8th Grade Unit of Study Exponents

Transcription

1 DRAFT 8th Grde Unit of Study Exponents Grde: 8 Topic: Exponent opertions nd rules Length of Unit: 6 dys Focus of Lerning Common Core Stte Stndrds: Expressions nd Equtions 8.EE Work with rdicls nd integer exponents. 8.EE. Know nd pply the properties of integer exponents to generte equivlent numericl expressions. For exmple, EE. Use numbers expressed in the form of single digit times n integer power of 0 to estimte very lrge or very smll quntities, nd to express how mny times s much one is thn the other. For exmple, 8 estimte the popultion of the United Sttes s 0 nd the popultion 9 of the world of 7 0, nd determine tht the world popultion is more thn 0 times lrger. 8.EE. Perform opertions with numbers expressed in scientific nottion, including problems where both deciml nd scientific nottion re used. Use scientific nottion nd choose units of pproprite size for mesurements of very lrge or very smll quntities (e.g. use millimeters per yer for sefloor spreding). Interpret scientific nottion tht hs been generted by technology. Mthemticl Prctices:. Mke sense of problems nd persevere in solving them.. Reson bstrctly nd quntittively.. Construct vible rguments nd critique the resoning of others.. Model with mthemtics. 5. Use pproprite tools strtegiclly. 6. Attend to precision. 7. Look for nd mke use of structure. 8. Look for nd express regulrity in repeted resoning. Enduring Understnding(s): Students will understnd tht ) Number sense resoning genertes rules for multiplying nd dividing powers with the sme bse. ) The rules for multiplying nd dividing powers with the sme bse genertes the mening of nd rules for zero nd negtive exponents. ) Plce vlue nd bse-0 is integrl to understnding scientific nottion. ) Scientific nottion is used to represent lrge nd smll numbers. 5) Opertions with scientific nottion cn be used to solve rel world problems. Essentil Questions: These questions will guide student inquiry. ) Why is it helpful to use exponents? ) How re exponents useful to model rel world situtions? ) When is it pproprite to express numbers in scientific nottion? ) How cn we estimte relly lrge nd relly smll quntities using exponents? 5) How does understnding the exponent rules help you solve rel world problems involving scientific nottion?

2 Student Performnce Knowledge: Students will understnd/know Appliction: Students will be ble to The definition of bse, power, nd coefficient The exponent in n exponentil term expresses how mny times the bse is to be multiplied (with positive integers only) The rules for multiplying nd dividing powers with the sme bse lwys work The rule for rising n exponentil term to nother exponent lwys works The proof of x 0, when x 0 using the properties of exponents Negtive exponents cn be written s positive exponents Scientific nottion is used to represent lrge nd smll numbers The rules for opertions with exponents re used to perform opertions with numbers expressed in scientific nottion Assessments (ttched) Expnd, simplify, nd evlute expressions involving exponents, including products nd quotients rised to powers Prove the rules for opertions of exponents with the sme bse (below) by using the definition of n exponent: m n mn ) m mn b) n m n mn c) ( ) Generte nd use the rules for multiplying nd dividing powers with the sme bse Generte nd use the rules for zero exponents nd negtive exponents ) 0 m mn b) n Express lrge nd smll numbers in scientific nottion Perform opertions with numbers expressed in scientific nottion, nd choose units of pproprite size to represent given mesurements. Solve rel-world problems involving numbers expressed in scientific nottion. Formtive nd Interim Assessments: o Illustrtive Mthemtics: 8.EE Extending the Definitions of Exponents, Vrition (Lesson ) o Illustrtive Mthemtics: 8.EE Gintburgers (Lesson 6) o Illustrtive Mthemtics: 8.EE Ants versus Humns (Lesson 6) o Smrter Blnced Smple Item: MAT.08.CR.000EE.B.9.C.TB (Lesson 6) Post Assessment (Culminting Tsk): o Blood in the Humn Body

3 Lerning Experiences (Lesson Plns Attched) Dys Lesson Sequence Mterils Lesson : Definition of n Exponent Students will know: The definition of bse, power, nd coefficient The exponent in n exponentil term expresses how mny times the bse is to be multiplied (with positive integers only) Students will be ble to: Expnd, simplify, nd evlute expressions involving exponents, including products nd quotients rised to powers. Lesson : Properties for Opertions of Exponents with the Sme Bse Students will know: The rules for multiplying nd dividing exponents with the sme bse lwys work The rule for rising n exponentil term to nother exponent lwys works Students will be ble to: Prove the rules for opertions of exponents with the sme bse (below) by using the definition of n exponent: ) m n mn m mn m n mn b) c) ( ) n Lesson : Properties for Zero nd Negtive Integer Exponents Students will know: The proof of x 0, when x 0 using the properties of exponents (see lesson ). Negtive exponents cn be written s positive exponents using the rules for multiplying nd dividing exponents with the sme bse. Students will be ble to: Use the rules tht they generted in Lesson (for multiplying nd dividing exponents with the sme bse) to generte properties of zero nd negtive exponents. ) 0 m b) m Lesson : Properties - Review nd Assessment Interim Assessment: Students will: Illustrtive Propose, justify nd communicte solutions Mthemtics: Extending the Definitions of Exponents (This Interim Assessment item is ttched to the Lesson lesson pln) Lesson 5: Expressing Number in Scientific Nottion Students will know: Scientific nottion is used to represent lrge nd smll numbers Students will be ble to:

