Calculating with Irrational Numbers
SKILL BUILDER ACTIVITY
Calculating with Irrational Numbers In this activity, students analyze how calculations are impacted by rounding, evaluating how common notations of pi affect precision.
BEFORE THE ACTIVITY Objectives Students will be able to • Compare and explain relationships among data • Cite evidence and develop logical arguments • Draw conclusions
What You Will Need • Skill Builder Activity Student Handout (p. 12) • 1 calculator per student, preferably with a pi key
DURING THE ACTIVITY
Build Background: Understand the Skill The following skills may require review before starting. • Place value, rounding to the ten-thousandths place • Order of operations Distribute the Calculating with Irrational Numbers student handout to students. Ask students to identify examples of calculations they perform regularly. For each, have students explain why they feel it is or is not important to get exact answers when calculating. Explain that for certain calculations, we tend to round and not worry about exact answers. For some calculations, however, such as budgeting or adjusting recipes, we tend make certain our answers are exact. It is important to know when exact, accurate calculations are required, as well as how using rounded numbers affects the answer.
Establish Relevance: Why the Skill Matters Read Why the Skill Matters on the student handout. Explain to students that depending on the context, precision when calculating can be extremely important. Discuss how for some careers, such as engineers, and construction workers, the safety of the products they build depends on precise measurements and calculations.
Model the Skill Display the following: π = 3.1415926535897…, asking students to identify what kind of number pi is (irrational). Explain that since pi is an irrational number, we must round when performing calculations that involve pi, such as finding the area of a circle. Draw a circle on the board, identify the radius, and explain the formula for finding
Calculating with Irrational Numbers
INSTRUCTOR PLAN the area of a circle. Model how to find area using two common rational numbers used for pi, 3.14 and 22/7. Activity Directions Have students use a calculator to find the area of a circle with a radius of 3 ft., one student using 3.14 as pi, the other using 22/7. Be certain students square just the radius (3), not the product of π x 3. Have students check their answers with those on the student handout. Factor for Pi Radius Area of Circle Difference 28.2600 ft2
3.14 22 7
3 ft.
78.5000 ft2
3.14 22 7
5 ft.
153.8600 ft2 7 ft.
9 ft.
0.0714 ft2
2
0.1400 ft2
0.2314 ft2
254.5714 ft
379.9400 ft2
3.14 22 7
154.0000 ft2 254.3400 ft2
3.14 22 7
2
78.5714 ft
3.14 22 7
0.0257 ft2
28.2857 ft2
11 ft.
380.2857 ft2
0.3457 ft2
Discussion Questions As students answer the discussion questions, they should note that as the radius increases, the difference between the two calculations using different numbers for pi also increases. This is because 22/7 (a repeating, rational number) is approximately 0.0029 larger than 3.14 (a terminating, rational number). As these numbers are multiplied by larger numbers, the differences between the resulting calculations increase as well. If students have a pi key on their calculator, they should note that since 22/7 is closer to pi than 3.14, the resulting calculations using 22/7 will always be closer to the areas calculated using the pi key as well. Summary Questions Ask students to estimate the smallest whole-number radius is for which the difference in area between using 3.14 and 22/7 for pi would be at least 1 ft2, then have students calculate the actual answer. (Answer: 19 ft radius; area when using 3.14 is 1,133.5400 ft2 ; area when using 22/7 is 1,134.5714 ft2) Finally, ask students to discuss why they think it is or is not possible to have a 100% accurate calculation when using an irrational number in calculating. Students should conclude that the lower the place value used when rounding to calculate, the more accurate the resulting calculation; however, it technically cannot be exact. Lesson 1.1 1
SKILL BUILDER ACTIVITY
STUDENT HANDOUT
Calculating with Irrational Numbers In this activity, you will work with a partner to analyze how precision is impacted when calculating with irrational numbers.
Why the Skill Matters The importance of being precise in calculations often depends on the situation. For example, when completing your federal tax forms, you are allowed to round amounts 50 cents or above up to the nearest dollar and amounts less than 50 cents down to the nearest dollar. When you round numbers and then make calculations using those rounded numbers, the answers that you reach most likely will not be 100% precise. In most cases, calculating with rounded numbers results in answers that are greater than or less than the answer you’d reach using the actual numbers. However, when a situation requires more precision–such as finding the length of a board needed to install a shelf– it is important that your measurements and any calculations you make with those measurements are as precise as possible so the board fits. Consider calculating the area of a circle, where the irrational number pi (π) is used within the formula. Since pi is an irrational number, we must round it in order to calculate. Two common rational numbers used as a rounded number to calculate with pi are 3.14 and 22/7.
