math and the minds eye activities 11

' Ｇ｛ｅ［ｾ［ｮｕｾｩＧＳ［Ｌ［Ｌ｜ｹｾＧＧＧＮＧＧＯＮﾷＮﾷＮﾷＮＧﾷＮ ＼ ｣ Ｇ Ｇ ｩ Ｎ ｊ ｊ ｲ Ｑ ｩ ｪ ｩ Ａ ｾ ｊ Ｏ 1v1::.1rn and the - '' -: ·- '•'• " - ' ｍｩｮ､Ｇｾｅｹ･ａ｣ｴｩｶｩｴｩ･ｳﾷﾷＮﾷﾷ aphing ebraic " " ' ""'·o n8hip - Unit XI I Math and the Mind's Eye Activities Graphing Algebraic Relationships An Introduction to Graphs, Part I ath and the Mind's Eye materials The values of rhe arrangements in extended sequences are graphed The graphs are examined for information abour these values are intended for use in grades 4-9 They are written so teachers can adapt An Introduction to Graphs, Part II them to fit student backgrounds and Extended sequences of arrangements are augmented so their graphs become continuous grade levels A single activity can be extended over several days or used in part An Introduction to Graphs, Part Ill Further investigations with continua of arrangements Introduction to Graphing Calculators, Part I Graphing calculators are introduced to provide an alternative to the "by-hand" method of plotting graphs, and as a way to represent continua of arrangements in more detail Connections between Algebra Piece, graphing, and symbolic representations of patterns arc developed A catalog of Math and the Mind's Eye materials and teaching supplies is available from The Math Learning Center, PO Box 3226, Salem, OR 97302, 503370-8130 Fax: 503-370-7961 Introduction to Graphing Calculators, Part II We continue our explorations with graphing calculators, investigating systems of equations Connections among the Algebra Piece, graphing, tabular, and symbolic representations of equations are reinforced Math and the Mind's Eye Copyright© 1996 llu: IVhth learning Centcr The Math Learning Center grams permission to da.•;sroom teachers to reproduce d1c student Ktiviry pages in appropriate quantities for their chu;sroom usc These materials wcre prepared with the support of National Science Foundation Grant :viDR-840371 ISBN 1-88Cl13!-39-2 Unit XI • Activity An Introduction to Graphs, Part I Prerequisite Activity Unit IX, Picturing Algebra Materials Red and black bicolored counting pieces, Algebra Pieces, masters and activity sheets as noted Actions Comments Distribute counting pieces to the students Have them form an extended sequence of counting piece arrangements which fits the following data, where n is the arrangement number and v(n) is the value of arrangement n n -2 v(n) -3 -1 -1 1 Various extended sequences of arrangements are possible Shown below are two possibilities with the formulas for v(n) they suggest Be sure the students indicate how they are numbering the arrangements in their sequences Ask the students to write a formula for v(n) Discuss Arrangement number, n: ••• -2 -1 (-1) + 0+1 • ••• • • ••• •••• Value, v(n) (-2)+(-1) 1+ 2+3 v(n)= n+ (n+ 1) Arrangement number, n: Value, v(n) -2 -1 •• ••• •• • ••• ••••• II 2(-2) + 2(-1) + 2(0) + 2(1) + ••• 2(2) + v(n) = 2n+ 1 Unit XI ã Activity â Copyright 1996, The Math Learning Center Actions Comments Give each student a copy of Activity Sheet XI-1-A Have the students form an extended sequence of counting piece arrangements which fits the data displayed in graphical form on the sheet Ask them to determine v(-4), v(-3), v(3) and v(4) for their sequence and, if possible, add this information to their graph Arrangement number, n: -3 -2 -1 Shown below is one extended sequence that fits the data For this sequence, v(-4) = -13, v(-3) = -10, v(3) = and v(4) = 11 In the latter three cases, this information has been added to the graph shown below The value for v(-4) lies outside the range of the graph ••• •• ••• •• • • •• •• • • •• • •••• ••• •• • •• ••• •••• ··m •• ' • :: kt %0:: Value, v(n) -10 -7 -4 -1 v(n) •v v ｾ n Ｍ ｾ - I - I - n v "- a , v(n) Ask the students to find a formula for v(n) for thesequence they constructed in Action and record it in the space provided on the activity sheet Ask the students for their observations about the graph Discuss = 3n-1 v(n) The form of the arrangements shown in Comment suggests the formula v(n) = 3n - I The students may have other formulations Some possible observations: • The points of the graph lie on a straight line • The points are equally spaced • To get from one point to the next, go square to the right and up • The increase from point to point is always the same • There are only points on the graph when n is an integer Continued next page Unit XI • Activity Math and the Mind's Eye Comments Actions ill full full full m full fll fll fll Arrangement number ill full m m m ill full ill full ill ••• •• •• •• •• •• •• Continued It may be instructive to compare the graph with the sequence obtained when the arrangements of the original sequence are rearranged into columns, as shown below The columns contain a minimal number of tile (so no column contains both red and black tile) Black columns extend above a base line and red columns extend below You may want to ask the students how the numbers in their formula for v(n) relate to the graph In the above formula, 3, the coefficient of n, is the amount the height increases as n increases by The constant term, -1, is the value of the Oth arrangement It indicates where the graph coincides with the vertical axis [II [II • -3 -2 -1 Some students may draw a line connecting the points of the graph, implying there are arrangements for non-integral values of n The students may suggest ways for constructing such arrangements (See the next activity, An Introduction to Graphs, Part II.) However, for the extended sequence shown above, there are only points on the graph for integral values of n Show the students the following Algebra Piece arrangement Tell them it is the nth arrangement for an extended sequence of tile patterns Ask the students to form the -3rd to 3rd arrangements of this sequence A transparency master of the arrangement is attached (Master 1, top half) If you use this transparency, not show the students the bottom half of the transparency until they have built several arrangements Below are arrangements number -3 through A master for a transparency of these arrangements is attached (Master 1, bottom half) Recall that a-n-frame contains red tile if n is positive and black tile if n is negative It contains no tile if n is Arrangement number Unit XI • Activity -3 -2 -1 •• ••••• •• •••• •• ••• ••• • •• ••• ••• ••• ••• •••• Math and the Mind's Eye Comments Actions Distribute copies of Activity Sheet XI-1-B to the students For the sequence of Action 4, ask the students to record a formula for v(n ), construct its graph and record their observations about the graph Discuss A master for the Activity Sheet is attached The formula for v(n) can be written in various forms One possibility is v(n) = 4- n Another is v(n) = + (-n) The completed graph is shown below v(n) ﾷ ｾ ov - v u ｾ r- -n - I -·- - - I -2- n I ｾ : I v(n) Repeat Actions and for the following Algebra Piece arrangement, using Activity Sheet XI-1-C in place of Activity Sheet XI-1-B I I v(n) =4- n A transparency master of the arrangement is attached (Master 2, top halt) If you use this transparency, not show the students the bottom half ofthe transparency until they have built several arrangements Below are arrangements number -3 through A master for a