4 Express lrge nd smll numbers in scientific nottion Lesson 6: Using Scientific Nottion to Solve Rel-World Problems Students will know: Scientific nottion is used to represent lrge nd smll numbers The rules for opertions with exponents re used to perform opertions with numbers expressed in scientific nottion Students will be ble to: Express lrge nd smll numbers in scientific nottion Perform opertions with numbers expressed in scientific nottion, nd choose units of pproprite size to represent given mesurements. Solve rel-world problems involving numbers expressed in scientific nottion. Review Students will: Propose, justify nd communicte solutions Culminting Tsk Students will: Show their knowledge nd understnding of exponents. Online Illustrtive Mthemtics Resources Formtive Assessments: Illustrtive Mthemtics: 8.EE Gintburgers Illustrtive Mthemtics: 8.EE Ants versus Humns Smrter Blnced Smple Item: MAT.08.CR.000EE.B. 9.C.TB (these formtive ssessment items re ttched to the Lesson 6 lesson pln) Post Assessment Exponents Blood in the Humn Body Text Prentice Hll Mthemtics. Cliforni Algebr. Boston: Person Eduction, Inc Inside Mthemtics/MARS tsks ; Ntionl Librry of Virtul Mnipultives Shoseki, Tokyo. Mthemtics Interntionl: Grde 8. 0 (Jpnese Text) Vn de Wlle, John, nd LouAnn Lovin. Teching Student-Centered Mthemtics: Grdes 5-8. Vol.. Boston: Person, 006. Progressions for the Common Core Stte Stndrds in Mthemtics Smrter Blnced Assessment Consortium

5 Lesson : Definition of n Exponent Unit: Exponents Approx. time: dys Lesson : Definition of n Exponent A. Focus nd Coherence Students will know The definition of bse, power, nd coefficient The exponent in n exponentil term expresses how mny times the bse is to be multiplied (with positive integer exponents only) Students will be ble to Expnd, simplify, nd evlute expressions involving exponents, including products nd quotients rised to powers. Student prior knowledge: Multipliction of fctors Which mth concepts will this lesson led to? Multiplying exponents with the sme bse Dividing exponents with the sme bse Products nd quotients rised to exponents Solving rel-world problems involving exponents CCSS-M Stndrds: 8.EE. Know nd pply the properties of integer exponents to generte equivlent numericl expressions. B. Evidence of Mth Prctices Wht will students produce when they re mking sense, persevering, ttending to precision nd/or modeling, in reltion to the focus of the lesson? Students will precisely rticulte the definition of n exponent through vrious exmples; for exmple, they will sy tht mens times ; mens times times ; nd ( ) mens times times. They will sy tht negtive nd frctionl bses work the sme wy, (e.g., (/) nd (- ) ), long with bses contining vribles nd coefficients (e.g., x ) Students will precisely rticulte the definitions of the words bse nd exponent. They will sy tht the bse is the number we re multiplying nd the exponent tells us how mny times we multiply the bse by itself (when the exponent is positive integer). Students will be ble to predict how exponents my be useful to model rel world situtions. (i.e. Posting picture on online socil network sites nd the number of people who could ultimtely view tht picture if it continues to get shred) Essentil Question(s). Why is it helpful to use exponents?. How re exponents useful to model rel world situtions?. How is (three times four) different from (three to the fourth power)? Formtive Assessments ( Ticket-out-the-door Questions ) ) Write in expnded form. ) Write number in expnded form nd show wht it looks like in exponentil form. Identify which term is the exponent nd which term is the bse. ) Why cn exponents be useful in rel-life situtions? Anticipted Student Preconceptions/Misconceptions Students my confuse the bse nd the exponent, for exmple they my incorrectly multiply the exponent insted of the bse, i.e. =. Students my think tht exponents imply multipliction of the bse nd the exponent, i.e. =. Students my incorrectly put the expnded form of number into exponent form, i.e. = 8 Mterils/Resources Individul whitebords to collect student feedbck

6 C. Rigor: fluency, deep understnding, ppliction nd dul intensity Wht re the lerning experiences tht provide for rigor? Wht re the lerning experiences tht provide for evidence of the Mth Prctices? (Detiled Lesson Pln) Wrm Up Using individul whitebords How else could you write +++++? ANSWER: 5 (or 5 ) Lesson Prt I: Definition of Bse nd Exponent; Writing Expressions in Exponentil nd Expnded Form. ) Techer: Put the numbers 6 nd on the bord. Hve students predict some possible vlues for solutions given those two numbers, without ny given opertions. Students should come up with: 6+ = 8; 6x = ; 6 = ; 6/ =. Students my not know ny more. Techer sys, I hve nother opertion 6, wht do you think its vlue might be? Let students tke guesses. Hve students tke guesses until they discover 6, nd tell them tht 6 is correct. Show them tht 6 = 6 6 = 6. Do the sme with,, 0, ect., llowing students to tke guesses for the vlue of ech expression, nd letting students see the pttern of multiplying the bse by itself. ) Introduce the vocbulry bse nd exponent (these will probbly be fmilir to them from previous grde levels). Show them tht the big number is the bse nd the little number is the exponent. Pose question to clss: Wht does the bse tell us nd wht does the exponent tell us in? Direct students to think bout their nswers from # (i.e. = ). Write definition on bord for students to copy down in their notes or mth dictionry: The exponent tells us how mny times the bse is to be multiplied by itself, when the exponent is positive integer. ) Introduce the terms exponentil form nd expnded form Exponentil form is when the term hs bse nd n exponent, like expressions on the left side of the tble. Expnded form of is when the fctors re written out with multipliction, like the expressions on the right side of the tble. Exponentil Form Expnded Form Use white bords to collect student feedbck. Hve students write the expnded form of the following expressions: ) 5 b) 6 c) Hve students write the exponentil form of the following expressions: ) x b) c) d) (-) (-) (-) *Students my hve questions bout (d). Hve students use the definitions of exponent nd bse to reson bout rewriting this expression in the sme wy s the expressions with positive bses. Ask students for number to use s bse (e.g. 7 ) nd positive integer to use s n exponent (e.g 5 ). Pose question: Suppose my bse is 7 nd my exponent is 5, write the expnded form of 7 5? *Do more problems s necessry.

