Algebra
Chapter 1 Answers, Data and Relationships
Cars, credit cards, banking,... all of these areas can be confusing for adults. How can you find the best financial deals as you begin paying for your own purchases? Is credit a good way to pay for goods and services? Let's investigate using the Internet and mathematics!
Part A, Interpreting Graphs of Data
1. Decide whether to get a new or used car by looking at the Which is better: new or used? Web site. Use the default values which should be set to:
Age of Used Vehicle: 3 years
Price of New Vehicle: $15,000
Future Depreciation: Average
Hit the "Calculate" button.
| New |
Used |
|
| Interest rate (%) | 8 |
10 |
| Loan term (months) | 24 |
24 |
| Down payment ($) | 2,000 |
2,000 |
| Years you own car | 4 |
4 |
| Maintenance costs ($/month) | 30 |
120 |
a. Scroll down and look at the line graph titled "Yearly Cost." Why does the average yearly cost of owning a new vehicle decrease sharply during the first 2 years?
[Posible answers include: While a new car depreciates in value, its overall value is still higher than the used car/s value. Repair costs may be less on a new car since it is under warranty.]b. What can you tell about the yearly cost of the new and used vehicles after you have owned them for 10 years?
[The average yearly cost of the new car is lower.]c. Go back to the data box given in a and change the loan term to 5 years. Explain what happens to the graph titled "Yearly Cost."
Hint: Remember to translate years into months.
[Possible answers include: the new car's average yearly cost decreased sharply for 5 years when the monthly payments end. After almost 4 years, the average yearly cost of the new car is lower than the cost of the used car.]
2. Think about how long it will be before you will have paid off the cost of the car.
a. If you can pay off your car within a few years, will it be cheaper for you to get a new car or a used car? [new]
b. If it will take you more than just a few years to pay off your car, will it be cheaper for you to get a new car or a used car? [used]
Part B, Adding with Integers and Matrices
3. Different credit cards charge different rates of interest and annual fees. To help you decide which factors matter the most, look at the credit card page of the Financial calculator Web site.
a. Use the default values, which should be set to:
Amount Owed Now: $5000
Future Monthly Charges: $275
Future Monthly Payments: $350
Annual Rate: 17%
Annual Fee: $25Scroll down and look at the "Schedule of Payments." Write the first six months as a matrix with the headings Balance Owed, Monthly Purchase, Monthly Payments, Interest and Fees, and Cumulative Interest and Fees. Label your work Matrix A.
[5000 275 350 95 95
5020 275 350 71 166
5016 275 350 71 238
5013 275 350 71 309
5009 275 350 70 380
5005 275 350 70 451
Matrix A]b. What is the dimension of Matrix A? [6 x 5 or 6 by 5]
4. Change the interest rate to 6% and hit "Calculate."
a. Now look at the "Schedule of Payments" and write the data for the first 6 months as a matrix with the same headings you used for Matrix A. This time, label your work Matrix B.
[5000 275 350 50 50
4975 275 350 24 74
4924 275 350 24 99
4874 275 350 24 124
4824 275 350 24 148
4773 275 350 23 172
Matrix B]b. What is the dimension of Matrix B? [6 x 5 or 6 by 5]
5. Compare Matrices A and B
a. Calculate the Matrix A - B and label it.
[0 0 0 45 45
45 0 0 47 92
92 0 0 47 139
139 0 0 47 185
185 0 0 46 232
232 0 0 47 279
Matrix A - B]b. Explain what Matrix A - B represents.
[The first column represents the change in the balance owed when the interest rate drops from 17% to 6%. The second column, all zeros, shows that there is no change in the monthly purchases. The third column, also all zeros, indicates that the monthly payment is the same. The fourth column shows the difference between the interest and fees when the interest rate drops from 17% to 6%. The last column shows the difference in cumulative interest and fees dropping to a final savings of $279 over the 6-month period.]c. What is the dimension of Matrix A - B? [6 x 5 or 6 by 5]
6. Notice that the dimensions of the matrices in 5, 6, and 7 are all the same. Must this always be the case to add or subtract? Explain.
[Yes. To add or subtract, matrices must be in the same dimension just as numbers must be in the same column.]
Part C, Multiplying with Integers and Matrices
7. The nation's money is not always worth exactly the same amount. Compared with the money of other nations, the dollar may be worth more or less.
a. Suppose the U.S. dollar were devalued so that $10 would be worth only $1. How would the devaluation affect the numbers in Matrix A?
[Each entry would be one-tenth of its original value.]b. Write a problem showing how multiplying your matrix by a scalar would account for this devaluation.
