Ivestigatio Moitorig Ivetory Name Period Date Art Smith has bee providig the prits of a egravig to FieArt Gallery. He plas to make just 2000 more prits. FieArt has already received 70 of Art s prits. The Little Prit Shoppe also wishes to order prits. Art agrees to supply FieArt with 0 prits each moth ad Little Prit Shoppe with 10 prits each moth util he rus out. Step 1 As a group, model what happes to the umber of umade prits, the umber of prits delivered to FieArt, ad the umber delivered to Little Prit Shoppe i a spreadsheet like the oe below. [ See Calculator Note 1C for differet ways to create this table or spreadsheet o your calculator. ] Moth Umade Prits FieArt Little Prit Shoppe 1 2000 70 0 2... Step 2 Use your table from Step 1 to aswer these questios: a. How may moths will it be util FieArt has a equal umber or a greater umber of prits tha the umber of prits left umade? b. How may prits will have bee delivered to the Little Prit Shoppe whe FieArt has received twice the umber of prits that remai to be made? Discoverig Advaced Algebra Ivestigatio Worksheets LESSON 1.1 5
Ivestigatio Moitorig Ivetory (cotiued) Step 3 Write a short summary of how you modeled the umber of prits ad how you foud the aswers to the questios i Step 2. Compare your methods with the methods of other groups. 6 LESSON 1.1 Discoverig Advaced Algebra Ivestigatio Worksheets
Ivestigatio Lookig for the Reboud Name Period Date You will eed: a ball, a motio sesor Whe you drop a ball, the reboud height becomes smaller after each bouce. I this ivestigatio you will write a recursive formula for the height of a real ball as it bouces. Step 1 Step 2 Set up your calculator ad motio sesor ad follow the Procedure Note to collect boucig-ball data. [ See Calculator Note 1D for calculator istructios o how to gather data. ] The data trasferred to your calculator are i the form (x, y), where x is the time sice you pressed the trigger, ad y is the height of the ball. Trace the data graphed by your calculator to fid the startig height ad the reboud height after each bouce. Record your data i the table. Bouce umber Reboud height (m) Bouce umber 0 5 1 6 Reboud height (m) Collectig Data 1. Hold the motio sesor above the ball. 2. Press the trigger, the release the ball. 3. If the ball drifts, try to follow it ad maitai the same height with the motio sesor.. If you do ot capture at least 6 good cosecutive bouces, repeat the procedure. 2 7 3 8 9 Discoverig Advaced Algebra Ivestigatio Worksheets LESSON 1.2 7
Ivestigatio Lookig for the Reboud (cotiued) Step 3 Step Graph a scatter plot of poits i the form (bouce umber, reboud height). Record the graphig widow you use. [ See Calculator Notes 1E, 1F, 1G, ad 1H to lear how to eter, plot, trace, ad share data. ] Compute the reboud ratio for cosecutive bouces. reboud ratio reboud height previous reboud height Step 5 Decide o a sigle value that best represets the reboud ratio for your ball. Use this ratio to write a recursive formula that models your sequece of reboud height data, ad use it to geerate the first six terms. Step 6 Compare your experimetal data to the terms geerated by your recursive formula. How close are they? Describe some of the factors that might affect this experimet. For example, how might the formula chage if you used a differet kid of ball? 8 LESSON 1.2 Discoverig Advaced Algebra Ivestigatio Worksheets
Ivestigatio Lookig for the Reboud With Sample Data Name Period Date Whe you drop a ball, the reboud height becomes smaller after each bouce. I this ivestigatio you will write a recursive formula for the height of a ball as it bouces. Step 1 Step 2 A group of studets set up their calculator ad motio sesor ad followed the Procedure Note to collect boucigball data. [ See Calculator Note 1D for calculator istructios o how to gather data. ] The data were trasferred to the calculator i the form (x, y), where x is the time sice they pressed the trigger, ad y is the height of the ball. They traced the data graphed by the calculator to fid the startig height ad the reboud height after each bouce. These data were recorded i the table. Bouce umber Reboud height (m) Bouce umber Reboud height (m) 0 1.081 5 0.166 1 0.830 6 0.119 2 0.578 7 0.08 3 0.377 8 0.06 0.25 9 0.059 Collectig Data 1. Hold the motio sesor above the ball. 2. Press the trigger, the release the ball. 3. If the ball drifts, try to follow it ad maitai the same height with the motio sesor.. If you do ot capture at least 6 good cosecutive bouces, repeat the procedure. Discoverig Advaced Algebra Ivestigatio Worksheets LESSON 1.2 9
Ivestigatio Lookig for the Reboud (cotiued) With Sample Data Step 3 Step Graph a scatter plot of poits i the form (bouce umber, reboud height). Record the graphig widow you use. [ See Calculator Notes 1E, 1F, 1G, ad 1H to lear how to eter, plot, trace, ad share data. ] Compute the reboud ratio for cosecutive bouces. reboud ratio reboud height previous reboud height Step 5 Decide o a sigle value that best represets the reboud ratio for the ball. Use this ratio to write a recursive formula that models the sequece of reboud height data, ad use it to geerate the first six terms. Step 6 Compare the experimetal data to the terms geerated by your recursive formula. How close are they? Describe some of the factors that might affect this experimet. For example, how might the formula chage if a differet kid of ball was used? 10 LESSON 1.2 Discoverig Advaced Algebra Ivestigatio Worksheets