Activity Directions
Discussion Questions • What pattern occurs within the “Difference” column of the chart as the circle radius increases? Why do you think this occurs? • If your calculator has a pi key, calculate the area of each circle in the chart again using the pi key. Compare your answers to the area calculations reached with 3.14 and 22/7. What conclusions can you draw based on this comparison?
Summary Questions • What do you think the smallest whole number radius is for which the difference in area between using 3.14 and 22/7 for pi would be at least 1 ft2 ? Explain your reasoning. Then, test your prediction. • Is it possible to ever have a fully accurate answer when calculating with irrational numbers? With repeating decimals? With terminating decimals? Explain your reasoning.
2 Lesson 1.1
Factor for Pi Radius Area of Circle Difference 28.2600 ft2
3.14 22 7
3 ft.
________ ft2
3.14 22 7
5 ft.
7 ft.
9 ft.
________ ft2 ________ ft2
3.14 22 7
________ ft2 ________ ft2
3.14 22 7
________ ft2 ________ ft2
3.14 22 7
28.2857 ft2
11 ft.
________ ft2
0.0257 ft2
______ ft2
_____ ft2
_____ ft2
_____ ft2
Tip Some calculators may not understand how to separate out calculations that involve multiple steps and multiple operations. One way to ensure this does not happen is to first square the radius (3 x 3…, 5 x 5…, etc.), then multiply by the factor you are using for pi. Calculating with Irrational Numbers
Copyright © McGraw-Hill Education. Permission is granted to reproduce for classroom use.
1. Use a calculator to complete the chart to the right. To do this, have one person calculate the area for each circle using 3.14 as pi, and have the other person calculate the area using 22/7 as pi. 2. For each calculation, round your answer to the nearest ten thousandth. Rewrite all hundredths numbers as equivalent ten-thousandths numbers. 3. Once you have found the area for each circle using both 22/7 and 3.14, find the difference between the two area calculations and write that number in the “Difference” column.
SKILL BUILDER ACTIVITY
INSTRUCTOR PLAN
Calculate Percent Change
U.S. Population
In this activity, students will analyze U.S. census data.
BEFORE THE ACTIVITY
1990
248,709,873
2000
281,421,906
2010
308,745,538
Objectives Students will be able to: • Compare and explain relationships among data (DOK 2) • Explain a procedure (DOK 2) • Justify a prediction based on data (DOK 3)
Percent Change, p
Percent, 1+p
13.2%
1.132
From 2000 to 2010
9.7%
1.097
From 1990 to 2010
24.1%
1.241
From 1990 to 2000
What You Will Need
Projected U.S. Population
• Calculate Percent Change Handout (p. 24) • 1 calculator per student
2020
DURING THE ACTIVITY Build Background: Understand the Skill You may wish to review the following skills before starting: • Calculating percent change • Finding the percent of a number Distribute the Calculate Percent Change student handout. Invite students to identify examples of percentages they encounter regularly. Have students discuss their strategies for performing quick percent calculations.
Establish Relevance: Why the Skill Matters Read Why the Skill Matters on the student handout. Explain to students that it is important to be knowledgeable and understand how to read data. Discuss the census and its importance in determining the number of representatives for each state and determining federal funding.
Model the Skill Display the formula for percent change on the board: new amount - original amount percent change = ____________________________ . Using original amount three-digit numbers for the original and new amounts, show students how to calculate percent change. Remind students to multiply the quotient by 100 to write it as a percent.
343,942,529
Discussion Questions The students should be able to form the conclusion that when dealing with population data, it is useful to use percent change since the raw numbers are very large. It is easier to think of population change as a percent increase or decrease rather than as a change in a large number of people. Students should recognize that percent change will be positive if the new amount is larger than the original amount; it will be negative if the new amount is less than the original amount. The three numbers that students found, 1 + p, are related in the fact that the product of the percentages from 1990–2000 and 2000–2010 equal the percentage from 1990–2010. Students should discuss using a proportion to find the increase in population from 2010 to 2020 and then adding it to the 2010 population to find the predicted population for 2020. When comparing predictions for the percent change of population from 2020 to 2050, students’ methods may vary but answers should reasonably reflect the data. Summary Questions Students should discuss the limitations of population growth and its effect on the community. Encourage the students to search the census website to learn more about population growth within their state.