[Students' problems will vary.]c. Find the answer to the problem you wrote in b.
[Answers to the students' problems will vary.]
Part D, Measures of Central Tendency
8. Work with a partner to examine the mathematics of the Coin flipping Web site.
a. Flip a coin 10 times for each trial. (You will each have 100 flips.) In the chart, record the number of "tails" for each trial.
Game 1
TRIALS1
2
3
4
5
6
7
8
9
10
Total
You Partner
[Students' results will vary.]b. Find the mean, median, and mode for both your scores and your partner's scores. [Students' results will vary.]
c. Explain which measure of central tendency best reflects your average coin-flipping score. [Students' responses will vary.]
Part E, Making Connections
9. Work with a partner and the Coin flipping Web site for 10 more flipping trials. Record your scores on a new chart.
| Game 2 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
Total |
| You | |||||||||||
| Partner |
a. Make a matrix of your first 10 trials from 8a (Game 1).
[Responses will vary.]b. Make a matrix of your second 10 trials from 9 (Game 2).
[Responses will vary.]
10. Create a new matrix showing your scores from Game 2 minus your scores from Game 1.
a. Explain what a number of +2 in the eleventh column would indicate.
[A number of +2 would indicate that coin showed "tails" two more times in Game 2.]b. What would a number of 0 in the eleventh column indicate?
[A number of 0 in the eleventh column would indicate that the coin showed "tails" the same number of times in Games 1 and 2.]c. What would a number of -2 in the eleventh column indicate?
[A number of -2 would mean that the coin showed "tails" two fewer times during Game 2.]
11. Look at the information gathered from the second game you and your partner played.
a. What type of graph could be drawn from this data? Explain.
[Possible answers include: A vertical bar graph in two different colors could represent each player's score.]b. Explain how you could represent both games on the same graph.
[Possible answers include: A bar graph with a bar to represent each player's total score for the two games.]
Some high school seniors must take the SAT test to get into college. What do schools find out about students by looking at their SAT scores? Let's investigate using the Internet and mathematics!
Part A, Working with Pairs of Data
1. Look at the scores on the math and verbal sections of the SAT Web site.
a. Look first at the data from verbal section of the 1997-98 SAT I. Is there a positive association, a negative association, or no association between the score and the percentile? [There is a positive association of the data.]
b. Look at the data from the math section of the 1997-98 SAT I. Is there a positive association, a negative association, or no association between the score and the percentile? [There is a positive association of the data.]
c. The mean score on the verbal section of the SAT I is at about what percentile? How do you know?
[An answer of 47 up to 49 in a good approximation. An exact answer is that the mean SAT I Verbal score is 505, which is 5/50 or one-tenth of the increase from the 47th percentile to the 64th percentile. This is approximately one-tenth of 17 percentiles, or 1.7. This increases the percentile from a 47 to approximately 49.]d. The mean score on the math section of the SAT I is at about what percentile? How do you know?
[A good approximation is 45 or up to 47. An exact answer is that the mean SAT I Math score is 511, which is 11/50 of the increase from the 45th percentile to the 62nd percentile. This is approximately 11/50 of 17 percentiles, or 3.74. ]
2. A scatter graph of the SAT scores will show the relationship of the data.
a. If you were to make a scatter plot with the SAT I verbal scores on the horizontal axis and the percentiles on the vertical axis, what would your graph look like?
[The graph would be a scatter plot with increasing data as you move from left to right.]b. If you made a scatter plot with the SAT I math scores on the horizontal axis and the percentiles on the vertical axis, what would your graph look like?
[The graph would be a scatter plot with increasing data as you move from left to right.]
Part B, Graphing Data on a Coordinate Plane
3. Plot the data from 2a on coordinate axes. Label each ordered pair.
4. Plot the data from 2b on coordinate axes. Label each ordered pair.
5. Which quadrant(s) do you use to plot the data? [Quadrant I]
6. What would it mean if you plotted data in Quadrant III?
[Data plotted in Quadrant III would not make any sense. This would indicate a negative score and a negative percentile. It is not possible to score below 200 on the SAT I.]
7. Using your graphs, near what percentile would you expect to find a score of 625 on the SAT I verbal section? How did you find your answer?
[Locate the score 625 on the x-axis. Read up to the graph and then over to the left to the Verbal Percentile on the y-axis. The score is close to the 80th percentile.]
8. Using your graphs, near what percentile would you expect to find a score of 710 on the SAT I math section? How did you find your answer?
[Locate the score 710 on the x-axis. Read up to the graph and then over to the left to the Verbal Percentile on the y-axis. The score is close to the 95th percentile.]
