Algebra I Part 1

Algebra I Part 1

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Unit 1, Activity 1, Where Do I Belong?

Blackline Masters, Algebra I–Part 1 Page 1 Louisiana Comprehensive Curriculum, Revised 2008

Directions: Place an “X” in the appropriate box to indicate what set each of the numbers in the first column belong. Be ready to explain your reasoning to the rest of the class.

Number Natural Whole Integer Rational Irrational

52−

81

11

-2

169

2

945

459−

π459−

3.14

01

10

2π

39−

Unit 1, Activity 1, Where Do I Belong? with Answers


Directions: Place an “X” in the appropriate box to indicate what set each of the numbers in the first column belong. Be ready to explain your reasoning to the rest of the class.

Number Natural Whole Integer Rational Irrational

52−

X

81

X X X X

11

X

-2

X X

169

X

2 X

X X X

945

X X X X

459−

X

π459−

X

3.14

X

01

10

X X X X

2π

X

39−

X X

Unit 1, Activity 5, What Method Should I Use?


Directions: For each problem shown, decide whether it is most appropriate to solve it using ESTIMATION (not exact); MENTAL MATH (exact); Paper/Pencil (exact); or using TECHNOLOGY (exact) to obtain the solution. In the second column, write a math log (a short essay explaining your reasoning behind your choice) and be prepared to share your thoughts on each of the problems with your classmates. Problem Math Log: Explain the method you chose to use and why you chose it.

1. John went to the store to buy some groceries. As he went through the store he wanted to make sure he had enough money to pay for everything. If he had $20 and bought milk for $2.89, bread for $1.19, soft drinks for $3.99, and ice cream for $4.59, did he have enough money to purchase the items?

2. Val wanted to find the average of all her grades

in math for the six weeks. Her grades were: 95, 87, 63, 54, 72, 98, and 71. What was her average for the six weeks?

3. Craig worked at a store as a cashier when the electricity went out. A customer had a bill for $1.95 and had given him a $5 bill. How much money should he give back to the customer?

4. There are four rows of corn in a garden and each row has 20 plants on it. How many corn plants are there in the garden?

5. Laquisha ran 4 ½ miles on Monday, 3 ¾ miles on Tuesday, and 4 1/8 miles on Wednesday. How many miles did she run altogether?

Unit 1, Activity 7, Changing Forms


Directions: For each number shown in the table, write the value that is missing in either standard form or scientific notation. Then solve the three problems below.

Standard Form Scientific Notation 5400

7.1 x 10 3− .0000872

540 3 x 10 12− 4.53 x 10 7

Problems:

1. In 2008, the population of the United States will be about 2.88 x 10 8 people. Spending for health care is expected to reach $5800 per person. About how much money will be spent on health care in the United States in 2008? Write your answer using scientific notation.

2. A computer can perform 4.76 x 10 9 instructions per second. How many instructions is that per day? Use scientific notation in your answer.

3. Order the following numbers from least to greatest and explain how you got your answer to this problem.

50.2 x 10 3− ; 4.9 x 10 1− ; .52 x 10 3− ; 491 x 10 2−

Unit 1, Activity 7, Changing Forms with Answers


Directions: For each number shown in the table, write the value that is missing in either standard form or scientific notation. Then solve the three problems below.

Standard Form Scientific Notation 5400 5.4 x 10 3 .0071 7.1 x 10 3−

.0000872 8.72 x 10 5− 540 5.4 x 10 2

.000000000003 3 x 10 12− 45300000 4.53 x 10 7

Problems:

1. In 2008, the population of the United States will be about 2.88 x 10 8 people. Spending for health care is expected to reach $5800 per person. About how much money will be spent on health care in the United States in 2008? Write your answer using scientific notation.

(2.88 x 10 8 people) x (5.8 x 10 3 dollars per person) =1.6704 x 10 12 dollars

2. A computer can perform 4.76 x 10 9 instructions per second. How many instructions is that per day? Use scientific notation in your answer.

4.76 x 10 9 instructions x 3600 sec x 24 hours = 4.11264 x 10 14 instruct/day sec hour day

3. Order the following numbers from least to greatest and explain how you got your answer to this problem.

50.2 x 10 3− ; 4.9 x 10 1− ; .52 x 10 3− ; 491 x 10 2− Rewrite each number in standard form so you can see the order: .52 x 10 3− ; 50.2 x 10 3− ; 4.9 x 10 1− ; 491 x 10 2−

Unit 1, Activity 13, Operations with Radicals


Directions: For each of the problems below, perform the indicated operation and simplify the answer fully. 1. 3 5• 2. 5 15• 3. 76 + 74 4. 113 - 118 5. 83 - 25 6. 59 + 34 7. 453 + 207 8. 509 - 24

Unit 1, Activity 13, Operations with Radicals with Answers


Directions: For each of the problems below, perform the indicated operation and simplify the answer fully. 1. 3 5• 2. 5 15• Ans: 15 Ans: 35 3. 76 + 74 4. 113 - 118 Ans: 710 Ans: 35− 5. 83 - 25 6. 59 + 34 Ans: 2 Ans: Cannot be simplified 7. 453 + 207 8. 509 - 24 Ans: 523 Ans: 241

Unit 1, Activity 15, Patterns in the Real World


Directions: In groups of three, read each problem carefully and answer the questions presented. Work together! Keep in mind the difficulties you have as you attempt the problems and how you overcame them to produce your solutions as you will be writing about this at the end of this activity.

1. A man is offered a $10,000 starting salary, with an annual raise of $800. a. List his annual salaries for the first five years. Start with $10,000.

Year 1 2 3 4 5 Salary

b. Determine his salary in year 20. c. Write an expression showing his salary in year n.

2. A water lily has an area of 8 square inches. These lilies reproduce so fast that the area they cover will double every week.

a. If two lilies are introduced into a pond, list the total area covered at the end of each of the first five weeks.

Week 1 2 3 4 5 Area

b. Determine the area covered by the 20th week. c. Write an expression showing the area covered in the nth week.

3. Suppose you save the given amounts of money over a five-week period. Week 1st 2nd 3rd 4th 5th Money 28¢ 45¢ 62¢ 79¢ 96¢ a. Find the amount you would save in the 52nd week. b. Write an expression showing the amount you would save in the nth week.

Unit 1, Activity 15, Patterns in the Real World with Answers


Directions: In groups of three, read each problem carefully and answer the questions presented. Work together! Keep in mind the difficulties you have as you attempt the problems and how you overcame them to produce your solutions as you will be writing about this at the end of this activity.

1. A man is offered a $10,000 starting salary, with an annual raise of $800. a. List his annual salaries for the first five years. Start with $10,000.

Year 1 2 3 4 5 Salary $10,000 $10,800 $11,600 $12,400 $13,200

b. Determine his salary in year 20. $25,200 c. Write an expression showing his salary in year n.

S = 800(n – 1) + 10,000 where S is salary, in dollars, and n is number of years.

2. A water lily has an area of 8 square inches. These lilies reproduce so fast that the area they cover will double every week.

a. If two lilies are introduced into a pond, list the total area covered at the end of each of the first five weeks.

Week 1 2 3 4 5 Area 16 sq. in. 32 sq. in. 64. sq. in. 128 sq. in. 256 sq. in.

b. Determine the area covered by the 20th week. 8,388,608 sq. in. c. Write an expression showing the area covered in the nth week.

A = 8* 2 n , where A is the area, in square inches, and n is the number of weeks.

3. Suppose you save the given amounts of money over a five-week period. Week 1st 2nd 3rd 4th 5th Money 28¢ 45¢ 62¢ 79¢ 96¢ a. Find the amount you would save in the 52nd week. $8.95 b. Write an expression showing the amount you would save in the nth week.

A = 28 + 17 (n – 1), where A is the amount saved, in cents, and n is the number of weeks.

Unit 1, Activity 17, Tables to Graphs


Directions: The table below shows the cost associated with renting a truck for a single day from Move-4-Less Truck Rental based upon the number of miles the truck is driven. Use this table to answer the questions that follow.

Move-4-Less Truck Rental Prices

# of Miles

0 1 2 3 4 5 6 7 8

Cost to Rent

$45.50 $46.00 $47.00 $48.50

1. Fill in the table of values based upon the pattern you see in the data. 2. Based upon the pattern in the table, how much would it cost to rent the truck for a

single day even if you didn’t have any mileage associated with the rental?

3. Explain in words what the pattern is in the table and what the cost associated with renting a truck from Move-4-Less entails.

4. Based upon your explanation in question 3, write an equation relating the cost, C, in dollars of renting a truck for a single day if you were to drive m miles.

5. Look at the numberless graphs shown below. Based upon the data in the table, which graph do you think best matches the situation presented here? Explain in words why you think the graph you have chosen best matches, and explain why each of the others are not a match to the data shown. (a) (b) (c)

Miles

Cos

t (do

llars

)

Miles

Cos

t (do

llars

)

Miles

Cos

t (do

llars

)

Unit 1, Activity 17, Tables to Graphs with Answers


Directions: The table below shows the cost associated with renting a truck for a single day from Move-4-Less Truck Rental based upon the number of miles the truck is driven. Use this table to answer the questions that follow.

Move-4-Less Truck Rental Prices

# of Miles

0 1 2 3 4 5 6 7 8

Cost to Rent

$45.00 $45.50 $46.00 $46.50 $47.00 $47.50 $48.00 $48.50 $49.00

1. Fill in the table of values based upon the pattern you see in the data. 2. Based upon the pattern in the table, how much would it cost to rent the truck for a

single day even if you didn’t have any mileage associated with the rental? It would cost $45.00.

3. Explain in words what the pattern is in the table and what the cost associated with

renting a truck from Move-4-Less entails. The initial cost of renting the truck is $45.00 and with each mile you travel it adds $0.50 more to the cost of the truck rental. The pattern in the table reflects this amount added as each mile is traveled.

4. Based upon your explanation in question 3, write an equation relating the cost, C,

in dollars of renting a truck for a single day if you were to drive m miles. C = 45 + .5m

5. Look at the numberless graphs shown below. Based upon the data in the table,

which graph do you think best matches the situation presented here? Explain in words why you think the graph you have chosen best matches, and explain why each of the others are not a match to the data shown. (a) (b) (c)

The correct graph is (b) since it reflects an initial cost not equal to $0 and an increase in cost as the miles increase.

Miles

Cos

t (do

llars

)

Miles

Cos

t (do

llars

)

Miles

Cos

t (do

llars

)

Unit 2, Activity 1, What Does It Mean To Be Accurate?


