Teaching Guide Zero and Negative Exponents

Transcription

1 Teching Guide Zero nd Negtive Eponents Prepring for Your Clss Common Vocbulr Zero eponent, negtive eponent Instruction Tips In the more complicted eponentil epressions, it m be helpful to students if ou indicte the chnge from one step to the net in color. If students do this in their notes too, it will be esier for them to identif the individul steps s the progress through reding their notes. Once students lern the rule for negtive eponents, most of them hve ver difficult time discerning between epressions like nd. Epressions like these two seem to be nturll ctegorized s equivlent when the re not. For students, looks similr to, nd so it is probble tht the re both initill processed the sme w during the lerning process. Do our best to emphsize the difference t ever opportunit, s this is hrd concept to fortif. Nturl instincts led students to believe tht should be (lthough it is not). This is n unfortunte remnnt of their lerning the multipliction propert of. Since we lern first tht n number times zero is zero, we begin to ssocite n epression with zero in it tht involves multipliction with n nswer of zero. Consider tht is n epression with zero in it, nd we define eponents in terms of the number of fctors (multipliction), so it is not surprising tht students defult to this ide tht should be. Ever time this mistke rises in clss, remind students of the pttern development on the Guided Lerning Activit tht clerl shows tht hs to be. After we tech students tht the epressions with negtive eponents cn be moved cross the frction br, the inevitbl wnt to tke the coefficients long for the ride. For emple, the mjorit of students will rewrite s insted of. To replce this fult thinking, mke sure to emphsize the eponents on ever fctor; thus becomes. Wh do we rewrite epressions without negtive eponents? First, nd of most importnce for lgebr students, it is much more intuitive to evlute n epression like thn it is to evlute the epression. The second reson pplies to the future clsses tht students might tke; students in Clculus must be ble to convert flwlessl between epressions tht do nd do not involve negtive eponents in order to ppl the power rule for derivtives. Consider these emples from clculus section: d d o ( ) d d d d o ( ) d d A summr of eponent rules is provided in lrge font t the end of this guide. Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

2 Teching Your Clss Guided Lerning Activit: Zero nd Negtive Eponents. The concepts of zero nd negtive eponents re not nturll intuitive. Becuse it is best for lerning if students discover these rules for themselves, this Guided Lerning Activit will guide students to the rules for zero eponents nd negtive eponents. After the clss completes this worksheet, ou should be ble to discuss the more difficult ides in this section. (EXP-) Another set of lerning emples for zero nd negtive eponents Simplif ech emple two ws. nd nd nd (thus ) nd (thus ) nd (thus ) Other simplifiction involving negtive eponents Consider this emple: so A generl rule: n n so n n n n n Consider this emple: A generl rule: n n (with nd, n nturl number) Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

3 Emples: Rewrite without negtive eponents. Simplif if possible. 6 6 b b ( ) vs. vs. ( 6) vs. 6 vs. 6 6 ( ) ( ) Simplifing epressions tht involve mn eponent rules For most of these emples, there re t lest two ws to show the simplifiction. Becuse students ll lern differentl, it is importnt to show multiple ws to perform the simplifictions. Some students will nturll grvitte to one method over the others. or + or 7 7 ( ) 8 ( ) or 8 ( ) ( b ) 9 b b replcing fctors of with 7 6 b or 6 ( b ) 8 ( ) b b b or Student Activit: Mtch Up on Trickier Eponent Rules. This ctivit works well in groups or with students working in pirs t whitebord (hve students drw n empt grid on the bord for their nswers). Some of the boes in the grid involve remembering how to dd or subtrct like terms involving eponents. (EXP-) Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

