and the exponent in the second expression is a 3. The exponent is 1 less in the second row.) Record.

SOLVE Problem (WG, CP, IP) S73 (Answers on T177.)

Have students turn to S73 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to apply the properties of integer exponents to generate equivalent numerical expressions. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer}

Introduction to Exponents (M, GP, CP, WG, IP) S72, S73, S74, S75 (Answers on T176,T177, T178, T179.)

M, GP, CP, WG: Students will examine and explore patterns with bases and exponents. Be sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer}




• Have student pairs look at the value of the expression in Row 1 and compare it to the value in Row 2. Explain the relationship between the two values. [Value of the expression in Row 1 (16) was divided by 2 or is twice the value of the expression in Row 2(8).] Record.

Step 4: Have student pairs identify the value of the expression in Row 3. (4) Discuss what they notice about the changes in the expression from Row 2 to Row 3 looking specifically at the change in the exponent and base.

• Partner A, as you look at the two expressions, 23 and 22, identify and describe the bases. (The base is a 2 and stays the same.) Record.

• Partner B, as you look at the two expressions, 23 and 22, identify and describe the exponents. (The exponent in Row 2 is a 3 and the exponent in Row 3 is a 2. The exponent is 1 less in the third row.) Record.

• Have student pairs look at the value of the expression in Row 2 and compare it to the value in Row 3. Explain the relationship between the two values. [Value of the expression in Row 2 (8) was divided by 2 or is twice the value of the expression in Row 3 (4).] Record.

Step 5: Have student pairs discuss what the value of the expression in Row 4 will be if this pattern continues. What is the value of the expression in Row 4? (2) Justify your thinking. (The base stays the same, but the exponent is one less than in Row 3. This means that the value of the expression is 2 which is half of the value of the expression in Row 3.)

Step 6: Complete the last three expressions in the table following the questioning in Step 3.

Step 7: Have student pairs discuss the conclusion at the bottom of the chart and be prepared to justify their answers.

• Partner A, what is your conclusion after completing this table? (When working with a common base, as the exponent decreases by 1, the value of the expression is divided by the value of the base.) Record.

• Partner B, justify your thinking and give an example. (Sample answer is given, but the example may vary. )

Step 8: Direct students’ attention to page S74. • Guide students through the chart using a base of 3 using the questions

from Step 3. • Focus on the conclusion and justification of the thinking to support

students in establishing a pattern for the expressions.



MODELING

Product of Powers

Step 1: Direct students’ attention to the graphic organizer on S76. Have students look at the expression in Row 1.

• Partner A, what operation is represented in the expressions that are listed in the chart in Column 1? (multiplication)

• Partner B, look at the first base and exponent in the Expression column and explain how many twos are multiplied in the expression 23? (3 twos, or 2 • 2 • 2) Write out the multiplication expression in the “Expand” column.

• Partner A, look at the second base and exponent in the Expression column and explain how many twos are multiplied with 22? (2 twos, or 2 • 2) Record in the “Expand” column.

Step 3: Partner B, if we multiply all of the twos together, what is the total number of twos multiplied? (3 + 2 = 5) Record.

• Partner A, using the exponent expressions from the first column, how can we write out the product? (23 • 22 = 25) Record. Explain your thinking. (When we wrote out the expansion in Column 2 we multiplied 5 twos which can be written as 25.)

• Partner B, what is the value of 25 as a numeral? (32) Record.

Step 4: Have student pairs look at the second expression. • Partner B, explain how many twos are multiplied in the expression 24.

(4 twos, or 2 • 2 • 2 • 2) Write out the multiplication expression in the “Expand” column.

IP, CP, WG: Have students select a value from 4 – 8 and complete the chart on S75. Students may refer to the charts on S73 or S74 to support them as they work through the chart. Review student responses focusing on the conclusion and justification as that will be standard no matter what value students use to create the chart. {Verbal Description, Graphic Organizer}

Product of Powers (M, GP, CP, WG, IP) S76 (Answers on T180.)

