CONCEPTS SEC MATH
CONCEPTS SEC MATH EMAT 3500
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This 42 page Class Notes was uploaded by Shirley Spencer on Saturday September 12, 2015. The Class Notes belongs to EMAT 3500 at University of Georgia taught by Olive in Fall. Since its upload, it has received 58 views. For similar materials see /class/202320/emat-3500-university-of-georgia in Mathematics Education at University of Georgia.
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Finding Roots Maximum Rectangle Area Vim will need Quadratic functions can model relationships other than projectile motion In Mam this activity you will nd an equation relating the area of a rectangle to its Width You will also look at realiworld meanings for the xiintercepts and the vertex of a parabola MAKE A CONJECTURE Suppose you have 24 meters of fencing material and you Want to use it to enclose a rectangular space for your vegetable garden 01 What dimensions should you use for your garden to have the largest area possible for your vegetables INVESTIGATE 1 Open the TliNspire document Maximum Rectangle Areatns on your handheld and go to page 12 Em thls zmble gi he 2 You should see a rectangle with a xed perimeter of 24 centimeters Drag vertex an e roun s lenglhs and mm in C or D to see different dimensions of the rectangle Record the lengths and Whole numbers Widths on page 13 Get at least eight different rectangles It is okay to have Widths that are greater than their corresponding lengths l1ii21314gtiuo AUTO REAL A 5 Widths lengths D C E gfh8 cm k Wldlll4 m pznmtlzv24 cm 3 Calculate the area of each rectangle 02 From the table What is your guess for the largest area El 4 Go to page 14 to see a scatter plot ofyour widths areas data 03 From the scatter plot view has your guess changed If so What your new guess Exploring Algebrai Wllh TerspireW 5 Ouadralios and Higheeregree Polynomials 105 z 09 Key Cuiiicuium Pres Finding Roots Maximum Rectangle Area contlnued Try to gel somevery small and very large 106 In the next problem you will use the same rectangle but the lengths widths and areas will be captured automatically when you drag the vertex Values ol with Box 4 Write an expression for lengths in terms of widths You can determine this algebraically or use the scatter plot of wid Zen data on page 24 To nd the slope for your expression using two points on the scatter plot doubleeclick the xecoordinate of the given point and enter a new width The cursor will jump to the nearest point on the scatter plot Go to page 23 to see the scatter plot ofthese widths areas data 5 Go to page 21 and drag vertex C or D around to gather many data points As you drag the data will be captured on page 22 RAD AUTO REAL I15 Using your expression for the length from Q4 write an equation for the area of the garden in terms of the width l Go back to page 23 and enter this equation on the scatter plot choose Text from the Actions menu and click in an empty space to open a text box Type your equation using x and y and press Press to put the text tool away then drag the equation to an axis Press to draw the graph 13 a Trace to nd the exact largest area V a a o 21 22 23 241 RAD AUTO REAL M 40 Widlfrsmreas 3910 20 choose Graph Trace from the Trace menu Place a point on the graph you may have to press A or V to trace the graph instead of the scatter plot Press to put the trace tool away Drag the point toward the top of the graph until an M for maximum appears What is the maximum value of area At what width does this occur I 4 Locate the points where the graph crosses the xeaxis by doubleeclicking the yecoordinate of the trace point and entering a new one To get the other xeintercept move toward it and repeat the process 3918 Explain the meaning of the xeintercepts in this situation 5 Ouadrancs and ngheeregree Polynomlals Explorlng Algebr cm a l Wlth TersplreW 009 Key Cuvmulum Pres Finding Roots Maximum Rectangle Area lam rl f7 m Objectives Students will write equations that model data from a geometric situationr Activity Time 25 minutes Materials Maximum Rectangle Area tns Mathematics Prerequisites Students should have some number sense involved with measurement and drawing of rectangles they should be familiar with nding the perimeter and area of a rectangle TliNspire Prerequisites Students should be able to open and navigate TIrNspire documents graph functions and enter data See the Tip Sheets TliNspire Skills Students will trace functionsr Notes This activity can be done in multiple ways depending on the skill level of students and amount of time you allocater Before students open the TIrNspire document you might have them ll out a mble of lengths widths and areas by hand If students have trouble with this part they can also draw the rectangles Use 247cm lengths of string for kinesthetic learnersr A width or length of zero is not acceptable as a measurement but these are useful values to list Regardless of how students get the data have them nd and record the lengthwidth and area of several possible rectanglesr Students could also graph the data by hand instead of using the scatter plot on page 1A of the TIrNspire documentr MAKE A CONJECTURE 01 There could be a variety of guesses for the maximum area INVESTIGATE 2 When students are dragging the vertex of the rectangle to make it change make sure they don t stop when the width becomes greater than the length If they stop too soonthey will only get half of the parabola Instead of referring to the values as strictly length and width you might refer to them as two consecutive sides 3 Students can calculate the areas by hand or by typing the formula into the formula cell for column Cr Exploiiiig Algebra l With TerSplleTM 2009 Key Cuiiicuium Piass Activity Notes Flanami hk anrlF vi Kami bk 2 The guess will be determined by which values students used as widthsr Sample data 5 5 5 i3 i 5775 i l 35 7 A 8727 8 3 9 21 C l35 03 This answer might change depending on how close the values are that were chosen from earlier stepsr iiiiiziia mionng 3 AU V wdthur2u rm 5 The recmngle s dimensions of length and width are shown rounded to the nearest whole integer although the able captures more exact values 04 length 12 widthr Some students might bene t from solving the equation Zlength Zwidth 24 for length Make sure that the pattern makes sense to students 05 area width 12 width or area 12width widt 06 The maximum area is 36 cm2 at a width of 6 cmr The 6 by 6 rectangle is actually a square 07 0 0 and 12 0 08 The rectangle has no area if the width is 0 cm or 5 Quadratics arid HigheirDegiee Polyiiomials 107 Finding Roots Maximum Rectangle Area Activity Notes continued DISCUSSION QUESTIONS EXTENSION How will the equation for we change if he Have students graph other relationships in the dam such perimeter is changed as widths perimeters lengths areas and so on and nd HOW can you nd the vertex of the parabola ifyou equations for these relationships Students will need to know only the Xiimercepts create a new list for perimeters What does the vertex of the parabola have to do with the maximum of the function Is the maximum area the best answer to the original problem What assumptions have been made about the situation 108 5 Quadratics and Higheeregree Poiynomiais Expioring Aigebrai with TirNspireTM 2009 Key Curricuium Press Factored Form Roots and Lines INVESTIGATE Exploring Algebrai Wllh TersplreW 2009 Key Cumculum Press The factored form ofa quadratic equation is y ax rl x r2 The form helps identify the roots of the equation rl and r2 This activity will help you discover connections between a quadratic equation in factored form and its graph 1 Open a new document on your handheld and add a Graphs amp Geometry page Equot Graph the equations flx x i 4 and fzx x 3 w Find the xiintercepts of each equation choose Intersecu39on Points from the Points 8 Lines menu Place a point at the intersection of each line with the xiaxis Press to put the tool away 01 What is the xiintercept of each equation 4 Now graph the equation f3x x x 02 Describe this graph m FAD AUTO