7 ) Hve students expnd expressions with bses tht re not positive integers: ) (-) b) (/) c) x Use white bords to collect student feedbck. Prt II: Powers Rised to Exponents As whole group, expnd nd simplify: *Refer bck to the definitions of bse nd exponent. ) ( ). The bse is nd the exponent is. This mens times itself times. Expnded form: = = 6 b) (5 ) 5 Wht is the bse? Wht is the exponent? c) Wht is the bse? Wht is the exponent? Pose question to clss: Wht do you notice? (Don t introduce rule of multiplying powers together when you hve n exponent rised to nother exponent - this will hppen in the next lesson). Hve students write the following expressions in exponentil form Use white bords to collect student feedbck. (Include bses tht re negtive numbers, frctions, nd vribles, for exmple): ) b) ( ) ( ) c) 5x 5x 5x 5x 5x 5x *Do more problems s necessry. Closure Tlk to your neighbor bout wht you lerned tody using our new vocbulry nd explin wht it mens. Hve students shre their explntions with the whole group. Give the formtive ssessment on hlf sheet to be turned in s ticket out the door: Ticket-out-the-door questions: ) Write in expnded form. ) Write number in expnded form nd show wht it looks like in exponentil form. Identify which term is the exponent nd which term is the bse. ) Why cn exponents be useful in rel-life situtions? Suggested Homework/Independent Prctice Attched worksheet

8 Nme: Lesson : Homework Worksheet Write in expnded form. 5. (-) Dte:. (½) 5. (x) 6 Write in exponentil form x x x x x x x x x 9. In the term 5, wht is the bse nd wht is the exponent? Wht does the bse in Wht does the exponent in 5 tell us? 5 tell us? 0. Wht does n exponent of men? (For exmple, 5 ). Wht is the difference between 5 nd 5?. Write number in expnded form nd show wht it looks like s n exponentil term. Identify which term is the exponent nd which term is the bse.

9 Lesson : Properties for Opertions of Exponents with the Sme Bse Unit: Exponents Approx. time: Lesson : Properties for Opertions dys of Exponents with the Sme Bse A. Focus nd Coherence Students will know The rules of multiplying nd dividing exponents with the sme bse lwys work The rule for rising n exponentil term to nother exponent lwys works Students will be ble to Prove the rules for opertions of exponents with the sme bse (below) by using the definition of n exponent: m n mn m mn b) n c) m n ( ) mn Student prior knowledge: Definitions of exponent nd bse Know how to expnd, simplify, nd evlute expressions involving exponents, including products nd quotients rised to powers Simplifying frctions by cnceling pirs of fctors in the numertor nd denomintor CCSS-M Stndrds: 8.EE. Know nd pply the properties of integer exponents to generte equivlent numericl expressions. B. Evidence of Mth Prctices Wht will students produce when they re mking sense, persevering, ttending to precision nd/or modeling, in reltion to the focus of the lesson? Students will generte the rules for opertions of exponents with the sme bse by noticing repeted clcultions. Students will use the cdemic vocbulry pproprite for this lesson when responding to verbl questions during the lesson nd in their Ticket-Out-the-Door written responses (power, bse, exponent, fctors, simplify, expnded form, etc.) Students will use the definition of n exponent to explin nd justify/prove why the rules of exponents with the sme bse lwys work. Which mth concepts will this lesson led to? Properties for zero nd negtive integer exponents Frctionl exponents Scientific nottion Guiding Question(s). How cn we multiply or divide two powers with the sme bse without expnding the expression?. How cn we rise power to power without expnding the expression? Formtive Assessments Chorl Response to Exmples,, & (see lesson) Ticket-Out-the-Door Prompt: Write written explntion on how to multiply nd divide powers with the sme bse, without expnding the expression. Anticipted Student Preconceptions/Misconceptions When multiplying powers with the sme bse, students multiply the exponents or multiple the bses. When dividing powers with the sme bse, students divide the exponents or divide the bses. When rising power to power, students perform the exponent opertion on the bse (e.g. ( ) = 7 9 ) Mterils/Resources Document cmer, individul whitebords nd mrkers, mth textbook (for homework only)

10 C. Rigor: fluency, deep understnding, ppliction nd dul intensity Wht re the lerning experiences tht provide for rigor? Wht re the lerning experiences tht provide for evidence of the Mth Prctices? (Detiled Lesson Pln) Wrm Up Students nswer wrm-up problems individully, then techer leds whole-clss discussion:. Expnd x 5. Expnd 5 7. Expnd (5 ) (for discussion: Do you think it would be helpful to find shortcut for #?). (Wht cn we substitute for the blnk to mke this eqution true?) x x x 5. Simplify x x Lesson Prt : Multiplying Powers with the Sme Bse Techer guides discussion, sking the whole clss questions nd providing wit time for students to think. ) Wht is the expnded form of this expression? (5 55) (5 555) 5 5 b) Hve students do more exmples like the one in prt () until they strt to see pttern. c) Wht do you notice bout multiplying powers with the sme bse? Is there shortcut tht we cn use, without hving to write the exponents in expnded form? Does the bse chnge? Does the exponent chnge? How? Wht is the rule? After sufficient discussion from prt (c), hve students copy the rule for multiplying powers with the sme bse in their notebooks or mth journls: When multiplying powers with the sme bse, dd the exponents: m n mn d) Prctice for Prt (); Students try problems on their own, on individul whitebords for immedite feedbck x x x *Do more problems s necessry Find two powers tht will mke the eqution true: 5. Explin your resoning to prtner. Hve few students shre their responses nd explntions with the whole clss. 7

