Mathematics Success – Grade 8 T152 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 – S89 Pull a Power Cards Pages 1 and 2 (1 copy of each per student pair) T178 – T179 Pull a Power Number Tiles (1 copy per student pair) T180 Calculator Scissors [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, raising a power to a power, product of powers, zero power, quotient of powers [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 T166.) 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]
Transcript
Mathematics Success – Grade 8T152
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) T178 – T179Pull a Power Number Tiles (1 copy per student pair) T180CalculatorScissors
[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, raising a power to a power, product of powers, zero power, quotient of powers
[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)
[WARM-UP] (IP, WG) S71 (ANSWERS ON T166.)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 8 T153
MODELING
Introduction to Exponents
Step 1: Direct students’ attention to the top of S72. • Havestudentslookatthetopleftbox. • PartnerA,explainthenumericalexpressionthatiswritteninthebox.
(Possible answers: 23, a number raised to a power, exponents) • PartnerB,whatdowecallthesmall“3”thatisraisedtotherightof
three times: two times two times two) • PartnerB,whatisthevalueof23? (8)
Step 2: Direct students’ attention to the graphic organizer on S73. • PartnerA,identifytheexponentialexpressioninRow1.(24) • PartnerB,explain themeaningof thisexpression. (2•2•2•2=16)
Step 3: Have student pairs discuss what they notice about the changes in the expressionfromRow1toRow2lookingspecificallyatthechangeintheexponent and base.
• PartnerA,asyoulookatthetwoexpressions,24 and 23, identify and describe the bases. (The base is a 2 and stays the same.) Record.
• PartnerB,asyoulookatthetwoexpressions,24 and 23, identify and describetheexponents.Theexponentinthefirstexpressionisa4and the exponent in the second expression is a 3. (The exponent is 1 less in the second row.) Record.
SOLVE Problem (WG, GP) S73 (Answers on T168.)
HavestudentsturntoS73intheirbooks.ThefirstproblemisaSOLVEproblem.Youareonly 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 T167, T168, T169, T170.)
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}
LESSON 7: Properties of Exponents
Mathematics Success – Grade 8T154
• HavestudentpairslookatthevalueoftheexpressioninRow1andcompare it to the value in Row 2. Explain the relationship between the twovalues.[ValueoftheexpressioninRow1(16)wasdividedby2or 23(8) is half of 24 (16).]Record.
Step 4: Have student pairs identify the value of the expression in Row 3. (4) Record. Discuss what they notice about the changes in the expression from Row
2toRow3,lookingspecificallyatthechangeintheexponentandbase. • PartnerA,asyoulookatthetwoexpressions,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.
• HavestudentpairslookatthevalueoftheexpressioninRow2andcompare 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 22(4) is half of 23 (8).]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 22(4).) Record.
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.
• PartnerA,whatisyourconclusionaftercompletingthistable?(Whenworking with a common base, as the exponent decreases by 1, the value of the expression is divided by the value of the base.) Record.
• PartnerB,justifyyourthinkingandgiveanexample.(Sampleansweris given, but the example may vary. )
Step 8: Direct students’ attention to page S74. • Guidestudentsthroughthechartusingabaseof3usingthequestions
from Step 3. • Focusontheconclusionandjustificationof thethinkingtosupport
students in establishing a pattern for the expressions.
LESSON 7: Properties of Exponents
Mathematics Success – Grade 8 T155
LESSON 7: Properties of Exponents
MODELING
Product of Powers
Step 1:Directstudents’attentiontothegraphicorganizeronS76.Havestudentslook at the expression in Row 1.
• PartnerA,whatoperationisrepresentedintheexpressionsthatarelisted in the chart in Column 1? (multiplication)
• PartnerB,lookatthefirstbaseandexponentintheExpressioncolumnand explain how many twos are multiplied in the expression 23? (3 twos, or 2 • 2 • 2)Write out themultiplication expression in the“Expand”column.