What Does It Mean to Be Accurate?

Accuracy - How close a measurement is to the accepted “true” value. Example: If a 100-gram weight is placed on a scale and the scale reads 100 grams, then it is said to be accurate. Try this experiment with the different scales that are available. Place a 100-gram weight on each of the scales provided. Place a sticky note on the scales (if any) that are accurate. If none are accurate, which one is most accurate or closest to the actual value. Groups of 4: Have each student in your group weigh himself/herself on the bathroom scale provided and record the different measurements. Student 1 Student 2 Student 3 Student 4 Is it possible to know if the scale is accurate? When is it possible to know if the measuring instrument you are using is giving an accurate measurement? Accuracy in Time If you have a watch, record the time at the exact second the teacher prompts you. Time_________________ Record the different times on the board. Which watch is most accurate? The website http://www.time.gov has the official U.S. time, but even its time is accurate to within .7 seconds. Ask your teacher or another student to check the time on the website at the exact time you and your classmates check your watches to see whose watch is most accurate. If your watch is set with the school bell, how accurate is that time?

http://www.time.gov

Unit 2, Activity 1, What Does It Mean To Be Accurate?


Exercises: Determine if it is possible to get an accurate measure from the information given. 1. Jordan measures a piece of wood to be 4 ½ feet long. Is his measurement accurate? 2. Jerry bought a 5-pound bag of sugar. When he got home he measured the bag on a scale that he had calibrated with a 5-pound weight. The bag actually weighed 4.75 pounds. Which measurement is more accurate? 3. Alex checked the time on his watch at exactly 3:52:04. The time on the world universal website was exactly the same. Is his watch accurate? 4. Trevor measured the temperature outside to be 82.67 degrees. Joey also measured the temperature at the same time and got 83. 04. Whose measurement is more accurate? 5. When is it possible to know if a measurement is accurate?

Unit 2, Activity 1, What Does It Mean To Be Accurate? with Answers


Accuracy - How close a measurement is to the accepted “true” value. Example: If a 100-gram weight is placed on a scale and the scale reads 100 grams, then it is said to be accurate. Try this experiment with the different scales that are available. Place a 100-gram weight on each of the scales provided. Place a sticky note on the scales (if any) that are accurate. If none are accurate, which one is most accurate or closest to the actual value. Groups of 4: Have each student in your group weigh himself/herself on the bathroom scale provided and record the different measurements. Student 1 Student 2 Student 3 Student 4 Is it possible to know if the scale is accurate? No, because we do not know our true weight. When is it possible to know if the measuring instrument you are using is giving an accurate measurement? Only when you know the actual true value. Accuracy in Time If you have a watch, record the time at the exact second the teacher prompts you. Time_________________ Record the different times on the board. The website http://www.time.gov has the official U.S. time, but even its time is accurate to within .7 seconds. Ask your teacher or another student to check the time on the website at the exact time you and your classmates check your watches to see whose watch is most accurate.. If your watch is set with the school bell, how accurate is that time?

http://www.time.gov

Unit 2, Activity 1, What Does It Mean To Be Accurate? with Answers


Exercises: Determine if it is possible to get an accurate measure from the information given. 1. Jordan measures a piece of wood to be 4 ½ feet long. Is his measurement accurate?

It is impossible to tell since the actual measurement is not known. His measuring tool may not be accurate.

2. Jerry bought a 5-pound bag of sugar. When he got home he measured the bag on a scale that he had calibrated with a 5-pound weight. The bag actually weighed 4.75 pounds. Which measurement is more accurate?

4.75 pounds is more accurate since the scale was calibrated with an actual weight. 3. Alex checked the time on his watch at exactly 3:52:04. The time on the world universal website was exactly the same. Is his watch accurate?

Yes, his watch is accurate 4. Trevor measured the temperature outside to be 82.67 degrees. Joey also measured the temperature at the same time and got 83. 04. Whose measurement is more accurate?

It is impossible to tell who is more accurate since the actual temperature is not known. 5. When is it possible to know if a measurement is accurate? It is only possible to tell if a measurement is accurate if the actual value is known.

Unit 2, Activity 2, What is Precision?


What is Precision? Precision of an instrument – the degree of refinement of a measuring instrument or the number of digits in a reading taken from that measurement In this activity, we will be focusing on the precision of an instrument. Look at the different rulers shown below and determine the nearest unit of measure that can be obtained using that ruler.

If you were to measure a toothpick, which ruler would give the most precise measure? Using the different rulers provided, measure the toothpick. Ruler 1 _______________ Ruler 2 _______________ Ruler 3________________

Unit 2, Activity 2, What is Precision?


Ruler 4________________ Which of the measurements is the most precise? Take your four-sided meter stick and record the measurement of the length of the sheet of paper that you were given. Measure the sheet of paper with each of the four sides and record your measurements below. Side 1 _____________________ Side 2______________________ Side 3______________________ Side 4______________________ Record your measurements on the class chart. Which side of the meter stick gave the most precise measure of the length of the sheet of paper?

Unit 2, Activity 5, Target


If you were trying to hit a bull’s eye (the center of the target) with each of five darts, you might get results such as in the models below. Determine if the results are precise, accurate, neither or both.

This is a random-like pattern, neither precise nor accurate. The darts are not clustered together and are not near the bull’s eye.

This is a precise pattern, but not accurate. The darts are clustered together but did not hit the intended mark.

Unit 2, Activity 5, Target


This is an accurate pattern, but not precise. The darts are not clustered, but their average position is the center of the bull’s eye.

This pattern is both precise and accurate. The darts are tightly clustered, and their average position is the center of the bull’s eye.

Unit 2, Activity 5, Precision vs. Accuracy


Directions: Answer the following questions based upon what you know about precision and accuracy. Remember, precision has two aspects to consider…repeatability of the actual measurement and the number of digits in the measurement. Example 1: Using the table below, answer the following questions. Assume that each data set represents 5 measurements taken from the same object.

• Which of the following sets of data is more precise, based on its range? • Do you know which data set is more accurate? Explain.

Set A Set B 14.32 36.56 14.37 36.55 14.33 36.48 14.38 36.53 14.35 36.55

Example 2: The data tables below show measurements that were taken using three different scales. The same standard 100 gram weight was placed on each scale and measured 4 different times by the same reader using the same method each time.

Trial # Weight on Scale 1 Weight on Scale 2 Weight on Scale 3 1 101.5 100.00 100.10 2 101.5 100.02 100.00 3 101.5 99.99 99.88 4 101.5 99.99 100.02

Average Weight

• Determine the average weight produced by each scale. Use this average as the

actual weight of the 100g mass determined by each scale. Write down the results for each scale used.

• Which scale was the most precise? Explain how you know. • Which scale was the least precise? Explain how you know. • Which scale was the most accurate if we consider the true value of the weight to

be 100 grams? Explain your answer.

Unit 2, Activity 5, Precision vs. Accuracy


Example 3: Below is a data table produced by 4 groups of students who were measuring the mass of a paper clip which had a known mass of 1.0004 g.

• Determine the average weight produced by each group’s measurements and fill in the results in the table. Use this average as the weight of the paper clip for each group.

• Which of the group’s measurements represents a properly accurate and precise measurement of the mass of the paper clip?

• Which of the group’s measurements was the least accurate? Explain why. • Which of the group’s measurements had an accurate answer, but not a precise

answer? Explain.

Trial # Group 1 (g) Group 2 (g) Group 3 (g) Group 4 (g) 1 1.01 3.863287 10.13252 2.05 2 1.03 3.754158 10.13258 0.23 3 0.99 3.186357 10.13255 0.75

Average Weight

Unit 2, Activity 5, Precision vs. Accuracy with Answers


Directions: Answer the following questions based upon what you know about precision and accuracy. Remember, precision has two aspects to consider…repeatability of the actual measurement and the number of digits in the measurement. Example 1: Using the table below, answer the following questions. Assume that each data set represents 5 measurements taken from the same object.

Which of the following sets of data is more precise, based on its range? (Solution: Data Set A has a range of .06 while Data Set B has a range of .08, thus the more precise data set is Set A. Note that both sets of data were measured to the hundredth…but it is impossible to know hundredth of what since there are no units associated with the measurement.)

• Do you know which data set is more accurate? Explain. (Solution: There is no way of knowing which is more accurate since in both cases there is no indication of the true measure of the object being measured.)

Set A Set B 14.32 36.56 14.37 36.55 14.33 36.48 14.38 36.53 14.35 36.55

Example 2: The data tables below show measurements that were taken using three different scales. The same standard 100 gram weight was placed on each scale and measured 4 different times by the same reader using the same method each time.

Trial # Weight on Scale 1 Weight on Scale 2 Weight on Scale 3 1 101.5 100.00 100.10 2 101.5 100.02 100.00 3 101.5 99.99 99.88 4 101.5 99.99 100.02

Average Weight

101.5g 100.00g 100.00g

• Determine the average weight produced by each scale. Use this average as the actual

weight of the 100g mass determined by each scale. Write down the results for each scale used.

• Which scale was the most precise? Explain how you know. (Solution: In terms of precision, there are two aspects to consider…if we look at repeatability, then the most precise measurements were made with Scale 1 since the range of values is smaller than in the other scales. In terms of the measurement tool that was used, it would appear that Scales 2 and 3 have units to the hundredth of a gram, and so in terms of the measurement device being used, Scales 1 and 2 are the most precise.)

Unit 2, Activity 5, Precision vs. Accuracy with Answers


• Which scale was the least precise? Explain how you know. (Solution: In terms of repeatability, Scale 3 since the range of values is larger. In terms of the measurement tool being used then Scale 1 would be least precise since it is only able to be read to the nearest tenth of a gram as opposed to the nearest hundredth in the other two scales. )

• Which scale was the most accurate if we consider the true value of the weight to be 100 grams? Explain your answer. (Solution: If we look at the average weights to be the weight given by each scale, then both Scale 2 and Scale 3 are equally accurate.)

Example 3: Below is a data table produced by 4 groups of students who were measuring the mass of a paper clip which had a known mass of 1.0004 g.

• Determine the average weight produced by each group’s measurements and fill in the results in the table. Use this average as the weight of the paper clip for each group.

• Which of the group’s measurements represents a properly accurate and precise measurement of the mass of the paper clip? (Solution: Both Group 1 and Group 4 had an average weight in line with the true weight of the mass; however, Group 4 did not have a precise measurement—the readings have too wide a range. The average just happened to come out to a value close to the true weight; therefore, only Group 1 data represents both an accurate and precise measurement.)