11 Prt : Dividing Powers with the Sme Bse Techer guides discussion, sking the whole clss questions nd providing wit time for students to think. 6 ) Hve students guess how to simplify. A few students explin how they simplified (choose students who simplified in different wys) For exmple, Or, 6 Or, 6 b) Hve students do more exmples of simplifying frctions like the one in prt () until they see pttern 8 0 x 8 x c) Wht do you notice bout dividing powers with the sme bse? Is there shortcut tht we cn use, without hving to write the exponents in expnded form? Does the bse chnge? Does the exponent chnge? How? Wht is the rule? After sufficient discussion from prt (c), hve students copy the rule for dividing powers with the sme bse in their notebooks or mth journls: When dividing powers with the sme bse, subtrct the exponents: m mn n d) Prctice for Prt (); Students try problems on their own, on individul whitebords for immedite feedbck ? 0 x x ( ) ( ) *Do more problems s necessry? 5 Find two powers tht will mke the eqution true: 9. Explin your resoning to prtner.? Hve few students shre their responses nd explntions with the whole clss.

12 Prt : Rising Power to Power Techer guides discussion, sking the whole clss questions nd providing wit time for students to think. ) Let s look bck to Wrm-Up Question #: Expnd (5 ) How did you get it? If 5 is the bse nd is the exponent, you cn do 5 multiplied by itself times, then continue to expnd the exponents: ( 5 ) (5 555) (5 555) (5 555) 5 b) Hve students simplify more expressions, like the exmple in prt (), until they see pttern ( 6 ) ( ) 6 (x ) c) Wht do you notice bout powers rised to other powers? Is there shortcut tht we cn use, without hving to write the exponents in expnded form? Does the bse chnge? Does the exponent chnge? How? Wht is the rule? After sufficient discussion from prt (c), hve students copy the rule for rising power to power in their notebooks or mth journls: When rising power to power, multiply the exponents m n mn ( ) d) Prctice for Prt (); Students try problems on their own, on individul whitebords for immedite feedbck. ( 5 7 ) 7 (( 7) ) 9 6? ( ) (x 5 ) 5 Find two exponents tht will mke the eqution true: ( x? )? x. Explin your resoning to prtner. Hve few students shre their responses nd explntions with the whole clss. Closure Ticket-Out-The-Door Prompt: Write n explntion on how to multiply nd divide powers with the sme bse. Possible extension questions:. Simplify n expression nd provide written explntion for ech step. Why do the powers hve to hve the sme bse to perform the opertions you lerned tody? Suggested Homework/Independent Prctice Some homework problems from the textbook (for procedurl fluency) Using only three vribles x, y, z, write the three rules of opertions of exponents with the sme bse.

13 Unit: Exponents Lesson : Properties for Zero nd Negtive Integer Exponents A. Focus nd Coherence Students will know Lesson : Properties for Zero nd Negtive Integer Exponents Approx. time: dys The proof of x 0, when x 0, using the properties of exponents Negtive exponents cn be expressed using positive exponents using the rules for multiplying nd dividing exponents with the sme bse Students will be ble to Use the rules tht they generted in Lesson (for multiplying nd dividing exponents with the sme bse) to generte the property of zero exponents: 0 Use the rules tht they generted in Lesson (for multiplying nd dividing exponents with the sme bse) to generte the property of negtive exponents: m m Student prior knowledge: The rules for multiplying nd dividing exponents with the sme bse lwys work m m n mn mn ) b) n Identify multiple representtions for nmes of. Which mth concepts will this lesson led to? Understnding the reltionship between squre roots nd cube roots nd the frctionl exponents of ½ nd /. Propose, justify nd communicte solutions involving properties of exponents. CCSS-M Stndrds: 8.EE. Know nd pply the properties of integer exponents to generte equivlent numericl expressions. B. Evidence of Mth Prctices Student cn explin with justifiction why ny number to the zero power is one becuse they understnd the rules of dditive lws of exponents m 0 m nd multiplictive identity: nd m m, so 0 Students cn use the dditive lw of exponents nd the multiplictive identity to prove/justify tht, for exmple: 0 0 becuse (dditive lw of exponents) 9 = 9 = (multiplictive identity) Students cn use the subtrctive lw of exponents nd division to prove tht 0 for exmple: 0 nynumber, nd itself So, 0 Students cn explin why negtive exponent is equivlent to its reciprocl becuse they understnd the rules of dditive lws of exponents nd they ve lredy proven tht 0. Students m m 0 will show tht, nd since 0, m m m. This implies tht m nd must be reciprocls. For exmple: is the unknown, but we know 0 8 = =, which is 8 Essentil Question(s): ) Why is using exponents helpful? ) How does understnding the exponent rules help you solve rel-world problems? ) Multipliction is to repeted ddition, s Exponents re to.? ) How does understnding the dditive lw of exponents help you develop rules for zero nd negtive exponents?.