4 Student Activit: Double the Fun on Eponent Rules. In this ctivit, students prctice simplifing eponentil epressions using direct ppliction of eponent rules nd b moving fctors to void working with negtive eponents. Simplifing the epressions both ws is n ecellent w for students to lern to check their work nd trul understnd eponent rules. (EXP-) Student Activit: Mthemticl Heteronms. In this ctivit, students prctice simplifing sets of epressions tht look ver similr, but ield ver different results. Students rell struggle with this ctivit if the hve not thoroughl seprted similr-looking mthemticl concepts in their minds. (EXP-6) Student Activit: Eponents Using Clcultor. Students prctice entering epressions tht involve eponents on their clcultor. (EXP-7) Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

5 A Summr of Eponent Rules Let nd b represent integers nd let there be no divisions b zero.. Product Rule:. Quotient Rule: b b + b b b. Power Rule: ( ). Power of Quotient: b. Power of Product: ( ) 6. Eponent of or : 7. Negtive Eponent: nd 8. Negtive Eponent in the Denomintor: 9. Negtive Eponent on Quotient:. Moving Eponents Across the Frction Br: Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

6 Guided Lerning Activit Zero nd Negtive Eponents To develop the concepts of zero nd negtive eponents, we will first look t severl ptterns of vlues.. Fill in the blnks to complete ech sequence below: 8,7,9,,,, 6,8,,,,, How do ou get the net term? How do ou get the net term?,,,,, How do ou get the net term?,,,,, How do ou get the net term?,,,,,. Now complete ech tble below b finishing the ptterns: How do ou get the net term? Bsed on our observtions in the ptterns bove, finish the eponent rules below: Rule for Zero Eponents: (for ) It m seem like nothing, but it mkes me ONE with the Universe. Rule for Negtive Eponents: n (for ). Appl the rules ou hve just developed to write ech of these eponentil epressions without zero or negtive eponents. 6 z ( ) Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

7 Student Activit Mtch Up on Trickier Eponent Rules Mtch-up: Mtch ech of the epressions in the squres of the grid below with n equivlent simplified epression from the top. If n equivlent epression is not found mong the choices A through D, then choose E (none of these). A B 9 C 9 D E None of these Mrk m words! You hrness tht negtive power of ours, nd ou cn mke it to the top just like me! ( ) ( ) ( ) ( ) ( ) b 7 ( 9 ) b b 7 ( z) ( ) z 8 8 ( )( ) ( z ) z Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

8 Student Activit Double the Fun on Eponent Rules Directions: For mn problems involving the simplifiction of eponents, there re t lest two ws to tckle the problem. For ech of the problems below, tr the problem both ws our nswers should be the sme for both methods. If the re not, ou ll hve to go bck nd look for mistke. The first one hs been done for ou. Simplif this epression Appl eponent rules directl ( 6) ( ) First move fctors to void working with negtive eponents b 8 b ( ) ( 6 ) 6 ( ) Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

9 Student Activit Mthemticl Heteronms Directions: In writing, there re words tht re spelled the sme but hve different pronuncitions nd different definitions; these re clled heteronms. Mn mthemticl epressions look similr but re rell ver different (lmost like mthemticl heteronms). In ech set of epressions below, p close ttention to the use of prentheses nd the mthemticl opertions nd nottion.. ( ) ( ) ( ). ( ). ( ) ( ). ( ) ( ) ( ) ( ). ( ) Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning

10 Student Activit Eponents Using Clcultor When ou input eponents into clcultor, ou must be creful to tell the clcultor which prt is the bse nd which prt is the eponent. Ech clcultor requires specific set of kestrokes to evlute eponentil epressions nd this cn become quite complicted when the epression lso involves negtives. We will fill out the tble below, first performing ech clcultion b hnd, then finding the proper kestrokes to do the evlution on our clcultor. First ou need to locte our eponent button. It m look like ^ or. Reminders: There is difference between the minus ke nd the negtive ke on our clcultor. Frctions need to be plced inside prentheses for proper evlution. Epression. Evlute b hnd Clcultor Kestrokes to get equivlent result b. c. d. e. f. g. ( ) h. i. ( ) j. + k. ( + ) l. Algebr Activities Instructor's Resource Binder, M. Andersen, Copright, Cengge Lerning