• Which of the group’s measurements was the least accurate? Explain why. (Solution: Group 3 had the least accurate answer for the weight of the paper clip since its average value is farthest from the actual value of the paper clip.)

• Which of the group’s measurements had an accurate answer, but not a precise answer? Explain. (Solution: Group 4 had an accurate weight (if the average is used) but was not precise at all.)

Trial # Group 1 (g) Group 2 (g) Group 3 (g) Group 4 (g) 1 1.01 3.863287 10.13252 2.05 2 1.03 3.754158 10.13258 0.23 3 0.99 3.186357 10.13255 0.75

Average Weight 1.01 3.601267 10.13255 1.01

Unit 2, Activity 6, Absolute Error


Absolute Error At each measurement station, perform the indicated measurements and answer the questions below. Measurement: Mass

OBSERVED VALUE

ACCEPTED VALUE

ABSOLUTE ERROR

Scale 1 Scale 2 Scale 3

Which scale is more accurate?_______________________ Why?________________________________________________________________ Measurement: Volume

OBSERVED VALUE

ACCEPTED VALUE

ABSOLUTE ERROR

Beaker 1 Beaker 2

Measuring Cup Which measuring instrument is more accurate?_______________________ Why?________________________________________________________________ Measurement: Length

OBSERVED VALUE

ACCEPTED VALUE

ABSOLUTE ERROR

Meter stick Ruler 1 Ruler 2

Which measuring instrument is more accurate?_______________________ Why?________________________________________________________________

Unit 2, Activity 6, Absolute Error


Measurement: Time

OBSERVED VALUE

ACCEPTED VALUE

ABSOLUTE ERROR

Wrist watch Calculator Cell Phone

Which measuring instrument is more accurate?___________________________ Why?________________________________________________________________ Is it always possible to determine if a measuring instrument is accurate? Why or why not? Explain how to determine if a measuring instrument is accurate or not.

Unit 3, Activity 8, Going on Vacation


Directions: Read the following problem and, with a partner, answer the questions that follow it.

Mr. Waters needs to rent a car to go on a trip he has planned. In order to rent the car, Mr. Waters will have to pay a flat fee of $45 plus an additional rate of $20 per day.

1. Which two variables are related in this situation? Which is the dependent variable and

which is the independent variable?

2. What is the cost for the car rental if Mr. Waters rents only 1 day? 2 days? 5 days? 9 days? Make a table of values relating the information and use the data to make a line graph using grid paper. Label the graph appropriately with an appropriate scale and title.

3. Write an equation that relates the cost for renting a car for x days.

4. Determine whether there exists a direct or inverse relationship between the two variables in this situation, and explain how you determined your answer.

5. Is the graph of the data you found linear? Explain.

6. Interpret the real-world meaning of the point that intercepts the vertical axis of the graph you created.

Unit 3, Activity 8, Going on Vacation with Answers


Directions: Read the following problem and, with a partner, answer the questions that follow it.

Mr. Waters needs to rent a car to go on a trip he has planned. In order to rent the car, Mr. Waters will have to pay a flat fee of $45 plus an additional rate of $20 per day.

1. Which two variables are related in this situation? Which is the dependent variable

and which is the independent variable? Answer: The two quantities related in this situation are the cost in dollars to rent the car and the time in days for which the car is to be rented. The cost is the dependent variable since the cost depends on the number of days (time is the independent variable) the car will be rented for.

2. What is the cost for the car rental if Mr. Waters rents only 1 day? 2 days? 5 days? 9 days? Make a table of values relating the information and use the data to make a line graph. Label the graph appropriately with an appropriate scale and title.

Solution: The data table is shown below. Check student’s graphs.

Days Rented (d)

1 2 5 9

Cost (C) 65 85 145 225 3. Write an equation that relates the cost for renting a car for x days.

Solution: C = $45 + $20x

4. Determine whether there exists a direct or inverse relationship between the two variables in this situation, and explain how you determined your answer.

Solution: There is a direct relationship since the cost increases as the time increases.

5. Is the graph of the data you found linear? Explain. Solution: Yes, the data is linear since it forms a straight line and has a constant increase.

6. Interpret the real-world meaning of the point that intercepts the vertical axis of the graph you created.

Solution: The point at which the graph intercepts the vertical axis is the initial cost to rent the car—the flat fee.

Unit 3, Activity 9, Analyzing Distance/Time Graphs


Directions: Each graph below displays the distance three different cars traveled over a certain time period. Analyze the graphs and answer the questions below. Car A Car B Car C

1. Identify the dependent and independent variables for each graph.

2. Determine which car was traveling fastest and which car was traveling slowest, and explain how this relates to the steepness of the graph.

3. Determine the rate of speed for each of the three cars, and explain how you obtained your answers.

4. Create a line graph of a car D that travels at a rate of 50 miles per hour for 4 hours, and turn the graph in to the teacher.

1 2 3 4 Time (hours)

50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)


50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)


50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)

Unit 3, Activity 9, Analyzing Distance/Time Graphs with Answers


Directions: Each graph below displays the distance three different cars traveled over a certain time period. Analyze the graphs and answer the questions below. Car A Car B Car C

1. Identify the dependent and independent variables for each graph. Solution: The independent variable is the time and the dependent variable is the distance.

2. Determine which car was traveling fastest and which car was traveling slowest, and explain how this relates to the steepness of the graph.

Solution: Fastest car is B; slowest car is A. The steeper the graph is, the faster the speed.

3. Determine the rate of speed for each of the three cars, and explain how you obtained your answers.

Solution: Car A (5 mph); Car B (12.5 mph); Car C (10 mph); check students explanations.

4. Relate slope of a line with the concept of a rate of change. 5. Create a line graph of a car D that travels at a rate of 50 miles per hour for 4

hours, and turn the graph in to the teacher. Solution: Check student graphs. Possible graph is shown below.


50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)


50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)


50 40 30 20 10

Dis

tanc

e Tr

avel

ed

(mile

s)


250 200 150 100 50

Dis

ttanc

e tra

vele

d (m

iles)

Unit 3, Activity 10, What does Slope Tell Us about a Graph?


Directions: In this activity, you will interpret the meaning of slope as a rate as it applies to a real-life situation.

Karl’s Weight (kg) 67 71 74 76 74 68 Year 1991 1992 1993 1994 1995 1996

1. Using the table above, make a line graph of the data. Remember a graph does not

have to start at zero. Choose an appropriate scale and label all axes.

2. What two quantities are related in the graph that was drawn using the data? Describe the relationship in words.

3. During which year(s) did John’s weight increase at the greatest rate? What was this rate of increase? Explain how you determined this value.

4. During which year(s) did John’s weight decrease at the greatest rate? What was this rate of decrease? Explain how you determined this.

5. Look at the graph and explain what the steepness of the segments on the graph (the slope) tells us about the data in real-world terms.

6. Does this graph represent a direct relationship, an indirect relationship, or a combination of the two? Explain your answer.

Unit 3, Activity 10, What does Slope Tell Us about a Graph? with Answers


Directions: In this activity, you will interpret the meaning of slope as a rate as it applies to a real-life situation.

Karl’s Weight (kg) 67 71 74 76 74 68 Year 1991 1992 1993 1994 1995 1996

1. Using the table above, make a line graph of the data. Remember a graph does not

have to start at zero. Choose an appropriate scale and label all axes. Check student graphs.

2. What two quantities are related in the graph that was drawn using the data? Describe the relationship in words.

Solution: Karl’s weight depends on the year he was weighed.

3. During which year(s) did John’s weight increase at the greatest rate? What was this rate of increase? Explain how you determined this value.

Solution: From 1991 to 1992, Karl’s weight increased at a rate of 4 kg per year.

4. During which year(s) did John’s weight decrease at the greatest rate? What was this rate of decrease? Explain how you determined this.

Solution: Karl’s weight decreased at a rate of 6 kg per year from 1995 to 1996.

5. Look at the graph and explain what the steepness of the segments on the graph (the slope) tells us about the data in real-world terms.

Solution: The steepness is associated with the rate of change of Karl’s weight gain or loss. A bigger rate of gain or loss is associated with a larger degree of steepness.

6. Does this graph represent a direct relationship, an indirect relationship, or a combination of the two? Explain your answer.

Solution: The graph shows a direct relationship when there is a weight gain as time increases and an indirect relationship when there is a weight loss as time increases. Thus it shows a combination of the two.

Unit 3, Activity 11, When Will They Meet?


Directions: Read the problem presented here and then answer the questions that follow: Lester left home at 9 a.m. one morning to go on a business trip. He immediately got on the interstate and drove at a constant rate of 65 mph. Assume Lester drove on a straight road with no traffic that would prevent him from having to slow down and that he had enough gas to travel for 8 hours without stopping. One hour after Lester left home on his business trip, his wife, Gertrude, realized that he had forgotten his briefcase. She immediately got on the interstate and began to try to catch up to him, traveling at a constant rate of 75 mph in the process.

Questions:

1. Which two quantities are related in this problem? Tell which quantity represents the dependent variable and which quantity represents the independent variable.

2. Make a data table that depicts the distance Lester is away after each hour from home. Use this data table to plot points to create a line graph showing the data. Is the data shown on the graph a direct or indirect relationship? Is it linear? Explain.

3. Create a similar data table that depicts the distance Gertrude is from home during the same time frame. Remember, Gertrude left one hour later than Lester. Using this data, plot points on the same graph as Lester’s. Label each line to show the distinction between Gertrude and Lester. Use different colors to make the distinction.

4. Using the double line graph, determine when (what time) and where (how many miles from home) Gertrude will finally catch up to her husband Lester?

5. Look at the steepness of the line that represents Lester’s motion and compare it to the steepness of the line that represents Gertrude’s motion? How does the steepness of the line relate to the speed that they traveled? Explain.

Unit 3, Activity 11, When Will They Meet? with Answers


Directions: Read the problem presented here and then answer the questions that follow: Lester left home at 9 a.m. one morning to go on a business trip. He immediately got on the interstate and drove at a constant rate of 65 mph. Assume Lester drove on a straight road with no traffic that would prevent him from having to slow down and that he had enough gas to travel for 8 hours without stopping. One hour after Lester left home on his business trip, his wife, Gertrude, realized that he had forgotten his briefcase. She immediately got on the interstate and began to try to catch up to him, traveling at a constant rate of 75 mph in the process. Questions:

1. Which two quantities are related in this problem? Tell which quantity represents the dependent variable and which quantity represents the independent variable. Answer: The two quantities here are the distance from home and the time. Distance from home is dependent upon the time traveled so distance is dependent and time is independent variable.