14 Formtive Assessments: Ask the following questions s Ticket-Out-The-Door closure item, for students to reflect on the dy s lerning: 0 ) In, where did the exponent in the nswer come from? ) Why is 0? Is this lwys true? ) Prove tht - Anticipted Student Preconceptions/Misconceptions: 0 0 nd other responses - 8 nd other responses Mterils/Resources: Worksheet/workout pper, notebooks, individul whitebords nd mrkers C. Rigor: fluency, deep understnding, ppliction nd dul intensity Wht re the lerning experiences tht provide for rigor? Wht re the lerning experiences tht provide for evidence of the Mth Prctices? (Detiled Lesson Pln) Wrm Up (Dy ) Use exponent rules to simplify: ) = ANS: ) = ANS: ) ANS: ) ANS: 5) Wht re your thoughts bout the nswer to #? Wrm Up (Dy ) Use exponent rules to simplify: ) = ANS: ) ANS: ) ANS: ) ANS: 5) Wht re your thoughts bout the nswer to #? Lessons Dy : Zero Exponents ) Debrief wrm up nswers nd hve whole-group clss discussion bout #. ) Pose question to the clss: Wht re some guesses of the vlue of? Chrt student guesses on the bord. ) Use exponent rules to prove why 65 0 = For exmple: (dditive lw of exponents) There is one unknown tht we re trying to solve for (65 0 =?) x x = (multiplictive identity)

15 ) Show nother wy: Use exponent rules to prove why 0 = :, nd So, 5) Hve students prove tht ech of the following equls by using the strtegies from # nd/or # 0 ) 7 0 b) c), ) Hve students reson bout wht must equl (when is ny non-zero number). 7) Put the rule for zero exponents on the bord for clss to copy down in notebooks Zero s n Exponent For every nonzero number, 0 Dy : Negtive Exponents: ) Debrief wrm up nswers nd hve whole-group clss discussion bout #. ) Pose question to clss: Wht re some guesses of the vlue of? Chrt student responses on the bord m ) Hve students use the dditive lw of exponents to prove tht m Exmple: is the unknown but we know = 8 =, hs to be the multiplictive inverse! So 8 Hve students do vrious exmples like the one bove until they strt to see pttern. Exmple : Wht is? 0 We know tht ( see bove to continue ) Exmple : Wht is? Exmple : Wht is 9 So So So n ) Hve students reson bout wht must equl 5) Put the rule for negtive exponents on the bord for the clss to copy down in their notebooks. Negtive Exponents For every nonzero number, nd integer n, n n

16 Closure Use the formtive ssessment questions s Ticket-Out-The-Door reflection from tody s work: 0 ) In, where did the exponent in the nswer come from? ) Why is 0? Is this lwys true? ) Prove tht - Suggested Homework/Independent Prctice Use the property for multiplying powers with the sme bse: ) 5 0 = ) ( 5) 0 = ) 0 ) 5) 7 7 n 6) n m n mn to prove the following:

17 Lesson : Properties Review nd Assessment Unit: Exponents Approx. Lesson : Properties Review nd time: Assessment dy A. Focus nd Coherence Students will Review the properties of exponents m n mn ) m mn b) n c) 0 n d) n CCSS-M Stndrds: 8.EE. Know nd pply the properties of integer exponents to generte equivlent numericl expressions. B. Evidence of Mth Prctices Students will use words, symbols, nd numbers to explin nd justify why the properties of exponents work. Students will rticulte their thoughts ccurtely nd precisely in writing. Propose, justify, nd communicte solutions to n Interim Assessment tsk regrding properties of exponents Mterils/Resources: Interim Assessment worksheet nd nswer key (ttched) C. Rigor: fluency, deep understnding, ppliction nd dul intensity Wht re the lerning experiences tht provide for rigor? Wht re the lerning experiences tht provide for evidence of the Mth Prctices? (Detiled Lesson Pln) Wrm Up Review homework from Lesson : m n mn Use the property for multiplying powers with the sme bse: to prove the following: ) 5 0 = ) ( 5) 0 = ) 0 ) 5) 7 7 n 6) n Lesson Administer Interim Assessment Tsk: Extending the Definition of Exponents (ttched) Students will complete this tsk independently. Closure Ticket-Out-The-Door written reflection: Wht do you still need help with regrding exponent opertions? Suggested Homework/Independent Prctice None

18 Lesson Interim Assessment: Extending the Definition of Exponents Nme: Dte: Mrco nd Seth re lb prtners studying bcteril growth. They were surprised to find tht the popultion of the bcteri doubled every hour. ) The tble shows tht there were,000 bcteri t the beginning of the experiment. Wht ws the size of the popultion of bcteri fter hour? After,, nd hours? Enter this informtion into the tble: Hours into study 0 Popultion in thousnds b) If you know the size of the popultion t certin time, how do you find the popultion one hour lter? c) Mrco sid he thought tht they could use the eqution P t to find the popultion t t time t. Seth sid he thought tht they could use the eqution P. Decide whether either of these equtions produces the correct popultions for t =,,, nd. d) Wht number would you use to represent the time hour before the study strted? hours before? hours before? Complete the first row of the tble showing the hours prior to the beginning of the study. e) Assuming the popultion doubled every hour before the study begn, wht ws the popultion of the bcteri hour before the students strted their study? Wht bout hours before? Complete the second row of your tble to show the popultion of the bcteri during the hours prior to the beginning of the study. f) Use Seth s eqution to find the popultion of the bcteri hour before the study strted. Use the eqution to find the popultion of the bcteri hours before the study strted. Are these the sme vlues you found in prt (e) nd entered into your tble?