transparency of these arrangements is attached (Master 2, bottom halt) Arrangement number -3 ii\HIHI!IIHIHil ••• -2 -1 • • • til Dill • • lllllllll • • llltll • • • llllll • • • tllDtll llllllllllll lllDtll IIHIJIIl • • llltlllll •••••• •••••• •••••• ••••• ••••• •••• lllllllllllllll ﾷ ﾷ ﾷ ｾ ｾ ｾ Ｍ ｾ ｾ ｾ ﾷﾷﾷﾷＭｾｾｾ Continued next page Unit XI • Activity Math and the Mind's Eye Comments Actions Continued Here are two possibilities for v(n): v( n) = n - 2n -3 v(n) = (n + I)(n- 3) Edge pieces may help the students see the latter formulation: v(n) ｾ ｾ ｾ l"v I > 3J II> 3J II> 3J I I I I I I I I I I I I I I 3J 3J 3J 3J II> II> II> II> II> II> v 1(n)=6n-2 v 2(n) = n + 7n- Other formulations are possible For example, v2(n) = (n + 8)(n- I) Edge pieces may help the students see this formulation 3] 3J 3J II> Ｇ , , Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ ｌ ｉ ｾ ｉ Ｑ ｾ Ｑ Ｑ ｾ > 3] Ｑ • Ｍ Ｍ ｊ ｩ ｬ ｬ II II ｾ I I ＺＺｪｾ Math and the Mind's Eye Comments Actions Ask the students to graph v1(n) and v2 (n) on Activity Sheet XI-1-D Have them indicate the points on the graph ofv 1(n) with an X and those on the graph of vin) with an o Ask the students to examine the graphs and record their observations Discuss A master of Activity Sheet XI-1-D is attached The completed graph is shown below It may facilitate discussion to refer to points of the graphs by their coordinates The coordinates of a point of a graph are a pair of numbers the first of which locates the point horizontally, the second vertically For example, (-4, -20), (0, -2) and (4, 22) are coordinates of points on the graph of v(n) 3v ,,., ｾ ｾ v 1(n) ｵ Notice that two points, (2, 10) and (-3, -20), are on both graphs This tells us that the 2nd arrangements of the two sequences have the same value, namely 10, and the -3rd arrangements also have the same value, namely -20 It also tells us the equation 6n- = n + 1n- has two solutions, n = and n = -3 ｾ ,.,, ﾷ ｾ n 1- 0- 9- - - Ｍ ｾ - - n -2 - The students may wish to form the 2nd arrangement of each sequence and verify that they have the same value Likewise for the -3rd arrangement Arrangements number -3 through for both sequences are shown below A transparency of these arrangements is attached (Master 4) _Q_ 1-"' ,., v(n) v2 (n)=n -7n-8 Arrangement number -3 IIHI!Il ill ill ill • • • lfllfllfl ｉ ｬ ｬ ｾ Ill ill lflfil Ill rill!! IIIII illl!ilfllfl l!ilfllfl lflll1lfl 111111111111 II -2 lfllfllfllfl lfl!llllllfl lflllilfllli IIi Ill !filii !llllillilli llilfllfllfl lli!llllllfl lllllilfllfl lfllfl!ll !llllilli llillilli !filii IIi llilfllfl llilll!ll Ill lUI IIlii IIl -1 ill Ill ill Ill 111111 lfllli Unit XI • Activity II Ill Ill Ill Rlli IIlii 111111 Ifill Ill ill !lAllA Ill Ill •••• •••• •••• ••• ••• •• Ill IIi II Ill Ill Ill IIJ Ill •• •• •• •• •• •• •••• •• ••• ••• •••m ••m ••m ﾷﾷ ﾷﾷ ｾｾ ｾｾ ﾷﾷｾｾ ••m ••m •••m •••m ﾷﾷﾷｾｾ •••m ﾷﾷﾷｾｾ •••m ﾷﾷﾷｾｾ •••m ﾷﾷﾷｾｾ •111 ﾷｾｾ •m •m ﾷｾｾ ﾷｾｾ •m •m • ｾ ｲ ｾ !lA ﾷｾｾ ••111 ﾷﾷｾｾ • • ill • • IIJ • • Ill ﾷﾷﾷｾｾ •••111 • • • IIJ Math and the Mind's Eye Comments Actions Ask the students to use the pieces to build the nth arrangement of both extended sequences, v1(n) and vin), and to then solve the equation 6n- = n2 + ?n- using the pieces Ask students to share their solution One possibility is given below (See also Unit IX, Activities and 7) Thus the students will have three different ways to represent the solution to an equation like 6n- = n2 + 7n- 8: using the pieces, using a graph, or comparing arrangements of extended sequences You might point out these three representations to the students and ask them which one they like best at this point, and why 6n-2 NNNNNN