Activity Directions As the students begin the activity, be sure they are using .gov sites to retrieve the data. Explain the importance of using reputable sources when researching on the Internet.
Calculate Real-World Percentages
Lesson 2.223
CCA_MATH_IRG_C2_L2_SB_136703.indd 23
Program: CCA
13/01/14 2:10 PM
Mathematics
Vendor: Six Red Marbles Instructor Resource Guide
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SKILL BUILDER ACTIVITY
STUDENT HANDOUT
Calculate Percent Change In this activity, you will analyze U.S. census data.
Why the Skill Matters Upon reading a newspaper or the headlines on the Internet, you are bound to encounter various statistics involving percentages, such as department store ads, election results, and even census data. As an informed citizen, it is important for you to be able to interpret what the statistics mean and how they may affect you or your community. In this activity, you will study the change in population of the United States by looking at U.S. census data. The effects of the census can change the number of representatives your state will have and how much federal funding your state will receive.
Activity Directions
U.S. Population
Use the Internet to find the population of the United States for the census years 1990, 2000, and 2010. For the following questions, round all answers to the nearest tenth of a percent.
2000 2010
Percent Change, p
Percent, 1+p
From 1990 to 2000 From 2000 to 2010 From 1990 to 2010 Copyright © McGraw-Hill Education. Permission is granted to reproduce for classroom use.
1. Find the percent change from 1990 to 2000 and 2000 to 2010. 2. Find the percent change in population from 1990 to 2010. 3. For each percent change, p, as a decimal, rewrite the new decimal 1 + p. 4. Assume that the percent change from 2010 to 2020 will be 11.4%. According to this assumption, what will be the population in 2020?
1990
Projected U.S. Population 2020
Discussion Questions • When dealing with population data, why might it be useful to use percent change instead of raw numbers? • How do you know if a percent change is positive or negative? • Look at the 1 + p numbers in your table. How are these 3 numbers related? How would you expect them to be related? • Explain how your prediction for the year 2020 was calculated.
Tip When calculating percent change, remember to follow the order of operations. You must perform the subtraction before the division. To do this on a calculator, you can enter the subtraction, press ENTER or =, and then divide, or you can use parentheses around the subtraction.
Summary Questions • Is it possible for the population to continue to increase at its current rate? Explain. • Use the Internet to research more about population change in your state over the last 20 years.
24Lesson 2.2
Calculate Real-World Percentages
CCA_MATH_IRG_C2_L2_SB_136703.indd 24
Program: CCA
13/01/14 3:07 PM
Mathematics
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SKILL BUILDER ACTIVITY
INSTRUCTOR PLAN
In this activity, students use a system of linear equations to analyze costs, revenues, and profits when starting a business.
BEFORE THE ACTIVITY Objectives • Model real-world problems with equations and graphs DOK 2 • Interpret mathematical results in the context of the situation they represent DOK 3
the x-values where the lines are equal, and decide which side on that value the line is larger.) Activity Directions Have students complete the activity in pairs. Remind students to use the context of the problem to label the axes appropriately. Guide students to choose a reasonable scale and interval for each axis. As students complete the activity, circulate to be certain their equations and graphs are accurate. Cost: y = 0.5x + 45 Revenue: y = 3x Profit: y = 2.5x – 45
What You Will Need • Analyze Costs, Revenues, and Profits Student Handout (p. 80) • Graph paper (p. 340) • Red and blue colored pencils • Rulers
DURING THE ACTIVITY Build Background: Understand the Skill You may wish to review the following skills before starting: • Writing linear equations to model real-world problems • Graphing linear equations Distribute the Analyze Costs, Revenues, and Profits student handout to students. Ask students to share any experiences they have had starting or running a business or any plans they might have to do so. Discuss why you must spend money to make money in any business.
