Student Activity: Show Me the Money!

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1 1.2 The Y-Intercept: Student Activity Student Activity: Show Me the Money! Overview: Objective: Terms: Materials: Procedures: Students connect recursive operations with graphs. Algebra I TEKS b.3.b Given situations, the student looks for patterns and represents generalizations algebraically. c.1 The student understands that linear functions ban be represented in different ways and translates among their various representations. recursion, increasing, decreasing graphing calculators Have students complete the activity. Have the groups display their results on 1 grid paper. Have two groups discuss their results. Emphasize that the repeated addition of the same number every time results in a constant rate of change, and hence a linear function. 1. A recursive routine to model Susan s savings: 25 Enter Ans + 2.5, Enter, Enter, etc. 2. Time Money (weeks) 0 $ $ $ $ $ $ $ $42.50 M o n e y $45 $40 $35 $30 $25 $20 $15 $10 $5 $ Time (weeks) 3. Addition 4. linear, increasing 5. A recursive routine to model Manuel s spending: 1090 Enter Ans - 30, Enter, Enter, etc. 1

2 1.2 The Y-Intercept: Student Activity 6. Time (weeks) Money Saved $1,120 0 $ $ $ $1000 M o n e y $1,060 $1,000 $940 4 $970 5 $940 6 $910 7 $880 $ Time (weeks) 7. Subtraction, which can also be thought of as repeated addition of a negative number. 8. linear, decreasing. Emphasize that repeated subtraction is the same as repeated addition of a negative number. Assessment Answers: 1. c 2. f 3. d 4. b 5. g 6. e 7. a 8. h Summary: Using a recursive routine, students generate points on a graph and make generalizations. Repeated addition results in a linear graph. Repeated addition of a positive number is an increasing line. Repeated addition of a negative number is an decreasing line. 2

3 1.2 The Y-Intercept: Leaders Notes 1.2 The Y-Intercept Overview: Objective: Terms: Materials: Participants use real life experiences to build the concepts of y-intercept as the starting point and slope as a rate of change. Algebra I TEKS (c.1.c) The student translates among and uses algebraic, tabular, graphical, or verbal descriptions of linear functions. (c.2.a) The student develops the concept of slope as rate of change and determines slopes from graphs, tables, and algebraic representations. (c.2.b) The student interprets the meaning of slope and intercepts in situations using data, symbolic representations, or graphs. y-intercept, slope, rate of change, increasing, decreasing, recursion graphing calculators Procedures: Participants should be seated at tables in groups of 3 4. Depending on the participants, briefly talk through or work through the Student Activity, which connect recursion with graphing. Activity 1: The Birthday Gift Work through Activity 1 with participants, modeling good pedagogy: ask leading questions, use appropriate wait time, have teachers present their work, etc. Introduce the scenario. 1. Guide participants in filling in the table, using language similar to the following: At time zero, Susan started with $25. Time (weeks) Process Amount Saved 0 $25 $25 After the 1 st week, Susan had the $25 she started with and $2.50. Time (weeks) Process Amount Saved 0 $25 $25 1 $ 25 + $ $27.50 After the 2 nd week, Susan had the $25+$2.50 from week 1 and another $2.50. In other words, Susan had the $25 she started with and two $2.50 s. Time (weeks) Process Amount Saved 0 $25 $25 1 $ 25 + $ $ $ 25 + $ $ = $ ( $ 2. 50) $

4 1.2 The Y-Intercept: Leaders Notes Note that the above step is not a natural step for many students. They are more apt to operate recursively on the previous term, adding $2.50 to $ Teachers need to be aware that this is difficult for some students. Time (weeks) Process Amount Saved 0 $25 $25 1 $ 25 + $ $ $ 25 + $ $ = $ ( $ 2. 50) $ = (. ) $ $ = + ( ) 2. Write the sentence in words and then abbreviate to variables: After t weeks, Susan will have the $25 she started with and t ($2.50 s). Because of convention, mathematicians write 2.5t, instead of t (2.5). Time (weeks) Process Amount Saved t 25 + t( 2. 5) = t t 3. Use questions to lead participants to find a suitable viewing window. What does x represent in this problem? [Elapsed time in weeks] What values make sense for x in this problem? [Answers will vary. Sample answer. Zero weeks to 10 weeks.] What does y represent in this problem? [Total money saved] What values make sense for y in this problem? [Answers will vary. Sample answer. No money to $60.] 4. Sample answer. The variable x stands for elapsed time in weeks so zero to 10 weeks shows a reasonable amount of time. The variable y stands for total money saved, so $0.00 to $60.00 will show all the savings and the x- axis ( )= Susan will have $42.50 after 7 weeks t = After 46 weeks, Susan will have more than $139.99, enough to buy the ring. You may have to open up your window. We did as follows: 4