19 Lesson Answer Key Extending the Definition of Exponents ) After hour the popultion is thousnd, fter hours it s 8 thousnd, fter hours it s 6 thousnd, nd fter hours it s thousnd. Hours into study 0 Popultion in thousnds 8 6 b) If you know the popultion t certin time, you cn multiply by to get the size of the popultion n hour lter. c) Seth s eqution is correct becuse it gives the correct popultion sizes for hours,, nd. The eqution doubles the number every hour into the study. d) I used to represent the time hour before the study, to represent hours before the study, nd to represent hours before the study. Hours into study 0 Popultion in thousnds 8 6 e) Hours into study 0 Popultion in thousnds 8 6 double double double f) Seth s eqution gives the sme vlues tht I found in prt (e) nd put in the tble. t I used P nd I plugged in t =,, nd. At t = At t = At t = P = P P 8

20 Lesson 5: Expressing Numbers in Scientific Nottion Unit: Exponents Lesson 5: Expressing Number in Scientific Nottion Approx. time: dys A. Focus nd Coherence Students will know Scientific nottion is used to represent lrge nd smll number CCSS-M Stndrds: 8.EE. Use numbers expressed in the form of single digit times n integer power of 0 to estimte very lrge or very smll quntities, nd to express how mny times s much one is thn the other. For exmple, estimte the popultion of the United 8 Sttes s 0 nd the popultion of the world of 9 7 0, nd determine tht the world popultion is more thn 0 times lrger. B. Evidence of Mth Prctices Wht will students produce when they re mking sense, persevering, ttending to precision nd/or modeling, in reltion to the focus of the lesson? Students will be ble to Express lrge nd smll numbers in scientific nottion Student prior knowledge: Definition of n exponent Rules for multiplying nd dividing powers with the sme bse The proof of x 0 = Rewriting negtive exponents s positive exponents Plce vlue chrt Students will explin when it is pproprite to express number in scientific nottion. Students will justify why their expression written in scientific nottion is equivlent to the expnded form of the number (e.g. I know tht 7,000,000 is 7 x 0 6 becuse 0 6 represents,000,000 nd 7 times,000,000 is 7,000,000 ). Students will precisely rticulte how to properly red numbers written in scientific nottion (e.g. 7 x 0 6 is seven times ten to the sixth ) Which mth concepts will this lesson led to? Applying scientific nottion to rel world situtions Essentil Question(s):. When is it pproprite to express numbers in scientific nottion?. How cn we estimte numbers written in scientific nottion? (Know certin plce vlues) The distnce from plnet X to plnet Y is 6.95 x 0 6 miles. Approximtely how mny miles is tht? Students knowing tht the exponent of 6 is for million, so tht s bout 7 million miles. Formtive Assessments: Ticket-Out-The-Door writing prompt: Why do scientists nd mthemticins sometimes use scientific nottion s opposed to stndrd form? Anticipted Student Preconceptions/Misconceptions: Formt for writing in Scientific Nottion Concept of bse 0 Possible unresolved issues with positive, negtive nd zero powers Adding zeros s opposed to moving the deciml point Why we use X vs. multipliction point or prentheses Forgetting tht the first number must be greter thn or equl to nd less thn 0 Mterils/Resources Individul whitebords, mrkers, nd ersers

21 C. Rigor: fluency, deep understnding, ppliction nd dul intensity Wht re the lerning experiences tht provide for rigor? Wht re the lerning experiences tht provide for evidence of the Mth Prctices? (Detiled Lesson Pln) Wrm Up Which expression represents the gretest mount? How do you know? 8. x 0 9 x x 0 (Answer: 7.56 x 0 ) Lesson Prt Understnding Scientific Nottion Whole-Clss Discussion round the following tlking points: Mthemticins nd scientists used wht s clled Scientific Nottion to express relly lrge or relly smll numbers. o For exmple, the erth is,600,000,000 yers old. Tht s relly big number with lot of zeros! There is wy to write tht number without ll of the zeros, which is wht we will be lerning bout tody. o For exmple, the mss of n nt is kg. Tht s relly smll number with lot of zeros! There is wy to write tht number without ll of the zeros, which is wht we will be lerning bout tody. When do you think it is pproprite to express numbers in scientific nottion? o Give n exmple of when you might use relly big number or relly smll number to represent something in the rel world. Prt Writing Numbers in Scientific Nottion Hve students write down: ) 95 followed by 7 zeros, b) 95 followed by 5 zeros, c) 95 followed by 5 zeros. Whole-Clss Discussion round the following questions: ) Is there wy to write these numbers without ll the zeros? ) Wht does it men when we hve bse of 0? (E.g. 0, 0, 0 6, etc.) Cn we use tht informtion to help us write these numbers without ll of the zeros?. Any guesses? Write down student guesses nd listen to student explntions. Work out problems s group until correct nswers hve been discovered. ) How to red numbers in scientific nottion:. 9.5 x 0 5 is 9 nd 5 hundredths times 0 to the 5 th power b. 6 x 0 is six times 0 to the rd power c. More exmples s needed ) Wht s bigger: 9 x 0, 5 x 0, x 0, or x 0?. How do you know? b. Do you hve to expnd ech of the numbers in order to know which one is bigger? c. Cn you just look t the power of 0?