Establish Relevance: Why the Skill Matters Read Why the Skill Matters on the student handout. Tell students that if they own a business or plan to own one someday, they will need to know how to analyze how their business spends and makes money. The same methods they use in this activity can also be applied to their personal budgets—to analyze their earnings and expenses.
Money Spent and Received (in $)
Analyze Costs, Revenues, and Profits
80 60
y
costs
40
break-even point
20 revenue O
10
20
30
x
Number of Stickers Produced and Sold Discussion Questions After students complete the activity, direct them to answer the discussion questions in pairs or small groups. Students should recognize that the break-even point at (18, 54) shows that they need to sell 18 stickers before they can start making a profit. At that point, costs and revenues are equal (both = $54). The region to the right of this point represents when the business makes money (profits) because revenues are greater than costs. The region to the left represents when the business loses money because costs are greater than revenues. If you lowered the price, the break-even point would be farther up and to the right on the graph because you would need to sell more stickers to cover your costs. The opposite would happen if you increased the price. Summary Questions Have students find the break-even point by solving the system of equations with substitution or elimination. Discuss how it is easier to determine the exact values for the break-even point when you solve the system algebraically, but it is easier to understand the relationship between costs and revenues when you graph the system. Have students graph the profit equation. Point out that the profit line crosses the x-axis at (18, 0), which is the x-value of the break-even point. For x < 18, y is negative because you lose money (negative profits). For x > 18, y is positive because you make money (positive profits).
Model the Skill Write the equations y = x + 3 and y = 2x on the board, asking students which line is larger than the other. Explain that one line is not always larger than the other because it depends on the value of x. Draw both lines on the same graph asking the students how they can determine when one line is larger than the other. (Determine Solve Systems of Linear Equations
Lesson 5.479
CCA_MATH_IRG_C5_L4_SB_136703.indd 79
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21/01/14 9:53 AM
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SKILL BUILDER ACTIVITY
STUDENT HANDOUT
Investigate Profits and Expenses In this activity, you will use a system of linear equations to analyze costs, revenues, and profits when starting a business.
Why the Skill Matters Every business owner must think about three values: costs, revenue, and profit. Costs are things you must spend money on, such as rent, equipment, supplies, advertising, or paying employees. Revenue is the money your business receives for selling products or services. Profit is the money you actually make. To make a profit, you must sell enough products or services to cover all of your costs. The point at which this happens is called the break-even point. To determine what the break-even point is, you can solve a system of linear equations.
Activity Directions
Discussion Questions
1. Read the problem.
• What does the break-even point represent for your business? • What do the red and blue shaded regions of your graph represent? • What would happen to your break-even point if you lowered the price of each bumper sticker? If you increased the price?
Bumper Sticker Business You spend $45 to start a business making and selling bumper stickers. It costs you $0.50 to make each sticker, and you charge $3 per sticker. How many bumper stickers do you need to make and sell in order to make a profit?
Summary Questions
2. Write linear equations to model your cost and revenue, y, for making and selling x bumper stickers.
• How can you find the break-even point without graphing the equations?
Revenue: 3. Graph the system of two equations. Use a red pencil for the cost line and label it “Cost.” Use a blue pencil for the revenue line and label it “Revenue.” 4. Label the point of intersection “break-even point.” 5. Shade the area between the two lines to the right of the break-even point blue. Shade the area between the two lines to the left of the break-even point red. 6. Profit equals revenue minus costs. Write a linear equation to model your profit, y, for making and selling x bumper stickers.
Copyright © McGraw-Hill Education. Permission is granted to reproduce for classroom use.
• If you graphed your profit equation, how would that graph model your break-even point?
Cost:
Tip When you use a system of equations to model a real-world problem, the variables can represent different, but related, values in each equation. In the cost equation, y represents the amount of money spent, and x represents the number of items produced. In the revenue equation, y represents the amount of money received, and x represents the number of items sold. In the system of equations, y represents the amount of money spent and received, and x represents the number of items produced and sold.
Profit: 7. Substitute the x-value of your break-even point into the profit equation and simplify. Explain the results.
80Lesson 5.4
Solve Systems of Linear Equations
CCA_MATH_IRG_C5_L4_SB_136703.indd 80
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13/01/14 3:20 PM
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Calculating with Irrational Numbers In this activity, students analyze how calculations are impacted by rounding, evaluating how common notations of pi affect precision.