I I ｎ ｎ ｾ II II ｭ ｬ ｬ ｬ ｭ •• ｉ Ｑ ｾ ＱＱ ｾ ＱｬﾷＮﾷﾷｦｾＤ Ｍ ｲｾ ｾ II II II I • ifiiD!illfilfili] ｾ II ｾ ｾ ｾ ｾ ｾ ｾ• = ｎ II ｭ Remove n-strips and shaded tiles from each arrangement Then we obtain the equivalent equation: n + n- = n2 + n- n2 + n- -"'IIｾ r-'1 r-111r-'1, I II II I II ｬ ｾ ｬ 11 I 11 I I(_-_:-_ 00 ］ R Ill ill n+3 i II II II Unit XI • Activity ］ Ｍ ｉ ｊ Ｂ Ｑ II II ｾ ｾ ｾ ｾ II II 00 I I < lJIJIJ I II< lJIIII I II< lJIJII n-2 I I 99 =0 Note: We can complete a rectangle by including opposite frames and more n-frames in our arrangement and maintain a net value of n2 + n- 199 • ｡ Ｚ Ｍ ｾ ｾ ｾ II This rectangle has edge piece values n - and n + Thus, (n - 2)(n + 3) = when n =-3 or n =2 (The area is if and only if one of the edges has value 0.) II< lJIJ IJ I ｾ lJIJII I ｾ _{]1111 Math and the Mind's Eye Comments Actions 10 (Optional) Show the students the following portions of extended sequences I and II Ask them to write formulas for v1(n ), the value of the nth arrangement of sequence I and vin), the value of the nth arrangement of sequence II Then have the students graph v 1(n) and vin) on Activity Sheet XI1-E Ask the students to record their observations Discuss Arrangement number -3 -2 •• ｉ ｬ ｬ ｾ ｾ ﾷ II -1 •• m••• ••• •••• ••••• •••••• From sequence I, one sees that v 1(n) = n + The students may readily see the pattern of sequence II, but have difficulty in writing a formula for vz(n) You can suggest to the students that they write the formula in two parts, one part for non-negative arrangement numbers and one part for negative arrangement numbers: Ill Ill •rm ••m •m •••m ••m Ill • • • • • • !Bl 10 Masters for Activity Sheet Xl-1-E and a transparency of the two extended sequences (Master 5) are attached Ill I] Ill Ill Ill •m ••m •••m •m ••m •••m I] ' Ill 2n- 3, n non-negative { vin) = Ill 2(-n)- 3, n negative ••• Notice that the case n = is included in the first part You may want to tell the students about the mathematical symbollnl, read absolute value of n, which is defined to be n, if n is non-negative (e.g., 131 = 3) and -n if n is negative (e.g., 1-31 = 3) Using this symbol, one has vz(n) = 21nl- for all n The graphs ofv 1(n) and vz(n) are shown below The points of the graph ofv 1(n) lie on a straight line The points of the graph of vz(n) lie on a V whose vertex is the point (0, -3) Note that the points (-2, 1) and (6, 9) lie on both graphs v n u ｾ u " n - - -p- ｾ - n Ｍ ｾ " fl v(n) X Unit XI • Activity v, (n) = n + o v2 (n) ={ 2n- 3, n pos or -2n- 3, n neg Math and the Mind's Eye -I : I ｉ Ｍ Ｍ Ｍ ｾ I I I I I ｾ L - I : T-Ｍ I I ｾ - - I Ｍ ｾ I I Ｍ Ｍ T-Ｍ I I -I- - I r- - - Ｍ ｾ I I I I t- - - Ｍ ｾ I I - - - I - - I I I_ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ r , Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ T-Ｍ Ｍ ｾ Ｍ Ｍ T-Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｉ -j - I I "- - - I I - - I - - Ｍ ｾ I I : I I _I_ I - -1- - - I I I I ｾ - - - :- - I I I _I_ I - - -I- - - I I I ｾ - - - :- - - I l - - I I -I_ I I I I ｾ - - I I I - - I I ｾ - - I I J - - - L - - - I I I ｾ - - -: - - - - - I I I _j I I I I - ｾ I I L - - - I I - - -:- - I _I- I I I I ｾ I L - - - I I _I_ I I I I I I - - -:- I I I l - - - I I _I_ I I - - - - I I I t- - - - I I - - I - - Ｍ ｾ I I -t I I I I - - I - - - -t - - - 1- - - -t - - - 1- - - -I - - - t - - - - - - - t - - - - - - - - t - - - - - - - t - - I I I I - - I - - - ｾ I I I I - - I - - - I - - ｾ I I I I I Ｍ ｾ - - - I - - I I I Ｍ ｾ I I - - - I - - Ｍ ｾ I I I I I I - - I - - Ｍ ｾ I I I I I - - I - - - I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I - - - - - - - + - - - - + - - - - - + - - - I - - - - + - - - r - - - i - - - - - - - - - - - + - - - - + - - - - - + - - I I I I I I Ｍ ｾ ｾ I I I I I I I - - -:- - - ! 