22 For students who my still be struggling with this concept nd need visul representtion, you cn drw visul of block, 0 x 0 squre, 0 x 0 x 0 cube, etc. This might help with comprisons when deciding which vlue is greter. Hve students repet this process with other exmples s needed. Py close to ttention to the first number in Scientific Nottion it must be greter thn or equl to nd less thn 0. Prt Writing Numbers in Stndrd Form Present problems in scientific nottion nd hve students work bckwrds to put them into stndrd form. Put the following numbers in stndrd form: ) 5. x 0 5 b) x 0 - c). x 0 7 d) x 0 e) More exmples s needed Closure Ticket-Out-The-Door writing prompt: Why do scientists nd mthemticins sometimes use scientific nottion s opposed to stndrd form? Suggested Homework/Independent Prctice Prentice Hll Mthemtics: Cliforni Algebr Textbook Chpter 7-, pges 7 #,, 8,,, 7, 9. (These problems get t the computtionl skills involved with writing numbers in scientific nottion nd in stndrd form; the following lesson will focus on the rel-world ppliction of scientific nottion).

23 Lesson 6: Using Scientific Nottion to Solve Rel World Problems Unit: Exponents Lesson 6: Using Scientific Nottion to Solve Rel World Problems Approx. time: dys A. Focus nd Coherence Students will know Scientific nottion is used to represent lrge nd smll numbers The rules for opertions with exponents re used to perform opertions with numbers expressed in scientific nottion Students will be ble to Perform opertions with numbers expressed in scientific nottion nd choose units of pproprite size to represent given mesurements. Solve rel-world problems involving numbers expressed in scientific nottion. Convert between units of mesurement using numbers expressed in scientific nottion CCSS-M Stndrds: 8.EE. Perform opertions with numbers expressed in scientific nottion, including problems where both deciml nd scientific nottion re used. Use scientific nottion nd choose units of pproprite size for mesurements of very lrge or very smll quntities (e.g., use millimeters per yer for sefloor spreding). Interpret scientific nottion tht hs been generted by technology. B. Evidence of Mth Prctices Wht will students produce when they re mking sense, persevering, ttending to precision nd/or modeling, in reltion to the focus of the lesson? Students will mke sense of word problems by ppropritely converting unit mesurements, nd determining the resonbleness of their results. Students will pply exponent opertions nd scientific nottion rules in solving rel world problems nd justify their solutions. Students will mke predictions nd inferences using dt with scientific nottion (in grphs, tbles, or digrms). Student prior knowledge: Express lrge nd smll numbers in scientific nottion Write mesurements in stndrd form nd/or scientific nottion. Perform opertions of exponents (multiply, divide, zero nd negtive exponents) Convert between different units of mesurement (i.e. dimensionl nlysis) Students will use pproprite vocbulry nd units of mesurements to represent predictions nd inferences. Which mth concepts will this lesson led to? Propose, justify nd communicte solutions to rel-world problems tht involve scientific nottion nd exponents Guiding Question(s). Why is using exponents helpful?. When is it pproprite to express numbers in scientific nottion?. How cn we estimte quntities of vrible lengths using exponents?. How does understnding the exponent rules help you solve rel-world problems involving scientific nottion? Formtive Assessments (see ttched) Ants vs. Humns (smple problem from Gintburgers (smple problem from Distnce to the Sun (smple problem from Smrter Blnced)

24 Anticipted Student Preconceptions/Misconceptions Clcultors cn hndle ll types of numbers Performing opertion does not involve grouping terms Products or quotients do not need to be chnged to scientific nottion When converting between stndrd form nd scientific nottion, there will be misconception in moving the deciml bsed on the sign of the exponent. Mterils/Resources Formtive ssessments (see ttched) C. Rigor: fluency, deep understnding, ppliction nd dul intensity Wht re the lerning experiences tht provide for rigor? Wht re the lerning experiences tht provide for evidence of the Mth Prctices? (Detiled Lesson Pln) Wrm Ups Dy Simplify. Express ll exponents s positive numbers: ( )( 5 ) = 5 = 0 0 (7 )(50 ) = = Dy Express the following numbers in scientific nottion: The ge of the erth:,600,000,000 yers The mss of n nt: 0.00 grms Dy or The current United Sttes popultion is bout,000,000 people. If n ipd costs $500.00, wht would be the totl cost of purchsing n ipd for ech person? A. Express the U.S. popultion in scientific nottion. B. Write n expression to represent the totl cost of purchsing ipds for everyone in the U.S. C. Wht is the totl cost of purchsing IPds for everyone in the U.S.? D. Explin how you know your nswer is correct (model, drw picture, etc.). Lesson (to be tught over dys) Direct Instruction: Tody you will use ll exponent rules to perform opertions of numbers written in scientific nottion such s the rules for multiplying nd dividing powers with the sme bse. Techer goes over the rules, then, invites the ttention of the students on the bord to follow through the review on how the rules will pply to the given exmples on the bord. Model Instruction: Give t lest two exmples on the white bord/projector of multiplying numbers expressed in Scientific Nottion nd lso dividing numbers expressed in Scientific Nottion: Exmple #) (5 x 0 )(5 x 0 ) = (5)(5)x(0 )(0 ) =5 x 0 5 =.5 x Exmple #) = 0.5 x 0 =.5 x 0 60