REAL g 03 The xiintercepts of the parabola look 21 3 1 29 y very close to the xiintercepts of the ghawnw lines Are they the same Use tracing to 2 it nd out choose Graph Trace from the 20 736 40 20 Trace menu Press V until the trace point is on the parabola When you are at the intercept you ll see a Z on the 1339 k screen Are the intercepts the same as those of the lines 04 Make a conjecture about the roots of a quadratic equation and the xiintercepts of its factors 5 To test your conjecture translate RAD AUTO REAL RAD AUTO REAL and rotate the lines To translate a J V line grab it near the middle To 0 rotate it grab it near the end z 7 05 Was your conJecture r1ght Rev1se 7 4 0 20 4 0 20 it if not 06 When does the parabola have only one xiintercept o q Can you move the lines so that the parabola has no xiintercepts Explain o 9 Explain anything else you notice about the relationship between the parabola and the lines 5 Ouadrancs and Higheeregree Polynomials 109 Factored Form Roots and Lines Activity Notes Fll nw mi l ik nrl Fri Kami hk Objectives Students will learn that the roots of a quadratic 05 Students may or may not have been right before equation can be found from its factored form They will Watch for confusion when students are rotating lines explore the relationship between the factors ofa quadratic This introduces the multiplier 91 of the factored form equation and its graph but the xrintercepts are still the same Encourage Activity Time 20 minutes students to doublecheck by solving the equations of the lines for the xrintercepts Malena None 06 The parabola has only one xrintercept when the lines Mathematics Prerequisites Students should be familiar have the same xrintercept with quadratic equations vertex form and general form 07 You cannot create apambola Wim no xiimemepm x im c epts Md the concept of mom using two lines as factors Explanations might include TliNspire Prerequisites Students should be able to 18 fact mat 18 lines 83d MW 0 H055 U18 xraXiS make a new document and graph equa 0n5 see the Tip somewhere unless they are horizontal in which case Sheets multiplying them results in a constant or a linear equation not a quadratic equation TliNspire Skills Students w111 construct intersection points trace graphs and translate and rotate lines a on Students may note that moving the lines close together makes anarrow parabola and moving them Notes You might do this activity as awholerclass far apart makes a wide one They may possibly also presentation or have students work in pairs notice that the parabola opens down if the lines intersect above the Joan39s INVESTIGATE 01 f x 73 f2x 4 DISCUSSION QUESTIONS 2 The graph is a parabola 39 At some point through the investigation ask students 03 The intermpm are me same x 73 and x 4 to name two numbers that have the product of zero Entertain all ideas being sure that products of 04 The roots ofa quadratic equation are the same as the opposites or reciprocals are rejected Introduce the xrintercepts of its factors term murpmductp39roperty 5 You might encourage students to translate the lines What can you conclude about xifyou know mat rst then rotate them If students move the lines so x 3x 7 4 that the parabola disappears they can drag the axes inward until the parabola reappears 110 5 Ouadratics and Higheeregree Polynomials Exploring Algebra l With TerspireW 2009 Key Cuiilculum Pres Quadratic Motion Rolling Gan Van will need motion sensor empty collee can or large paper roll long table EXPERIMENT Exploring Algebra1 Wllh TersplreW 2009 Key Cumculum Press In this activity you will do an experiment to nd a quadratic function to model data You will collect parabolic data and then nd an equation in vertex form that matches the graph in Open a new document on your Move the cursor to the play button in The data collected by the sensor will Set up the experiment as shown Prop up one end of the table slightly Position the motion sensor at the high end of the table and aim it toward the low end Practice rolling the can up the table directly in front of the sensor The can should roll up the table stop about 2 feet from the sensor and then roll back down handheld Plug the handheld into the sensor As soon as it is plugged in you will see a screen similar to the one shown here the small window When you are ready 35 4 3 to collect data click the play button and gently roll the can up the table Catch the can as it falls off the table The sensor should stop collecting data after 5 seconds AUTO REAL have the form time distance It is collected into two lists named runatimej and runadistim If you did the experiment correctly you should see a parabolic pattern in the graph If you need to repeat the experiment click the play button again and choose OK to rewrite the data 5 Ouadrancs and Higheeregree Polynomials A run run 111 uadlalic Motion Rolling Rn mmmmd INVESTIGATE WWWW n1 Use the table to nd the vertexofa parabola that t yourdata Ifyou need to Wan 322 mm quot W arrow up to select the column choose Resin from the Actions menu then choose ColumnWidth Press p to expand the column then press 5 Graphyourequauon Choose39l extfmm theAc ons menu P163510 o e the textbox Type yourequatron and pres agarn Then drag the equauon to an 2x1 to graph rt n2 How well does your funcuon t the data mt doesn t t well try draggrng rt to adjust rt What 1 your equatron n1 What 1 the rvalue lfx 7 5 Explarn what thr pornt mean 111 word EXPLORE MORE Expand your equatron rnto general form V m2 bx c and add rt to the graph Doe rt match your orrgrnal equatron N Add a Data amp Stamuc page make aecatter plot of the data and perform a quadratrc regremon on the data How does thr equatron compare wrth the other two How could you 1mpmve the t of the regremon equauon HZ a omnusammuterhaumz antnmns autumnmam WMTH39BDUEW o 2m9Ksmvnmn n5 Quadratic Motion Rolling Can Activity Notes lam l fv m F ll nKaml hk and F vl Kaml hk Objectives Students will use data collection devices to 03 For the sample data 3398amp After 75 s the can is collect realrworld data that can be modeled by quadratic 33984 m from the motion sensor and has fallen off equationsi They will then write quadratic equations to the table model real rworld datai DISCUSSION QUESTIONS 39 Why is the parabola right side up Activity Time 50 minutes Materials motion sensors empty coffee cans or large 39 How is this situation similar to projectile motion How is it different paper rolls long tables books Optional Rolling Can Sampletns Mathematics Prerequisites Students should be familiar EXPLORE MORE With quadratic equations including general form and L The general form of the equation given above is y 01st 7 01024 207 Graphs of general forms should match students original graphsi vertex formi TleNspire Prerequisites Students should be able to open and navigate a document See the Tip Sheets 2 The quadratic regression for the sample data is TleNspire Skills Students will use a motion sensor y 0 12 7 079 1 80 This equation probably Optimal StUdmtS Will send documents bemeen doesn t t the sample data as well as students other handhdds see the TIVNSpire Rderence Guide equations The t could be improved by deleting Notes You can do this activity in a variety of waysi the rst second of data that doesn t t the quadratic Option 1 Do the experiment and collect the data as a Pattern demonstration then send the data to students Option 2 If enough motion sensors are available have students mo AUTO REAL i j 3 do the experiment in groups of three or four then have 12XA279X130 them share their data within their group Option 3 Give ll 7 students the sample data from the Rolling Can Sampletns 087 document then proceed with nding the equation to t a the data 057 0 some 4 5 INVESTIGATE 01 Sample vertex 31 044 Students might scroll EXTENSION through the data to nd the lowe value The motion sensor collects velocity and acceleration 02 Sample equation y 0i16x 322 0i44i data as well as distance and time You might want to have students explore time velocity and time acceleration 5 I m R D AiLOREABL m0 g graphs and