reproducible activities - pearsonwps.pearsoncustom.com/wps/media/objects/2605/...prime and composite...

REPRODUCIBLE ACTIVITIES The activities in this section are designed to be self-contained units of work. They can be used in class or out of class, as assignments to be completed individually by students or in groups, or as long-term projects that can be completed by students individually or in groups. The activities are intended to enrich the presentation in the textbook and to guide the students through an investigative or discovery process. It is hoped that, after having completed an activity, a student will have a more in-depth understanding of the underlying mathematical principles and will have developed a richer number sense. The following activities are available in this supplement. The authors are continually exploring new activities and as they are refined they will be made available in future editions. More information on when and how to use each activity is given in the teaching tips. TRANSPOSING DIGITS (p. 22) Outcome: Verify that the difference between two numbers that have two digits transposed is divisible by nine.

FACTORING INTO PAIRS OF FACTORS (pp. 23-26) Outcome 1: Find all possible pairs of factors of a given number. Outcome 2: Distinguish between prime and composite numbers. Outcome 3: Find numbers that are perfect squares. Outcome 4: Find the square root of a number using a calculator.

ADDING AND SUBTRACTING INTEGERS (pp. 27-30) Outcome 1: Formulate rules for adding integers. Outcome 2: Discover the relationship between subtracting integers and adding integers.

LOCATING COORDINATES ON A SPHERE (pp. 31-33) Outcome: Find locations on a sphere using ordered pairs.

ARRANGING GLOBAL INTERACTIVE COMMUNICATIONS (p. 34) Outcome: To determine an optimum time for a global interactive communication.

PREPARING A REFERENCE CHART FOR GLOBAL COMMUNICATIONS (p. 35) Outcome: To gather and organize data to improve efficiency.

MULTIPLYING AND DIVIDING INTEGERS (pp. 36-39) Outcome 1: Formulate rules for multiplying integers. Outcome 2: Formulate rules for dividing integers.

COMMON FRACTIONS (p. 40) Outcome: Distinguish between proper and improper common fractions.

DECIMAL FRACTIONS (p. 41) Outcome: Examine decimal equivalents of common fractions.

FRACTIONS IN SIMPLEST FORM (p. 42) Outcome: Examine equivalent fractions in simplest form.

FRACTION RELATIONSHIPS (p. 43-46)

Outcome: Develop a procedure for comparing fractions to 1, 12

, or 14

by inspection.

College Mathematics for Technology, Sixth Edition Cleaves and Hobbs Reproducible Activities

2004 by Pearson Education, Inc. All rights reserved.

19

SIZE OF FRACTIONS (p. 47-49)

Outcome: Categorize fractions based upon their relationship to 14

, 12

, 34

, and 1.

ANALYZING NUTRITION LABELS (pp. 50-54) Outcome: Use proportions and percents to analyze nutrition labels.

WHAT PERCENT TAX DO I REALLY PAY? (pp. 55-57) Outcome 1: Determine the percent of tax withheld for a given taxable income. Outcome 2: Determine the percent of income tax for a given annual taxable income.

ESTIMATING MEASURES (pp. 58-59) Outcome: Estimate linear and circular measure in inches.

WHAT IS PI, π ? (p. 60) Outcome: Discover the relationship between the circumference and diameter of a circle.

CIRCLE, BAR, AND LINE GRAPHS (p. 61) Outcome 1: Interpret data from various types of graphs. Outcome 2: Plan, design, and conduct an investigation in which the findings are reported in writing and graphically.

CRITIQUING GRAPHS (p. 62) Outcome: Critique graphs found in recent publications.

ENGINEERING NOTATION (p. 63-65) Outcome: Change a number from ordinary notation to engineering notation.

ESTIMATING RADICALS (pp. 66-67) Squares and Square Roots Outcome 1: Distinguish between squares and square roots and rational and irrational numbers. Outcome 2: Determine two whole numbers between which a given irrational number will fall.

QUADRATIC EQUATIONS (pp. 69-70) Calculating Vehicular Speed from Skid Marks and Road Conditions Outcome: Use quadratic equations in career applications

RATIONAL EQUATIONS (p 68) Patient Charting Efficiency Outcome: Use rational equations in career applications WHAT IS THE NATURAL EXPONENTIAL e? (p. 71)

Outcome: Discover the effect of large values of n in the expression n

n

+

11 .

