Student Activity: Show Me the Money! - WordPress.com. Linear Functions 1.2 The Y-Intercept: Student Activity TEXTEAMS Algebra I: 2000 and Beyond Spring 2001 Student Activity: Show

II. Linear Functions 1.2 The Y-Intercept: Student Activity

TEXTEAMS Algebra I: 2000 and Beyond Spring 2001

Student Activity: Show Me the Money!Overview: Students connect recursive operations with graphs.

Objective: Algebra I TEKSb.3.B Given situations, the student looks for patterns and representsgeneralizations algebraically.c.1 The student understands that linear functions ban be represented indifferent ways and translates among their various representations.

Terms: recursion, increasing, decreasing

Materials: graphing calculators

Procedures: Have students complete the activity. Have the groups display their results on1” grid paper. Have two groups discuss their results. Emphasize that therepeated addition of the same number every time results in a constant rate ofchange, and hence a linear function.

1. A recursive routine to model Susan’s savings:25 EnterAns + 2.5, Enter, Enter, etc.

2.Time

(weeks)Money

0 $25.001 $27.502 $30.003 $32.504 $35.005 $37.506 $40.007 $42.50

$ 0$ 5

$10$15$20$25$30$35$40$45

0 1 2 3 4 5 6 7

Time (weeks)

M

o

n

e

y

3. Addition4. linear, increasing

5. A recursive routine to model Manuel’s spending:1090 EnterAns - 30, Enter, Enter, etc.

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II. Linear Functions 1.2 The Y-Intercept: Student Activity


6.Time

(weeks)MoneySaved

0 $10901 $10602 $10303 $10004 $9705 $9406 $9107 $880

$880

$940

$ 1,000

$ 1,060

$ 1,120

0 1 2 3 4 5 6 7

Time (weeks)

M

o

n

e

y

7. Subtraction, which can also be thought of as repeated addition of a negativenumber.

8. linear, decreasing. Emphasize that repeated subtraction is the same as repeatedaddition of a negative number.

Assessment Answers:1. c2. f3. d4. b5. g6. e7. a8. h

Summary: Using a recursive routine, students generate points on a graph and makegeneralizations. Repeated addition results in a linear graph. Repeatedaddition of a positive number is an increasing line. Repeated addition of anegative number is an decreasing line.

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II. Linear Functions 1.2 The Y-Intercept: Leaders’ Notes


1.2 The Y-InterceptOverview: Participants use real life experiences to build the concepts of y-intercept as the

starting point and slope as a rate of change.

Objective: Algebra I TEKS(c.1.C) The student translates among and uses algebraic, tabular, graphical, orverbal descriptions of linear functions.(c.2.A) The student develops the concept of slope as rate of change anddetermines slopes from graphs, tables, and algebraic representations.(c.2.B) The student interprets the meaning of slope and intercepts insituations using data, symbolic representations, or graphs.

Terms: y-intercept, slope, rate of change, increasing, decreasing, recursion

Materials: graphing calculators

Procedures: Participants should be seated at tables in groups of 3 – 4.Depending on the participants, briefly talk through or work through theStudent Activity, which connect recursion with graphing.

Activity 1: The Birthday GiftWork through Activity 1 with participants, modeling good pedagogy: askleading 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 Saved0 $25 $25

After the 1st week, Susan had the $25 she started with and $2.50.Time (weeks) Process Amount Saved

0 $25 $251 $ $ .25 2 50+ $27.50

After the 2nd 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 Saved0 $25 $251 $ $ .25 2 50+ $27.502 $ $ . $ . $ $ .25 2 50 2 50 25 2 2 50+ + = + ( ) $30.00

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Note that the above step is not a natural step for many students. They aremore apt to operate recursively on the previous term , adding $2.50 to $27.50.Teachers need to be aware that this is difficult for some students.

Time (weeks) Process Amount Saved0 $25 $251 $ $ .25 2 50+ $27.502 $ $ . $ . $ $ .25 2 50 2 50 25 2 2 50+ + = + ( ) $30.003 25 2 5 2 5 2 5 25 3 2 50+ + + = + ( ). . . . $32.504 25 2 5 2 5 2 5 2 5 25 4 2 50+ + + + = + ( ). . . . . $35.00

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 Savedt 25 2 5 25 2 5+ = +t t( . ) . 25 2 5+ . 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 to10 weeks shows a reasonable amount of time. The variable y stands fortotal money saved, so $0.00 to $60.00 will show all the savings and the x-axis.