M, GP, CP, WG: Students will be building on the foundation of exponential expressions from S73 – S75 to explore the product of expressions with exponents. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer}

*Teacher note: Throughout the lesson, students will be using the graphic organizer on S72 to organize the concepts relating to laws of exponents.



• Partner A, look at the second base and exponent in the Expression column and explain how many twos are multiplied with 23. (3 twos, or 2 • 2 • 2) Record in the “Expand” column.

Step 5: Partner B, if we multiply all of the twos together, what is the total number of twos multiplied? (4 + 3 = 7) Record.

• Partner A, using the exponent expressions from the first column, how can we write out the product? (24 • 23 = 27) Record. Explain your thinking. (When we wrote out the expansion in Column 2 we multiplied 7 twos which can be written as 27.)

• Partner B, what is the value of 27 as a numeral? (128) Record.

Step 6: Have student pairs complete the last row of the graphic organizer and review the answers as a whole group.

• Partner A, what do you notice about the base of the factors that are being multiplied? (They are both two, which means they are the same.) Record.

• Partner B, what happened with the bases in the final product? (The base for the final product was 2, which means it remained the same.) Record.

• Partner A, what happened with the exponents in the final product? (The exponents from the factors were added together.) Record.

IP, CP, WG: Have students complete the graphic organizer below Question 3 using the base of 3. Review the answers from the graphic organizer as a whole group and then answer Questions 4 – 7. {Verbal Description, Graphic Organizer}

Graphic Organizer – Product of Powers (M, GP) S72 (Answers on T176.)

• Direct students’ attention to page S72.• Partner A, when we are multiplying two exponential expressions and the bases

are the same, what did we do with the exponent. (added them)• Partner B, how could we write that? (23+2) Record.• Partner A, what is the product of 23 • 22? (25) Record. • Partner B, what did we do to the exponents when we found the product of

power? (Add the exponents.) Record.

Multiplying Expressions with Different Bases (M, GP, CP, WG, IP) S77(Answers on T181.)

M, GP, CP, WG: Students will explore multiplying expressions with different bases and exponents. Be sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer}



MODELING

Multiplying Expressions with Different Bases

Step 1: Direct students’ attention to the top of S77. • Have students discuss the first expression in the chart on S77. Make a

prediction about what will happen when we multiply expressions with different bases.

• Partner A, what is 23 expanded? (2 • 2 • 2) Record. • Partner B, what is 33 expanded? (3 • 3 • 3) Record. • Partner A, when these numbers are multiplied, what is the product as

a numeral? (8 • 27 = 216) Record. • Partner B, do we have the same bases? (No)

Step 2: Have student pairs discuss Question 2: What if we were to multiply the bases and then add the exponents?

• Let’s explore what will happen if we multiply the bases and add the exponents.

• Partner A, if we multiply the bases (2 • 3 = 6) and add the exponents (3 + 3 = 6), what is that value? (66) Record.

• Partner B, when we expand the expression 66, what is the value? (46,656) Record.

Step 3: Partner A, is the expression in Row 1 equivalent to the expression in Row 2? (No) Record. Explain your thinking. (In Row 1, we have to follow the order of operations, so we must evaluate any exponents before multiplying.) Record.

• Have student pairs discuss what conclusion can be drawn about simplifying exponential expressions. (When we simplify values, we must have the same base in order to add the exponent.) Record.

Step 4: Have student pairs look at Question 7 in the chart. • Partner A, can we simplify this expression using the base of 4? Justify

your answer. (Yes, we can simplify the expression because the bases are the same.)

• Partner B, explain how to simplify the expression. [The base will be 4 and we will add the exponents (8 + 5) so our simplified expression is 413.] (48+5 = 413) Record.

Step 5: Have student pairs look at Question 8 in the chart. • Partner A, can we simplify this expression using the bases of 6 And

7? Justify your answer. (No, because we proved in the example that you cannot multiply the bases and add the exponents to simplify the expression.)

• Have students write the word simplified in the box.