2. Make a data table that depicts the distance Lester is away after each hour

from home. Use this data table to plot points to create a line graph showing the data. Is the data shown on the graph a direct or indirect relationship? Is it linear? Explain.

T(hours) 1 2 3 4 5 6 7 8 D(miles) 65 130 195 260 325 390 455 520

The data is linear and it shows a direct relationship (as time increases, so does distance). See student graphs of data.

3. Create a similar data table that depicts the distance Gertrude is from home during the same time frame. Remember, Gertrude left one hour later than Lester. Using this data, plot points on the same graph as Lester’s. Label each line to show the distinction between Gertrude and Lester. Use different colors to make the distinction.

T(hours) 1 2 3 4 5 6 7 8 D(miles) 0 75 150 225 300 375 450 525 See student graphs of data.

4. Using the double line graph, determine when (what time) and where (how many miles from home) Gertrude will finally catch up to her husband Lester? Answer: Gertrude will catch up to her husband 6.5 hours after she left home (he will have been traveling for 7.5 hours) at a distance of 487.5 miles from home.

5. Look at the steepness of the line that represents Lester’s motion and compare it to the steepness of the line that represents Gertrude’s motion? How does the steepness of the line relate to the speed that they traveled? Explain. The steepness of the line relates to the speed. Since Gertrude’s speed is faster, the line is steeper relating her motion.

Unit 3, Activity 12, Direct and Inverse Variation-Part 1


Part 1: Direct Variation Directions: With a partner, answer the following questions and be prepared to discuss the results with your classmates. Many situations in everyday life involve a relationship between two measurable quantities. For example, your pay from a part-time baby-sitting job increases as you work more hours. Suppose the pay, P, in dollars is $3.85 per hour, h. Then the equation P = $3.85 h could be used to figure out the pay. 1. Question: In this example, what is the independent and what is the dependent variable? Independent Variable: ________ Dependent Variable: _________ 2. Make a table of values for 1, 5, 10, 20, 30, and 40 hours of work and the pay the person would receive. Label the table appropriately.

3. Draw a graph showing the data you have in the table. Label the graph appropriately.



4. Interpret the data in words….explain what the relationship is between the variables in the graph. If two variables, x and y, vary DIRECTLY, there is a non-zero number k such that the following is true: y = kx. The number k is referred to as the constant of variation. In this example, we say that “y varies directly as x” or “y is directly proportional to x.” Two quantities that vary directly are said to have DIRECT VARIATION. 5. Question: In this pay example, is this an example of direct variation and if so, what is the constant of variation? Explain how you know. 6. Would the following equation be a direct variation situation and if so, what is the constant of variation, k, in this example? F = ½ t 7. Determine which of the following equations exhibit direct variation situations. Circle the direct variation equations, and underline the constant of variation in those equations.

(a) W = 40d (b) D = 3.5 t (c) y = -3x (d) y = x5

8. Suppose the variables x and y vary directly. When x = 5, y = 20. Using this information, find the value of the constant of variation, k, for this problem, then write an equation which relates x and y. 9. Suppose we know m and n vary directly and the constant of variation, k = 3. What is the value of n when m = 10?



10. The model y = kx for direct variation can be rewritten as k = xy . The ratio form tells

you that if x and y have direct variation, then the ratio of y to x is the same for all values of x and y. Sometimes real-life data can be approximated by a direct variation model, even though the data may not fit the model exactly. Example: The forearm lengths and body heights in inches of 5 people are shown below. Forearm Length (F) in. 7.0 7.5 8.0 8.5 9.0 Body Length (B) in. 52.5 55.5 58.4 63.5 68.4

a. Plot the data in a graph on graph paper. Label all axes appropriately.

b. Figure out what the constant of proportionality, k, would be for this set of data.

c. Write a direct variation model that relates the body length B to forearm

length F.

d. Estimate the body length of a person whose forearm length is 10 inches. 11. Which of the following graphs show a direct variation situation? 12. In a direct variation situation, if k is negative, as x increases by a factor of four, what happens to the value of y? Explain. 14. Jan was studying direct variation in her math and science class. After studying the graphs associated with direct variation situations, Jan wrote the following statement:

In a direct variation situation, as the quantity of x increases, the quantity of y increases by that same factor. As the quantity of x decreases, the quantity of y decreases by that same factor. This will always happen in a direct variation situation.

Is Jan correct? Explain your answer to defend your choice.

Unit 3, Activity 12, Direct and Inverse Variation Part 1 with Answers


Part 1: Direct Variation Directions: With a partner, answer the following questions, and be prepared to discuss the results with your classmates. Many situations in everyday life involve a relationship between two measurable quantities. For example, your pay from a part-time babysitting job increases as you work more hours. Suppose the pay, P, in dollars is $3.85 per hour, h. Then the equation P = $3.85 h could be used to figure out the pay. 1. Question: In this example, what is the independent and what is the dependent variable? Independent Variable: _hours(h)_ Dependent Variable: _pay in dollars (P)____ 2. Make a table of values for 1, 5, 10, 20, 30, and 40 hours of work and the pay the person would receive. Label the table appropriately.

Hours worked (h) Pay in dollars (P) 1 3.85 5 19.25 10 38.50 20 77.00 30 115.50 40 154.00

3. Draw a graph showing the data you have in the table. Label the graph appropriately.

Pay

in d

olla

rs (P

)

Hours worked (h) 5 10 15 20 25 30 35 40

175 150 125 100 75 50 25



4. Interpret the data in words….explain what the relationship is between the variables in the graph. Answer: The data is linear and there is a direct relationship between the number of hours worked and the pay (as hours increase, so does pay). If two variables, x and y, vary DIRECTLY, there is a non-zero number k such that the following is true: y = kx. The number k is referred to as the constant of variation. In this example, we say that “y varies directly as x” or “y is directly proportional to x”. Two quantities that vary directly are said to have DIRECT VARIATION. 5. Question: In this pay example, is this an example of direct variation and if so, what is the constant of variation? Explain how you know. Answer: Yes, this is an example of direct variation where P=3.85h. The k here is 3.85 which represents the constant rate of pay per hour. 6. Would the following equation be a direct variation situation and if so, what is the constant of variation, k, in this example? F = ½ t Answer: Yes, this is direct variation with k = ½ in this situation. 7. Determine which of the following equations exhibit direct variation situations. Circle the direct variation equations, and underline the constant of variation in those equations.

(a) W = 40 d (b) D = 3.5 t (c) y = -3x (d) y = x5

k = 40 k = 3.5 k = -3 8. Suppose the variables x and y vary directly. When x = 5, y = 20. Using this information, find the value of the constant of variation, k, for this problem, then write an equation which relates x and y. Answer: y = 4x where k = 4. 9. Suppose we know m and n vary directly and the constant of variation, k = 3. What is the value of n when m = 10? Answer: if n = km, when k = 3 and m = 10, then n = 30.



10. The model y = kx for direct variation can be rewritten as k = xy . The ratio form tells

you that if x and y have direct variation, then the ratio of y to x is the same for all values of x and y. Sometimes real-life data can be approximated by a direct variation model, even though the data may not fit the model exactly. Example: The forearm lengths and body heights in inches of 5 people are shown below. Forearm Length (F) in. 7.0 7.5 8.0 8.5 9.0 Body Length (B) in. 52.5 55.5 58.4 63.5 68.4

a. Plot the data in a graph on graph paper. Label all axes appropriately. See student graphs.

b. Figure out what the constant of proportionality, k, would be for this set of

data. Answer: k is about 7.5.

c. Write a direct variation model that relates the body length B to forearm length F. Answer: B = 7.5F

d. Estimate the body length of a person whose forearm length is 10 inches. Answer: B is about 75 inches (or 6 ft. 3 in.)

11. Which of the following graphs show a direct variation situation? Direct Direct Not Direct 12. In a direct variation situation, if k is negative, as x increases by a factor of four, what happens to the value of y? Explain. Answer: If k is negative, then this is a decreasing function, and there is an inverse relationship associated with the graph, therefore as x increases by a factor of four, the value of y decreases by a factor of 4. 14. Jan was studying direct variation in her math and science class. After studying the graphs associated with direct variation situations, Jan wrote the following statement:

In a direct variation situation, as the quantity of x increases, the quantity of y increases by that same factor. As the quantity of x decreases, the quantity of y decreases by that same factor. This will always happen in a direct variation situation.

Is Jan correct? Explain your answer to defend your choice. Answer: This is only true when k is positive…not when k is negative. See student answers.



Part 2: Inverse Variation When k = xy, there is a different type of relationship that exists and this will be explored in this activity. If we use algebra, and rearrange the variables in terms of y, we get the

following: y = xk (notice that this is different than that of direct variation).

1. Suppose we have a value of 8 for the constant of proportionality, k. Our equation

would become y = x8 . Use this equation to fill in the table below and then plot the points

you found on graph paper and see what the resulting graph looks like.

x y = 8/x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2. Does the graph you created look linear? Explain. 3. What do you notice happens to the value of y as the value of x increases? Does its value increase or decrease? Compare the table values with the graph to see what this looks like graphically. 4. If the value of k was 20, would the basic shape of the graph been the same? Explain. 5. If the value of k was 20, how would this affect the answer to question 3? Explain.



Suppose we were to keep the area of a rectangle a constant 24 square units. If A = LW, and area is 24 square units, we get the equation: 24 = LW. 6. Is there a direct or inverse variation between length and width by looking at the equation? Explain. 7. Fill in the table below for the different values of Length associated with the values of width to keep the area a constant of 24 square units.

Width (W) Length (L) 1 2 3 4 6 8 10 12 24

8. Use the table to make a graph and display the results. Use the width as the independent variable and the length as the dependent variable. Is the graph linear? 9. As the width is increased, what happens to the value of the length? 10. In the Distance = Rate x Time equation, suppose the rate was held at a constant of 40 miles/hour. Would you say in this case that D and T vary directly or inversely with one another? Explain. 11. In the Distance = Rate x Time equation, suppose the Distance was held at a constant of 120 miles. Would you say in this case that R and T vary directly or inversely with one another? Explain. 12. Write an equation or formula that matches each relationship. Use variables that are appropriate to the given situation and k as the constant of proportionality.

• The circumference of a circle, C, varies directly as the diameter, D. __________ • The resistance of a wire, R, varies directly as its length, L. _________ • Under certain conditions, the volume, V, of a gas varies directly as the absolute

temperature, T. _________ • Under fixed conditions, the volume, V, of a gas varies inversely as the pressure, p.

__________



13. In the table below, determine if the relationship is direct or inverse variation and write an equation that matches each table.