compare them with the time distance graphi l gf a fgxg w j j 39 V I The graph of distance velocity is also interesting to I I if 70 4856 explorer 2 i 05 l4820l k 3 A 5 ll Exploring Algebra l With TersplreW 5 Quadratlcs and Higheeregree Polynomials 113 2009 Key Culllculum Pless Binomial Products Sales and Profits Vim will need You have developed a greatitasting nutrition drink You sell it in 127packs to mmmS 20 retail markets in your area Some of the discount stores resell the 127packs at a low price in order to sell a large number of packs Some health clubs sell drinks individually at a high price and sell only a few packs You have decided to sell your own product at a local festival but you need to choose a price MAKE A CONJECTURE 01 Is it better to sell many drinks at a lowprice or a few drinks at a high price Explain your ideas INVESTIGATE You decide to test your opinions by collecting data on last month s sales at each outlet and nding a model to represent the sales and pro ts 1 Open the TliNspire document Sales and Pro tstns on your handheld and go to page 12You will see data on the selling price per pack from each outlet the pro t they made on each pack and the total sales for the previous month 2 Start your research by looking for any patterns in these values Add two Data amp Statistics pages 13 and 14 and create scatter plots for sell 4mm pro t per and selliprice packsisold l 11 l 12 l 13 I 14 PM AUTO REAL E nun AUTO REAL O OO i6 0 do 0 g 0 O Ul 5 we 0639quot 08 a k 0 o 4 O O 0039 00 6 A 81bi ihi396i39s2bz 2li 3968ibi392i394f5i 82b2 22 a ll rice selliprice 3 Find the best line of t modeling each graph Vou may need to drag 02 What model did you use for the rst graph What can you learn from this we axes 0 see we graph s x7 and yiintercepts iniercepis 03 Give the model for the second graph and explain what the slope in this model tells you Exploring Algebral With TerspireW 5 Ouadralics and Higheeregree Polynomials 115 2009 Key Cuiiicuium Pres Binomial Products Sales and Profits continued Vou won t see the data at this pom Vou may want to hide the View menu Choose Graph Trace mm the Tr c m rm to mm the Mniercepis Hhe parabola 116 5 Ouadrancs and Higheeregree Polynomials I14 Albert s Market sold 127packs for 1350 making a pro t of1000 on each of the 45 packs they sold How would you calculate the amount of money Albert s Market made from this product last month 4 You want to know which stores made the most money Go to page 12 and add a new Variable called endpro t Give it a formula to calculate this Value To quickly enter Variables in a formula press and choose Link To 3915 What formula did you use for endpro Which outlet made the greatest pro t Because the best price according to the model may not be one of the prices any outlet charged you need to look for a formula 5 Add a Graphs amp Geometry page and create a third graph to study how end pro t relates to the selling price Choose Scatter Plot from the Graph Type menu Press to choose Variables and to m0Ve between them 6 Choose Function from the Graph Type menu and graph the function fx x2 Then choose Window Settings from the Window menu and enter a window that will allow you to see both the parabola and the data 4 15 PAD AUTO REAL Us Drag the parabola until it ts the data What model did you nd to t these data According to that model what price should you charge at the festiVal and what pro t will you receiVe Now that you haVe solVed the problem one way you wonder whether using algebra would giVe you a solution without dragging a function You decide to compare the three graphs Q7 What are the xeintercepts of the three models Explain any patterns you see us Expand the equation of your parabola to get the general form y axZ bx c Then multiply the right sides of your answers to Q2 and Q3 Explain any patterns you see 3919 How could you haVe found the model for selljrice as a function of endpro t without dragging the parabola What solution would you haVe gotten Exploring Algebra 1 With TerspireW 2009 Key Cuvmulum Pres Binomial Products Sales and Profits day a f7 m P A imth i 77 r Objective Students will explore how linear models can give information both graphic and symbolic about the quadratic model that is their product Activity Time 3H0 minutes Materials Sales and Pro tstns Mathematics Prerequisites Students should be able to multiply binomials and interpret intercepts on graphs TliNspire Prerequisites Students should be able to open and navigate a document create a scatter plot use movable lines de ne variables using formulas add function plots to a graph and trace functionsr See the Tip Sheets TliNspire Skills None Notes Step 3 and Q2 Q3 and Q6 give students a chance to nd a line or curve of t and interpret the meaning of each function s terms for this problem situationr Students are gaining experience applying the process of nding a mathematical model to t a situation solving the model then interpreting the result back into the problem situationr Q7 and Q8 give students further experience with looking for patterns by doing calculationsr For a Presentation Ask several students to interpret the meaning of the constants and the coef cients in the lines of t Before you create the graph in step 5 ask students what shape they think the points on the scatter plot will have MAKE A CONJECTURE 01 Answers will vary wider You neednot reach consensus at this time INVESTIGATE 3 Students may addmovable lines use one of the built in regressions or nd the equation of a line through two representative points If they write the equation in pointrslope form encourage them to change it to sloperintercept form to facilitate later calculationsr Exploriiig Algebra l With TerspireTM 2009 Key Cuiiicuium Piass Activity Notes 02 TIrNspire s linear regression gives y x i 350 though student values may differ slightly The yrintercept is the perrpack wholesale cost to the remilerr Each item pack costs each store 350 The xrintercept gives sales that would yield a pro t of or Selling packs at 350 would return no pro t y 100 x350 03 The regression yields y 3729 96r25though student values may differ slightly The slope is the rate at which the number of sales decreases as the price increases The remiler gets in fewer sales for each dollar increase in price liii2li3i4lgtmmom REAL D so 0 E y 372 x9625 2 so 20 A ibi zikilsi zbz zzh zeitprice 04 Multiply 10 per item by 45 items sold to get 450 pro t 05 endpru t pro tiper packsisold Don s Beverage 1 and 3 made a pro t of 504 for the month 5 Quadratics arid Higheeregree Poly omials 117 Binomial Products Sales and Profits onti ued 06 The sample t shown below is approximatelyy 731380 7 15Z 460though student values may differr The maximum vertex indicates that the best price is about 15 per pack which gives an end pro t near 4601 That also means that pro t per pack is about 15 3150 or about 1150 and that you will sell about 40 packs totalr Exact numbers based on a quadratic regression give a perrpack price of 14183 selling 40163 packs and making a pro t of 450331 mm AUTO REAL B R 21 ummndpm d x 25 8 7 Graph 1 intercept at x 315 representing zero pro tr Graph 2 intercept at x 2587 representing zero salesr Graph 3 based on a quadratic regression intercepts at x 3142 and x 2623 representing zero pro t for either of these reasons The zeros of 11 B 5 Quadrancs and H1gheeregree Po1ynom1a1s Activity Notes the pro t function are approximately the zeros of its factors The zeros are not be exactly the same due to the level of estimation involved but it is important that students understand both the logic behindthe relationships of the intercepts and the issues behind problems that show up with the numbers 08 The coef cients of the general form of the quadratic equation should be approximately the same as the coef cients of the product from Q2 and Q31 That is the pro t function is the product of the other two functions 09 The product of the two linear expressions is a quadratic whose zeros are those of the linear functions The problem could have been solved by graphing the product of the two linear