BEFORE THE ACTIVITY Objectives Students will be able to • Compare and explain relationships among data • Cite evidence and develop logical arguments • Draw conclusions
What You Will Need • Skill Builder Activity Student Handout (p. 12) • 1 calculator per student, preferably with a pi key
DURING THE ACTIVITY
Build Background: Understand the Skill The following skills may require review before starting. • Place value, rounding to the ten-thousandths place • Order of operations Distribute the Calculating with Irrational Numbers student handout to students. Ask students to identify examples of calculations they perform regularly. For each, have students explain why they feel it is or is not important to get exact answers when calculating. Explain that for certain calculations, we tend to round and not worry about exact answers. For some calculations, however, such as budgeting or adjusting recipes, we tend make certain our answers are exact. It is important to know when exact, accurate calculations are required, as well as how using rounded numbers affects the answer.
Establish Relevance: Why the Skill Matters Read Why the Skill Matters on the student handout. Explain to students that depending on the context, precision when calculating can be extremely important. Discuss how for some careers, such as engineers, and construction workers, the safety of the products they build depends on precise measurements and calculations.
Model the Skill Display the following: π = 3.1415926535897…, asking students to identify what kind of number pi is (irrational). Explain that since pi is an irrational number, we must round when performing calculations that involve pi, such as finding the area of a circle. Draw a circle on the board, identify the radius, and explain the formula for finding
Calculating with Irrational Numbers
INSTRUCTOR PLAN the area of a circle. Model how to find area using two common rational numbers used for pi, 3.14 and 22/7. Activity Directions Have students use a calculator to find the area of a circle with a radius of 3 ft., one student using 3.14 as pi, the other using 22/7. Be certain students square just the radius (3), not the product of π x 3. Have students check their answers with those on the student handout. Factor for Pi Radius Area of Circle Difference 28.2600 ft2
3.14 22 7
3 ft.
78.5000 ft2
3.14 22 7
5 ft.
153.8600 ft2 7 ft.
9 ft.
0.0714 ft2
2
0.1400 ft2
0.2314 ft2
254.5714 ft
379.9400 ft2
3.14 22 7
154.0000 ft2 254.3400 ft2
3.14 22 7
2
78.5714 ft
3.14 22 7
0.0257 ft2
28.2857 ft2
11 ft.
380.2857 ft2
0.3457 ft2
Discussion Questions As students answer the discussion questions, they should note that as the radius increases, the difference between the two calculations using different numbers for pi also increases. This is because 22/7 (a repeating, rational number) is approximately 0.0029 larger than 3.14 (a terminating, rational number). As these numbers are multiplied by larger numbers, the differences between the resulting calculations increase as well. If students have a pi key on their calculator, they should note that since 22/7 is closer to pi than 3.14, the resulting calculations using 22/7 will always be closer to the areas calculated using the pi key as well. Summary Questions Ask students to estimate the smallest whole-number radius is for which the difference in area between using 3.14 and 22/7 for pi would be at least 1 ft2, then have students calculate the actual answer. (Answer: 19 ft radius; area when using 3.14 is 1,133.5400 ft2 ; area when using 22/7 is 1,134.5714 ft2) Finally, ask students to discuss why they think it is or is not possible to have a 100% accurate calculation when using an irrational number in calculating. Students should conclude that the lower the place value used when rounding to calculate, the more accurate the resulting calculation; however, it technically cannot be exact. Lesson 1.1 1
SKILL BUILDER ACTIVITY
STUDENT HANDOUT
Calculating with Irrational Numbers In this activity, you will work with a partner to analyze how precision is impacted when calculating with irrational numbers.
Why the Skill Matters The importance of being precise in calculations often depends on the situation. For example, when completing your federal tax forms, you are allowed to round amounts 50 cents or above up to the nearest dollar and amounts less than 50 cents down to the nearest dollar. When you round numbers and then make calculations using those rounded numbers, the answers that you reach most likely will not be 100% precise. In most cases, calculating with rounded numbers results in answers that are greater than or less than the answer you’d reach using the actual numbers. However, when a situation requires more precision–such as finding the length of a board needed to install a shelf– it is important that your measurements and any calculations you make with those measurements are as precise as possible so the board fits. Consider calculating the area of a circle, where the irrational number pi (π) is used within the formula. Since pi is an irrational number, we must round it in order to calculate. Two common rational numbers used as a rounded number to calculate with pi are 3.14 and 22/7.