Table 1: R 2 4 10 T 50 25 10

The table is direct/inverse (circle one). Equation: ____________ Table 2:

C 5 10 15 N 1 2 3

The table is direct/inverse (circle one). Equation: ____________

14. If y varies directly as x and y = 50 when x = 5, find y when x = 10. 15. D varies directly as R. What change takes place in D if R is tripled and k is positive? Explain your thinking. 16. If s varies inversely as t, and if s = 6 when t = 2, find s when t = 3. 17. V varies inversely as P. If P is tripled, what change takes place in V? Explain your thinking.

Unit 3, Activity 12, Direct and Inverse Variation-Part 2 with Answers


Part 2: Inverse Variation When k = xy, there is a different type of relationship that exists, and this will be explored in this activity. If we use algebra, and rearrange the variables in terms of y, we get the

following: y = xk (notice that this is different than that of direct variation).

1. Suppose we have a value of 8 for the constant of proportionality, k. Our equation

would become y = x8 . Use this equation to fill in the table below, and then plot the points

you found on graph paper and see what the resulting graph looks like.

x y = 8/x 0 Does not exist 1 8 2 4 3 8/3 (2.7) 4 2 5 8/5 (1.6) 6 4/3 (1.3) 7 8/7(1.1) 8 1 9 8/9 (.9) 10 4/5 (.8) 11 8/11(.7) 12 2/3(.67) 13 8/13(.6) 14 4/7(.57) 15 8/15(.53) 16 1/2 (.5)

2. Does the graph you created look linear? Explain. Answer: Graph is not linear. See student explanations. 3. What do you notice happens to the value of y as the value of x increases? Does its value increase or decrease? Compare the table values with the graph to see what this looks like graphically. Answer: As x increases, y decreases but not at a constant rate as in the direct variation situations. 4. If the value of k was 20, would the basic shape of the graph been the same? Explain. Answer: The graph’s basic shape would remain the same. Investigate this with a graphing calculator. 5. If the value of k was 20, how would this affect the answer to question 3? Explain. Answer: The answer would remain the same.



Suppose we were to keep the area of a rectangle a constant 24 square units. If A = LW, and area is 24 square units, we get the equation: 24 = LW. 6. Is there a direct or inverse variation between length and width by looking at the equation? Explain. Answer: Inverse variation since the product of L and W is constant. 7. Fill in the table below for the different values of Length associated with the values of Width to keep the area a constant of 24 square units.

Width (W) Length (L) 1 24 2 12 3 8 4 6 6 4 8 3 10 2.4 12 2 24 1

8. Use the table to make a graph and display the results. Use the width as the independent variable and the length as the dependent variable. Is the graph linear? See student graphs; the graph is not linear. 9. As the width is increased, what happens to the value of the length? Answer: As the width is increased, the length decreases. 10. In the Distance = Rate x Time equation, suppose the rate was held at a constant of 40 miles/hour. Would you say in this case that D and T vary directly or inversely with one another? Explain. Answer: if the Rate=40mph constant, the equation becomes D = 40T or D/T = 40. Since the ratio of D to T is constant, then this would represent direct variation between D and T. 11. In the Distance = Rate x Time equation, suppose the Distance was held at a constant of 120 miles. Would you say in this case that R and T vary directly or inversely with one another? Explain. Answer: If D = 120 miles constant, then 120 = RT. Since the product of R and T is constant, then R and T would have an inverse variation. 12. Write an equation or formula that matches each relationship. Use variables that are appropriate to the given situation and k as the constant of proportionality.

• The circumference of a circle, C, varies directly as the diameter, D. __C=kD__ • The resistance of a wire, R, varies directly as its length, L. _R=kL___ • Under certain conditions, the volume, V, of a gas varies directly as the absolute

temperature, T. _V=kT___



• Under fixed conditions, the volume, V, of a gas varies inversely as the pressure, p. __V=k/p__

13. In the table below, determine if the relationship is direct or inverse variation and write an equation that matches each table.

Table 1: R 2 4 10 T 50 25 10

The table is direct/inverse (circle one). Equation: _____100 = RT__or T = 100/R or R = 100/T_____ Table 2:

C 5 10 15 N 1 2 3

The table is direct/inverse (circle one). Equation: _____5 = C/N or C = 5N_______

14. If y varies directly as x and y = 50 when x = 5, find y when x = 10. Answer: y = 100 when x = 10. 15. D varies directly as R. What change takes place in D if R is tripled and k is positive? Explain your thinking. Answer: If R is tripled then D is tripled since they are directly related and since k is positive, then we know that as one goes up so does the other. 16. If s varies inversely as t, and if s = 6 when t = 2, find s when t = 3. Answer: When t = 3, then s = 4 since k = 12. 17. V varies inversely as P. If P is tripled, what change takes place in V? Explain your thinking. Answer: Since there is inverse variation, we know that the product of V and P must remain constant. Thus if P is tripled, then V must decrease to1/3 of its original value.

Unit 4, Activity 4, Slope as a Rate of Change


Problem: Suppose it costs $3 for each bottle of cola purchased for the school fair. Using this information, perform the following tasks and answer the questions below.

1. Which statement below is more applicable to this situation? (Be prepared to defend your choice.) Statement 1: The total cost for the cola depends on the number of bottles purchased. Statement 2: The number of bottles purchased depends on the total cost of the cola.

2. If the total cost, C, of purchasing x bottles of cola were written as an equation, what would this equation be written as? ___________________

3. In this equation, what would the dependent variable be, and what would the independent

variable be?

4. Make an input/output table with ten points relating the cost and the number of bottles of cola bought for the school fair. Label the table appropriately.

5. Using graph paper, plot the points from the input/output table you made creating a graph

of the relationship in this situation. 6. Using any two points from this graph, determine the slope of the line you drew.

7. Does it matter which two points you choose, or will the slope be the same value no matter which two points you use? Explain.

8. In this real-life situation, what exactly does the slope of the graph mean in real-world terms? What units are associated with this slope?

Unit 4, Activity 4, Slope as a Rate of Change with Answers


Problem: Suppose it costs $3 for each bottle of cola purchased for the school fair. Using this information, perform the following tasks and answer the questions below.

1. Which statement below is more applicable to this situation? (Be prepared to defend your choice.) Statement 1: The total cost for the cola depends on the number of bottles purchased. Statement 2: The number of bottles purchased depends on the total cost of the cola. Answer: Statement 1 is correct since total cost depends on the number of bottles purchased.

2. If the total cost, C, of purchasing x bottles of cola were written as an equation, what would this equation be written as? __C = 3x__

3. In this equation, what would the dependent variable be, and what would the

independent variable be? C is dependent variable; x is independent variable.

4. Make an input/output table with ten points relating the cost and the number of bottles of cola bought for the school fair. Label the table appropriately.

x (number of bottles) C (cost in dollars) 0 0 1 3 2 6 3 9 4 12 5 15 6 18 7 21 8 24 9 27

5. Using graph paper, plot the points from the input/output table you made creating a

graph of the relationship in this situation. See student graphs! 6. Using any two points from this graph, determine the slope of the line you drew.

Slope = 3

7. Does it matter which two points you choose, or will the slope be the same value no matter which two points you use? Explain. It doesn’t matter which two points are used since they will all result in a slope of 3.

8. In this real-life situation, what exactly does the slope of the graph mean in real-world terms? What units are associated with this slope? The slope here is 3 dollars per bottle which is the actual cost for each bottle of cola purchased.

Unit 4, Activity 7, Match the Equation with the Graph


Directions: Based upon what you have learned about the slope-intercept form for an equation and the effects that m and b have on the graph, match the equations below with the numberless graphs drawn. Be prepared to defend your choices. Equation 1: y = -5x + 4 Equation 4: y = -2x Equation 2: y = ¼ x + 2 Equation 5: y = .2x – 3 Equation 3: y = 3x Equation 6: y = -x - 0.25

Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 Graph 6

Unit 4, Activity 7, Match the Equation with the Graph with Answers


Directions: Based upon what you have learned about the slope-intercept form for an equation and the effects that m and b have on the graph, match the equations below with the numberless graphs drawn. Be prepared to defend your choices. Equation 1: y = -5x + 4 Equation 4: y = -2x Equation 2: y = ¼ x + 2 Equation 5: y = .2x – 3 Equation 3: y = 3x Equation 6: y = -x - 0.25

Graph 1 (Equation 3) Graph 2 (Equation 4) Graph 3 (Equation 2) Graph 4 (Equation 1) Graph 5 (Equation5) Graph 6 (Equation 6)

Unit 5, Activity 3, What’s the Equation of the Line?


Directions: Working with your group, use the graph below to answer the following questions, and be ready to discuss this activity with your classmates. Janet sketched the graph of the line containing the points (-5, -2) and (8,3). Using this sketch, answer the following questions:

1. What is the slope of the line containing these two points?

2. Using what you learned in class, write an equation for the line which contains these two points. Explain in words and show mathematically what you would do to create such an equation. Put the equation in all three forms discussed in class (i.e., point slope; slope-intercept; standard form).

3. Do you think there is more than one line (and equation) that contains these two points?

Explain why or why not.

4. Determine the equation of the line containing the points (-3,5) and (2, -1). Put your answer in slope-intercept form.

(-5, -2)

(8, 3)

Unit 5, Activity 3, What’s the Equation of the Line? with Answers


Directions: Working with your group, use the graph below to answer the following questions, and be ready to discuss this activity with your classmates. Janet sketched the graph of the line containing the points (-5, -2) and (8,3). Using this sketch, answer the following questions:

1. What is the slope of the line containing these two points?

m = 135

2. Using what you learned in class, write an equation for the line which contains these two points. Explain in words and show mathematically what you would do to create such an equation. Put the equation in all three forms discussed in class (i.e., point slope; slope-intercept; standard form).

See students’ explanations. Students should explain that to come up with the equation, first the slope had to be found. Using the method discussed in class of point-slope form, the resulting slope and one of the two points are used to determine the equation of the line. The equations are as follows: Point-Slope: Slope-Intercept Standard

(y – 3) = 135 (x – 8) y =

135 x -

131 5x – 13y = 1

3. Do you think there is more than one line (and equation) that contains these two

points? Explain why or why not. Since we know that two points determine a distinct line, there is only one line (and equation) that fits these two points.

4. Determine the equation of the line containing the points (-3,5) and (2, -1). Put your answer in slope-intercept form.

y = -56 x +

57

(-5, -2)

(8, 3)

Unit 5, Activity 5, Fahrenheit and Celsius—How are they Related?