functions to get a price of 14169 with a pro t near 4661 Exp1ormg A1gebra 1 W111 TLNspHeW 2009 Key Cumcmmn P1855 Polynomial Factoring Maximum Area Van will need Mathematical analysts in business and industry collect data and create models 11 paper to nd maximums such as the maximum yield or the maximum pro t and minimums such as the minimum waste or the minimum cost In this activity you will solve a similar problem by folding a sheet of paper to nd the largest triangle EXPERIMENT On a sheet of85 X 11 in paper mark each inch from top to bottom along the left 11 in edge of the sheet N Fold the uppereright corner to one of the marks and crease the paper There is now a right triangle of a single thickness in the uppereleft corner of the page above the part of the edge that is folded The two legs of the triangle are along the side and the top of the paper 01 Which mark do you believe will result in the triangle with the largest area side 4 3 Open a new TleNspire document on your handheld Add a Lists amp Spreadsheet page and label the rst two columns side and top g Record in the table the lengths of the triangle s legs as you move the top right corner to marks along the left edge 02 How many marks can you actually use Explain INVESTIGATE Your goal is to nd the exact position for the fold that makes the triangle the largest 5 Create a new attribute for area using the formula 05 side top To help you see the data add a Data amp Statistics page and create a scatter plot of the side area data ExploringAlgebrai Wllh TersplreW 5 Ouadrancs and Higheeregree Polynomials 119 2009 Key Cumculum Pres Polynomial Factoring Maximum Area continued 120 5 Ouadrancs and Higheeregree Polynomials The graph looks somewhat quadratic The graph of a quadratic function has symmetry with the highest point halfway between the horizontal intercepts 3913 Think about how you gathered these data Where should the horizontal intercepts be That is which values of side would give you 0 area What point is halfway between the two side lengths with no area CM Do you believe these data are actually quadratic Why or why not The easiest tyqae of model to nd is linear Often in statistics you look for ways to change the data to unbend the curves then you reverse the process to bend the line after you have found a model This sequence is called linearization If 2 is a zero of a function meaning a horizontal intercept of the graph then x i z is a factor of the function Because you know two intercepts of this graph you know two factors You can create a data set of lower degree and therefore one that is more linear by dividing the data by one factor 6 Go to the Lists amp Spreadsheet page Create a new variable called factored and give it the formula of area divided by one of the factors you know Use side instead of x in your factor 7 If the values of factored are linear then the original data is quadratic Add another Data amp Statistics page and make a scatter plot of the side factored data I15 Is this graph linear or curved Is it increasing decreasing or both 8 Because the data are not yet linear divide factored by the other factor creating factored2 Create a scatter plot of this new variable versus side as Choose Add Movable Line from the 412 I M 5 mm Apr REAL Actions menu and adjust the line to amredz wayside 42 nd a linear model for the data points A 39 th39 h g 037 O O O 0 111 is grap 3 o E 06 O 0 9 To nd the model you re seeking E 39 for area work backward Start with gt09 the equation you found in Q6 and 7 I I gt I multiply it by each of the factors you 1 Aside 7 8 used to make the data linear Test this model by plotting it as a function on the scatter plot ofthe side area data Exploring Algebra 1 With TersplreW 2009 Key Cuvmulum Pres Polynomial Factoring Maximum Area continued 3917 What is your model for the area 10 Add a new Graphs amp Geometiy page and plot your function from Q7 using x instead of side Hide the entry line by choosing Hide Entry Line from the View menu 3918 Choose Graph Trace from the Trace menu What does tracing the graph tell you about how to fold the paper to get a triangle of maximum area According to your model What is that area EXPLORE MORE lam RAD APPRX REAL la 1 3 V 1709 3495 05 s 1 o 5 9 You found models for factoredZ and for area How can you adjust the model for factoredZ to get a model for factored Exploring Algebra 1 With Terspire M 2009 Key Cuiiiculum Pies 5 Ouadratics and Higheeregree Polynomials 121 Polynomial Factoring Maximum Area lam rlfv m P A lwith l P Objectives Students will use factoring as part of a process of modeling a thirdrdegree polynomial Students will explore the relationships among intercepts zeros and factors as theymaximize area in a paperrfolding activity Activity Time 2amp35 minutes Materials 85 gtlt 11 in paper rulers Optional Maximum Areatns Mathematics Prerequisites Students should be able to solve equations and multiply polynomialsr T 1 L IA k kl t and navigate a document de ne variables using formulas in the Lists amp Spreadsheet application make scatter plots use movable lines add function plots andtracer See the Tip Sheets TleNspire Skills None Notes This activity can start with the collection of data using a sheet of paper and a ruler or you can save time and use the sample data in Maximum Areatns You might start by demonstrating how to fold the paper and showing the location of the triangle that students need to measure If your time is limited andyou smrt with the data in Maximum Areatns rst demonstrate what is being measured If students are confused go back to the physical model perhaps labeling the side and the top As you visit working pairs nd one group that divided rst by side i 0 and another that used side i 85 Ask both pairs to be prepared to share For aPresentation If you only have access to one computer with presentation capability or one handheld and a projection device you can still ask students to gather the data Start a table on the TIrNspire to enter each group s side and top measurements for each inch mark then use the class average for the presentation You might plot the value x 85 and talk about Q4 and Q5 Before the student running the computer shows the graphs in steps 779 ask what students expect to see Ask the Explore More question 122 5 Quadratlcs and ngheeregree Poly omlals Activity Notes EXPERIMENT 01 Students will likely pick the 4 in mark or 425 in halfway between 0 in and 85 in This is a good guess but it is not as exact as the value they will derive laterr o r If students collect their own dam have them change their document settings to approximate answers Press choose System Info then Document Settings Tab down to Auto and click to choose Approximate Then tab to 0K and press r INVESTIGATE 02 8 There is no triangle at inch marks below 9 or 10 03 At 0 in and at 85 in If students have trouble encourage them to think about a triangle with no area 04 If the graph were symmetric then the maximum would be at 415 in but it is not The data are probably not quadraticr 05 The graph is not linear It will be decreasing whether students divide by the factor side i 0 or by the factor side i 85 06 Answers will vary depending on the accuracy of students original measurementsr One possible model is factoredZ 0r03side 025 07 Using the sample answer to Q6 area side 0 side 8r50r03 side i 025 EXDlOll g Algebra l Wlth TersplreW 2009 Key Cumculum P1855 Polynomial Factoring Maximum Area Activity Notes conti ue 08 Answers will Vary The maximum area occurs when EXPLORE MORE Z szde 15 about 4 9 Inqglvmg an area of about 7m t awed jammed sidg or awed jammed 4 3114115 L6 mo APPRX REAL 3 side i 85 depending on the last factor divided onto 4 777 V M 031547003 U l1XXX78S2030925 73 Eprrmg A gebra 1 wnh TLNspwreW 5 Quadratms a d ngheeregree Po ymomwa s 123 2009 Key Cumcumm Pvass Exponential Growth Interest You will need You have 600 that you want to save and you know that it can earn money for you 39quotteresuns When you let someone else use your money they must pay you rent That rent is called interest In this activity you will explore