COMPOUNDED AMOUNT AND COMPOUND INTEREST (p. 72) Outcome: Compare the two compound amount formulas for compound interest,

A = nt

nrP

+1 and A Pent= .

DEVELOPING TABLE VALUES (pp. 73-76) Outcome: Use formulas to expand tables.



20

INEQUALITIES (p. 77) Intelligence Quotient (IQ) Outcome: Use inequalities in career applications.

GRAPHING EQUATIONS AND INEQUALITIES PROJECT (pp. 78-79) Instructions for Group Projects for Five-Member Group

GRAPHING EQUATIONS AND INEQUALITIES (pp. 80-89) Graphing Activity 1 Outcome: Examine equations in the form y mx= .

Graphing Activity 2 Outcome: Examine equations in the form y mx b= + .

Graphing Activity 3 Outcome: Examine equations in the form y k= and x k= .

Graphing Activity 4 Outcome: Graph various inequalities and examine similarities and differences.

Graphing Activity 5 Outcome: Examine quadratic equations in the form y . ax= 2

Graphing Activity 6 Outcome: Examine quadratic equations in the form y a . x b= +2

Graphing Activity 7 Outcome: Examine quadratic equations in the form y x . b= +( )2

GRAPHICAL REPRESENTATION (p. 90) Commodities Market Investing Outcome: Use graphing in consumer applications.

SYSTEMS OF EQUATIONS (p. 91) Making Business Choices Outcome: Use systems of equations to make good business choices.



21

TRANSPOSING DIGITS Outcome: Verify that the difference between two numbers that have two digits transposed is divisible by nine. We have been told that the difference between two numbers that have two digits transposed is divisible by nine. Is this always true? Does it matter how many digits are in the number or if more than one pair of digits are transposed? Find the difference between the following numbers. To avoid using negative numbers, subtract the smaller number from the larger number. Then divide the difference by nine. 1. 58 and 85 2. 72 and 27 3. 36 and 63 4. 285 and 825 5. 285 and 258 6. 417 and 147 7. 417 and 471 8. 3842 nd 8342 9. 3842 and 3482 10. 3842 and 3824 11. 3842 and 8324 12. 13,574 and 31,574 13. 13,574 and 15,374 14. 13,574 and 13,754 15. 13,574 and 13,547 16. 13,574 and 31,754 15. 13,574 and 31.547 18. 13,574 and 15,347 Summarize your conclusions to the following questions: Is it always true that the difference between two numbers that have two digits transposed is divisible by nine? Does it matter how many digits are in the number? Does it matter if more than one pair of digits are transposed? Verify your conclusions with additional examples. Summarize what you learned in this activity:



22

FACTORING INTO PAIRS OF FACTORS Outcome 1: Find all possible pairs of factors of a given number. Numbers that are multiplied together are called factors. The result of the multiplication is called the product. In this activity we will limit ourselves to looking at pairs of factors, or two numbers that multiply together to give a particular product. Some numbers may have only one pair of factors while other numbers may have two, three, or more pairs of factors. We will list pairs of factors of a given number by following a pattern. Start with the number 1. Can it pair with another number to result in a product of the given number? Try 2. Try 3. Continue until you reach the given number. EXAMPLE 1. List all pairs of factors of 12.

12 1, 12 2, 6 3, 4 4, 3 6, 2 12, 1

Since multiplication is commutative, the pairs 3, 4 and 4, 3 are the same. Similarly 2, 6 and 6, 2 are the same and 1, 12 and 12, 1 are the same.

We can modify our procedure for listing all pairs of factors by replacing the statement, "Continue until you reach the given number;" with the following statement: Continue until the same pair of numbers appears but in the opposite order. For example, when listing the pairs of factors of 30, if one pair is 5 and 6 and another pair is 6 and 5, all remaining pairs of factors will also be duplicates of other pairs of factors that are already listed. Another way to determine when factors are beginning to repeat is when the first factor is larger than the second. This observation only applies when you begin with 1 and examine every number. 1. How many different pairs of factors are there for the number 12? List them. 2. State the divisibility rule that shows that 5 cannot pair with another whole number to give a product of 12. Look at Example 2.

EXAMPLE 2. List all pairs of factors of 210.