Directions: With your group members work through the following problems dealing with the relationship between the Fahrenheit and Celsius temperature scales. The relationship between temperatures measured in degrees Fahrenheit and Celsius is linear. Using what you learned about finding equations of lines given two points, use the following information: water freezes at 32°F and 0°C; water boils at 212°F and 100°C.

1. Use the information about freezing and boiling points to write two data points. Use Fahrenheit temperature as the independent variable and Celsius temperature as the dependent variable.

2. What is the slope of the line containing the two data points and explain what this

slope represents in real-life terms. 3. Write the linear equation which describes the relationship between the two

temperature scales. Again, use Fahrenheit as the independent variable and Celsius temperature as the dependent variable. Explain the variables you use in your equation and what they represent.

4. Determine the y-intercept for this line and explain in real-world terms what it represents.

5. Determine the x-intercept for this line and explain in real-world terms what it

represents.

6. Using the equation you created, find the Celsius temperature for a Fahrenheit temperature of 50°F. After you get your answer, look at an actual thermometer to see if this temperature matches with what you would expect.

7. If the equation were re-arranged so that Fahrenheit temperature was written in terms of the Celsius temperature, what equation would result?

Unit 5, Activity 5, Fahrenheit and Celsius—How are they Related? with Answers


Directions: With your group members work through the following problems dealing with the relationship between the Fahrenheit and Celsius temperature scales. The relationship between temperatures measured in degrees Fahrenheit and Celsius is linear. Using what you learned about finding equations of lines given two points, use the following information: water freezes at 32°F and 0°C; water boils at 212°F and 100°C.

1. Use the information about freezing and boiling points to write two data points. Use Fahrenheit temperature as the independent variable and Celsius temperature as the dependent variable.

Solution: (32,0) and (212,100)

2. What is the slope of the line containing the two data points and explain what this

slope represents in real-life terms.

Solution: m = 95 ; This slope represents the relationship between the change of

Fahrenheit temperature in relation to the change in Celsius temperature. For every 5°change in Celsius temperature, there is a 9°change in Fahrenheit temperature.

3. Write the linear equation that shows the algebraic relationship between the two temperature scales. Again, use Fahrenheit as the independent variable and Celsius temperature as the dependent variable. Explain the variables you use in your equation and what they represent.

Solution: Possible equations are C = 59

(F - 32°) or y = 59

(x - 32°) where y

represents Celsius temperature and x represents Fahrenheit temperature. 4. Determine the y-intercept for this line and explain in real-world terms what it

represents. Solution: The y-intercept for this line is –17.77 or –17 7/9 and it represents the Celsius temperature which relates to 0°F. (i.e., 0°F = -17.77°C)

5. Determine the x-intercept for this line and explain in real-world terms what it represents. Solution: The x-intercept for this line is –32 and it represents the Fahrenheit temperature that relates to a Celsius temperature of 0°.

6. Using the equation you created, find the Celsius temperature for a Fahrenheit temperature of 50°F. After you get your answer, look at an actual thermometer to see if this temperature matches with what you would expect. Solution: 50°F = 10°C

7. If the equation were re-arranged so that Fahrenheit temperature was written in terms of the Celsius temperature, what equation would result?

Solution: F = 95

C + 32

Unit 5, Activity 6, From Tables to Equations


Directions: Use the data table provided below to answer the questions presented. Work in pairs to perform the indicated tasks. N (number of sides of a polygon)

3 4 5 6

S (sum of the interior angles)

180 360 540 720

1. The data table shows how the sum of the interior angles of a polygon depends on

the number of sides of that polygon. Using the data table, create a graph (using graph paper) to display the data. Label the independent and dependent axes in an appropriate manner.

2. Determine a linear equation for the data presented. Explain what variables you are using and what they represent.

3. What does the slope in this problem indicate in real-world terms? What is the slope?

4. Does this graph have a y-intercept? If so, explain in real-world terms what it represents. If it doesn’t have a y-intercept, explain why.

5. Does this graph have an x-intercept? If so, explain in real-world terms what it represents. If it doesn’t have an x-intercept, explain why.

6. If a polygon has 10 sides, what will the sum of the angles of the polygon be?

Solve this problem using the linear equation you created in problem number 2.

Unit 5, Activity 6, From Tables to Equations with Answers


Directions: Use the data table provided below to answer the questions presented. Work in pairs to perform the indicated tasks. N (number of sides of a polygon)

3 4 5 6

S (sum of the interior angles)

180 360 540 720

1. The data table shows how the sum of the interior angles of a polygon depends on

the number of sides of that polygon. Using the data table, create a graph to display the data. Label the independent and dependent axes in an appropriate manner. Solution: The sum of the angles depends on the number of sides, so the y-axis (dependent) should be the sum of the angles and the x-axis (independent) should be the number of sides. See student graphs.

2. Determine a linear equation for the data presented. Explain what variables you are using and what they represent. Solution: S = 180n – 360 or S = 180 (n-2) or y = 180 (x – 2), where x is the number of sides and y is the sum of the angles.

3. What does the slope in this problem indicate in real-world terms? What is the slope? Solution: The slope is 180 and it represents the change in the angle measure sum as an additional side is added (i.e., 180 degrees per side).

4. Does this graph have a y-intercept? If so, explain in real-world terms what it represents. If it doesn’t have a y-intercept, explain why. Solution: Mathematically, the y-intercept would be –360 meaning the sum of the angles of a figure with no sides would be –360 degrees. Since this makes no sense in real world terms, this graph really does not have a y-intercept—the equation only makes sense for a polygon with three sides or greater.

5. Does this graph have an x-intercept? If so, explain in real-world terms what it represents. If it doesn’t have an x-intercept, explain why. Solution: The x-intercept for this graph is x = 2. Since x represents the number of sides of a polygon, if a polygon had 2 sides (which cannot occur in real-life), the sum of the angles would be 0°.

6. If a polygon has 10 sides, what will the sum of the angles of the polygon be?

Solve this problem using the linear equation you created in problem number 2. Solution: 1440°.

Unit 5, Activity 7, Wages vs. Hours Worked


Directions: Use the following information to answer the questions provided. Mark earns an hourly rate of $5.75 per hour. He wants to save enough money to buy a motorcycle worth $3000. He already has $450 saved in his account.

1. Complete the following table showing how much Mark will make at work after working N number of hours. N (number of hours worked)

1 2 3 4 5

P (money earned in dollars)

2. Using graph paper, sketch the graph of the data using the relationship between the

number of hours worked, N, and the money earned, P. Which variable is the dependent variable and which is the independent variable?

3. Is the data linear? If so, explain what characterizes this data as being linear.

4. What is the slope for the graph and what does it represent in real-world terms?

5. Suppose Mark saves all of his money that he makes at work to buy the motorcycle. If Mark wanted to write an equation showing the total amount of money, P, he will have (including the $450 he already has saved) after working N hours, what equation would he use?

6. If Mark works for 50 hours, how much money will he have saved altogether? Use your equation to find the total amount saved.

7. Using your equation, figure out how many hours Mark will have to work in order to save enough money to purchase the motorcycle?

Unit 5, Activity 7, Wages vs. Hours Worked with Answers


Directions: Use the following information to answer the questions provided. Mark earns an hourly rate of $5.75 per hour. He wants to save enough money to buy a motorcycle worth $3000. He already has $450 saved in his account.

1. Complete the following table showing how much Mark will make at work after working N number of hours. N (number of hours worked)

1 2 3 4 5

P (money earned in dollars)

$5.75 $11.50 $17.25 $23.00 28.75

2. Using graph paper, sketch the graph of the data using the relationship between the

number of hours worked, N, and the money earned, P. Which variable is the dependent variable and which is the independent variable? See students’ graphs; In this situation, since the money earned, P, depends on the number of hours worked, N, then P is the dependent variable and N is the independent variable.

3. Is the data linear? If so, explain what characterizes this data as being linear. Solution: The data is linear because of the constant rate of change.

4. What is the slope for this graph and what does it represent in real-world terms? Solution: The slope of the graph is $5.75 per hour, which is the rate of pay that Mark gets for working on the job ($5.75 per hour).

5. Suppose Mark saves all of his money that he makes at work to buy the motorcycle. If Mark wanted to write an equation showing the total amount of money, P, he will have (including the $450 he already has saved) after working N hours, what equation would he use? Solution: P = 450 +5.75N

6. If Mark works for 50 hours, how much money will he have saved altogether? Use your equation to find the total amount saved. Solution: $737.50

7. Using your equation, figure out how many hours Mark will have to work in order

to save enough money to purchase the motorcycle? Solution: If done on calculator the answer is 443.47 hours, but in reality, if he only works whole hours, Mark will have to work 444 hours in order to save enough money for the motorcycle (assuming $3000 is enough to include taxes!).

Unit 5, Activity 8, The Stock is Falling!


Directions: Use the following information to answer the questions provided. Lynn bought a stock at a price of $38. At the end of the first week, the price of the stock had fallen to $35. At the end of the second week, the stock had fallen to $32. At the end of the third week the stock had fallen to $29.

1. Create a table which models the situation presented here. 2. Using graph paper, graph the relationship between the price of the stock and the

number of weeks since the stock was bought. 3. Find and interpret the rate of change associated with this graph. Explain in real-

world terms what this information tells us about this situation.

4. Assuming this trend continues indefinitely; write an equation for the value of the stock after w weeks have gone by. Explain the variables you used when writing your equation. Solution: P = 38 – 3x, where P is the price of the stock in dollars after x weeks.

5. Find and interpret the meaning of the y-intercept for this equation.

6. Find and interpret the meaning of the x-intercept for this equation.

7. Use the equation you wrote to determine what the price of the stock will be at the end of the 10th week if the trend continues.

Unit 5, Activity 8, The Stock is Falling! with Answers


Directions: Use the following information to answer the questions provided. Lynn bought a stock at a price of $38. At the end of the first week, the price of the stock had fallen to $35. At the end of the second week, the stock had fallen to $32. At the end of the third week the stock had fallen to $29.

1. Create a table which models the situation presented here.

W (number of weeks since stock was purchased)

0 1 2 3

P (value of stock in dollars)

38 35 32 29

2. Using graph paper, graph the relationship between the price of the stock and the number of weeks since the stock was bought. See student graphs!

3. Find and interpret the rate of change associated with this graph. Explain in real-

world terms what this information tells us about this situation. Solution: Since the slope in this problem is –3, the rate of change associated with this situation indicates that there is a decrease of $3.00 per week in the value of the stock.