different types of interest INTRODUCTION A relative agrees to pay you interest at a rate of 10 This means that 10 of the 600 will be added to your savings each year This is called simple interest 01 How much will your savings increase each year A bank advertises that it will pay you 6 interest The bank s interest is compounded In other words each year the bank adds to your savings 6 of the growing amount of savings not just 6 of the original 600 02 How much will your savings increase the rst year The second year Your goal in this activity is to decide whether the banks offer can ever be better than your relative s offer INVESTIGATE Egg 1 Open the TI Nspire document 11 12 13 14 mm AFpr REAL j Interesttns on your handheld Go A year 5 Simple C compound D A to page 12 You will be making a ISeq U spreadsheet showing your earnings 1 0 over 50 years 2 1 3 2 2 First enter a formula that will generate 4 3 the years The formula for year looks 5 4 v like this Ayearseqnzrn 1lli050 seqnun 1 1 0 50 3 To calculate the simple interest enter 11 12 13 L4 mam Apr REAL D the formula A year B simple C compound D A 0 seqnu seqnun seqnun 1 60 600 50 1 0 600 The un 1 in the formula tells TI 2 i 660 r Nspire to use the previous value of the 3 2 720 variable The 600 tells TI Nspire to use the i 339 78039 value 600 for the initial case here it is the 4 840 V 0th case B simplequnyn 1J6o 600 50 03 What does the 60 in the formula do Exploring Algebra 1 with TI NspireTM 3 Exponential Equations 57 2009 Key Curriculum Press Exponential Growth Interest continued 4 For compound set up a formula to 1 12 L3 L4 APPRX REAL represent the banks offer You can use A year B simple C the formula seqnar1 1 06 ar1 1 600 50 04 Explain in your own words what this formula does 39 39 impound 05 Will your savmgs ever earn more money with the banks offer than with your relative s Explain You would like formulas that give you the value of your savings for any year without having to nd the values for all previous years You will use scatter plots to nd these formulas 5 At the top of page 13 make a scatter plot of the year simple data 6 The data points appear linear so you ll Q 12 13 14 Dario 10111231 REAL m if be looking for an equation of the form 39 U 110 a1b1x a 2400 E 4 szmple a 17 year Choose Plot E 5 Function from the Actions menu and m 120039 t enter the equation 1 00 Z ail bill 0 5 10 1395 20 2395 30 3395 10 4 5 5390 x Use the sliders at the bottom of the year page to adjust a and b to t the data O 0 points as well as possible 391 39 39 3945039 39 398039039 5 39 39339039 391539039 399390 3 b 06 What values of a and 17 give the best t How do these values relate to 600 of your initial savings and to the 10 simple interest rate 7 At the top of page 14 make a scatter plot of the year compound data and plot the line from page 13 07 Can you adjust a and b to make the 11 12 13 14 FRAD APPRX REAL cm graph t the compound interest data E 10000 3 Nhy 0139 not g 5000 8 a 4 Because compound interest cannot be 0 mutt modeled by a linear equation you will 0 5 10 15 2035mm 35 do 45 50 look for another type of equation that can L modelit I39MIIHII Q lllllll 0 2000 4000 0 150 b300 450 El 58 3 Exponential Equations Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Exponential Growth Interest continued 8 Perhaps this curve is part of a parabola Go to page 15 and make another scatter 1310000 plot of the year compound data Plot the function fx c1 d1 X2 To get the exponent press or J Adjust the c and d sliders to match the data as well as possible 15 gtRAD APPRX REAL El f3x c1d1xA2 I I l 3910 15 2O 25 3O 35 4O 45 50 year C I quotquotquot39 I39l39l39l39l39l39l 08 How well does this graph model the 0 5009 100m 0 4 8 1a 15 20 2 data Do you think compound interest growth can be modeled by a parabolic graph To zoom use the 9 Another curve is the graph Of an i 13 14 15 16 RAD APPRX REAL WindowZoom menu 10000 or drag the ends of equation in Wthh year is in the g the aXeS exponent Go to page 16 to explore an EL 5000 equation of the form compound m 8 nym You will need to plot the function fx m1 n1X This is called an exponential equation To adjust n you O mayneed to zoom in on values near 1 039 360 560 960 00 10 2 30 m n 09 What values of m and n give the best t for the data Do you think compound interest can be modeled by exponential equations 010 When compound interest in this situation is modeled by an equation of the form A mnquot how do you think m and n relate to the 600 initial investment and to the rate of 6 EXPLORE MORE Egg Insert a new Lists amp Spreadsheets application as page 17 What formulas give the following sequences 246816 3612241536 1123589 Exploring Algebra 1 with Tl NspireTM 3 Exponential Equations 59 2009 Key Curriculum Press Exponential Growth Interest Activity Notes Adapted from Exploring Algebra 1 with Fathom by Eric Kamischke Larry Copes and Ross Isenegger Objectives Students will learn that a linear equation models simple interest and an exponential equation models compound interest They will relate the values of a and b in the equations A a bx and A ab to the principal and interest rate respectively Activity Time 3 5 50 minutes Materials Interesttns Mathematics Prerequisites Students should be able to calculate a percentage of an amount write a percent as a decimal evaluate equations by substitution and substitute variable expressions into a formula TINspire Prerequisites Students should be able to open and navigate a document use formulas in the Lists amp Spreadsheet application create scatter plots plot an equation on a scatter plot adjust the scale of an axis and insert a new page See the Tip Sheets TINspire Skill Students will insert formulas using the seqn command Notes As you facilitate student work probe for understanding of the quantity 106 especially in Q9 You might have pairs who nish the main activity rst prepare to present the Explore More questions and answers For a Presentation As you lead a class discussion using a presentation computer or projected handheld emphasize Q5 Q9 INTRODUCTION o1 1096of600is60 02 Year 1 6 of 600 or 36 Year 2 6 of 636 or 3816 INVESTIGATE 03 The 60 adds 10 of 600 or 60 04 The formula starts with 600 It takes that value multiplies it by 006 and adds that to the value to get the next value It does this 50 times 60 3 Exponential Equations 05 The savings in the bank passes the savings with the relative in year 18 06 a 600 the initial amount b 60 the amount of interest being added each year Introduce the terms principal for the initial amount of 600 and interest rate for the 10 07 No the compound interest graph is not linear 08 The compound interest graph is not quite parabolic It ts well for the rst 20 years but does not increase fast enough after that 09 The best value for m is the vertical intercept 600 The best value for n is 106 An exponential equation ts well 010 m is the principal 600 11 is 1 more than the interest rate DISCUSSION QUESTION You entered several formulas at the beginning of this activity What does each part of the formula do Here is a quick look at the formula 2 seqnun 1 60 60050k de nes a formula seqn creates a sequence un 1 the previous term in the list un 1 and un 2 are the previous two terms 60 what to do to previous term 600 the initial term of the sequence The rst and second terms would be 1 b 50 the number of terms to list EXPLORE MORE seqnun 1 2 2 8 seqnun 1 2 3 10 seqnun 1 un 2 1 1 11 Students could also nd these formulas using cell operations and ll down Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Inverse Variation Boyle s Law You will need The amount of time a scuba diver can safely spend under water depends on the 303quot Law39tns amount of air in the diving tank Your friend has asked for your help in determining how much air is in a particular tank You can see that the volume of the tank is 0340 cubic feet but you also know that the tank is pressurized to pack more air into a smaller space As the air leaves the tank it eXpands You can see that the tank is full and that the pressure gauge reads 3500 pounds per square