210 1, 210 2, 105 3, 70 5, 42 6, 35 7, 30

10, 21 14, 15

3. Using individual and combined divisibility rules, state a reason why each given number cannot pair



23

Factoring into Pairs of Factors, page 2

with another whole number to give a product of 210. (a) 4 (b) 8

(c) 9 (d) 11 (e) 12 (f) 13

(g) 16 4. How do you know that ALL the pairs of factors of 210 have been listed?

5. For each number listed below, list all pairs of factors of the number. 15 24 18 36 13 16

20 22 17 28 30 23

25 1 6 8 40 48

Summarize what you learned about factor pairs.



24


Prime and Composite Numbers Outcome 2: Distinguish between prime and composite numbers. A whole number greater than 1 whose only pair of factors is 1 and the number is said to be prime. All other numbers are said to be composite. The smallest prime is 2. Then, all multiples of 2 (4, 6, 8, …) are composite. Next, 3 is prime and multiples of 3 (6, 9, 12,…) are composite.

6. Circle each prime number in the following list of numbers. Then, remove all multiples of the prime

by putting a slash through the number.

46 47 48 49 50 52 54 56 57 58 59 60

11631

617691

21732

627792

31833

637893

41934

647994

52035

658095

6213651668196

72237

678297

8233853688398

92439

698499

102540557085

100

112641

7186

122742

7287

132843

7388

142944

7489

153045

7590

7. List the prime numbers that are less than 100.

8. List all pairs of factors for the following numbers.

100 125 132 230 248 143

144 121 195 350 278 203

Summarize what you have learned about prime and composite numbers.

When a number has several different pairs of factors, it is often desirable to find a particular pair that has a certain property. Questions 8 and 9 give some practice in this concept. 9. Looking at the lists of factors for the numbers in Exercise 5, find a pair of factors that meet each stated condition. (a) factors of 24 whose sum is 11 (b) factors of 36 whose difference is 5 (c) factors of 18 whose sum is 9 (d) factors of 28 whose difference is 12 (e) factors of 36 that are the same number 10. Looking at the lists of factors for the numbers in Exercise 8, find a pair of factors that meet the stated conditions. (a) factors of 132 whose difference is 1 (b) factors of 230 whose sum is 33 (c) factors of 350 whose sum is 39 (d) factors of 195 whose difference is 80 (e) factors of 144 that are the same number



25


Perfect Squares

Outcome 3: Find numbers that are perfect squares.

When a number has a pair of factors that are the same number, such as 6 × 6 = 36, this product is called a perfect square. 11. Circle each perfect square in the following list of numbers.

46 47 48 49 50 52 53 54 56 57 58 59 60

11631

617691

21732

627792

31833

637893

41934

647994

52035

658095

6213651668196

72237

678297

82338

688398

92439

698499

102540557085

100

112641

7186

122742

7287

132843

7388

142944

7489

153045

7590

12. For each circled number in #10, give the pair of like factors. Example:, 1 = 1 × 1, 4 = 2 × 2, etc.

13. Extend your list of perfect squares to include all perfect squares between 100 and 1000.

Square Roots

Outcome 4: Find the sqaure root of a number using a calculator.

A perfect square has a factor pair of identical factors. The number in the identical factors is the square root of the perfect square. Other numbers also have square roots, but they are not whole numbers. To find the approximate value of a number that is not a perfect square, use your calculator and the square root key .

In listing all pairs of factors of a given number, how can we be certain we have them all? We have been testing every whole number until we get a pair with the larger factor first or until a pair of factors is repeated but in the opposite order. Let’s see if we can find another procedure.

14. Use your calculator to find the square root of each number in Exercise 8.

15. For each number in Exercise 8, compare the factor pair that has the largest first factor with the square root of the original number.

16. Make a generalization about when to be sure you have found all factor pairs of a number.

Test your generalization from Exercise 16 with the following numbers. Find all pairs of factors of each number.

17. 85 18. 120 19. 136 20. 225 Summarize what you learned in this activity.



26

ADDING AND SUBTRACTING INTEGERS Outcome 1: Formulate rules for adding integers.

The numbers illustrated on the following number line are integers.

+ + +++ +

0 1 2 3 4 5+1

+2

+3

+4

+5

The set of integers includes the set of whole numbers, 0, 1, 2, 3, …, and the opposite of each nonzero whole number,−1, −2, −3, ….

To formulate the rules for adding integers we will use markers that represent one positive unit and markers that represent one negative unit.

+

1 positive unit 1 negative unit

+ and equal 0.

One positive unit and one negative unit combineto be zero.