4. Assuming this trend continues indefinitely; write an equation for the value of the stock after w weeks have gone by. Explain the variables you used when writing your equation. Solution: P = 38 – 3w, where P is the price of the stock in dollars after w weeks.

5. Find and interpret the meaning of the y-intercept for this equation. Solution: The y-intercept is 38 and it represents the initial value of the stock ($38).

6. Find and interpret the meaning of the x-intercept for this equation. Solution: The x-intercept is 12 2/3 and it represents the number of weeks it would take for the value of the stock to be worthless (its value is $0). Note: Since the number of weeks is a discrete number, it would actually take 13 weeks for the value of the stock to be worth nothing.

8. Use the equation you wrote to determine what the price of the stock will be at the end of the 10th week if the trend continues. Solution: $8.00

Unit 6, Activity 2, Real-life Inequalities


Directions: With your group members, answer the following questions regarding real-life inequalities.

1. In some states, in order to drive you must be at least 16 years old to obtain a driver’s license. Write an inequality to express this situation.

2. A bus is being rented by the math club to go on a trip. The bus can seat at most 50 people. Write an inequality to express the amount of people the bus will hold.

3. In order to receive a free gift at the grand opening of Fashion World, the customer must buy at least $350 worth of merchandise. Write an inequality to express this situation.

4. In order to drive safely on the highway, the minimum speed is 50 mph and the maximum speed is 70 mph. Write a combined inequality to express this situation.

5. The local phone company allows a maximum of 700 minutes of call time under its calling plan. Express the number of minutes allowed under this plan as an inequality.

6. Jeremy was trying to determine the best calling plan to buy for his family. Under Plan A, the phone company charges $15 flat fee plus an additional $0.35 per minute for each minute of service. Under Plan B, there is a flat fee of $85 for unlimited minutes.

a. Write an inequality showing the relationship of the costs associated with Plan A and Plan B if the cost of Plan A is greater than Plan B.

b. Solve the inequality to determine the number of minutes Jeremy would have to talk in order for Plan A to cost more than Plan B.

Unit 6, Activity 2, Real-life Inequalities with Answers


Directions: With your group members, answer the following questions regarding real-life inequalities.

1. In some states, in order to drive you must be at least 16 years old to obtain a driver’s license. Write an inequality to express this situation. age≥ 16

2. A bus is being rented by the math club to go on a trip. The bus can seat at most 50 people. Write an inequality to express the amount of people the bus will hold. n≤ 50

3. In order to receive a free gift at the grand opening of Fashion World, the customer must buy at least $350 worth of merchandise. Write an inequality to express this situation. c≥ 350

4. In order to drive safely on the highway, the minimum speed is 50 mph and the maximum speed is 70 mph. Write a combined inequality to express this situation. 50 ≤≤ r 70

5. The local phone company allows a maximum of 700 minutes of call time under its calling plan. Express the number of minutes allowed under this plan as an inequality. m≤ 700

6. Jeremy was trying to determine the best calling plan to buy for his family. Under Plan A, the phone company charges $15 flat fee plus an additional $0.35 per minute for each minute of service. Under Plan B, there is a flat fee of $85 for unlimited minutes.

a. Write an inequality showing the relationship of the costs associated with Plan A and Plan B if the cost of Plan A is greater than Plan B. 15 + .35m>85

b. Solve the inequality to determine the number of minutes Jeremy would have to talk in order for Plan A to cost more than Plan B. m>200; Jeremy would have to talk for longer than 200 minutes for Plan A to cost more than Plan B.

Unit 6, Activity 7, Real-life Absolute Values


Directions: Use what you learned about absolute values to solve the following real-life problems.

1. When making a pencil at a pencil factory, a machine cuts the pencils being produced at a length of 12 cm (ideally). However, since the pencil machine is not exact, there is a certain amount of tolerance associated with each cut. The difference between the actual length, x, and the ideal length of 12 cm should be at most 0.1 cm. Any pencil that is produced beyond this tolerance is discarded. This situation can be expressed by the absolute equation: 12 0 1x .− = Solve the equation to determine the smallest and largest possible pencil that is allowed.

2. A jewelry company produces gold rings which should weigh 450g (ideal weight). The acceptable tolerance for any gold ring is within 5g of the ideal weight. A ring weighs x grams.

a. Write an absolute value inequality to show the acceptable weights for the gold ring.

b. Solve the inequality, and use the solution to express the acceptable weights for the gold ring.

Unit 6, Activity 7, Real-life Absolute Values with Answers


Directions: Use what you learned about absolute values to solve the following real-life problems.

1. When making a pencil at a pencil factory, a machine cuts the pencils being produced at a length of 12 cm (ideally). However, since the pencil machine is not exact, there is a certain amount of tolerance associated with each cut. The difference between the actual length, x, and the ideal length of 12 cm should be at most 0.1 cm. Any pencil that is produced beyond this tolerance is discarded. This situation can be expressed by the absolute equation: 1.12 =−x Solve the equation to determine the smallest and largest possible pencil that is allowed.

x = 12.1 (largest possible pencil length) x = 11.9 (smallest possible pencil length)

2. A jewelry company produces gold rings which should weigh 450g (ideal weight). The acceptable tolerance for any gold ring is within 5g of the ideal weight. A ring weighs x grams.

a. Write an absolute value inequality to show the acceptable weights for the gold ring.

5450 ≤−x

b. Solve the inequality, and use the solution to express the acceptable weights for the gold ring.

455445 ≤≤ x ; The acceptable weights of the ring must be from 445g to 455g.

Unit 7, Activity 1, Is There a Point of Intersection?


Directions: In the table below, you are given 7 sets of equations. Each set has an “Equation 1” and an “Equation 2”. For each set of equations, graph Equation 1 and 2 on the same coordinate grid using a graphing calculator. For each set, determine if the two lines intersect by looking at the graphs produced. Use the results of this investigation to answer the questions that follow.

Equation 1 Equation 2 Is there a point of intersection? Yes or No

Set 1 y = 3x – 4 y = 3x + 4 Set 2 y = 4x + 2 y = x + 2 Set 3 y = 6x – 1 y = 2x – 3 Set 4 y = 5x – 3 y = -2x Set 5 y = 6x – 3 y = 6x Set 6 y = 2x – 3 y = -2x + 4 Set 7 y = -2x + 1 y = -2x – 4

1. Which set of equations produced graphs that had a point of intersection?

2. Look at the slopes associated with each set of equations that produced a point of intersection. What do you notice about the slopes?

3. Jenny’s math teacher wanted her to determine if the two equations below would produce a point of intersection. Rewrite the two equations in slope-intercept form and compare their slopes. Explain how this can help Jenny determine if there will be a point of intersection without actually graphing the two equations?

Equation 1: 2x +3 y = 1 Equation 2: 4x +6 y = 1

Unit 7, Activity 1, Is There a Point of Intersection?


4. Look at the two equations shown below. Equation 1: y = - ½ x + 2 Equation 2: 2x + 4y = 8 Rewrite Equation 2 in slope-intercept form. What do you notice about the two equations after writing them both in slope-intercept form? If both equations are graphed on a single coordinate grid, will there be a single point of intersection, no point of intersection, or will there be another possibility? Explain your answer.

5. If the slopes of two equations are the same, will the graphs of the two equations produce

a point of intersection? Explain.

6. Using what you have learned, determine if there will be a point of intersection for the two equations below without actually graphing the two equations. Explain how you determined your answer. Afterwards, use the graphing calculator to check your prediction. Equation 1: 3x + 4y = 12 Equation 2: 2x + 4y = 8

Unit 7, Activity 1, Is there a Point of Intersection? with Answers


Directions: In the table below, you are given 7 sets of equations. Each set has an “Equation 1” and an “Equation 2”. For each set of equations, graph Equation 1 and 2 on the same coordinate grid using a graphing calculator. For each set, determine if the two lines intersect by looking at the graphs produced. Use the results of this investigation to answer the questions that follow.

Equation 1 Equation 2 Is there a point of intersection? Yes or No

Set 1 y = 3x – 4 y = 3x + 4 NO Set 2 y = 4x + 2 y = x + 2 YES Set 3 y = 6x – 1 y = 2x – 3 YES Set 4 y = 5x – 3 y = -2x YES Set 5 y = 6x – 3 y = 6x NO Set 6 y = 2x – 3 y = -2x + 4 YES Set 7 y = -2x + 1 y = -2x – 4 NO

1. Which set of equations produced graphs that had a point of intersection?

Sets 1, 5, and 7.

2. Look at the slopes associated with each set of equations that produced a point of intersection. What do you notice about the slopes? In all cases where the lines intersected, the slopes were different..

3. Jenny’s math teacher wanted her to determine if the two equations below would produce a point of intersection. Rewrite the two equations in slope-intercept form and compare their slopes. Explain how this can help Jenny determine if there will be a point of intersection without actually graphing the two equations?

Equation 1: 2x +3 y = 1 Equation 2: 4x +6 y = 1

31

32

+−= xy 41

32

+−= xy

Since the slopes are the same and their y-intercepts are different, the graphs will be parallel to one another and won’t have any point of intersection.

Unit 7, Activity 1, Is there a Point of Intersection? with Answers


4. Look at the two equations shown below. Equation 1: y = - ½ x + 2 Equation 2: 2x + 4y = 8 Equation 2: y = - ½ x + 2 Rewrite Equation 2 in slope-intercept form. What do you notice about the two equations after writing them both in slope-intercept form? They are the same equations. If both equations are graphed on a single coordinate grid, will there be a single point of intersection, no point of intersection, or will there be another possibility? Explain your answer.

If they are graphed on a single coordinate grid, since they are the same equations they will produce identical graphs and will overlap one another. In this case, they will have infinitely many points of intersection.

5. If the slopes of two equations are the same, will the graphs of the two equations

produce a point of intersection? Explain.

If the slopes of two equations are the same, one of two things will occur, depending on the y-intercept.

If the slopes are the same and the y-intercepts are different, the two lines will be parallel to one another and will have no point of intersection.

If the slopes are the same and the y-intercepts are the same, they are identical to one another and will have infinitely many points of intersection.

6. Using what you have learned, determine if there will be a point of intersection for the two equations below without actually graphing the two equations. Explain how you determined your answer. Afterwards, use the graphing calculator to check your prediction. Equation 1: 3x + 4y = 12 Equation 2: 2x + 4y = 8

If the equations are rewritten in slope intercept form, Equation 1 becomes

343

+−= xy and Equation 2 becomes 221

+−= xy . Since the slopes are

different from one another they will intersect at exactly one point.