inch psi To determine how much air the tank will supply while your friend is diving you need to know what the volume of the air will become as it is released from the tank and as the pressure becomes the normal air pressure of 147 psi Boyle s law relates these two qualities INVESTIGATE 1 You have found some data 011 the I 11 I 12 I 13 I 21 IIRAD AUTO REAL Internet and put it into a table to A pressure B volume C D quot help you look for patterns Open the document Boyles Lawtns and go to ll 48 page 12 to see the data 2 305625 45 3 31935 14 2 Go to page 13 and create a scatter 4 335 42 plot of the volume pressure data 5 353125 40 v at the right of the page The data A I k are not linear but perhaps they are exponential Click on the point 01 As the volume is halved from 48 ft3 to 11 l 12 13 21 PRAD AUTO REAL D volume 24 ft3 what happens to the pressure x 1000 O I 6 O 0 C5 0 values You can also quot7 lookinthetable 02 Does the pressure appr0X1mately 0 C53 0 0 double when the volume goes from 55 O O 30 ft3 to 15 ft Does the pressure 00 7 O O 0 always double when the volume is cut 9 3 0 j O O Q in half Show the evidence for your 39 conjecture You can use either the scatter plot or the table to nd this information Exploring Algebra 1 with TI NspireTM 3 Exponential Equations 61 2009 Key Curriculum Press Inverse Variation Boyle s Law continued To plot a value choose Plot Value from the Actions menu 62 3 Exponential Equations 03 If the equation were exponential would the pressure double when the volume is cut in half U Because the data do not appear to 11121314gtRAD AUTO REAL f1x k1xquotb1 be exponential you decide to try x 1000 modeling with a power equation of the form 1 k xb Unlike in an exponential equation the variable pressure will be in the base rather than D 001 Q pressure in the exponent Add the graph of the 30 power equation to your scatter plot Use the equation f1 x k1 xquotb1 4 Adjust the k and b sliders to t the data as well as you can 04 According to your model what pressure is needed to reduce the volume below 10 ft You may have found that a good value for b is approximately 1 Using the de nition of a negative exponent it can be simpler to write the equation as pressure The coef cient k of this equation is called the constant of variation Note that k pressure volume Finding the value of this constant in particular situations is important for answering pressure volume questions like the one in this activity 5 To see this relationship in the data table add a new variable constant with the formula pressure volume 6 Add a new Data amp Statistics page by pressing and choosing Data amp Statistics 05 Create a new dot plot of constant and 14 M40 4010 REAL plot the value meanconstant How does this mean relate to the values of the sliders Explain v1 meanmonstan 06 What is the constant of variation for 1403 the scuba tank you are examining Why 800 3 03 0 CI might it be different from the constant 0 O Q 030 CCCOCD C O 39 I 39 39 l 1396 1400 1404 1408 1412 1416 of variation in the table constant 07 What volume will the air have when it is released from the tank and the pressure becomes 147 psi Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Inverse Variation Boyle s Law continued EXPLORE MORE 1 To explore more equations of the form 12 13 14 21 AUTO REAL xy k or the equivalent go to the plot ofy on page 21 Explore the graph for the values of slider k near 0 Describe the graph when k lt 0 k 0 15 10 5 and k gt 0 J In Explore More 1 you studied a graph in which the product of x and 1 was the constant k Now eXplore graphs in which the sum of x and y is the constant k You may need to add a new Data amp Statistics page after 21 to do this Be as speci c as you can about what is always true about the graph and what changes as k changes U What if the difference were always k What if the quotient were k Try this Are you surprised by the results Exploring Algebra 1 with Tl NspireTM 3 Exponential Equations 63 2009 Key Curriculum Press Inverse Variation Boyle s Law Activity Notes Adapted from Exploring Algebra 1 with Fathom by Eric Kamischke Larry Copes and Ross Isenegger Objectives Students will model inversely proportional 01 The pressure roughly doubles from 29125 psi to quantities with equations of the form y 14 xy k and 588125 psi y kx l They will investigate the change in one variable as the other doubles relate y to a linear model and l 13 l 14 I 21 22 RAD AUTO REAL eX lorethe ra hofx k x 0 P g P y 100 100 Activity Time 30 40 minutes pressure 4 O I Materials Boyles Lawtns 00 7 39 Q 40 Mathematics Prerequisites Students should be able to 33 1amp3 k solve literal equations and evaluate a formula 1b 2b 3390 4390 539 volume TINspire Prerequisites Students should be able to create 02 As the volume goes from 30 ft3 to 15 ft3 the pressure roughly doubles from 470625 psi to 930625 psi Some students might say pressure is not quite doubled as volume goes from 15 ft3 to 30 ft3 The table demonstrates that the ratio of the pressures is very a scatter plot plot a function on a scatter plot use Tl Nspire to calculate a mean and change the scale of sliders and scatter plots TINspire Skills None Notes Boyle s law states that if a gas is kept at constant Close to 2 in 6Very case temperature the pressure and volume are inversely Volume Pressure Volume Pressure Ratio of proportional or have a constant product Robert Boyle fts Psi fts Psi Pressures published his ndings that pressure times volume is 48 29 1250 24 58 8125 2 019313 constant in his 1662 article A Defense of the Doctrine 46 30 5625 23 61 3125 2 006135 Touching the Spring and Weight of the Air 44 319375 22 640625 2005871 As you facilitate student work look for students who 42 335000 21 670625 2001866 have complete answers to Q2 and Q3 Have them share 40 35 3125 20 70 6875 2 001770 their results with the class A variety of answers to Q6 38 37 0000 19 74 1250 2 003378 can also be shared students taking physics will know that 36 393125 18 778750 198092 temperature is an important factor 39 39 39 34 416250 17 827500 1987988 INVESTIGATE 32 441875 16 878750 1988685 30 470625 15 930625 1977424 1 These data are Boyle s original The Internet source 28 50 3125 14 100 4375 1 996273 is given on the last page of the document Boyle s 26 54 3125 13 107 8125 1 985040 methodology and his published ndings are 39 39 39 24 588125 12 1175625 1998937 intriguing reading 64 3 Exponential Equations Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Inverse Variation Boyle s Law continued 03 No If the equation were exponential the pressure would double for constant changes in the volume not for proportional changes 9 Finding the best values is tricky It requires ne motor control the ability to adjust the scale of a slider and visual estimation skills This model with k 1253 and b i097 matches the graph remarkably well which might make you wonder whether Boyle s data were fudged There is some evidence that other historically important data can be statistically shown to be too close to the predictions to have been produced by experimentation 13 14 21 22 RAD AUTO REAL 1391 X 1000 0 D L 3 m to lt1 5 O 097 04 Using pressure 2 1253 volume a pressure of 160 psi will decrease the volume below 10 ft3 05 If slider k is set to the mean value 1408 the graph goes through the data points Doing a power regression on the data yields the equation y 39 Choose Regression from the Actions menu lt 13 I 14 l 21 22 RAD AUTO REAL x 1000 O O D 1 pressure xi C 6 pressure volume constant so 0343500 1190 Possible reasons the constant may differ from that of the table include differences in temperature and the nature of the gas being compressed 07 About 81 ft3 volume 735 Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Activity Notes EXPLORE MORE 1 When k lt 0 the branches are in the second and fourth quadrants This graph is called a hyperbolu Its branches approach but never cross its asymptotes which in this case are the x and y axes When k 0 the graph is the x axis with the point 0 0 removed because is unde ned When k gt 0 there will be two branches of the