5 1.2 The Y-Intercept: Leaders Notes As an extension, note that the question really asks for an inequality: t Susan s starting value is lower, so the line will start on the y-axis at 15 instead of 25. The y-intercept changed. The slope, or amount of money she saved every week, did not change. The two lines are parallel, with the new line translated down from the original. 8. Susan s starting value is higher, so the line will start on the y-axis at 40 instead of 25. The y-intercept changed. The slope, or amount of money she saved every week, did not change. The two lines are parallel, with the new line translated up from the original. 9. Susan s rate of saving has changed so the amount of money will not grow as fast, so the line will be less steep. Susan s rate of saving has changed. Her starting point, or the y-intercept, did not change. The new line is not parallel to the original line because the rate of saving has changed. 10. When the rate of saving changes, the slope of the line changes. 11. When the starting value in Susan s saving s plan changed, the starting point, or y-intercept of the line, changed. 12. The point (0, y) is where a line intersects the y-axis. This point represents the starting value of Susan s savings plan. Activity 2: Spending Money Have participants work through Activity 2 in their groups. Encourage them to practice the language they plan to use when teaching their students. 1. Time (weeks) Process Amount of Money 0 $1090 $ $ 1090 $ 30 $ $ 1090 $ 30 $ 30 = $ ( $ 30) $ = ( 30) $ = ( 30) $ Write the sentence in words and then abbreviate to variables: After t weeks, Manuel will have the $1090 he started with minus t ($30 s). Because of convention, mathematicians write 30, instead of t (30). Time (weeks) Process Amount of Money t 1090 t( 30) = t t 5

6 1.2 The Y-Intercept: Leaders Notes 3. Use questions to lead participants to find a suitable viewing window. What does x represent in this problem? [Elapsed time in weeks] What values make sense for x in this problem? [Answers will vary. Sample answer. Zero weeks to 10 weeks.] What does y represent in this problem? [Total amount of money] What values make sense for y in this problem? [Answers will vary. Sample answer. $800 to $1090.] 4. Sample answer. The variable x stands for elapsed time in weeks so zero to 10 weeks shows a reasonable amount of time. The variable y stands for total amount of money, so $800 to $1090 will show all his money ( 11)= 760. Manuel will have $760 after 11 weeks t = 0. After 36 weeks, Manuel will only have $10. He will cannot spend the whole $30 the next week, only $10 and then he will be out of money. You may have to open up your window. An example: 7. Manuel s starting value is higher, so the line will start on the y-axis at 1300 instead of The y-intercept changed. The slope, or amount of money he spent every week, did not change. The two lines are parallel, with the new line translated up from the original. 8. Manuel s starting value is lower, so the line will start on the y-axis at 890 instead of The y-intercept changed. The slope, or amount of money he spent every week, did not change. The two lines are parallel, with the new line translated down from the original. 6

7 1.2 The Y-Intercept: Leaders Notes 9. Manuel s rate of spending has changed so now the amount of money will not deplete as fast, so the line will be less steep. Manuel s rate of saving has changed. His starting point, or the y-intercept, did not change. The new line is not parallel to the original line because the rate of saving has changed. Activity 3: Money, Money, Money ork through Activity 3 with participants. 1. Use questions to find a suitable viewing window. What does x represent in this problem? [Elapsed time in weeks] What values make sense for x in this problem? [Answers will vary. Sample answer. Zero weeks to 38 weeks.] What does y represent in this problem? [Total amount of money] What values make sense for y in this problem? [Answers will vary. Sample answer. No money to $1100.] Sample answer. The variable x stands for elapsed time in weeks so zero to 38 weeks shows the time it takes Manuel to spend all of his money. The variable y stands for total amount of money, so $0.00 to $1100 will show both graphs ( )= Susan will have $42.50 after 7 weeks t = t. They never do have the same amount of money because they are saving or spending each week, not in the middle of the week. This is shown in the table as we choose the increment to be a week not a part of a week. After week 33, Susan has $ and Manuel has $100, which is the closest they get to each other. 4. From earlier work, we found that Manuel had only $10 to spend after 36 weeks. So we will say that after 36 weeks, Manuel is out of money. So the question is now, how much money does Susan have after 36 weeks? Susan has $115 after 36 weeks. To find this answer, we solved t = 0 and used the solution to solve ( 36)=