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_ I _ J _ - I I I I I I _: - ｾ Ｍ Ｍ _:_- -l _:_- -1 -: I L - - _I_ - - ! - - _I_ - - l - - _I_ - - - - -I_ - - J - - - L - - - ｾ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ I : - I I - - T - - I I I I - I I -+ I - - - r - - -1 - - r - - - I -1- I I - +- - - -1- - - + - - -I- - I -+ - - - - I I I I e -l , , I I - - - - - t- - ｾ -1- - - +- - - -1- I ｴ Ｍ Ｍ Ｍ I ｾ Ｍ Ｍ Ｍ Ｑ : I _J _ _ _ I 1 : - - Ｍ Ｍ Ｍ Ｍ Ｑ Ｍ Ｍ Ｍ Ｍ - -+ I I Ｍ Ｍ ｾ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ ｾ Ｍ - Ｍ - 1- I I Ｍ Ｍ Ｍ - Ｍ -+ I Ｌ - I I I 1- -1- + I 1 I I I I -1- j_ I 1 I - - Ｍ ｾ - I Ｍ - Ｍ e - - - Ｍ I ｾ Ｍ Ｍ i - ｾ Ｍ I - - T - - Ｍ ｾ I I I I I I I - T - I I I I -1- I I I Ｍ ｾ - I -+ I I I - I I I - I I I I I -1- J _ _ _ t- I I I - - - 1- - I I I I I I I I I I I I Ｍ I I I I I I I I Ｍ Ｍ -1- I - Ｍ ｾ Ｍ Ｍ Ｍ Ｍ I I I I t- -1- + I I I I I I I I I I + -I- - - - - - + - - -I I I I - - - I - - Ｍ ｾ I +- - - I _J _ _ _ I I I ｾ - I J _ _ _ L _ I _ _ _ L _ I _ l _ _ _ I _ l _ I _ J _ _ _ L J - I 1- I - I I I r - - I - - - - - Ｍ I I 1 -1 - : I Ｍ -I- - I ｾ Ｍ +- - - I Ｍ Ｍ Ｍ Ｍ ｲ Ｍ I I I -1- + I I I I - - I - - Ｍ ｾ I I Ｍ I I Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ I I -1- -I I I I I I - - T - - Ｍ ｾ I I I I - I 1 I - - - - I - _ L - - _I _ - _ -1.- _ _ _ I _ l _ _ _ I _ J _ _ _ I I I I I I I I I : I - - , - - - I : I I I I I I I I I I r- _I _ _ _ L I- - - ｾ - - I - Ｍ I - - - ｾ Ｍ - - T - ｾ I I I I I I _ I _ _ I _ l -,- - - ｾ I I I I I I - - - ,_ - - ｾ I I I Ｍ I I _ I _ I I I I I - - - '- - - - r - - , - - - - - - - - 1- - - - - ｾ - - - I I I I I _ I I I I I I J _ - - - - - - - - - I I I I I I I L _j _ _ _ L I I I I I I J _ - - - _, - - 1-cm grid paper - ｾ I I I Ｍ - -, - Ｍ - - T - ｾ I I I I I _ I _ L _ I _ l I I I - - - - ｾ I - - ｾ I I I - - - ,- - - ｾ Ｍ _ I - - - '- - - - - - - ｾ I I I I _ _l _ _ _ I I - - I - - - - I - - - y '' / 'u 10 / .-o I"''' / ., TU / - X -ｾ - I -j -ｾ Ｍ ｾ Ｍ ｾ / l7 v v -v I v -2 v I u 11 ｾ , ｾ ｾ I ｾ X ｾｾ ｾ I I I I Xl-3 Master v I I v v ·o ·g IU " -ftl: ' , 'I U ,I g "tl:.U ,, y ©1996, The Math Learning Center v{x)= -3x+ v(x) , ｾ I I \ \ IU ;:J \: \ ｾ " A ｾ ｾ ｾ (0, 5) \ -" \ \ (1, 2) I X Ｍ ｾ -ｾ -j ｾ -) -· Ｍ Ｎ ｾ Ｍ ｾ \ -1 ｾ ｾ I -, ｾ , ｪｾ ｾ ｾ X (2, -1) \ A Ｍ ｾ , ｾ \ Ｍ Ｎ Ｎ ［ ｾ Ｍ ｾ ｾ ' ｾ \ •g ,- -o - -;:J IU I I v(x) Xl-4 Master ©1996, The Math Learning Center Xl-4 Master 1) Y= -3X+ 3) Y= X+ 2)y=-X+5 4) Y= 3x+ 1) Y= -3X+ 3) Y= -3X+ 2) Y= -3x- 4) Y= -3x- ©1996, The Math Learning Center v(x) = x2- v(x) (-4, 12) (4, 12) I I IU .:::1 g \ I \\ , J u (-3, 5) I (3, 5) ｾ 1\ J "'' \ \ \ ｾ I I " v I (2, 0) (-2, 0) \ X -$ Ｍ ｾ -$ ｾ _, -$ Ｍ ｾ Ｍ ｾ \ \ ｴｾ 'I $ ｾ ｾｉﾻ II» IP s X I v , K: / -?) (0, Ｍ ｾ , ·u - 'I ·o • ［ ｴ ｾ , 'IU I I v(x) Xl-4 Master ©1996, The Math Learning Center 3) y = -(x2- 6) 1) Y= 4x2 Xl-4 Master 3) Y= -4x ©1996, The Math Learning Center v(x) X v(x) Xl-4 Coordinate Graph Paper ©1996, The Math Learning Center Xl-5 Master A) Y1 =4 + 2x Y2 =X+ B) Y1 =4- x Y2 =- + x C) Y1 =3x-2 Y2 =3X+ D) Y1 = 2x+ Y2 =4x 2- 3x+ ©1996, The Math Learning Center A man and his child are racing one another on a track The man can run 20 meters in seconds His child can run 20 meters in seconds They decide to give the child a 30 meter head start Make up some questions based on this situation Share and record your questions in your group Xl-5 Master ©1996, The Math Learning Center child seconds meters 10 15 30 30 + 20 =50 70 90 child seconds meters Xl-5 Master man seconds meters 20 40 60 man seconds meters 30 34 38 42 6213 131Ja 20 10 11 12 13 14 15 16 17 