For the purpose of formulating rules, consider addition to mean "put on."

EXAMPLE 1: Add +5 + −3. (Note: +5 + −3 and +5 + (−3) are the same.)

This means "put on" 5 positive units and "put on" 3 negative units.

+ + + + +0

0

0

Simplify your results by forming as many "zeros" as you can and removing all "zeros." If no zero can be formed or if all zeros have been removed, the remaining markers represent the simplified answer.

+5 + −3 = +2



27

Adding and Subtracting Integers, page 2

Use positive and negative markers to solve the following problems.

1. +3 + +1 2. +4 ( (+2) 3. +1 + 2 4. 3 + 4

5. Formulate a rule for adding two positive integers.

6. −3 + −1 7. −4 + (−2) 8. −1 + −2 9. −3 + (−4)

10. Formulate a rule for adding two negative integers.

11. Formulate a general rule for adding two integers with like signs.

12. +3 + −1 13. −3 + (+1) 13. 5 + (−3) 14. −5 + +3

16. Formulate a rule for adding integers with unlike signs.

17. Trade rules with your study partner and critique each other's work.

Outcome 2: Discover the relationship between subtracting integers and adding integers. To subtract means to "take off." Start with the number and type of markers indicated by the minuend (first number). If the type of marker that needs to be taken off is not showing, add (put on) enough zeros (1 plus marker and 1 minus marker) to complete the problem. EXAMPLE 2: 5 − (−3)

This means put on 5 positive markers and "take off" 3 negative markers. Since there are no negative markers showing , add 3 zeros.

+ + + + ++ + +

3 zeros

Now, "take off" 3 negative markers and count the remaining positive markers The result is + . 8



28


Use positive and negative markers to solve the following problems. 18. +3 − (+2) 19. −5 − (−4) 20. 4 − (−2) 21. −3 − (+1)

Compare the results of the following pairs of addition and subtraction problems.

22. +3 + (−2) +3 − (+2) 23. −5 + (+4) −5 − (−4) 24. 4 + (+2) 4 − (−2) 25. −3 + (−1) −3 − (+1) 26. Describe the relationship between addition and subtraction of integers.

Test your rules for adding and subtracting integers on the following problems.

27. 28. − + −3 5 − + +3 5 29. 3 5+ − 30. 3 5+ + 31. 32. − − −3 5 − − +3 5 33. 3 5− − 34. 3 5− + 35. 36. 3− 5 − −3 5 37. 3 5+ 38. 3 5−



29


EXTENSION: Can your rules be extended to include all signed numbers? Apply your rules to solve the following problems.

39. 40. − +283 156 315 614− 41. − −317 128

42. 43. 3 5 4 2. .− − +15 0 7. 44. − −34

12

Calculator: On many calculators, negative numbers are entered by (1) first entering the digits of

the number and then (2) making the number negative by using the "sign change" key + − .

On other calculators, the subtraction sign is used to enter negative numbers and the sign is entered in the natural order. Still others have a special negative key . (−)

−

EXAMPLE 3: 5 − (−3) =

Sign-change key:

Negative key:

+ −35 =

− 3

Display: 8.

Display: 8.(−)

−

= 45. Use a calculator to verify your answers to problems 39 - 44.

46. Trade rules with your study partner and critique each other's work. Summarize what you learned in this activity:



30

LOCATING COORDINATES ON A SPHERE Outcome: Find locations on s sphere using ordered pairs. Materials: World globe Two number lines can be placed together by aligning zero on each line and making the lines meet to form a square corner. These two number lines are often referred to as the rectangular coordinate system. The zero point where the two lines meet is called the origin. The horizontal line is called the x-axis and the vertical line is the y-axis.

A point in a rectangular coordinate system is located with two directional numbers. The first shows the amount of horizontal movement and the second shows the vertical movement. The point indicated by (−2, 3) means that you move two units from the origin to the left. Then, from that point, move three units up.

1. Locate the following points on a rectangular coordinate system.

A = (3, −1) B = (−2, −3) C = (−1, 5) D = (4, 2)



31

Locating Coordinates on a Sphere, page 2

On a world globe we have a spherical coordinate system for locating points. Using resources like an encyclopedia, a world atlas, or the Internet, develop a strategy for locating points on a world globe. 2. Locate the following reference points on a world globe.

