algebra i - part irichland.k12.la.us/documents/common core standards/cc... · 2012-10-01 ·...

Algebra I - Part I

Unit 1, Activity 1, Where Do I Belong?

Blackline Masters, Algebra I–Part 1 Page 1-1

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 10, 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 10, 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 12, 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 12, 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 3, How Many Significant Digits Are There?


Directions: In this activity, work with a partner and discuss the answers to the following questions presented. Write your answers in the space provided. Be ready to justify your answers in a whole-class discussion.

Examples of Significant Digits Example Number of

Significant Digits Justification for Answer

1278.50 m

8.002 g

43.050 m

5.420 x 10 5−

6271.91

543, 091

453 kg

5057 L

5.00 kg

0.007 L

Unit 2, Activity 3, How Many Significant Digits Are There? with Answers


Directions: In this activity, work with a partner and discuss the answers to the following questions presented. Write your answers in the space provided. Be ready to justify your answers in a whole-class discussion.

Examples of Significant Digits Example Number of

Significant Digits Justification for Answer

1278.50 m 6 Additional zeros to the right of decimal and a sig. dig. are significant.

8.002 g 4 Zeros between 2 sig. dig. are significant.

43.050 m 5 Additional zeros to the right of decimal and a sig. dig. are significant. Zeros between 2 sig. dig. are significant.

5.420 x 10 5− 4 Additional zeros to the right of decimal and a sig. dig. are significant.

6271.91 6 All non-zero digits are always significant.

543, 091 6 Zeros between 2 sig. dig. are significant.

453 kg 3 All non-zero digits are always significant.

5057 L 4 Zeros between 2 sig. dig. are significant.

5.00 kg 3 Additional zeros to the right of decimal and a sig. dig. are significant.

0.007 L

1

Placeholders are not sig.

Unit 2, Activity 6, Multiplication and Division Using Significant Digits


Directions: Part I—Read the following informational text to help you answer the questions that follow regarding multiplying and dividing numbers utilizing significant digits.

Informational Text: Suppose you used a calculator and determined that “25 feet divided by 6.0 equals 4.166666667 feet,” which is what your calculator will tell you. Since the measurement of 25 feet is measured to the nearest foot, you should wonder, “Can you get an answer that is accurate to a billionth of a foot?” The answer, of course, is no. Therefore, when dealing with measurement problems that involve multiplication and division operations, it is important to understand some “rules” for expressing answers to problems with the correct number of significant digits.

The rule for multiplying and dividing is this:

RULE: When multiplying or dividing, your answer may only show as many significant digits as the multiplied or divided quantity having the least number of significant digits.

Example 1: When dividing 25 by 6.0 in a measurement problem, since each number has two significant digits, the answer will have two significant digits. So even though the answer using a calculator would result in 25÷6.0 = 4.166666667, the proper way to express this answer would be to round off to two significant digits. Therefore the answer will be 25÷6.0 = 4.2.

Example 2: Suppose the length of a walkway was 81.7 meters and its width was measured to be 2.405 meters, what would its area be if you are using the correct number of significant digits in your calculation? In order to find the area, you would need to multiply 81.7m ×2.405 m. Your calculator says 196.4885. But since the number 81.7 has three significant digits and the number 2.405 has four significant digits, the rule says that the final answer should have the smaller of the number of significant digits, which in this case is 3 significant digits. Therefore, the answer must be rounded to 196 square meters (in order to have three significant digits). Additional examples are shown below:

Multiplication

The answer must be rounded off to 2 significant figures, since 1.6 only has 2 significant figures.

Division

The answer must be rounded off to 3 significant figures, since 45.2 has only 3 significant figures. Important: Any rounding takes place at the end of the calculation process.

Unit 2, Activity 6, Multiplication and Division Using Significant Digits


Directions: Part II—After reading the informational text, work with a partner on the questions below and be ready to discuss your answers with the rest of the class. You must use the informational text to justify your answers.

1. In your own words, explain the rule for multiplying and dividing two quantities when using significant digits.

2. A wall is 9.42 meters long and 2.3 meters tall. Calculate the area of the wall with the correct number of significant digits.

3. Janice rode 8 miles in 50 minutes. What is her

average speed in miles per minute, taking into account significant digits?

4. Using your calculator, find 34.78×11.7÷0.17, then

express the result with the correct number of significant digits.

5. What is the product of 45 x 3.00 if you use

significant digits in our answer?

Justification for Answers: 2. 3. 4. 5.

Unit 2, Activity 6, Multiplication and Division Using Significant Digits with Answers


Directions: Part II—After reading the informational text, work with a partner on the questions below and be ready to discuss your answers with the rest of the class. You must use the informational text to justify your answers.

1. In your own words, explain the rule for multiplying and dividing two quantities when using significant digits.

See student explanations.

2. A wall is 9.42 meters long and 2.3 meters tall. Calculate the area of the wall with the correct number of significant digits.

Answer: 22 square meters

3. Janice rode 8 miles in 50 minutes. What is her average speed in miles per minute, taking into account significant digits?