18 70 74 78 82 86 90 94 98 102 10 11 12 13 14 15 60 662/a 731Ja 80 861Ja 932/a 100 ©1996, The Math Learning Center - - -1- - - i - - Ｍ ｾ - - I - - Ｍ ｾ - - I - - - 1- - - I - - - r - - I - - - r - - -1- - - i - - Ｍ ｾ - - I - - - I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 - - - J - - _ L - - J - L _ _ _I _ L _ I _ l_ _ _ _ I _ l _ I_ _ _ J _ I _ J _ L J - L - - -1 I - - I - - - I I I I I I I I I I I I 1- - - - - - r - - - - - r - - , - - - r - I I I I I I I J I I_-I J - L-I I Ｍ I ｾ Ｍ Ｍ I T-Ｍ I Ｍ I I ｾ I I Ｍ Ｍ I I I Ｍ ｾ I I Ｍ Ｍ I I I I I I I I J I I I I I I I I I I I I I I I I I - I - - I I I I I _ I _ - - L - - -1 I I r - - I- - J - L _I_-I I I I I - - -t - - - r - - I - - - r - - I - - - t- - - -1- - - t- - - -1- - - -t - - -1- - - -t - - - r - - I I I -r - r - - - , - - - r - - , - - - r - - - Ｍ _ I _ - - L - - _ I _ - - l - - _1_ l - - _ I _ - I I I I I I I I I I I I I I I I T-Ｍ I I I t- - - - - I I I I _I _ I _ _I _ I I 160 I I I I I - - -f - - - f- - - -1 - - I I I I I - I I I I f- I I I I - - _I _ !_ _ I _ ｾ ｉ I +- - - -1- - - - I I I Ｍ Ｍ Ｍ I -1- I I I I I ｾ Ｍ Ｍ Ｍ ｉ Ｍ I Ｍ Ｍ ｾ Ｍ Ｍ I +- - - I I I I I +- - -I- - I I I -I- I I I _I _ I _ _I _ _! _ _ _ I _ _! _ _ _ I _ I I I I I I - - -1 - - - f- - - -1 - I I I - f- I I I -1- - - l- - - I I I I I I I I I +- - - I I -1 I I I I I T-Ｍ Ｍ +- - - -1- - - I I I -t - - - f- - - -1 - - - f- - - -1- - - - I ｾ Ｍ Ｍ Ｍ I ｉ Ｍ Ｍ Ｍ I ｾ Ｍ Ｍ Ｍ I ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ I Ｍ I I I I Ｍ Ｍ Ｍ ｾ Ｍ Ｍ I Ｍ I I +- - -I- - - I ｉ I 140 _I _ !_ -1 ｾ - I - - - -l - - I I I - f- I I I I - -1 - - - f- I I I -1- I - l- - - -1 - I I I Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ I 120 I I I I I I L I I I I I I I - - J - - - L - - J - - - L - - 1- - - I - -1- I - _j_ - -1- I - I - -1- I !- - I J - - - L - - J - - - L - - 1- - - L - - - - I I I I I I I I I I I I I - - I - - - I - - I - - - I - - I - - - I - Ｍ I I - - T - ｾ Ｍ I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I - - ｾ - - - ｾ - - ｾ - - - ｾ - - -:- - - ; - - -:- - - ｾ - - -:- - - ｾ - - -:- - - ｾ - - - ｾ - - ｾ - - -· - - ｾ 100 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I CJ I - - ｾ - - - ｾ - - ｾ - - - ｾ - - ｾ - - - ｾ - - -:- - - ｾ - - -:- - - ｾ - - - :- - - ｾ - - - ｾ - - ｾ U) a: w 80 1w - - ｾ - - - :- - I : ｾ - - ｾ - I ｾ - ｾ :E - I - - -: - I - - I ｾ - - : : : - ｾ I - : - -:- - I -: - ｾ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ : -:- - I : - I ｾ - - - :- - - It;! J_ ｾ I • - - ｾ I - -: - - - I cp I I I I ｾ - - -: - I ｾ Ｍ Ｍ ｾ Ｍ Ｍ : I m I - -Q- - -: I I I I I - -£:3 - - , - - - ｾ - - -: I - 1- - I : m • ｾ -: -liJ Ｍ ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ I - - T - - - 1- - - I - - - I - - I - - - I - - I - - - I - - -I ｾ - : - - ｾ I - ｾ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ : - -: I : I - - -: : : ｾ ｾ Ｚ ｾ ｾ ｾ :c ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ ｾ ｾ ｾ ｴ ｩ ｬ ｾ ｾ ｾ ｾ ｾ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ ｾ ｾ ｾ Ｚ 60 ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ @- - - + - _: -! _:_ ｾ Ｍ Ｍ Ｍ CJ : I I I.J.J - - -l I - I • : I " :_- - ｾ I I Ｍ Ｍ Ｍ ｾ I ｾ I Ｍ Ｍ Ｍ ｾ I ｾ I I f- I - - I -1 I - I f- - - • - • I I I -1- - - l- - - -1- - I I I I +- - I -I- - - I I I +- - I I I -I- - - -t - - - f- - - -1 - I I I I - - I I f- I I - -1- I I I I I I I I I I I I I I I - -l - - - • • - Man L I - -1 I - - L I I - -I- I - L I I I - -1- I - !- I I I £I; I I I I I I , I , , I D - Child - I '- - - L _J - L _j - , I I I I I SECONDS - -1- I - !