Square Roots Teacher Notes


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1 Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this topic. The Common Core Stte Stndrds do not mention simplifying rdicls, perhps ecuse tht topic hs lost its importnce in the ge of electronic lger. Geoord Are Becuse of the visul / geometric emphsis of the unit, it is criticl tht students hve strong sense of the mening of re. This lesson helps students grpple with it in concrete context, y doing simple computtions in order to find the res of figures on the geoord. Do not precede this lesson with review of re formuls. For mny of the weker students, these formuls get in the wy of understnding wht re even mens. However, s students work on the ctivity, it will proly e useful to discuss the re of rectngles, nd the fct tht right tringle cn e pictured s hlf of rectngle. These ides will come cross most effectively if they rise out of clss discussion. A common misunderstnding in this context is counting pegs to find the side of rectngle. Mke sure to confront this when it rises. This is converstion tht will need to hppen oneonone with mny students, ut it cn lso e reinforced y clss discussion. The ctivity is tken from Geometry Ls. The whole ook, including solutions nd techer notes, is ville for free downlod t If time is short, you my limit the ctivity to #14 nd Discussion questions A nd B. Squres nd Squre Roots This lesson introduces the centrl ide of the unit: the squre root of numer is the side of squre whose re is tht numer. In #1, most students will not find the first seven squres too difficult, ut the tilted squres will e considerly more chllenging. If students re stumped, or give outndout wrong nswers, do not revel the correct nswers right wy. Insted, suggest tht they strt y finding the re of the squres. If they need hints on how to do tht, you my suggest tht they sutrct the res of tringles from the lrger enclosing squres. Expressing the length of the sides s 2 nd 5 is sufficient for #1. Deciml pproximtions re requested in #2. In #4, the first tle cn e filled out with whole numers, ut the other two require decimls. You my chllenge students to fill out s much s possile without clcultor. Mny if not most students will e surprised tht in this tle, the squre roots of numers re greter thn the numers (except in the cse of the totl re.) You my hve clss discussion of this. #5 is n extr chllenge prolem. It is not necessry tht every student complete it. Squre Roots pge i
2 Henri Picciotto Geoord Squres This ctivity (once gin from Geometry Ls) follows up on #1 from the previous lesson. Side lengths cn e expressed s squre roots. (No need for deciml pproximtions.) It is not essentil tht students find ll 33 possiilities, ut try to mke sure tht ech student hs found the re nd side length for t lest few tilted squres. If time is short, skip Discussion questions BD. On the other hnd, if you wnt to do this lesson in depth, give it second or even third period, nd show the students how it leds to the Pythgoren theorem. The Exponent ½ This lesson (sed on Alger: Themes, Tools, Concepts 9.8) is review of the lws of exponents, leding to the ide tht the positive squre root of numer cn lso e thought of s tht numer rised to the power ½. #35 will certinly require clss discussion! Additionl Activities Mentl Arithmetic After the Squres nd Squre Roots lesson, s you work through the unit, you my sk students to pproximte squre roots without clcultor, nd then see how close they got with the help of clcultor. For exmple, you might sk: The squre root of 90 is etween which two whole numers? Do this often, s it is very importnt! Equl Squres A solid understnding of squre roots helps ly the groundwork for solving qudrtic equtions y the equl squres method, which is spelled out in the L Ger mterils, nd in Alger: Themes, Tools, Concepts 7.7 nd (ATTC is ville free t Extensions Geometric Puzzles Tngrms nd supertngrms re sed on right isosceles tringles, nd thus re intimtely involved with the squre root of 2. You cn find tngrm ctivities in Geometry Ls, nd supertngrm ctivities here: < And More For more squre root mterils, see: Geometry Ls 9.3, 9.4 Alger: Themes, Tools, Concepts 9.3, 9.4, 9.5, 9.8, 9.9 Squre Roots pge ii