IP, CP, WG: Have students complete Questions 9 – 12 in the graphic organizer. They should simplify the expressions if the bases are the same and if not, they should write the word “simplified”. After students have completed the graphic organizer, review the answers as a whole group. Students should be prepared to explain and defend their answers. {Verbal Description, Graphic Organizer}

Quotient of Powers with the Same Base and Exponent(M, GP, IP, CP, WG) S78 (Answers on T182.)

M, GP, CP, WG: Have students turn to S78 in their books. Use the following activity to help students explore the quotient of powers when the base and exponents are the same. Make sure students know their designation as Partner A or Partner B, {Verbal Description, Graphic Organizer}

MODELING

Quotient of Powers with the Same Base and Exponent

Step 1: Direct students’ attention to the top of S78. Have partners discuss Questions 1 – 3 and then review the answers as a group.

• Partner A, what is the opposite of multiplication? (division) Record. • Partner B, what is the opposite of addition? (subtraction) Record. • Partner A, when we multiplied expressions with common bases, what

operation did we use with the exponents? (addition) Record. • Discuss with your partner your prediction about what operation will be

used with the exponents when we divide. (Answers may vary.)

Step 2: Now, let’s see if your prediction is correct. • Let’s expand the expression in Column 2.

• Partner A, what is the fraction value? ( 2 • 2 • 2 • 22 • 2 • 2 • 2 = 1616 ) Record.

• Partner B, explain how we can simplify the fraction in Column 3. (We can find the greatest common factor and simplify by that value.)

• Partner A, what is the value? (1616 ÷ 1616 = 11 = 1) Record.

Step 3: In Column 4, explain how we can simplify using the exponential expressions.

• Partner B, explain how to simplify the expression. (Divide the numerator and denominator by the greatest common factor.)

• Partner A, what is the quotient? (24

24 ÷ 24

24 = 1) Record.

• Partner B, what was the quotient for each of the expressions? (The numerators and denominators simplify to 1 in all the fractions.) Record.



• Partner A, if both the numerator and denominator are 1, what is the fraction equal to? (1) Record.

Step 4: Have students complete the expansion and simplifying of the exponent expressions in Rows 2 and 3 and review their findings as a whole group.

Step 5: Have student pairs discuss Questions 9 and 10. • Partner B, what do you notice about the final exponent after subtraction

in the last column? (All of the exponents are 0.) Record. • What can we conclude about any number raised to a power of 0 based on

our chart? (Any number raised to the power of 0 is equal to 1.) Record.

Graphic Organizer – Zero Power (M, GP) S72 (Answers on T176.)

• Direct students’ attention to page S72. Have students look at the box entitled “Zero Power”.

• Partner A, when we are dividing two exponential expressions and the bases are the same, what did we do with the exponents? (subtracted them)

• Partner B, when the exponents are the same, what is the difference of the two exponents? (0)

• Partner A, what is the quotient of 24

24 ? (24-4 = 20 = 1) Record.

• Partner B, what is the value of any base with an exponent of 0? (1) Record.

Quotient of Powers (M, GP, IP, CP, WG) S79, S80 (Answers on T183, T184.)

M, GP, CP, WG: Have students turn to S79 in their books. Use the following activity to help students explore the quotient of powers. Make sure students know their designation as Partner A or Partner B, {Verbal Description, Graphic Organizer}

MODELING

Quotient of Powers

Step 1: Direct students’ attention to the top of S79. Have partners discuss Questions 1 – 4 and then review the answers as a group.

• Partner A, when we multiplied two expressions with the same base, what happened to the base? (It stayed the same.) Record.

• Partner B, when we divided the expressions with the same base, what happened to the base? (It stayed the same.) Record.

• Partner A, when we multiplied two expressions with the same base, what happened to the exponents? (We added the exponents.) Record.



Step 2: Because addition and subtraction are opposite operations, and if we add exponents when we multiply, what operation do you think we will use when dividing expressions with exponents? (subtraction) Record.

Step 3: Now, let’s see if your prediction is correct. • Let’s expand the expression in Column 2. • Partner A, what is the value when written as a fraction? (164 ) Record.

• Partner B, explain how we can simplify the fraction in Column 3. (We divide the numerator and denominator by the greatest common factor.) Record.