Ivestigatio Doses of Medicie Name Period Date You will eed (optioal): a bowl, a supply of water, a supply of tited liquid, measurig cups graduated i milliliters, a sik or waste bucket Our kideys cotiuously filter our blood, removig impurities. Doctors take this ito accout whe prescribig the dosage ad frequecy of medicie. I this ivestigatio you will simulate what happes i the body whe a patiet takes medicie. To represet the blood i a patiet s body, use a bowl cotaiig a total of 1 liter (L) of liquid. Start with 16 milliliters (ml) of tited liquid to represet a dose of medicie i the blood, ad use clear water for the rest. Step 1 Suppose a patiet s kideys filter out 25% of this medicie each day. To simulate this, remove 1 _, or 250 ml, of the mixture from the bowl ad replace it with 250 ml of clear water to represet filtered blood. Use the table to record the amout of medicie i the blood over several days. Repeat the simulatio for each day. Day Amout of medicie (ml) 0 16 1 2 3... Step 2 Write a recursive formula that geerates the sequece i your table. Step 3 How may days will pass before there is less tha 1 ml of medicie i the blood? Step Is the medicie ever completely removed from the blood? Why or why ot? Discoverig Advaced Algebra Ivestigatio Worksheets LESSON 1.3 11
Ivestigatio Doses of Medicie (cotiued) Step 5 Sketch a graph ad describe what happes i the log ru. y Amout of medicie (ml) x Day A sigle dose of medicie is ofte ot eough to treat a patiet s coditio. Doctors prescribe regular doses to produce ad maitai a high eough level of medicie i the body. Next you will modify your simulatio to look at what happes whe a patiet takes medicie daily over a period of time. Step 6 Start over with 1 L of liquid. Agai, all of the liquid is clear water, represetig the blood, except for 16 ml of tited liquid to represet the iitial dose of medicie. Each day, 250 ml of liquid is removed ad replaced with 23 ml of clear water ad 16 ml of tited liquid to represet a ew dose of medicie. Complete this table, recordig the amout of medicie i the blood over several days. Day Amout of medicie (ml) 0 16 1 2 3... 12 LESSON 1.3 Discoverig Advaced Algebra Ivestigatio Worksheets
Ivestigatio Doses of Medicie (cotiued) Step 7 Write a recursive formula that geerates this sequece. Step 8 Do the cotets of the bowl ever tur ito pure medicie? Why or why ot? Step 9 Sketch a graph ad explai what happes to the level of medicie i the blood after may days. y Amout of medicie (ml) x Day Discoverig Advaced Algebra Ivestigatio Worksheets LESSON 1.3 13
Ivestigatio Match Them Up Name Period Date Match each table with a recursive formula ad a graph that represet the same sequece. Write your matches i the blaks. Thik about similarities ad differeces betwee the sequeces ad how those similarities ad differeces affect the tables, formulas, ad graphs. 1. 2. 3.. 5. 6. 0 8 0 0.5 0 2 0 2 0 8 0 1 1 1 1 1 2 2 1 6 1 3 1 2 2 2 2.5 5 8 3 2 2 6 0.125 3 3.625 7 12 5 2 9 0.015625 8 5 3.8125 10 18 7 6 8 A. u 0 8 B. u 0 8 C. u 0 0.5 1 2 where 1 0.5 1 where 1 2 1 where 1 D. u 0 2 E. u 0 F. u 0 2 1 2 where 1 1 where 1 0.5 1 2 where 1 i. ii. iii. 8 8 8 0 2 6 8 0 2 6 8 0 2 6 8 iv. v. vi. 8 8 8 0 2 6 8 0 2 6 8 0 2 6 8 1 LESSON 1. Discoverig Advaced Algebra Ivestigatio Worksheets
Ivestigatio Match Them Up (cotiued) Write a paragraph that summarizes the relatioships betwee differet types of sequeces, recursive formulas, ad graphs. What geeralizatios ca you make? What do yootice about the shapes of the graphs created from arithmetic ad geometric sequeces? Discoverig Advaced Algebra Ivestigatio Worksheets LESSON 1. 15
Ivestigatio Life s Big Expeditures Name Period Date I this ivestigatio you will use recursio to explore loa balaces ad paymet optios. Your calculator will be a helpful tool for tryig differet sequece models. Part 1 You pla to borrow $22,000 from a bak to purchase a ew car. You will make a paymet every moth to the bak to repay the loa, ad the loa must be paid off i 5 years (60 moths). The bak charges iterest at a aual rate of 7.9%, compouded mothly. Part of each mothly paymet is applied to the iterest, ad the remaider reduces the startig balace, or pricipal. Step 1 What is the mothly iterest rate? What is the first moth s iterest o the $22,000? If you make a paymet of $300 at the ed of the first moth, the what is the remaiig balace? Step 2 Record the balaces for the first 6 moths with mothly paymets of $300. How may moths will it take to pay off the loa? Moth Balace 1 2 3 5 6 16 LESSON 1.5 Discoverig Advaced Algebra Ivestigatio Worksheets
Ivestigatio Life s Big Expeditures (cotiued) Step 3 Experimet with other values for the mothly paymet. What mothly paymet allows you to pay off the loa i exactly 60 moths? Step How much do you actually pay for the car usig the mothly paymet you foud i Step 3? (Hit: The last paymet should be a little less tha the other 59 paymets.) Part 2 Use the techiques that you discovered i Part 1 to fid the mothly paymet for a 30-year home mortgage of $16,000 with a aual iterest rate of 7.25%, compouded mothly. How much do you actually pay for the house? Moth Balace 1 2 3 5 6 Discoverig Advaced Algebra Ivestigatio Worksheets LESSON 1.5 17