Equator Prime Meridian International Date Line

3. How are these points similar to the x- and y- axes on a rectanglur coordinate system? Equator: Prime Meridian: International Date Line: 4. What point on a world globe is similar to the origin on a rectangular coordinate system? 5. Locate the following reference points on a world globe. North Pole South Pole 6. How are the directions North, South, East, and West used to identify locations on a world globe? North: South: East: West:

7. How do the four directions compare to the positive and negative directions on the x- and y-axes on a coordinate plane?

8. How is locating points on a sphere different from locating points on a coordinate plane? 9. Locate the following cities using the given coordinates. Juneau, Alaska: 58N/135W Rio De Janeiro, Brazil: 23S/43W London, United Kingdom: 51N/0 Phnom Penh, Cambodia: 12N/105E



32

Locating Coordinates on a Sphere, page 3

10. Write your strategy for locating a point on a globe. 11. Why is this skill useful in the world of business? Next, find familiar locations on a globe and write these coordinates. 12. Give the coordinates of the following cities.

Latitude Longitude Memphis ______ ______

Seattle ______ ______

Miami ______ ______

Tokyo ______ ______

Nairobi ______ ______

13. What are some resources available for finding the coordinates of global locations? 14. Look up the coordinates of five cities in a reference and locate the cities on a world globe.

City, Country Coordinates City 1: ______ ______

City 2: ______ ______

City 3: ______ ______

City 4: ______ ______

City 5: ______ ______

Summarize what you learned in this activity:



33

ARRANGING GLOBAL INTERACTIVE COMMUNICATIONS Outcome: To determine an optimum time for a global interactive communication. Materials: World globe, world time zone chart An interactive teleconference must be arranged for five directors at FedEx. Each director works a normal 8:00 AM - 5:00 PM day. Select a time for the teleconference that fits within everyone’s normal working hours, if possible. If not, select a time that will be most convenient for the largest number of directors. Location of Directors: Memphis, TN Toronto, Canada Frankfurt, Germany Tokyo, Japan San Juan, Puerto Rico Describe your plan for accomplishing this task. Time for Teleconference for Each Director: Memphis, TN: ____________ Toronto, Canada: ____________ Frankfurt, Germany: ____________ Tokyo, Japan: ____________ San Juan, Puerto Rico: ____________ Comment on any inconveniences this teleconference might cause. Summarize what you learned in this activity:



34

PREPARING A REFERENCE CHART FOR GLOBAL COMMUNICATIONS Outcome: To gather and organize data to improve efficiency. Materials: World globe, world time zone chart Businesses that operate throughout the world have to be constantly aware of time differences when communicating with personnel at other locations. An employee at FedEx has been asked to devise a reference guide that other employees may use to quickly determine the time in another part of the world. The guide will give the number of hours to add or subtract from the local time to determine the time at each listed city. The cities that are to be referenced are given below. Design a matrix that organizes this information. Memphis, TN Spokane, WA Anchorage, AK Brusslels, Belguim Lima, Peru Maui, HI Sumatra, Indonesia Manila, Phillipines Ho Chi Minh City, Vietnam Kuala Lumpur, Malaysia

T O FROM

Can the strategy used in developing this ten-city guide be used to develop a guide that includes more cities? Explain. Summarize what you learned in this activity:



35

MULTIPLYING AND DIVIDING INTEGERS Outcome 1: Formulate rules for multiplying integers. In using markers to illustrate multiplication, interpret the first sign as "put on" for positive and "take off" for negative. The number following the sign tells how many GROUPS you "put on" or "take off." The second number tells HOW MANY and WHAT TYPE markers are in each group.

EXAMPLE 1: 5(−2)

This means "put on" 5 groups of markers with each group containing two negative markers.

The result is − . 10

EXAMPLE 2: −3(−2)

This means "take off" 3 groups of markers with each group containing two negative markers. You cannot "take off" markers that are not there. The only way to have markers to take off is to put on enough zeros to have the markers you need to take off. Put on zeros.

+ + ++ + +

Now, "take off" 3 groups of − . 2

+ + ++ + +

The result is +6.

Use positive and negative markers to solve the following problems. 1. 5(2) 2. −5(−2) 3. 4(3) 4. −4(−3)



36

Multiplying and Dividing Integers, page 2

5. Formulate a rule for multiplying integers with like signs. 6. 5(−2) 7. −5(2) 8. 4(−3) 9. −4(3) 10. Formulate a rule for multiplying integers with unlike signs. Use your rules to solve the following problems. 11. 2(−5) 12. −6(−2) 13. −4(5) 14. 3(−4) EXTENSION: Can your rules be extended to include all signed numbers? Apply your rules to solve the following

problems. 15. −283(156) 16. 315(−614) 17. −317(−128) 18. −118(−153) 19. 3.5(−4.2) 20. −15(0.7)

21.