- I - -1- I - -l I - - L I - J - I I - l- - - -1 I I I - - I I L I -: - ｾ Ｍ Ｍ -: ｾ Ｍ Ｍ I • - - ｾ -: I ｾ Ｍ Ｍ Ｍ : -· - ｾ Ｍ Ｍ -: - ｾ Ｍ Ｍ -: - + -: - + -: - ｾ Ｍ Ｍ Ｍ : - ｾ Ｍ Ｍ Ｍ 20 Ｍ Ｍ I -9 -: -: ｾ Ｍ Ｍ -: -: -: -: -: -: -: - ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ -: - ｾ Ｍ Ｍ -: j, ｾ 40 I I - -I- I I I - I L _ I I -_I_ I I j_ _ I _ - - J - - - L - - _ J - - - L - - _ j - - - L - - -1 I I I I I I I I I I I I I I I ' - - - I - Xl-5 Master - - - - - - ｾ - - .- - - _,_ - - _,- - - - - - ,_ - - - - - ,_ - - _ - - - , - - , - - - , _ - - _ - - - 1- - - - ©1996, The Math Learning Center Miles Driven Cost RAW Cost WHT 10 20 $1.50 $3.00 $11 $12 50 $7.50 $15 $15.50 $20 Graph Key e100 Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ ' ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ' ｾ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ ｲ Ｍ l - - - ｾ - I I I - 1- - I - - ｾ - - - - - 1- - I -l I - I I I I I I I - - - - I - - I Ｍ - I ｾ Ｍ Ｍ Ｍ Ｍ Ｍ ' ｾ Ｍ Ｍ Ｍ Ｍ ' Ｍ Ｍ Ｑ Ｍ ｉ Ｍ Ｍ Ｍ ｲ ' Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ ｾ Ｍ I I I I I I I I I I I I I j - Ｍ Ｍ -I- I I I - I I I T - - - Ｍ - - j - L I - ｾ - - T- - 1- I I I 1- - - I I I I - - - - T - - - ｾ I I -1 I I I I I I I I _j - L _I I I I - - -I_- I Ｍ Ｍ ｾ Ｍ Ｍ Ｍ I I -1- I I - ｾ Ｍ Ｍ - - I - Ｍ - I I I 1- - - ｾ - - 1- - - I I I - - - -l I I I I I I - - - I - I I - L - I - - I I I I I I I Ｍ ｾ - - T - I L I I T - I - • I - l- I I I I 1- - - -1 - I I - I I I I I I Ｍ ｾ - - 1 L I I I - -1 I I I I I I I I I I I I _I - I I _ L -1 - I I I I I I I I I I I I I I I -r - - - r - - , - - - r - - , - - - r - - - - I I -1 - - I - - - 1- - - I - - - I - - -1- - - I - - - I Ｍ I I - - - - - - I - - Ｍ ｾ I I -I - I I I I ｾ ｾ I I .! - - I Ｍ - I - - - - - 1- I T -Ｍ -1 I I I I I l I I I I Ｍ - I I I I I I I I I I I I I I I I I I I I I I I I I I _ ·- _j _ _ _ L _ _ _I _ _ _ L _ _ _ I _ _ _ l _ I_ _ _ l _ I_ _ _ I _ I_ _ _ _j _ _ _ L _ - I I_ _ _ - I I L I I I -1 - I I I -I- - - - - r - - , - - - r - - , - - - - - Ｍ ｾ :-$-29 Ｍ I - I - I - I I -I- I I _ L I l- I - I I 1- - I - - - - I - - Ｍ ｾ I -I I L I I - l - - - I - - I - - I - Ｍ -Cost of RAW I I I 1-$39 Ｍ Cost of WHT _I_ - - l - - _I_ - - l- - _ I _ - - ! - - -I_ - - ! 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I ｟ I I I ｾ Ｍ Ｍ I Ｍ Ｍ Ｎ ｟ 1SO ' I I I I I MILES DRIVEN - - - - - - - - - - - - - - - - - - - Xl-5 Master ©1996, The Math Learning Center seconds X 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Xl-5 Master height in feet Y1 80x-16x2 = 36 64 84 96 100 96 84 64 36 -44 Y2 ' elevated tee height in feet 80x-16x2 + 20 = 20 56 84 104 116 120 116 104 84 56 20 -24 ©1996, The Math Learning Center Name Activity Sheet XI-S - Pose some questions about each of the following situations and then answer them using any approach that you wish Write up a problem solving summary for each situation 1) The Rent-a-Wreck (RAW) and the We Hardly Try (WHT) car rental companies are competing for business and change their prices as follows: WTH has an initial rental charge of $10 and then charges $.1 0/mile RAW eliminates the initial rental charge altogether but charges $.15/mile 2) The Saucy Pizza company is currently charging $7 for their pizzas The ingredients and labor for each pizza they make cost $2.50 Each day that the company operates costs them $100 in overhead (lights, water, heat, rent, etc.) 3) The U-Drive-lt golf range claims that the height (H) in feet above the ground of a golf ball at ( T) seconds after their pro hits it can be determined from the expression H= BOT -16T2 They also claim that when the ball is hit from their elevated loft tees that the height can be determined by H= BOT -16T2 + 20 ©1996, The Math Learning Center ; -I I I ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ T-Ｍ Ｍ ｾ Ｍ Ｍ T-Ｍ Ｍ ｾ I I Ｍ Ｍ Ｍ ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ "l - I I Ｍ ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ T-Ｍ Ｍ ｾ Ｍ Ｍ T-Ｍ Ｍ ｾ Ｍ Ｍ T -I Ｍ I I I I I I I I I I _j _ _ _ L _ I _ L _ I _ l _ I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I _ l _ I _ ! _ L _j _ _ _ L _ I _ L _ I _ _ _ j_ _ _ _ I _ l - - L I - r Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ I I I r- I I I I ! _ L I I L-I ｾ I I I -I I I _! _ _ _ ｾ I I I I I - - - -1 : I I I 1- I _ I I - - - I I : I I - Ｍ ｾ Ｍ Ｍ Ｍ I I I - - I l I I I I - - - T Ｍ ｾ Ｍ Ｍ Ｍ I I I I _ I _ _ _ l I ｾ I L I I I I _ I _ I I -: I ｾ T Ｍ I I I I I _ I _ I I - - -: I I I I I - - - - I 1" I I I_ _ I _ _ _ I I -: - I I I J _ I I I _ l ! ｾ I I I I I I I ｾ Ｍ Ｍ I I 1_ _ _ - - -: I j I I I I I I I I I J _ L I I I I ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ ｾ I j _ L L I I I I ｾ I r Ｍ ｾ Ｍ Ｍ Ｍ I I I I _ I _ l I I I Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ I I ! _ I "l - I I - ｉ I I I ｾ I I T Ｍ I I I I _ I _ _ _ I I - -: - I ｾ I ｾ Ｍ Ｍ Ｍ ｉ Ｍ Ｍ -I I I I I l _ I I ｾ - -:- - - I I I _ I _ I I I 1 I ｾ Ｍ Ｍ - - -:- - - -: I I I L l _ j _ I I I I I I I I I I I l I I I I I + - - - I - - - - + - - - I - - - - l - - - - - - - - - - - - - - - - - + - - - - I I I I I I I I I I I I I I I I I I I I I !_ _ 1_ _ _ !_ _ I _ ｾ _ I _ ! _ I_ I _ I_ _ I _ ｾ _ _ _ 1_ _ _ I I I I I I I I I I I I I I I I + I I I - I I I -1 I I I I I I I + - - - - - + - - -I I I I I I I !_ _ I _ _ _ ｾ _ I I I I I I - - - - - - - - - - - - - - - I - - - - + - - - I - - - - - - - - - - - - - - - - - I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ T - - Ｍ ｾ Ｍ Ｍ Ｍ T Ｍ ｾ Ｍ Ｍ Ｍ ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ I - - - ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｉ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ I - - - - - - - - I - - - - + - - -I I I I I I I I I I I I I I I T - - Ｍ ｾ Ｍ Ｍ Ｍ T Ｍ ｾ Ｍ Ｍ Ｍ ｉ Ｍ Ｍ Ｍ ｾ I I I L I I I - -1- I I I -1 - - I - L - - I I I T - - -1- - I I I Ｍ ｾ - - I -l- - -1- I I I - 1- - - I I I T - - I .! - I I I I - - I - 1- - I I I - 1- - - I .! 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I I I I I I I I I I I I I I L I- - : I I r Ｍ ｾ - - I T Ｍ ｾ I I - - T- - Ｍ ｾ I Ｍ Ｍ ｉ I Ｍ Ｍ Ｍ ｾ I Ｍ I I I I I I I I I _j _ _ _ L _ I _ L _ I _ l _ I _ l _ L I I I I I L I - r Ｍ I : I I I I _ I _ I I I I I I ｾ Ｍ Ｍ Ｍ I I I I I T Ｍ ｾ Ｍ Ｍ Ｍ I I I I I I T Ｍ ｾ Ｍ Ｍ Ｍ I I I I I I I L _ I _ j_ _ _ _ I _ j_ _ _ _ I _ I I I I I I I I I I I I I I "T Ｍ I I Ｍ Ｍ I I I I -I I - I I Ｍ ｾ Ｍ Ｍ Ｍ I T - - Ｍ ｾ I I T Ｍ ｾ Ｍ Ｍ Ｍ I I I I ｾ Ｍ Ｍ Ｍ ｉ Ｍ Ｍ Ｍ I I I I I I I I - r Ｍ I I I ｾ Ｍ Ｍ Ｍ I T Ｍ ｾ Ｍ Ｍ Ｍ I I I I I I I I I l" - I I I I I : I I I I l _ L _j _ _ _ L _ I _ L _ I _ _ _ l _ I _ l _ I I I I I I I I I I I T Ｍ ｾ Ｍ Ｍ Ｍ I I I "T - I l _ I _ J _ L I I I I I I I I _j _ _ _ L _ I _ L _ _ _ _ _ _ I I I I I I l l _ _ _ I ｉ ｟ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ ﾷ Ｍ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ 1-cm grid paper 1 ... The Math Learning Center II Xl-1 Master ©1996, The Math Learning Center • • • • • • M N 111 1111 1111 1 111 1111 1111 1 111 1111 1111 1 111 1111 1111 1111 1111 111 1111 1111 1111 1111 1111 11 111 1111 1111 1111 1111 111. .. 111 1111 1111 1111 1111 111 1111 1111 1111 1111 1111 11 111 1111 1111 1 111 1111 1111 1 111 1111 1111 1111 1111 1 111 1111 1111 1111 1111 1 , 111 1111 1111 1 rill , I • 111 1 111 1 111 1 C1) c E :::s r::: r::: C1) E C1) 0)... ©1996, The Math Learning Center • • • • • • 111 111 II II II M II 111 1 111 1 111 1 111 111 II II N 111 1 111 1 , 111 111 1111 II 111 111 rJ , 111 111 I NI 111 1 111 111 llrJII 111 111 B B 111 1 111 1 Q)