Part C, Making Connections
9. Ten students took the SAT I test. Their scores are on the following chart.
| Test | A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
| Verbal | 429 |
312 |
455 |
405 |
795 |
480 |
500 |
750 |
507 |
625 |
| Math | 648 |
482 |
455 |
629 |
410 |
425 |
610 |
762 |
509 |
400 |
a. Make a scatter plot of the scores. Put the verbal score on the horizontal axis and the math score on the vertical axis. Which student would you consider to be the "average" student? Explain your reasoning.
[Student I has a verbal score of 507 which is 2 points away from the mean verbal score as well as a math score of 509 which is 1 point away from the mean math score.]
b. Which point represents the student you would consider to be an outlier? Explain your reasoning.
[Student H is an outlier. Both of the scores are significantly higher than the mean scores and are away from the other data points.]
10. Draw a verticle line through the horizontal axis of your scatter plot to represent the mean verbal score and draw a horizontal line through the vertical axis to represent the mean math score. What can you tell about each student now?
[Student H is above the mean on both the verbal and the math tests. Students A, D, and G are above the mean on the math test but below the mean on the verbal test. Students B, C, and F are below the mean on both tests. Students J and E are above the mean on the verbal test and below the mean on the math test. Student I is near the mean on both tests.]
11. Find the mean scores of the ten students in the chart above.
a. What are the mean scores of the ten students on both the verbal and the math sections of the SAT I?
[The mean SAT I Verbal Score of the ten students is 525.8. The mean SAT I Math Score of the ten students is 533.]b. Did these ten students do better or worse than the national average? [better]
Part A, Probability and Experiments
1. Sometimes sailors use flags to communicate with other ships. Look at the Alphabet flags Web site to see the flags they use.
a. How many different alphabet flags are there?
[26 -- one for each letter of the alphabet.]b. If these flags were in a bag and you reached in and randomly took one out, what is the probability that it would be rectangular? Write your answer as a ratio and as a percent. [24/26 = 92.3%]
c. What is the probability that you would choose a flag with red on it? Write your answer as a ratio and as a percent. [13/26 = 50%]
d. What is the probability that you would choose a flag with blue on it? Write your answer as a ratio and as a percent. [15/26 = 57.7%]
e. Which is greater: The probability that a flag has blue on it or the probability that a flag has red on it?
[The probability that a flag has blue on it is 15/26 or 57.7%. The probability that a flag has red on it is 13/26 or 50%. Therefore, the probability that there is blue on a flag is greater.]
Part B, The Theory of Probability
2. Look at the Bolivian flag Web site.
a. How many different colors are on the flag? [3]
b. In how many different ways could you rearrange the colors? [3! = 6]
3. Look at the Flags of the world Web page.
a. How many stars are on the flag? [6]
b. In how many different ways could you rearrange these stars?
[6! or 720 different ways.]c. What is the principle that allows you to rearrange the stars?
[The counting principle.]
4. Look at the Honduran flag site.
a. How many stars does it have? [5]
b. In how many ways could you rearrange these stars? Explain.
[Technically you could rearrange the stars in 5! = 120 different ways, but because they are all the same size and color, there is only one way to arrange them.]
5. Look at the maps of the Americas.
a. According to the North American Web site, how many countries are on the continent of North America? [3]
b. According to the Central American Web site, how many countries are in Central America? [7]
c. According to the South American Web site, how many countries are on the continent of South America? [12]
d. If you wanted to fly a flag from each region -- North America, Central America, and South America-- how many different combinations could you choose to fly? [3 x 7 x 12 = 252]
e. If you wanted to fly a flag from Central America, South America, and North America, how many different combinations could you choose?
[7 x 12 x 3 = 252]f. Does the order in which you choose the flags matter? Explain.
[No. You will still have 252 different combinations of three flags.]
Part C, Making Connections
6. Look at the Middle East Web site.
a. How many countries are in the Middle East? [14]
b. If you wanted to display two different flags from this region, how many different combinations could you choose? [14 x 13 = 182]
c. If you wanted to display three different flags from this region, how many different combinations could you choose? [14 x 13 x 12 = 2184]
7. Explore the International marine signal flags Web site with a partner.
a. Write a probability investigation that you can share with another team. Make sure to solve your own problem, give the theoretical probability, and explain your answer. [Answers will vary.]
b. Switch with another team and solve their investigation. Give the theoretical probability and explain your answer. [Answers will vary.]
c. Check the work of the team that solved your original problem. Were they correct? Why or why not? [Answers will vary.]