−−21

43

Calculator: On some calculators, the times sign for multiplication must be entered, even if

parentheses are used. On other calculators the times sign can be used OR the parentheses, but both are not necessary.

EXAMPLE 3: −5(−3) =

Some calculators:

Other calculators: EXE−

+ −

3

5 =

−

5 3

Display: 15.

−

×

−

3

5

+ −

× EXE

Display: 15.

Display: 15.

( )

22. Use a calculator to verify your answers to problems 15 - 21. EXTENSION: How are your rules affected when you multiply more than two factors? 23. (−2)(−3)(−4) 24. (2)(−3)(−4) 25. (−2)(3)(−4) 26. (−2)(−3)(4) 27. Trade rules with your study partner and critique each other's work. Summarize what you learned about multiplying integers.



37


Outcome 2: Formulate rules for dividing integers. To use markers to illustrate division, the sign of the dividend tells you to "put on" or "take off" an indicated number of markers. The sign of the divisor tells what type of marker to "put on" or "take off." The number indicated by the divisor tells how to group the remaining markers. The answer is the NUMBER OF GROUPS remaining and the TYPE OF MARKERS in the groups. EXAMPLE 4: 8 2 ÷

This means to "put on" 8 positive markers. Then, group the markers in groups of 2. How many groups are there and what kind of markers are in the group?

+ + ++ + + + +

1 2 3 4 The result is + 4

EXAMPLE 5: − ÷ 8 2

This means to "take off" 8 positive markers. Before you can "take off" markers, you must "put on" enough zeros.

+ + ++ + + + +8 zeros

Now, "take off" 8 positive markers. Next, group the remaining markers in groups of 2. How many groups are there and what type of markers are in the groups?

1 2 3 4 The result is − . 4

EXAMPLE 6: −8 ÷ −2

This means to "take off" 8 negative markers. Before you can "take off" markers, you must "put on" enough zeros.

+ + ++ + + + +8 zeros

Now, "take off" 8 negative markers. Next, group the remaining markers in groups of 2. How many groups are there and what type of markers are in the groups?

+ + ++ + + + +

1 2 3 4 The result is + . 4



38


Use positive and negative markers to solve the following problems. 1. 2. 6 3÷ 10 5÷ 3. − ÷ −6 3 4. − ÷ −10 5 5. Formulate a rule for dividing integers with like signs. 6. 7. 6 3÷ 10 5÷ 8. − ÷ −6 3 9. − ÷ −10 5 10. Formulate a rule for dividing integers with unlike signs. Use your rules to solve the following problems. 11. 12. 15 3÷ − − ÷9 3 13. − ÷ −8 4 14. − ÷ −6 2 EXTENSION: Can your rules be extended to include all signed numbers? Apply your rules to solve the following

problems.

16. 17. − ÷630 35 1035 23÷ − 18. − ÷ −4 2 0 5. .

19. − ÷ −34

12

Calculator: On most calculators, the "divided by" key is the sign to use for division. The dividend is ÷

entered first. The display may show the “divided by” symbol ÷ or a slash /.

EXAMPLE 7: − ÷ −15 3

Sign change key:

Negative key:

+ −

3

5 =

15

Display: 5.3 + −

Display: 5.

÷

÷(−) − =

20. Use a calculator to verify your answers to problems 16-19.

21. Trade rules with your study partner and critique each other's work. Summarize what you learned in this activity:



39

COMMON FRACTIONS Outcome: Distinguish between proper and improper common fractions. Complete the following chart by writing the common fraction in each block with the numerator given at the top of the column and with the denominator given at the left of each row. Do not simplify any of the fractions. This chart will be used as a guide for completing charts that follow.