Activity Directions
Discussion Questions • What pattern occurs within the “Difference” column of the chart as the circle radius increases? Why do you think this occurs? • If your calculator has a pi key, calculate the area of each circle in the chart again using the pi key. Compare your answers to the area calculations reached with 3.14 and 22/7. What conclusions can you draw based on this comparison?
Summary Questions • What do you think the smallest whole number radius is for which the difference in area between using 3.14 and 22/7 for pi would be at least 1 ft2 ? Explain your reasoning. Then, test your prediction. • Is it possible to ever have a fully accurate answer when calculating with irrational numbers? With repeating decimals? With terminating decimals? Explain your reasoning.
2 Lesson 1.1
Factor for Pi Radius Area of Circle Difference 28.2600 ft2
3.14 22 7
3 ft.
________ ft2
3.14 22 7
5 ft.
7 ft.
9 ft.
________ ft2 ________ ft2
3.14 22 7
________ ft2 ________ ft2
3.14 22 7
________ ft2 ________ ft2
3.14 22 7
28.2857 ft2
11 ft.
________ ft2
0.0257 ft2
______ ft2
_____ ft2
_____ ft2
_____ ft2
Tip Some calculators may not understand how to separate out calculations that involve multiple steps and multiple operations. One way to ensure this does not happen is to first square the radius (3 x 3…, 5 x 5…, etc.), then multiply by the factor you are using for pi. Calculating with Irrational Numbers
Copyright © McGraw-Hill Education. Permission is granted to reproduce for classroom use.
1. Use a calculator to complete the chart to the right. To do this, have one person calculate the area for each circle using 3.14 as pi, and have the other person calculate the area using 22/7 as pi. 2. For each calculation, round your answer to the nearest ten thousandth. Rewrite all hundredths numbers as equivalent ten-thousandths numbers. 3. Once you have found the area for each circle using both 22/7 and 3.14, find the difference between the two area calculations and write that number in the “Difference” column.
SKILL BUILDER ACTIVITY
INSTRUCTOR PLAN
Calculate Percent Change
U.S. Population
In this activity, students will analyze U.S. census data.
BEFORE THE ACTIVITY
1990
248,709,873
2000
281,421,906
2010
308,745,538
Objectives Students will be able to: • Compare and explain relationships among data (DOK 2) • Explain a procedure (DOK 2) • Justify a prediction based on data (DOK 3)
Percent Change, p
Percent, 1+p
13.2%
1.132
From 2000 to 2010
9.7%
1.097
From 1990 to 2010
24.1%
1.241
From 1990 to 2000
What You Will Need
Projected U.S. Population
• Calculate Percent Change Handout (p. 24) • 1 calculator per student
2020
DURING THE ACTIVITY Build Background: Understand the Skill You may wish to review the following skills before starting: • Calculating percent change • Finding the percent of a number Distribute the Calculate Percent Change student handout. Invite students to identify examples of percentages they encounter regularly. Have students discuss their strategies for performing quick percent calculations.
Establish Relevance: Why the Skill Matters Read Why the Skill Matters on the student handout. Explain to students that it is important to be knowledgeable and understand how to read data. Discuss the census and its importance in determining the number of representatives for each state and determining federal funding.
Model the Skill Display the formula for percent change on the board: new amount - original amount percent change = ____________________________ . Using original amount three-digit numbers for the original and new amounts, show students how to calculate percent change. Remind students to multiply the quotient by 100 to write it as a percent.
343,942,529
Discussion Questions The students should be able to form the conclusion that when dealing with population data, it is useful to use percent change since the raw numbers are very large. It is easier to think of population change as a percent increase or decrease rather than as a change in a large number of people. Students should recognize that percent change will be positive if the new amount is larger than the original amount; it will be negative if the new amount is less than the original amount. The three numbers that students found, 1 + p, are related in the fact that the product of the percentages from 1990–2000 and 2000–2010 equal the percentage from 1990–2010. Students should discuss using a proportion to find the increase in population from 2010 to 2020 and then adding it to the 2010 population to find the predicted population for 2020. When comparing predictions for the percent change of population from 2020 to 2050, students’ methods may vary but answers should reasonably reflect the data. Summary Questions Students should discuss the limitations of population growth and its effect on the community. Encourage the students to search the census website to learn more about population growth within their state.