graph one in the rst quadrant and one in the third J The equation is y k x so the graph is a straight line with y intercept k and slope 1 It crosses the y axis above the origin when k gt 0 at the origin when k 0 and below the origin when k lt 0 DJ If the constant difference equals x y then y x k if the constant difference equals y x then y x k The graph of each is a straight line with slope 1 k is either the y intercept or its opposite If the constant quotient equals then y ix if the constant quotient equals then y kx In both cases the graph is a line through the origin with slope either k or its reciprocal EXTENSIONS 1 Have students research the pressure experienced by divers How deep would a diver need to be in order to be subject to twice the pressure experienced at sea level A diver at depth 103 m under fresh water experiences a pressure of about 2 atmospheres 1 atm for the air and 1 atm for the water 3 Exponential Equations 65 Inverse Variation Boyle s Law continued 2 66 Pose this problem A dam analyst uses the TIeNspire to t the exponential equation volume 4641098052Pm 9 29 to the data in the table Make a convincing argument for why either the exponential or the power model is better You may want to discuss the vertical intercept and its role in the model as well as the halving time for volume The equation volume 4641098032Pm 9 29 goes through the point 29 4641 which is approximately pressure and volume in the rst case The volume is decreasing approximately 2 per increase of 1 unit of pressure at the beginning 3 Exponentiai Equations Activity Notes of the experiment but it decreases more slowly in the later values Not only does a graph of the form pv k t the points better but it also ensures that the graph has no vertical intercept as pressure decreases to 0 volume expands to in nity The exponential model implies that when there is no pressure the volume will be about 826 which is physically incorrect In the activity volume is halved as pressure is doubled in Boyle s model as opposed to having a xed halving period as in the exponential function model Expioring Aigebrai witn TirNspireTM 2009 Key Curricuium Press Exponential Models Radioactive Decay You will need Radioactive Decaytns 0 paper plate 0 protractor 0 supply of small counters EXPERIMENT Exploring Algebra 1 with TI NspireTM 2009 Key Curriculum Press The particles that make up the atoms of some elements like uranium are unstable Over a period of time speci c to the element the particles change and the atom eventually becomes a different element This process is called radioactive decay In this experiment your counters represent atoms of a radioactive substance 1 A Lil Count the number of counters Open the Take a paper plate and draw an angle from g the center of your plate as illustrated TI Nspire document Radioactive Decaytns on your handheld and go to page 12 Enter the number of counters as the number of atoms after 0 years of decay Pick up all of the counters Drop the counters randomly on the plate Be sure the counters are scattered evenly over all parts of the plate do not aim for the center Counters that fall inside the angle represent atoms that have decayed Decide how you are going to handle counters that fall on the lines of your angle and that miss the plate they need to be accounted for also Count and remove the counters that fall inside the angle these atoms have decayed Subtract the decayed atoms from the previous value and enter the number of counters remaining after 1 year of decay Pick up the remaining E W A counters Repeat steps 3 and 4 until you have I 11 I 12 I 13 I 14 RAD Apr REAL fewer than ten atoms that have not A years B atoms C D E F decayed Each drop will represent another year of decay Enter the 1I 2 number of atoms rema1n1ng each time 3 on your handheld 4 a 5 A 3 Exponential Equations 67 Exponential Models Radioactive Decay continued INVESTIGATE You ll use your handheld to make a model for your experiment Let x represent elapsed time in years and let 1 represent the number of atoms remaining 6 On your handheld make a scatter plot m 12 13 14 RAD APPRX REAL of the data To do this go to page 13 and O O 7 O 0 add the appropriate variables to the plot O O O 01 What do you notice about the shape of O O the graph O O 8 Egg 7 Calculate the ratios of atoms remaining O 0 between successive years That is divide the number of atoms remaining 739 1 after 1 year by the number of atoms remaining after 0 years and so on To do this go to page 12 Arrow to the top of column C and type the variable name ratio In cell C1 type the formula b2b1 and press To ll in the rest of the values highlight the rst cell and choose Fill Down from the Data menu You will see the cell surrounded by dashed lines Press V to highlight all but the last row of data you entered then press You should have a ratio for each row except the last 11121314 Ran APPRXREAL a 11 12 13 14 RAD APPRXREAL El Ead39oagt ye oatatt sample data with 201 A aaa39oagt39tta Decay Sample data with 201 a counters and a 68 angle 9 counters and a 68 angle v A years 3 atoms C ratio D E A A years 393 atoms C ratio D E A O O F 39 0 201 321292 1 o 201 231343 2 1 14 2 1 147 31632 3 2 120 3 2 120 333333 v 4 3 94 7quot CI I 221372 a 02 How do the ratios compare 8 Find a representative ratio by calculating either the mean or median To use the Calculator application go to page 14 Type either meanratio or medianratio 03 Which representative ratio did you choose and why 04 At what rate did your atoms decay 05 Write an exponential equation that models the relationship between the amount of time elapsed and the number of atoms 68 3 Exponential Equations Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Exponential Models Radioactive Decay mutqu 9 Gtaph the equattohwtth theecattet plot choose Plot Funcn39on from the Actions menu and type the eq ttoh Mu t the two values unul you ate eatuhed To do 111 doublerchck the equattoh oh the edeeh ahd edtt the Value Record your nal equattoh tn the humbet In your equattoh ah equattoh that would model the decay onoo counter uethg a oehttat angle om as what are eome of the factor that mtght cauee dtffetehoee between actual data and Value ptedtcted by your equauoh EXPLORE MORE Hut mm my wtth the other that you found autumn Ahmt wnttrtttspttw 3 Buntentat Equatuts m 0 mm my mnulmt at Exponential Models Radioactive Decay Activity Notes Adapted from DiscoveringAlgebm by Jerald Murdock Ellen Kamischke and Eric Kamischke Objectives Students will write exponential equations that model real world decay data Activity Time 50 60 minutes Materials Radioactive Decaytns paper plates protractors counters Skittles MampM bits candy corn dried beans etc Optional Radioactive Decay Sampletns Mathematics Prerequisites Students have had some experience with exponential equations and function notation TINspire Prerequisites Students should be able to open and navigate a document graph an equation create a scatter plot and use the grab tool See the Tip Sheets TINspire Skills Students will nd the mean or median and type formulas into cells of the Lists amp Spreadsheet application Notes There is a TI Nspire document with sample data for your use All the screen shots below are with the sample data You may choose to have students work collaboratively with a partner or as a whole class EXPERIMENT 1 Students should use protractors to make an angle of less than 90 but not too small They can approximate the center by using a ruler to nd the midpoints of several diameters DJ The objective is to have the counters spread evenly and quickly An acceptable plan would be that counters on the line and counters that fall outside the plate but lie within the extended rays of the angle are accepted as being within the angle 4 Remind students that they are interested in the number of atoms that do not decay 70 3 Exponential Equations INVESTIGATE 6 Here is a graph of the sample data 11 12 13 14 APPRXREAL 0 00000 IO 01 Students should notice a curved exponential pattern 7 You may have to assist students with this step Some may just type in the