• PartnerA, lookatthesecondbaseandexponent intheExpressioncolumn and explain how many twos are multiplied with 22? (2 twos, or2•2)Recordinthe“Expand”column.
Step 2: Partner B, if we multiply all of the twos together, what is the total number oftwosmultiplied?(3+2=5)Record.
• PartnerA,usingtheexponentexpressionsfromthefirstcolumn,howcan we write out the product? (23 •22=25) Record. Explain your thinking. (When we wrote out the expansion in Column 2 we multiplied 5twoswhichcanbewrittenas25.)
• PartnerB,whatisthevalueof25 as a numeral? (32) Record.
Step 3: Have student pairs look at the second expression. • PartnerB,explainhowmanytwosaremultipliedintheexpression24.
IP, CP, WG: Have students select a value from 4 – 8 and complete the chartonS75.StudentsmayrefertothechartsonS73or S74 to support them as they work through the chart. Review student responses focusing on the conclusion andjustificationasthatwillbestandardnomatterwhatvalue students use to create the chart. {Verbal Description, Graphic Organizer}
Product of Powers (M, GP, CP, WG, IP) S76 (Answers on T171.)
M, GP, CP, WG: Students will be building on the foundation of exponential expressionsfromS73–S75toexploretheproductofexpressions 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.
Mathematics Success – Grade 8T156
LESSON 7: Properties of Exponents
• PartnerA, lookatthesecondbaseandexponent intheExpressioncolumn and explain how many twos are multiplied with 23. (3 twos, or 2•2•2)Recordinthe“Expand”column.
Step 4: Partner B, if we multiply all of the twos together, what is the total number oftwosmultiplied?(4+3=7)Record.
• PartnerA,usingtheexponentexpressionsfromthefirstcolumn,howcan 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.)
• PartnerB,whatisthevalueof27 as a numeral? (128) Record.
Step 5: 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 thatare being multiplied? (They are both two, which means they are the same.) Record.
• PartnerA,whathappenedwiththeexponents in thefinalproduct?(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 T167.)
are the same, what did we do with the exponent. (added them)• PartnerB,howcouldwewritethat?(23+2) Record.• PartnerA,whatistheproductof23•22? (25) Record. • PartnerB,whatdidwedototheexponentswhenwefoundtheproduct of
powers? (Add the exponents.) Record.
Multiplying Expressions with Different Bases (M, GP, CP, WG, IP) S77(Answers on T172.)
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}
Mathematics Success – Grade 8 T157
LESSON 7: Properties of Exponents
MODELING
Multiplying Expressions with Different Bases
Step 1: Direct students’ attention to the top of S77. • HavestudentsdiscussthefirstexpressioninthechartonS77.Makea
prediction about what will happen when we multiply expressions with different bases.
• Partner B,whenwe 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 aboutsimplifying 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. • PartnerA,canwesimplifythisexpressionusingthebaseof4?Justify
youranswer.(Yes,wecansimplifytheexpressionbecausethebasesare the same.)
Step 5: Have student pairs look at Question 8 in the chart. • PartnerA,canwesimplifythisexpressionusingthebasesof6and
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.)
• Havestudentswritethewordsimplifiedinthebox.
Mathematics Success – Grade 8T158
LESSON 7: Properties of Exponents
IP, CP, WG: Have students complete Questions 9 – 12 in the graphic organizer. They should simplify the expressions if the bases are thesameandifnot,theyshouldwritetheword“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 T173.)
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.
Quotient of Powers (M, GP, IP, CP, WG) S79, S80 (Answers on T174, T175.)
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.
• PartnerA,whenwemultipliedtwoexpressionswiththesamebase,what happened to the base? (It stayed the same.) Record.
• PartnerB,whenwedividedtheexpressionswiththesamebase,whathappened to the base? (It stayed the same.) Record.
• PartnerA,whenwemultipliedtwoexpressionswiththesamebase,what happened to the exponents? (We added the exponents.) Record.