• Partner A, what is the value? (164 ÷ 44 =

41) Record.

Step 4: Have students turn to page S80. • In Column 4, what operation is shown in the expression? (division)

Record. • Partner B, what is the greatest common factor? (22) • Simplify the fraction by dividing by the greatest common factor.

( 24

22 ÷ 22

22 = 22

1 = 22) Record.

• Have students complete the expressions in Rows 2 and 3 in the chart and review the answers as a whole group.

• Partner A, explain what happened with the bases in the final quotient? (The base for the final quotient was 2, which means it remained the same.) Record.

• If we added the exponents when we multiplied expressions with the same base, what operation would you predict we would use when dividing integers with the same base? (subtraction) Record.

• Based on our discovery from evaluating exponential expressions with multiplication and addition, what is your conclusion about the bases? (The bases must be the same to add or subtract the exponents.) Record.

IP, CP, WG: Have students complete Questions 12 - 17 in the graphic organizer. They will simplify the expressions using division. After students have completed the problems with partners, review the answers as a whole group. Students should be prepared to explain and defend their answers. {Verbal Description, Graphic Organizer}



Graphic Organizer – Quotient of Powers (M, GP) S72 (Answers on T178.)

• Direct students’ attention to page S72. Have students look at the box entitled Quotient of Powers.

• Partner A, when we are dividing two exponential expressions and the bases are the same, what did we do with the exponents. (subtracted them) Record.

• Partner B, what is the base? (2) Record.

• Partner A, what is the quotient of 24

22 ? (24-2 = 22 = 4) Record.

Quotient of Powers with Negative Exponents (M, GP, CP, IP, WG) S81 (Answers on T185.)

M, GP, CP, WG: Students will be working with quotients of powers where the exponent in the numerator is less than the exponent in the denominator. They will be continuing to apply what they have learned about Quotients of Powers but will have solutions with exponents that are negative. Be sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer}

MODELING

Quotient of Powers with Negative Exponents

Step 1: Direct students’ attention to the graphic organizer on S81. • Partner B, identify the operation used in the table. (division) • Have students expand the numerator and denominator for the first

row. What is the value? ( 2 • 22 • 2 • 2 • 2 =

416 ) Record.

Step 2: Partner A, in Column 3, explain how we can simplify the fraction. (Divide the numerator and the denominator by the greatest common factor of 4.) Record.

• Partner B, what is the simplified value? (14) Record.

• Have student pairs simplify the fraction in Column 4 using the original expression. What is the simplified expression? ( 22

24 ÷ 22

22 = 122 =

14)

• What do you notice about the quotient in Column 4? (It is the same as Column 3: 1

4) Record.

Step 3: What operation did we use with the exponents with division of expressions when the base is the same? (subtraction) Record.

• Have student pairs discuss how to write the expression in Column 5 with the base of 2 and show the subtraction of the exponents. (22-4 = 2-2) Record.



Graphic Organizer – Quotient of Powers with Negative Exponents (M, GP) S72 (Answers on T176.)

• Direct students’ attention to page S72. Have students look at the box entitled Negative Power

• Partner A, when we are dividing two exponential expressions and the bases are the same, what did we do with the exponents. (subtracted them)

• Partner B, what is the base? (2)• Partner A, what is the quotient of 2

2

24 ? (22-4 = 2-2 = 122 =

14) Record.

Raising a Power to a Power (M, GP, CP, IP, WG) S82 (Answers on T186.)

M, GP, CP, WG: Students will explore raising a power to a power with the properties of exponents. Be sure that students understand their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer}

MODELING

Raising a Power to a Power

Step 1: Direct students’ attention S82. • Partner A, explain the meaning of the exponent outside of the

parentheses in the first example. (The two tells us that we need to multiply two of the values inside the parentheses.) Record.

• Partner B, write the expanded form in the second column. [(53 • 53) which is equal to (5 • 5 • 5) • (5 • 5 • 5)] Record.

• Partner A, what are the total number of fives we are multiplying? (6) Record.