Unit 7, Activity 5, Starting a Business


Directions: Work on the following problem with a partner. Be ready to discuss the work with your classmates.

Problem: Mrs. Lowenstein started a business selling designer hats. The costs associated with starting her business included paying $900 for a professional sewing machine and $18 for the materials used to make each hat. Mrs. Lowenstein surveyed the local stores and decided to sell her hats at a price of $30 per hat.

1. Write an equation to represent the total cost, C, in dollars to make x hats.

2. Write an equation to represent the revenue, R, Mrs. Lowenstein will receive for selling x hats.

3. Using graph paper, make a graph showing the cost equation and the revenue equation on the same graph. Make the scale on the graph so that the point where the revenue and cost equations intersect can be determined.

4. From the graph, what appears to be the point where the two graphs intersect? Explain the real-life interpretation of this point.

5. Using either substitution or elimination, find the exact point of intersection for the two graphs. Explain in real-world terms what this information tells us in this situation.

6. Looking at the graph, determine when will Mrs. Lowenstein make money and when will she lose money.

Unit 7, Activity 5, Starting a Business


7. Determine what the slope is and explain what it represents in real-world terms for each of the graphs.

Unit 7, Activity 5, Starting a Business with Answers



Problem: Mrs. Lowenstein started a business selling designer hats. The costs associated with starting her business included paying $900 for a professional sewing machine and $18 for the materials used to make each hat. Mrs. Lowenstein surveyed the local stores and decided to sell her hats at a price of $30 per hat.

1. Write an equation to represent the total cost, C, in dollars to make x hats. Solution: C = 18x + 900

2. Write an equation to represent the revenue, R, Mrs. Lowenstein will receive for selling x hats.

Solution: R = 30x

3. Using graph paper, make a graph showing the cost equation and the revenue equation on the same graph. Make the scale on the graph so that the point where the revenue and cost equations intersect can be determined.

Solution: See student graphs!

4. From the graph, what appears to be the point where the two graphs intersect? Explain the real-life interpretation of this point.

Solution: Students answers may vary. Students should see that this point indicates where the revenue and cost are equal. When the revenue and cost are the same, this is referred to as the “break-even point.”

5. Using either substitution or elimination, find the exact point of intersection for the two graphs. Explain in real-world terms what this information tells us in this situation.

Solution: The point of intersection is (75, 2250). The x value (75) represents the number of hats that must be sold in order to break even. The amount of revenue and cost at this point are equal ($2250) to one another. There is not a profit or a loss at this point.

6. Looking at the graph, determine when will Mrs. Lowenstein make money and when will she lose money.

Unit 7, Activity 5, Starting a Business with Answers


7. Determine what the slope is and explain what it represents in real-world terms for each of the graphs.

Solution: The slope of the cost equation is 18, and it represents the rate of change of the cost to make each additional hat. The slope of the revenue equation is 30, and it represents the rate of change of the revenue for each hat sold.

Unit 7, Activity 6, Pizza Parlor



Problem:

Mr. Moreau is opening up a pizza parlor. To begin his business, he first has to purchase a pizza machine for $50.00. In addition to this initial cost, each pizza costs $2.00 for the ingredients to make each pizza.

1. Write an equation which could be used to model the total cost, C, to produce x

pizzas.

2. Explain what the independent and dependent variables are in this situation.

3. If Mr. Moreau wants to make a profit in his business, should he sell his pizza for less than $2.00 per pizza, exactly $2.00 per pizza, or more than $2.00 per pizza? Explain your reasoning.

4. Suppose Mr. Moreau sells his pizza for $12.00 per pizza, write an equation to represent the revenue, R, for selling x pizzas.

5. On the same graph, display the cost for the pizzas and using another color, plot the line showing the money Mr. Moreau collects (income or revenue) from the pizzas he sells.

6. At the point where the lines intersect, what information does this point of intersection tell us?

7. How many pizzas would Mr. Moreau have to sell before he starts making a profit? Explain how you know.

Unit 7, Activity 6, Pizza Parlor with Answers



Problem:

Mr. Moreau is opening up a pizza parlor. To begin his business, he first has to purchase a pizza machine for $50.00. In addition to this initial cost, each pizza costs $2.00 for the ingredients to make each pizza.

1. Write an equation which could be used to model the total cost, C, to produce x

pizzas. C=2x+50

2. Explain what the independent and dependent variables are in this situation.

Since the cost depends on the number of pizzas produced, Cost or C is the dependent variable, and the number of pizzas produced, x, is the independent variable.

3. If Mr. Moreau wants to make a profit in his business, should he sell his pizza for less than $2.00 per pizza, exactly $2.00 per pizza, or more than $2.00 per pizza? Explain your reasoning. Students should realize that he has to charge more than it cost to make so more than $2.00 per pizza is the only logical answer here.

4. Suppose Mr. Moreau sells his pizza for $12.00 per pizza, write an equation to

represent the revenue, R, for selling x pizzas. R = 12x

5. On the same graph, display the cost for the pizzas and using another color, plot

the line showing the money Mr. Moreau collects (income or revenue) from the pizzas he sells. See student graphs.

6. At the point where the lines intersect, what information does this point of intersection tell us? The intersection point is where the cost equals the revenue. In this case, that point is at(10,120) and so this means that when 10 pizzas are sold, the cost to produce them and the revenue being taken in are both $120.

7. How many pizzas would Mr. Moreau have to sell before he starts making a profit? Explain how you know. Mr. Moreau must sell greater than 10 pizzas to make a profit since this is the break-even point. Less than 10 pizzas results in a profit loss.

Unit 7, Activity 7, Which is the Better Offer?


Directions: With a partner answer the questions that follow related to the problem shown.

The Brown family is moving into a new home. They need to rent a moving van for a day but are unsure as to which is the best offer. One van company, DirtCheap Vans, charges $59.65 a day plus an additional 45¢ per mile. The other company, BestVans, charges $88.50 a day plus an additional $0.37 per mile.

1. Write a cost equation representing the cost for DirtCheap Vans based on driving x on a single day.

2. Write a cost equation representing the cost for BestVans based on driving x miles on a single day.

3. Using a graphing calculator, graph the two cost equations and determine the point where the two lines intersect.

4. Use the elimination or substitution method to verify the results you found using the graphing calculator.

5. Explain in real-life terms what this point of intersection represents.

Unit 7, Activity 7, Which is the Better Offer?


6. If you were the Brown family, how would you determine which Van company to choose so that you get the least expensive option? Explain fully.

Unit 7, Activity 7, Which is the Better Offer? with Answers


Directions: With a partner answer the questions that follow related to the problem shown.

The Brown family is moving into a new home. They need to rent a moving van for a day but are unsure as to which is the best offer. One van company, DirtCheap Vans, charges $59.65 a day plus an additional 45¢ per mile. The other company, BestVans, charges $88.50 a day plus an additional $0.37 per mile.

1. Write a cost equation representing the cost for DirtCheap Vans based on driving x on a single day. DirtCheap—C=59.65 +.45x

2. Write a cost equation representing the cost for BestVans based on driving x miles on a single day. BestVans—C=88.50 + .37x

3. Using a graphing calculator, graph the two cost equations and determine the point where the two lines intersect. The point of intersection is approximately (360.63, 221.93)

4. Use the elimination or substitution method to verify the results you found using the graphing calculator. Students should see that these methods produce similar results. Check student work.

5. Explain in real-life terms what this point of intersection represents. This means that if the Brown family travels 360.63 miles, the cost will be the same and will be approximately $221.93 for both plans.

Unit 7, Activity 7, Which is the Better Offer? with Answers


6. If you were the Brown family, how would you determine which Van company to choose so that you get the least expensive option? Explain fully.

The mileage the Brown family travels will affect which choice is best. If the Brown family plans on traveling more than 360 miles, then BestVans is the better choice. If the Brown family plans on traveling less than 360 miles, then the DirtCheap plan is cheaper.

Unit 8, Activity 1, Vocabulary Self-Awareness for Matrices


Directions: Complete the following self-assessment activity. For each term or topic listed, you will determine your own level of knowledge or comfort level using +, 0, or – signs. The left hand column is to be filled in first, before instruction of the topics takes place. At the end of the unit you will re-assess your level of understanding by filling in the right hand column. If you fully understand a topic or term, place a + sign in the column. If you have a limited understanding of a topic or term, place a 0. If you have no understanding at all for a particular topic or term, place a – sign. Comfort Level (Prior to Instruction) +, -, or 0

Topic or Term Comfort Level (After instruction) +, -, or 0

What is a matrix? What is a matrix used for in real life? Performing operations on matrices by

addition, subtraction, and multiplication

What is an inverse matrix? Solving systems of equations using

matrices

Unit 8, Activity 1, Movie Cinema Matrix


Chart of Items sold at Movie Cinema On Monday

Matrix

Snacks Popcorn Drinks 1:00 $4 $20 $25 3:00 $8 $24 $18 5:00 $10 $34 $28 8:00 $34 $38 $55

4 20 25 8 24 18 10 34 28 34 38 55

M =

Unit 8, Activity 7, Solving Systems Using Matrices


Directions: With your group, answer the following questions about the problem presented.

Jan bought 3 fries, 2 drinks, and 4 hot dogs for a total price of $8.95. Mary bought 2 fries, 6 drinks, and 5 hot dogs for a total of $12.85. Kyle bought 4 fries, 5 drinks, and 9 hot dogs for a total of $19.00.

a. Write three equations with three variables to represent this situation. Identify the variables used.

b. Create a matrix equation that can be used to solve for the three variables.

c. Using a calculator, solve the system using matrices to determine the price for a single fry, single drink, and a single hot dog.

Unit 8, Activity 7, Solving Systems Using Matrices with Answers


Directions: With your group, answer the following questions about the problem presented. Jan bought 3 fries, 2 drinks, and 4 hot dogs for a total price of $8.95. Mary bought 2 fries, 6 drinks, and 5 hot dogs for a total of $12.85. Kyle bought 4 fries, 5 drinks, and 9 hot dogs for a total of $19.00.

a. Write three equations with three variables to represent this situation. Identify the variables used.

Solution: 3f + 2d +4h = 8.95 2f + 6d + 5h = 12.85 4f + 5d + 9h = 19.00

b. Create a matrix equation that can be used to solve for the three variables. Solution:

c. Using a calculator, solve the system using matrices to determine the price for a single fry,

single drink, and a single hot dog. Solution: Fries cost $0.55; Drinks cost $0.75; Hot dogs cost $1.45

3 2 4 2 6 5 4 5 9

A =

F D H

X =

8.95 12.85 19.00

B =

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Algebra I Part 1