25 Exmple #) Convert the following units: 500 inches = feet 0,000 miles = inches 65 kilogrms = grms liters = milliliters Provide prctice problems for students to work on in pirs or groups, similr to exmples,, nd. Guided Prctice: Solve word problems tht llow students to perform opertions with numbers expressed in scientific nottion. Do problem s whole group. Exmple: The verge mss of n dult humn is bout 65 kilogrms while the verge mss of n nt is pproximtely x 0 - grms. The totl humn popultion in the world is pproximtely 6.8 billion, nd it is estimted there re currently bout 0,000 trillion nts live. Mke decision bout the following: The totl mss of nts is greter thn the totl mss of humns; or The totl mss of humns is greter thn the totl mss of nts How do you know? Explin your resoning. Other ides for rel-world ppliction problems: Clculting msses of protons, electrons nd neutrons in toms. Clculting ir pollutnts nd mking predictions of the ir qulity. Clculting distnce between plnets nd strs. Clculting the number of toms nd molecules in given mss of n element or compound nd relte to stoichiometric problems to determine yields in the industry. Clculting crbon footprints. Group/Prtner work: Hve students do formtive ssessment questions in prtners or smll groups (see ttched) o Possible Sttions Activity where ech sttion contins different rel-world ppliction problem involving opertions with scientific nottion. Hve students work in groups t their sttion, then rotte to different sttion (depending on time llotted). Whole group shre out nd debrief Closure Presenttions from group work Suggested Homework/Independent Prctice Prentice Hll Mthemtics: Cliforni Algebr Textbook Chpter 7-, pges 7 #6, 8,,,, 5

26 Lesson 6 Formtive Assessment Questions for group work: ) Ants vs. Humns The verge mss of n dult humn is bout 65 kilogrms while the verge mss of n nt is pproximtely x 0 - grms. The totl humn popultion in the world is pproximtely 6.8 billion, nd it is estimted tht there re currently bout 0,000 trillion live. Bsed on these vlues, how does the totl mss of ll living nts compre to the totl mss if ll living humns? ) Gintburgers This hedline ppered in newspper: Every dy 7% of Americns et t Gintburger resturnts. Decide whether this hedline is true or flse using the following informtion. There re bout 8 x 0 Gintburger resturnts in Americ. Ech resturnt serves on verge.5 x 0 people every dy. There re bout x 0 8 Americns. Explin your resoning nd show clerly how you figured it out. ) Distnce to the Sun The verge distnce from Jupiter to the Sun is bout 5 x 0 8 miles. The verge distnce from Venus to the Sun is bout 7 x 0 7 miles. The verge distnce from Jupiter to the Sun is bout how mny times s gret thn the verge distnce from Venus to the Sun?

27 8 th Grde Exponents Culminting Tsk Nme Post Assessment Blood in the Humn Body Dte Tody you will be sked to solve severl questions tht require the use of the properties of exponents. Before you begin, plese provide convincing rgument to show why the following rules re true. ) Prove tht b b m n b mn ) Prove tht b 0 Use the properties of exponents, s they pply, to solve the following problems. ) There re.7 x 0 8 hemoglobin molecules in single humn red blood cell, nd there re bout 5.0 x 0 6 humn red blood cells in one cubic mm of blood. How mny hemoglobin molecules re in one cubic mm of blood? Show your work.

28 ) Irving clims he used property of exponents to solve question. Wht property of exponents did Irving use? How do you know? 5) A red blood cell hs dimeter of pproximtely 7.5 x 0 - cm. Suppose one of the rteries in your body hs dimeter of.56 x 0 - cm. Irving sys tht 6.08 x 0 red blood cells would fit cross the rtery. Do you gree or disgree with Irving? Why or why not? 6) If you donte blood regulrly, the Americn Red Cross hs hd policy of witing 56 dys between dontions. They re currently ressessing their policy. One pint of blood contins bout.x0 red blood cells. Your body normlly produces bout x0 6 red blood cells per second. Wht is your recommendtion for witing period between blood dontions? Do you gree or disgree with the current policy of the Americn Red Cross? 56 dy witing period

29 8 th Grde Exponents Culminting Tsk Blood in the Humn Body Scoring Rubric Answer Points Totl Points. Smple Proof: Using the definition of n exponent, you cn expnd the numertor nd the denomintor nd simplify the frction by mking ones, for 5 b bbbbb exmple b. Or you cn subtrct the exponents, b bb 5 =, to represent tht the b s in the denomintor cncel (i.e. mke one ) with of the b s in the numertor. -. Smple Proof: Knowing the property for multiplying powers with the sme bse: m n b b mn 0 b, I could write n eqution to solve for b 5 0 b b 50 b Or... b b 5 0 b 0 b 5 must equl one becue nythingmultiplied by one Thereforeb 0 is itself Smple Proof: Knowing the property for dividing powers with the sme bse: m b mn 0 b, I could write n eqution to solve for b n b 5 5 b 55 0 b b b. Anything divided by itself equls, so, so 5 5 b b 0 b.. (.7 x 0 8 )(5.0 x 0 6 ) =.5 x 0 5 hemoglobin molecules Irving used the dditive property of exponents Or b m b n = b m+n Student recognizes tht in their work for problem they multiplied two powers nd therefore used the dditive property of exponents.

30 5. (.56 x 0 - )/( 7.5 x 0 - ) = 6.08 x 0 Student disgrees with Irving. Irving sid tht (.56 x 0 - )/( 7.5 x 0 - ) = 6.08 x 0, so it looks like he divided correctly t first to get.608 x 0, but then didn t correctly put it in scientific nottion. The correct nswer, 6.08 x 0, is ctully 0 times less thn Irving s nswer. 6. The humn body cn produce ( x 0 6 )(60)(60)() =.78 x 0 red blood cells per dy It would tke (. x 0 )/(.78 x 0 ) = dys to replenish the red blood cells lost for donting one pint of blood. Students disgree with the current policy of witing 56 dys between dontions. Students recommend witing period of dys or more between dontions - TOTAL 5