Mathematics Success – Grade 8T160
LESSON 7: Properties of Exponents
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’sexpandtheexpressioninColumn2. • PartnerA,whatisthevaluewhenwrittenasafraction?(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.
• PartnerA,whatisthevalue?(164 ÷ 44 =
41) Record.
Step 4: Have students turn to page S80. • InColumn4,whatoperationisshownintheexpression?(division)
Record. • PartnerB,whatisthegreatestcommonfactor?(22) • Simplify the fraction by dividing by the greatest common factor.
( 24
22 ÷ 22
22 =22
1 =22) Record.
• HavestudentscompletetheexpressionsinRows2and3inthechartand review the answers as a whole group.
• Ifweaddedtheexponentswhenwemultipliedexpressionswiththesame base, what operation would you predict we would use when dividing integers with the same base? (subtraction) Record.
• Basedonourdiscoveryfromevaluatingexponentialexpressionswithmultiplication 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}
Mathematics Success – Grade 8 T161
LESSON 7: Properties of Exponents
Graphic Organizer – Quotient of Powers (M, GP) S72 (Answers on T167.)
• Directstudents’attentiontopageS72.HavestudentslookattheboxentitledQuotient of Powers.
• PartnerA,whenwearedividingtwoexponentialexpressionsandthebasesarethe same, what did we do with the exponents. (subtracted them) Record.
• PartnerB,whatisthebase?(2)Record.
• PartnerA,whatisthequotientof24
22 ? (24-2=22=4)Record.
Quotient of Powers with Negative Exponents (M, GP, CP, IP, WG) S81 (Answers on T176.)
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. • PartnerB,identifytheoperationusedinthetable.(division) • Havestudentsexpandthenumeratoranddenominatorforthefirst
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.
Raising a Power to a Power (M, GP, CP, IP, WG) S82 (Answers on T177.)
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
parenthesesinthefirstexample.(Thetwotellsusthatweneedtomultiply two of the values inside the parentheses.) Record.
• PartnerA,whatdoyounoticeaboutthefinalexponentaftersubtractionin 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 negativeexponents? (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.
Mathematics Success – Grade 8 T163
LESSON 7: Properties of Exponents
Step 2: When we multiplied expressions with the same base in previous examples, what happened to the base? (It stayed the same.) Record.
• Havestudentsdiscusswhattheypredictwillhappentothebasewhenwe raise it to a power. (It will stay the same.) Record.
• Explain your thinking. (Weare using only one value as the base.)Record.
column? (They aremultiplied together to get the final exponent.)Record.
• PartnerA,whenweraiseapowertoanotherpower,we(multiply)theexponents 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 T167.)
• Directstudents’attentiontopageS72.HavestudentslookattheboxentitledPower Raised to a Power.
• PartnerA,whenweareraisingapowertoapower,whathappenstothebase?(It is the same.)
Pull a Power Activity (M, GP, CP, IP, WG) S83, S84, S85, S86 (Materials on T178, T179, T180, T181.)
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}
Mathematics Success – Grade 8T164
LESSON 7: Properties of Exponents
MODELING
Pull a Power Activity
Step 1: Direct students’ attention S83. • Explaintostudentsthattheywillbecompletinganactivitythatwill
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.
• BeginbypassingoutasetofPullaPowerCards,bothpages1and2 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.
• Thegamebeginswithstudentsplacingalloftheirnumbertilesupsidedown in a pile, as well as two piles for the Pull a Power Cards. (One pileforthecardslabeled“A”andonepileforthecardslabeled“B.”)These will also be faced upside down.
• PlaybeginsbyPartnerApullingacardfromhisorherpile.Ontherecording 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,PartnerB,will choosewherePartnerA’snumber tileswillbeplaced 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.
• PartnerAwillsimplifytheexpressionusingthepropertiesofexponentsand place a solution in the box in the bottom right corner.
• Afterstudentshavecompletedtheactivity,askforrequestsofstudentsto 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.
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