3 LAB 8.4 Geoord Are Nme(s) Equipment: Geoord, dot pper The re of the geoord figure t right is Find other geoord figures with re 15.The oundries of the figures need not e horizontl or verticl. Find figures tht re different from the ones your neighors find. Record your solutions on dot pper. It is esiest to find res of geoord rectngles with horizontl nd verticl sides.the next esiest figures re the esy right tringles, such s the two shown t right. 2. Find the res of these tringles. If you cn find the re of esy right tringles, you cn find the re of ny geoord figure! 3. Find the res of the figures elow. 4. Find the re of the figure elow. (It my e more difficult thn Prolem 3. Hint: Use sutrction.) Geometry Ls Section 8 Perimeter nd Are Henri Picciotto,
4 LAB 8.4 Geoord Are (continued) Nme(s) 5. Find the res of the figures elow. 6. How mny noncongruent geoord tringles re there with re 8? Limit yourself to tringles tht cn e shown on n geoord nd hve horizontl se. Record your findings on dot pper. 7. Puzzle: Find the geoord figure with the smllest re in ech of these ctegories.. Acute tringle. Otuse tringle c. Right tringle d. Squre e. Rhomus (not squre) f. Rectngle (not squre) g. Kite h. Trpezoid i. Prllelogrm Discussion A. A common mistke in finding geoord res is to overestimte the sides of rectngles y 1 (for exmple, thinking tht the rectngle t right is 4 5).Wht might cuse this mistke? B. Explin, with illustrted exmples, how the following opertions my e used in finding the re of geoord figure: division y 2; ddition; sutrction. C. Wht hppens to the re of tringle if you keep its se constnt nd move the third vertex in direction prllel to the se? Explin, using geoord or dot pper figures. D. Use geoord figures to demonstrte the re formuls for vrious qudrilterls. 114 Section 8 Perimeter nd Are Geometry Ls 1999 Henri Picciotto,
5 Henri Picciotto Squres nd Squre Roots To find the re of squre, if you know its side, you multiply the side y itself: you squre the side. To find the side of squre, if you know its re, you tke the squre root of the re. 1. For ech squre, write three equtions. Your equtions should hve the following forms: side side = re, side 2 = re, nd re = side.. c. d. e. f. g. h. i. Squre Roots pge 3
6 Henri Picciotto 2. Estimte x nd y s deciml numers (they re the sides of those squres).. Without clcultor. With clcultor When the side of squre is whole numer, the re is clled perfect squre. For exmple, 49 is perfect squre, ecuse 49 = 7, nd 7 is whole numer. 2 is not perfect squre, ecuse 2 = 1.414, nd is not whole numer. 3.. Nme five perfect squres tht re greter thn 49.. Is 0 perfect squre? c. Is 49 perfect squre? Squre Roots pge 4
7 Henri Picciotto 4. Fill out the tles with the res nd sides of squres,, c, d. If the nswer is not whole numer, enter deciml rounded to the nerest 1/100. Are Totl 100 Side 5 c d Are Totl 1 Side 0.30 c d 5. Chllenge: Totl c d Are Side Squre Roots pge 5
8 LAB 8.5 Geoord Squres Nme(s) Equipment: Geoord, dot pper 1. There re 33 differentsize squres on n geoord.with the help of your neighors, do the following.. Find ll the squres.. Sketch ech squre on dot pper, indicting its re nd the length of its side. Discussion A. How cn you mke sure tht two sides of geoord squre relly form right ngle? B. How cn you orgnize your serch so s to mke sure you find ll the squres? C. Is it possile to find squres tht hve the sme re, ut different orienttions? D. In the figure t right, find the following in terms of nd.. The side of the outside squre. The re of the outside squre c. The re of ech tringle d. The re of the inside squre e. The side of the inside squre Geometry Ls Section 8 Perimeter nd Are Henri Picciotto,
9 Adpted from Alger: Themes, Tools, Concepts Anit Wh nd Henri Picciotto The Exponent ½ 1. Write s single power: (4 3 ) 2 c Find x = 2 x = x 6 c. (2 4 ) 2 = 2 x Squre Roots pge 7
10 Adpted from Alger: Themes, Tools, Concepts Anit Wh nd Henri Picciotto 3. Find x.. 9 x 9 3 = x 9 x = 9 2 c. 9 x 9 x = 9 1 d. B x B x = B 1 4. Find x.. (9 x ) 2 = 9 6. (9 x ) 2 = 9 1 c. (B x ) 2 = B 6 d. (B x ) 2 = B 1 Squre Roots pge 8
11 Adpted from Alger: Themes, Tools, Concepts Anit Wh nd Henri Picciotto 5. Prolems 24 suggest mening for the exponent ½. Develop n explntion for wht n exponent of ½ mens. 6. Using this mening of the exponent ½, find the following. If the nswer is not whole numer, use squre root symol c d. 2 Squre Roots pge 9
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