Numerator Denominator

0

1

2

3

4

5

6

7

8

9

0 00

10

20

1 01

11

2 02

3

4

5

6

7

8

9

Each fraction on this chart is called a common fraction. When the numerator is smaller that the denominator the common fraction is called a proper fraction. When the numerator is the same as or larger than the denominator the common fraction is called an improper fraction. 1. Lightly shade the portion of the chart containing improper fractions. 2. Describe in words the portion of the chart that was shaded. Summarize what you learned in this activity:



40

DECIMAL FRACTIONS Outcome: Examine decimal equivalents of common fractions. Complete the following chart by using the common fractions chart and writing each common fraction in the equivalent decimal form. Use your calculator to change each fraction to the equivalent decimal form. To change a common fraction to an equivalent decimal fraction, divide the numerator by the denominator. Write exactly what is displayed in your calculator window in each appropriate block. Numerator Denominator

0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

1. Describe your entries in the first row on the chart. What does this mean? Why? 2. Describe your entries in the first column on the chart. What does this mean? Why? 3. Describe the location of common fractions whose decimal equivalent is 1. When is a common fraction equivalent to 1? 4. Describe the location of common fractions whose decimal equivalent is less than 1. When is a common fraction less than 1?

5. Describe the location of common fractions whose decimal equivalent is greater than 1. When is a common fraction greater than 1?

6. List any common fractions that have the same decimal equivalent.

For example, 1 0 5= , 2

. 2 , 4

0 5= . 36

0 5= . , 4 . 8

0 5= .



41

FRACTIONS IN SIMPLEST FORM Outcome: Examine equivalent fractions in simplest form. Complete the following chart by using the common fractions chart and writing each common fraction in the simplest form. Proper fractions should be reduced to lowest terms by dividing the numerator and denominator by the same amount. Improper fractions should be changed to an equivalent whole or mixed number by dividing the numerator by the denominator. Numerator Denominator

0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

1. Describe your entries in the first row on the chart. What does this mean? Why? 2. Describe your entries in the first column on the chart. What does this mean? Why? 3. Describe the location of common fractions equivalent to 1. When is a fraction equivalent to 1? 4. Describe the location of all proper fractions in simplest form. 5. Describe the location of all improper fractions in simplest form. 6. How can you tell if a common fraction is a proper fraction or an improper fraction? 7. How can you tell if an improper fraction in simplest form will be a whole number or a mixed number? 8. List any common fractions that are equivalent to the same fraction in simplest form. For example,

24

12

= , 3 , 6

12

=4 . 8

12

=



42

FRACTION RELATIONSHIPS

Outcome: Develop a procedure for comparing fractions to 1, 12

, or 14

by inspection.

Complete this activity by using a calculator to determine the decimal equivalent of a fraction. To find the decimal equivalent, divide the numerator by the denominator.

1. Circle the fractions from the list below that have a value less than 1.

2

3 7

8 3

2 4

5 8

5 3

3 9

10 12

6 9

8 4

9 6

6 8

2

0

8 5

1 6

7 15

16 9

6 4

9 2

1 15

4 7

3 2

5 7

10 6

5

2. Circle the fractions from the list below that have a value greater than 1. 1

4 5

6 4

5 5

4 8

8 7

3 7

15 11

12 3

8 9

4 1

6 1

2

9

3 11

12 4

9 3

2 4

2 8

11 14

6 5

5 8

2 5

7 7

5 9

8

3. Write a rule for deciding when a fraction is less than 1.

4. Write a rule for deciding when a fraction is greater than 1.

5. Write a rule for deciding when a fraction is equal to 1.

6. Classify the following fractions as less than 1, greater than 1, or equal to 1, according to your rules in #3, #4, and #5. Check your answer using a calculator. 34

28 23

25 85

90 48

12

6372

4747

4550

2523



43

Fraction Relationships, page 3

The decimal equivalent of 12

0 5= . . Complete the following using a calculator to examine the decimal equivalent of

each fraction.

7. Circle the fractions from the list below that have a value less than . 12

34

13

58

715

2540

1823

916

45

25

35

712

5

12 4

9 6

10 6

7 5

7 3

12 9

20 8

16 5

11 19

40 23

42

8. Circle the fractions from the list below that have a value greater than 12

.

35

14

56

710

2545

1533

1120

25

815

5

13 11

20 7

9 5

16 2

7 2

9 5

12 6

12 18

35

9. Circle the fractions from the list below that have a value equal to 12

.

48

35

1224

1525

1015

1428

1325

10. Write a rule for deciding when a fraction is less than 12

.

11. Write a rule for deciding when a fraction is greater than 12

.

12. Write a rule for deciding when a fraction is equal to 12

.