• Partner A, what do you notice about the final exponent after subtraction in the last column? (The exponent is negative.) Record.

Step 4: Look back at Column 4. When an exponent is negative, where is the location of the factors that did not cancel? (They are in the denominator.) Record.

• Partner B, what can we conclude about integers with negative exponents? (The exponent will be positive when placed in the denominator.) Record.

Step 5: Have student pairs complete the division of the expressions in Rows 2 and 3 in the graphic organizer and then review the answers as a whole group.



Step 2: When we multiplied expressions with the same base in previous examples, what happened to the base? (It stayed the same.) Record.

• Have students discuss what they predict will happen to the base when we raise it to a power. (It will stay the same.) Record.

• Explain your thinking. (We are using only one value as the base.) Record.

Step 3: How can we write the expression using the exponent of 6? (56) Record. • Partner B, what pattern do you notice with the exponents in the last

column? (They are multiplied together to get the final exponent.) Record.

• Partner A, when we raise a power to another power, we (multiply) the exponents and the (base) remains the same. Record.

Step 4: Have students complete the expressions in Rows 2 and 3 and review the answers as a whole group.

IP, CP, WG: Have students complete Questions 7–10 at the bottom of S82. Students will practice problems with negative powers and a power raised to a power. When students have completed the problems, review the solutions as a whole group. {Verbal Description, Graphic Organizer}

Graphic Organizer – Power Raised to a Power (M, GP) S72 (Answers on T178.)

• Direct students’ attention to page S72. Have students look at the box entitled Power Raised to a Power.

• Partner A, when we are raising a power to a power, what happens to the base? (It is the same.)

• Partner B, what is the base? (5) Record.• Partner A, what is the value of (53)2? (53 • 53 which equals 56) Record.• Partner B, what operation do we apply to the exponents? (multiplication) Record.

Pull a Power Activity (M, GP, CP, IP, WG) S83, S84, S85, S86 (Materials on T187, T188, T189, T190.)

M, GP, IP, CP, WG: Students will apply their knowledge of laws of exponents by completing this activity. Students will be given cards to practice simplifying expressions. Each student’s cards will be different based on the numbers they choose, so an answer key is not provided. Allow students to use a calculator to verify solutions as they go along. {Verbal Description, Graphic Organizer}



MODELING

Pull a Power Activity

Step 1: Direct students’ attention S83. • Explain to students that they will be completing an activity that will

allow them to use all of the rules that they learned today and recorded in their graphic organizer. Explain that these rules are referred to as laws of exponents or properties of exponents.

• Begin by passing out a set of Pull a Power Cards, both pages 1 and 2 and a page of the Pull a Power Number Tiles. Teacher may precut these cards or you may allow students to do it. Each pair of students needs both pages of cards and the number tiles.

• The game begins with students placing all of their number tiles upside down in a pile, as well as two piles for the Pull a Power Cards. (One pile for the cards labeled “A” and one pile for the cards labeled “B.”) These will also be faced upside down.

• Play begins by Partner A pulling a card from his or her pile. On the recording sheet, Partner A will record the problem from the card in the main section of one of the four boxes on the page. Next, Partner A will pull three number tiles from the pile and write them in the three small boxes provided inside that same box in the top left corner.

• Next, Partner B, will choose where Partner A’s number tiles will be placed within the problem so that there will now be three exponents as part of the problem. Partner B, will write the values from the three number tiles directly in Partner A’s problem.

• Partner A will simplify the expression using the properties of exponents and place a solution in the box in the bottom right corner.

• Play continues by roles reversing and Partner B choosing three number tiles and a card from the “B” pile.

• Students will alternate roles until each student has completed four problems, filling up their recording sheet.

• After students have completed the activity, ask for requests of students to share some of their problems with the numbers they chose and have the class help check the work to be sure that each student’s work is correct.

• An example of the Recording Sheet is shown below

Simplified Expression

Three Number Tiles

Expression/Problem

LESSON 7: Properties of Exponents [OBJECTIVEntnmath.kemsmath.com/Level H Lesson Notes/Grade 8-...

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