Answer: 0.2 miles per minute

4. Using your calculator find 34.78×11.7÷0.17, then express the result with the correct number of significant digits.

Answer: 2.4 x 10 3

5. What is the product of 45 x 3.00 if you use significant digits in our answer?

Answer: 1.4 x 102

Justification for Answers: 2. Using a calculator, the product of 9.42 x 2.3 = 21.66, but the 2.3 has only two significant digits, so the result must only have two significant digits in it. Thus, round 21.66 to 22. 3. Dividing 8 miles by 50 minutes results in 0.16 miles per minute. You have to round to 1 significant digit since 8 has only 1 significant digit. There the answer must be expressed as 0.2 (not 0.16). 4. Although the calculator answer is 2393.682353, since the number with the least number of significant digits is 0.17 (only 2 significant digits), the final answer must also be rounded to have only two significant digits, which would be 2400 written in scientific notation as 2.4 x 10 3 . 5. Using a calculator, the product is 135. However, since one of the factors was 45 (which has only two significant digits) the final answer should only have two significant digits. Thus you round 135 to 140 and write the answer using scientific notation as 1.4 x 102

Unit 2, Activity 9, Absolute Error in Measurement


Directions: With a partner, answer the following questions related to the concept of absolute error.

1. Jordan measures a piece of wood to be 4 ½ feet long. What is the absolute error related to his measurement?

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. What is the absolute error of the bag’s weight as described on the package?

3. Trevor used his digital thermometer to measure the high temperature at the Lafayette airport to be 82.7 degrees. The official temperature recorded for the Lafayette airport that day was 82.5 degrees. What is the absolute error of the thermometer Trevor used?

4. An object that is known to be a mass of 100-gram is placed on three different scales. The scales’ measures were recorded in the table shown below. Use the results to answer the questions below.

a. Which scale had the largest absolute error associated with it?

b. Which scale had the smallest absolute error associated with it?

c. If you were picking a scale to use to measure masses in a science experiment,

which scale would you want to use? Explain why.

Scale 1 102.1g Scale 2 101.2g Scale 3 98.9 g

Unit 2, Activity 9, Absolute Error in Measurement with Answers


Directions: With a partner, answer the following questions related to the concept of absolute error.

1. Jordan measures a piece of wood to be 4 ½ feet long. What is the absolute error related to his measurement?

Answer: It is impossible to tell since the actual measurement is not known.

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. What is the absolute error of the bag’s weight as described on the package?

Answer: Absolute error is 0.25 pounds.

3. Trevor used his digital thermometer to measure the high temperature at the Lafayette airport to be 82.7 degrees. The official temperature recorded for the Lafayette airport that day was 82.5 degrees. What is the absolute error of the thermometer Trevor used?

Answer: Absolute error is 0.2 degrees

4. An object that is known to be a mass of 100-gram is placed on

three different scales. The scales’ measures were recorded in the table shown below. Use the results to answer the questions below.

a. Which scale had the largest absolute error associated with it?

Answer: Scale 1 b. Which scale had the smallest absolute error associated with it?

Answer: Scale 3 c. If you were picking a scale to use to measure masses in a science experiment,

which scale would you want to use? Explain why.

Answer: The best scale to use would be the scale with the least amount of error associated with it (scale 3).

Scale 1 102.1g Scale 2 101.2g Scale 3 98.9 g

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 students’ 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 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 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 students’ 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 4, Activity 2, Anticipation Guide: What about “m”?


Directions: Prior to the lesson, read each statement below and circle if you agree or disagree with each statement. Once the lesson has been fully taught and discussed, revisit the questions and modify your thinking if necessary and explain why you changed your thinking in the space provided under each statement.

1. If the equation y = 4x were graphed, the slope would be negative, and the graph would be increasing.

AGREE DISAGREE

2. The steepness for the graph of y = 5x would be more than the steepness for the graph of the equation y = -5x.

AGREE DISAGREE

3. The steepness of a graph is different than the slope of a graph. AGREE DISAGREE

4. If a linear graph is decreasing, the value of “m” must always be negative. AGREE DISAGREE

5. If a linear graph is horizontal, the value of “m” must be zero. AGREE DISAGREE

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 = -2x + 4 Equation 4: y = -2x Equation 2: y = 3x + 2 Equation 5: y = 3x – 3 Equation 3: y = 3x Equation 6: y = -2x - 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 = -2x + 4 Equation 4: y = -2x Equation 2: y = 3x + 2 Equation 5: y = 3x – 3 Equation 3: y = 3x Equation 6: y = -2x - 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 were 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 –1797 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.

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 students’ 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

Blackline Masters, Algebra I–Part 1 Page 6- 1

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 number 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 number 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, 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, 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 7, Activity 1, 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 students’ 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. Answer: The steepness of the line relates to the speed. Since Gertrude’s speed is faster, the line is steeper relating her motion.

Unit 7, Activity 2, 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 2, 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 2, 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 2, 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 6, 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.

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

Unit 7, Activity 6, 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 students’ 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.

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 7, 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.

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 7, 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.

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 costs 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 students’ 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), 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 8, 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 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.

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 8, 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 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 students’ 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.

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 =

algebra i - part irichland.k12.la.us/documents/common core standards/cc... · 2012-10-01 ·...

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