5. 25 2 5 7 42 50+ ( ) =. . . Susan will have $42.50 after 7 weeks.

6. 25 2 5 139 99+ =. .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:

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As an extension, note that the question really asks for an inequality:25 2 5 139 99+ ≥. .t .

7. Susan’s starting value is lower, so the line will “start” on the y-axis at 15instead of 25. The y-intercept changed. The slope, or amount of moneyshe saved every week, did not change. The two lines are parallel, with thenew line translated down from the original.

8. Susan’s starting value is higher, so the line will “start” on the y-axis at 40instead of 25. The y-intercept changed. The slope, or amount of moneyshe saved every week, did not change. The two lines are parallel, with thenew line translated up from the original.

9. Susan’s rate of saving has changed so the amount of money will not growas 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 notparallel 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 topractice the language they plan to use when teaching their students.1.

Time (weeks) Process Amount ofMoney

0 $1090 $10901 $ $1090 30− $10602 $ $ $ $ $1090 30 30 1090 2 30− − = − ( ) $10303 1090 30 30 30 1090 3 30− − − = − ( ) $10004 1090 30 30 30 30 1090 4 30− − − − = − ( ) $35.00

2. 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 Moneyt 1090 30 1090 30− = −t t( ) 1090 30− t

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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 to10 weeks shows a reasonable amount of time. The variable y stands fortotal amount of money, so $800 to $1090 will show all his money.

5. 1090 30 11 760− ( ) = . Manuel will have $760 after 11 weeks.

6. 1090 30 0− =t . After 36 weeks, Manuel will only have $10. He willcannot spend the whole $30 the next week, only $10 and then he will beout 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 at1300 instead of 1090. The y-intercept changed. The slope, or amount ofmoney 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 at890 instead of 1090. The y-intercept changed. The slope, or amount ofmoney he spent every week, did not change. The two lines are parallel,with the new line translated down from the original.

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9. Manuel’s rate of spending has changed so now the amount of money willnot deplete as fast, so the line will be less steep. Manuel’s rate of savinghas changed. His starting point, or the y-intercept, did not change. Thenew line is not parallel to the original line because the rate of saving haschanged.

Activity 3: Money, Money, Moneyork 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 to38 weeks shows the time it takes Manuel to spend all of his money. Thevariable y stands for total amount of money, so $0.00 to $1100 will showboth graphs.

2. 25 2 5 7 42 50+ ( ) =. . . Susan will have $42.50 after 7 weeks.

3. 25 2 5 1090 30+ = −. t t . They never do have the same amount of moneybecause they are saving or spending each week, not in the middle of theweek. This is shown in the table as we choose the increment to be a weeknot a part of a week. After week 33, Susan has $107.50 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. Sothe question is now, how much money does Susan have after 36 weeks?Susan has $115 after 36 weeks.To find this answer, we solved 1090 30 0− =t and used the solution tosolve 25 2 5 36 115+ ( ) =. .

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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, b4. iii, c5. iv, a6. 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 moneyspent per week. Encourage participants to use the word “rate”.]

• What changes in the situation resulted in a change in the starting point ofthe line? [Changing the starting amount of money, initial amount ofmoney.]

• Look at your function rules. What does the constant represent in thisproblem? [The initial, or starting, amount of money]

• Look at your function rules. What does the coefficient of x represent inthis problem? [Encourage the words “rate of spending”]

• Look at your function rules. If the coefficient of x is negative, what doesthis represent in this problem? [Spending]

• Look at your function rules. If the coefficient of x is positive, what doesthis represent in this problem? [Saving]

Summary: Using real life situations, participants investigate the effects of changing thestarting point and the rate of change of a line.

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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 $2.50. 1. 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 TEXTEAMS Algebra I: 2000 and Beyond Austin ISD Mathematics Department

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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 TEXTEAMS Algebra I: 2000 and Beyond Austin ISD Mathematics Department

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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 $2.50. 1. 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.1.2 - Activity 1 TEXTEAMS Algebra I: 2000 and Beyond Austin ISD Mathematics Department

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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

$139.99 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.1.2 - Activity 1 TEXTEAMS Algebra I: 2000 and Beyond Austin ISD Mathematics Department

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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

$1090

2. 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.1.2 - Activity 2 TEXTEAMS Algebra I: 2000 and Beyond Austin ISD Mathematics Department

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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.1.2 - Activity 2 TEXTEAMS Algebra I: 2000 and Beyond Austin ISD Mathematics Department

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