formulas for every cell which is not an ef cient approach 02 The ratios should be approximately the same For the sample data they range between 073 and 09 03 Students could give reasons for selecting the mean the median or another value In these sample data the mean is about 0802 and the median is about 0804 04 For these sample data using 0802 the rate of decay is 198 per year If you are not using the sample data you might use this opportunity to discuss various student answers You may need to encourage students to look at their constant multiplier in the form of 1 i r That is a ratio ofiwould be 1 i 025 05 For this sample y 2011 0198 0 61 For the sample data the equation does not appear to t very well Students may nd that the multiplier A seems to adjust the curve s position vertically and r seems to change the steepness of the curve The equation y 2011 022x ts the sample data better Ii1li2li314 RADAPPRXREAL I3 180 g 1202 O E f1x 2011193quot X 60 0 I I I I I 2 l5 8 10 12 14 year Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Exponential Models Radioactive Decay continued 11121314 RAD APPRXREAL 6 8 years 07 To help students With this question ask What is the ratio of your angle measure to the Whole plate The ratio of the angle measure to 3600 should be approximately the same as r In the sample data 366 z 019 which is close to the r value used in Q5 08 y 40017 09 Factors given might include how evenly the counters are distributed on the plate What you do when a counter is on an angle line and how you treat counters that fall outside the plate Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press Activity Notes EXPLORE MORE To nd and graph the exponential regression from the Data amp Statistics application choose Regression from the Actions menu then choose Exponential Regression For the sample data the regression equation is y 17722080x This equation is close to the predicted equation With a smaller A value and a very similar r value Comparisons to students adjusted equations Will vary 11 121314 RAD APPRXREAL E 180 j y 1772280quotx g 120 8 5 60 0 I I I I l I I O 2 6 10 12 14 years 3 Exponential Equations 71 HalfLife Pendulums You will need Pendulumstns 0 soda can or bottle of water 0 string 0 meterstick EXPERIMENT You may have to collect data for every fourth or fifth swing to get an accurate reading INVESTIGATE Exploring Algebra 1 with TI NspireTM 2009 Key Curriculum Press 1 Make a pendulum with a soda can half lled 2 Pull the can back about 1 m from its resting 3 Open the TI Nspire document 0 1 2 01 Go to page 13 and make a scatter plot 3 4 5 4 Go to page 14 Here you will see a 02 Describe this graph as fully as you can In this activity you will do an experiment write an equation that models the decreasing exponential pattern and nd the halflzfe the amount of time needed for a substance or an activity to decrease to one half its starting value with water tied to at least 1 m of string use the pull tab on the can to connect it to the string Place the meterstick underneath or behind the pendulum so you can take readings 1113 if position then release it Measure how far the I can swings from the resting position for as Resting many SWll lgS as you can position 12 13 14 FRAD AUTO REAL A myswing B mydis C D E E V Pendulumstns on your handheld and go to page 12 Enter your data in the Lists 8r Spreadsheet application of your data What type of pattern does the graph seem to show 11121314gtRAD AUTO REAL scatter plot of data on 19 pendulum 85 3 swings collected using a CBRZ These 39 39 1 I data show the full swing of the I pendulum 00393513 How does this graph compare with the I graph of your data Explain l r esdism 3 Exponential Equations 73 HalfLife Pendulums continued Choose Function from the Graph Type menu to graph Double click the equation to edit it EXPLORE MORE 5 The maximum swings from the sample 4 12 13 L4 15 RAD JAUTO REAL data are on page 15 Go to page 16 and make a scatter plot of these data A swing B maxdis C capturedis D 03 Does this graph look similar to the graph of your data Explain 04 Find an equation in the form 1 A1 rquot that models the sample mwaAQ II 4 w IJ k 38459 86 58 76861 6 7 56849 7481 01 AIll data Graph this equation with the scatter plot Adjust the values until you have as good a t as possible What is your function 05 Find the half life of your data To nd the half life approximate the value of x that makes 1 equal to A 06 What does the half life mean for the situation in your experiment 07 Find the maximum distance after one half life two half life cycles and three half life cycles How do these values compare 08 Write a summary of your results Include descriptions of how you found your exponential model what the rate r means in your equation and how you found the half life 74 3 Exponential Equations a Data 8r Statistics scatter plot and choose Regression from the Actions menu Compare this equation with the one you found Perform an exponential regression with your handheld go to page 17 which shows Exploring Algebra 1 with Tl NspireTM 2009 Key Curriculum Press HalfLife Pendulums Activity Notes Adapted from DiscoveringAlgebrn by Jerald Murdock Ellen Kamischke and Eric Kamischke Objectives Students will write exponential equations that model real world decay data and nd the half life for the equation Activity Time 50 60 minutes Materials Pendulumstns soda cans or bottles of water string metersticks Mathematics Prerequisites Students have had some experience with exponential equations and function notation TINspire Prerequisites Students should be able to open and navigate a document graph an equation and modify it create a scatter plot and use the grab tool See the Tip Sheets TINspire Skills None Notes You have four options for this experiment depending on your time constraints and classroom setup Option 1 Have students collect data enter the data into the handheld analyze the data and compare them with the sample data Option 2 Do a demonstration for the whole class and collect data using the meterstick enter the data into the handheld analyze the data and compare them with the sample data Option 3 Demonstrate the basic experiment so students understand the situation then go right to the sample data Option 4 Discuss the experiment then go right to the sample data You could theoretically also collect data similar to the sample data using a CBR2 or other data collection device However the process of manually capturing the maximum swing points requires patience and precision and may not be worth class time You may choose to have students work collaboratively with a partner or as a whole class EXPERIMENT 01 Students will probably recognize an exponential pattern 02 The location of the pendulum bob is harmonic but its maximum distance from the resting position is roughly exponential Students should notice the sinusoidal pattern in their own words and that the graph is decaying or getting smaller They should Exploring Algebra 1 with TI NspireTM 2009 Key Curriculum Press notice that the top points of the graph look like their graph 5 Discuss with students any patterns they see with the data table 03 If students collected data accurately this graph should resemble their data They may note that this graph shows every swing whereas theirs shows every ve swings or so 04 Sample equation 13141516 RAD AUTO REAL D X sMngrraxdfsK 05 Using the sample equation the half life is about 82 swings One way to nd this is to graph y2 A and nd the intersection with y1 A1 r To nd the intersection choose Intersection Points from the Points 8 Lines menu and select the two functions lt13141516DRAD AUTO REAL l 8 V fslxrss9916X 785 x7 821705 03925 x sMngmaxdis 06 Sample answer The half life of the pendulum swing is approximately 82 swings This means that on the 82nd swing the pendulum s maximum distance is half of its original maximum distance 07 Using the sample equation approximately 0393 m 0197 m and 0098 m With each consecutive half life the value of y will be half the previous value of y 09 Student summaries will vary 3 Exponential Equations 75