8 1.2 The Y-Intercept: Leaders Notes Answers to Reflect and Apply: 1. a. Yen started with $20. b. Lira started with $0.00. c. Lira is saving $30 a month. d. Mark is saving $10 a month. 2. a. Frank started with $80. b. Ruble started with $40. c. Peso is spending $30 a month. d. Ruble is spending $10 a month. 3. ii, b 4. iii, c 5. iv, a 6. i, d Use the following questions to summarize and connect activities: What changes in the situation resulted in a change in the steepness of the line? [Changing the rate of spending per week, the amount of money spent per week. Encourage participants to use the word rate.] What changes in the situation resulted in a change in the starting point of the line? [Changing the starting amount of money, initial amount of money.] Look at your function rules. What does the constant represent in this problem? [The initial, or starting, amount of money] Look at your function rules. What does the coefficient of x represent in this problem? [Encourage the words rate of spending ] Look at your function rules. If the coefficient of x is negative, what does this represent in this problem? [Spending] Look at your function rules. If the coefficient of x is positive, what does this represent in this problem? [Saving] Summary: Using real life situations, participants investigate the effects of changing the starting point and the rate of change of a line. 8

9 Student Name: Class Period: Date: Student Activity: Show Me the Money! Susan s grandmother gave her $25 for her birthday. Instead of spending the money, she decided to start a savings program by depositing the $25 in the bank. Each week, Susan plans to save an additional $ Write a recursive routine to model Susan s savings plan. 2. Fill in the table and sketch a graph to model Susan s savings plan: Time (weeks) Money 3. What operation did you repeat in your recursive routine? 4. How does repeated addition look in a graphical representation? I.1.2 Student Activity Austin ISD Mathematics Department TEXTEAMS Algebra I: 2000 and Beyond 9

10 Manuel worked all summer and saved $1090. He plans to spend $30 a week. 5. Write a recursive routine to model Manuel s spending plan. 6. Fill in the table and sketch a graph to model Manuel s spending plan: Time (weeks) Money 7. What operation did you repeat in your recursive routine? 8. How does repeated subtraction look in a graphical representation? I.1.2 Student Activity Austin ISD Mathematics Department TEXTEAMS Algebra I: 2000 and Beyond 10

11 Student Name: Class Period: Date: Activity 1: The Birthday Gift Susan s grandmother gave her $25 for her birthday. Instead of spending the money, she decided to start a savings program by depositing the $25 in the bank. Each week, Susan plans to save an additional $ Make a table of values for the situation. Time (Weeks) Process Amount Saved $25 2. Write a function rule for the amount of money Susan will have after x weeks. 3. Find a viewing window for the problem situation. Sketch your graph: Note your window: Xmin: Xmax: Xscl: Ymin: Ymax: Yscl: 4. Justify your window choices. I Activity 1 Austin ISD Mathematics Department TEXTEAMS Algebra I: 2000 and Beyond 11

12 Use your graph and table to find the following: 5. How much money will Susan have after 7 weeks? Write this equation. Show how you found your solution. 6. Susan wants to buy a school ring. When will she have enough money to buy the $ ring? Write this equation. Show how you found your solution. 7. How will the line change if Susan deposits only $15 of the $25? Graph the line. What changed? What did not change? 8. How will the line change if Susan deposits the $25 from her grandmother plus another $15 she already had? Graph the line. What changed? What did not change? 9. How will the line change if Susan deposits the $25 from her grandmother, but decides she can only save $2.00 a week? Graph the line. What changed? What did not change? 10. What changes in the situation resulted in a change in the steepness of the line? 11. What changes in the situation resulted in a change in the starting point of the line? 12. Write the coordinates of the point where a line intersects the y-axis. This point is called the y-intercept. What do these coordinates represent in this problem? I Activity 1 Austin ISD Mathematics Department TEXTEAMS Algebra I: 2000 and Beyond 12

13 Student Name: Class Period: Date: Activity 2: Spending Money Manuel worked all summer and saved $1090. He plans to spend $30 a week. 1. Make a table of values for the situation. Time (Weeks) Process Amount Saved $ Write a function rule for the amount of money Manuel will have after x weeks. 3. Find a viewing window for the problem situation. Sketch your graph: Note your window: Xmin: Xmax: Xscl: Ymin: Ymax: Yscl: 4. Justify your window choices. II Activity 2 Austin ISD Mathematics Department TEXTEAMS Algebra I: 2000 and Beyond 13

14 Use your graph and table to find the following: 5. How much money will Manuel have after 11 weeks? Write this equation. Show how you found your solution. 6. When will Manuel be out of money? Write this equation. Show how you found your solution. 7. How will the line change if Manuel had initially earned $1300? Graph the line. What changed? What did not change? 8. How will the line change if Manuel spent $200 on school clothes and started the year with only $890? Graph the line. What changed? What did not change? 9. How will the line change if Manuel starts with the $1090, but decides he will only spend $25 a week? Graph the line. What changed? What did not change? II Activity 2 Austin ISD Mathematics Department TEXTEAMS Algebra I: 2000 and Beyond 14