Activity Directions As the students begin the activity, be sure they are using .gov sites to retrieve the data. Explain the importance of using reputable sources when researching on the Internet.
Calculate Real-World Percentages
Lesson 2.223
CCA_MATH_IRG_C2_L2_SB_136703.indd 23
Program: CCA
13/01/14 2:10 PM
Mathematics
Vendor: Six Red Marbles Instructor Resource Guide
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SKILL BUILDER ACTIVITY
STUDENT HANDOUT
Calculate Percent Change In this activity, you will analyze U.S. census data.
Why the Skill Matters Upon reading a newspaper or the headlines on the Internet, you are bound to encounter various statistics involving percentages, such as department store ads, election results, and even census data. As an informed citizen, it is important for you to be able to interpret what the statistics mean and how they may affect you or your community. In this activity, you will study the change in population of the United States by looking at U.S. census data. The effects of the census can change the number of representatives your state will have and how much federal funding your state will receive.
Activity Directions
U.S. Population
Use the Internet to find the population of the United States for the census years 1990, 2000, and 2010. For the following questions, round all answers to the nearest tenth of a percent.
2000 2010
Percent Change, p
Percent, 1+p
From 1990 to 2000 From 2000 to 2010 From 1990 to 2010 Copyright © McGraw-Hill Education. Permission is granted to reproduce for classroom use.
1. Find the percent change from 1990 to 2000 and 2000 to 2010. 2. Find the percent change in population from 1990 to 2010. 3. For each percent change, p, as a decimal, rewrite the new decimal 1 + p. 4. Assume that the percent change from 2010 to 2020 will be 11.4%. According to this assumption, what will be the population in 2020?
1990
Projected U.S. Population 2020
Discussion Questions • When dealing with population data, why might it be useful to use percent change instead of raw numbers? • How do you know if a percent change is positive or negative? • Look at the 1 + p numbers in your table. How are these 3 numbers related? How would you expect them to be related? • Explain how your prediction for the year 2020 was calculated.
Tip When calculating percent change, remember to follow the order of operations. You must perform the subtraction before the division. To do this on a calculator, you can enter the subtraction, press ENTER or =, and then divide, or you can use parentheses around the subtraction.
Summary Questions • Is it possible for the population to continue to increase at its current rate? Explain. • Use the Internet to research more about population change in your state over the last 20 years.
24Lesson 2.2
Calculate Real-World Percentages
CCA_MATH_IRG_C2_L2_SB_136703.indd 24
Program: CCA
13/01/14 3:07 PM
Mathematics
Vendor: Six Red Marbles Instructor Resource Guide
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SKILL BUILDER ACTIVITY
INSTRUCTOR PLAN
In this activity, students use a system of linear equations to analyze costs, revenues, and profits when starting a business.
BEFORE THE ACTIVITY Objectives • Model real-world problems with equations and graphs DOK 2 • Interpret mathematical results in the context of the situation they represent DOK 3
the x-values where the lines are equal, and decide which side on that value the line is larger.) Activity Directions Have students complete the activity in pairs. Remind students to use the context of the problem to label the axes appropriately. Guide students to choose a reasonable scale and interval for each axis. As students complete the activity, circulate to be certain their equations and graphs are accurate. Cost: y = 0.5x + 45 Revenue: y = 3x Profit: y = 2.5x – 45
What You Will Need • Analyze Costs, Revenues, and Profits Student Handout (p. 80) • Graph paper (p. 340) • Red and blue colored pencils • Rulers
DURING THE ACTIVITY Build Background: Understand the Skill You may wish to review the following skills before starting: • Writing linear equations to model real-world problems • Graphing linear equations Distribute the Analyze Costs, Revenues, and Profits student handout to students. Ask students to share any experiences they have had starting or running a business or any plans they might have to do so. Discuss why you must spend money to make money in any business.
Establish Relevance: Why the Skill Matters Read Why the Skill Matters on the student handout. Tell students that if they own a business or plan to own one someday, they will need to know how to analyze how their business spends and makes money. The same methods they use in this activity can also be applied to their personal budgets—to analyze their earnings and expenses.
Money Spent and Received (in $)
Analyze Costs, Revenues, and Profits
80 60
y
costs
40
break-even point
20 revenue O
10
20
30
x
Number of Stickers Produced and Sold Discussion Questions After students complete the activity, direct them to answer the discussion questions in pairs or small groups. Students should recognize that the break-even point at (18, 54) shows that they need to sell 18 stickers before they can start making a profit. At that point, costs and revenues are equal (both = $54). The region to the right of this point represents when the business makes money (profits) because revenues are greater than costs. The region to the left represents when the business loses money because costs are greater than revenues. If you lowered the price, the break-even point would be farther up and to the right on the graph because you would need to sell more stickers to cover your costs. The opposite would happen if you increased the price. Summary Questions Have students find the break-even point by solving the system of equations with substitution or elimination. Discuss how it is easier to determine the exact values for the break-even point when you solve the system algebraically, but it is easier to understand the relationship between costs and revenues when you graph the system. Have students graph the profit equation. Point out that the profit line crosses the x-axis at (18, 0), which is the x-value of the break-even point. For x < 18, y is negative because you lose money (negative profits). For x > 18, y is positive because you make money (positive profits).
Model the Skill Write the equations y = x + 3 and y = 2x on the board, asking students which line is larger than the other. Explain that one line is not always larger than the other because it depends on the value of x. Draw both lines on the same graph asking the students how they can determine when one line is larger than the other. (Determine Solve Systems of Linear Equations
Lesson 5.479
CCA_MATH_IRG_C5_L4_SB_136703.indd 79
Program: CCA
21/01/14 9:53 AM
Mathematics
Vendor: Six Red Marbles Instructor Resource Guide
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SKILL BUILDER ACTIVITY
STUDENT HANDOUT
Investigate Profits and Expenses In this activity, you will use a system of linear equations to analyze costs, revenues, and profits when starting a business.
Why the Skill Matters Every business owner must think about three values: costs, revenue, and profit. Costs are things you must spend money on, such as rent, equipment, supplies, advertising, or paying employees. Revenue is the money your business receives for selling products or services. Profit is the money you actually make. To make a profit, you must sell enough products or services to cover all of your costs. The point at which this happens is called the break-even point. To determine what the break-even point is, you can solve a system of linear equations.
Activity Directions
Discussion Questions
1. Read the problem.
• What does the break-even point represent for your business? • What do the red and blue shaded regions of your graph represent? • What would happen to your break-even point if you lowered the price of each bumper sticker? If you increased the price?
Bumper Sticker Business You spend $45 to start a business making and selling bumper stickers. It costs you $0.50 to make each sticker, and you charge $3 per sticker. How many bumper stickers do you need to make and sell in order to make a profit?
Summary Questions
2. Write linear equations to model your cost and revenue, y, for making and selling x bumper stickers.
• How can you find the break-even point without graphing the equations?
Revenue: 3. Graph the system of two equations. Use a red pencil for the cost line and label it “Cost.” Use a blue pencil for the revenue line and label it “Revenue.” 4. Label the point of intersection “break-even point.” 5. Shade the area between the two lines to the right of the break-even point blue. Shade the area between the two lines to the left of the break-even point red. 6. Profit equals revenue minus costs. Write a linear equation to model your profit, y, for making and selling x bumper stickers.
Copyright © McGraw-Hill Education. Permission is granted to reproduce for classroom use.
• If you graphed your profit equation, how would that graph model your break-even point?
Cost:
Tip When you use a system of equations to model a real-world problem, the variables can represent different, but related, values in each equation. In the cost equation, y represents the amount of money spent, and x represents the number of items produced. In the revenue equation, y represents the amount of money received, and x represents the number of items sold. In the system of equations, y represents the amount of money spent and received, and x represents the number of items produced and sold.
Profit: 7. Substitute the x-value of your break-even point into the profit equation and simplify. Explain the results.
80Lesson 5.4
Solve Systems of Linear Equations
CCA_MATH_IRG_C5_L4_SB_136703.indd 80
Program: CCA
13/01/14 3:20 PM
Mathematics
Vendor: Six Red Marbles Instructor Resource Guide
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