13. Classify the following fractions as less than 12

, greater than 12

, or equal to 12

according to the

rules written in #10, #11, and #12. 17

30 25

60 31

55 23

46

24

50 17

35 19

38 33

64

14. Check your answers in #13 by using a calculator. 15. Did the rules written in #10, #11, and #12 work? If not, modify your rules and try #13 again.



44


The decimal equivalent of 14

0 25= . . Complete the following using a calculator to examine the decimal equivalent of

each fraction.

16. Circle the fractions from the list below that have a value less than 14

.

415

520

310

624

725

630

1140

1250

17. Circle the fractions from the list below that have a value greater than 14

.

4

12 5

18 3

10 6

25 7

28 8

3 11

50 12

40

18. Circle the fractions from the list below that have a value equal to 14

.

3

15 6

24 2

9 9

36 6

20 9

40 11

44 12

48

19. Write a rule for deciding when a fraction is less than 1 . 4

20. Write a rule for deciding when a fraction is greater than 14

.

21. Write a rule for deciding when a fraction is exactly 14

.

22. Classify the following fractions as less than 14

, greater than 14

, or equal to 14

according to the

rules written in #19, #20, and #21. 17

30 25

60 31

55 23

46

24

50 17

35 19

38 33

64

23. Check your answer in #22 by using a calculator.

24. Did the rules written in #19, #20, and #21 work? If not, modify your rules and try #22 again.



45


25. Write five different fractions for each of the following categories. (a) less than 1 (b) greater than 1

(c) equal to 1 (d) less than 1 2

(e) greater than 12

(f) equal to 12

(g) less than 14

(h) greater than 14

, but less than 21

(i) equal to 14




46

SIZE OF FRACTIONS

Outcome: Categorize fractions based upon their relationship to 14

, 12

, 34

, and 1.

1 Put each fraction from your common fractions chart in the proper box. Do not simplify any of the

fractions. If you need help deciding in which box a fraction belongs, use your calculator or your decimal fractions chart. When you finish, there should be a total of 100 fractions on the chart.

Undefined Exactly

0 Greater than 0,

but less than 14

(0.25)

Exactly 14

(0.25) Greater than 1

4 (0.25)

but less than 12

(0.5)

Exactly 12


2(0.5),

but less than 34

(0.75)

Exactly 34


4(0.75),

but less than 1

Exactly 1 Greater than 1



47

Size of Fractions, page 2

List the similarities of the fractions contained in each box. Make a rule for determining when any fraction belongs in each box.

2. Undefined -

3. Exactly 0 -

4. Greater than 0, but less than 14

-

5. Exactly 1 - 4

6. Greater than 1 , but less than 4

1 - 2

7. Exactly 1 - 2

8. Greater than 1 , but less than 2

34

-

9. Exactly 3 - 4

10. Greater than 3 , but less than 1 - 4

11. Exactly 1 -

12. Greater than 1 -



48

Size of Fractions, page 3

14. Put each of the following fractions below in the proper box by using your own rules from #13. DO NOT use a calculator. 10

15 4

12 7

12 15

16 18

23 4

15 31

50 35

70 3

10 12

16

14

10 9

15 13

26 9

10 15

12 12

12 8

16 20

30 2

12 0

14

13

26 2

8 5

12 1

14 20

25 18

24 9

8 7

14 7

28 14

28

21

28 9

11 11

9 0

21 21

18 21

24 27

27 25

100 17

34 23

87

Undefined Exactly

0 Greater than 0,

but less than 4

1 (0.25) Exactly

4

1 (0.25) Greater than

4

1 , (0.25)

but less than 21 (0.25)

Exactly

2

1 (0.5) Greater than

2

1 (0.5),

but less than 4

3 (0.75)

Exactly

4

3 (0.75) Greater than (0.75),

but less than 1

Exactly 1 Greater than 1

15. Check the fractions in each box using a calculator and examining the decimal equivalents of each fraction. 16. If your rules did not work in all cases, modify yours rules. Check with your classmates to see the similarities and differences in your rules. Summarize what you learned in this activity:



49

ANALYZING NUTRITIONAL LABELS Outcome: Use proportions to analyze nutrition labels. In 1990 Congress passed the Nutrition Labeling and Education Act. This act requires that nutritional labels following the new standards should appear on all processed food products. Nutrition information for fresh produce, meat, poultry, and fish will appear in the areas of the store where they are displayed. To interpret the information given on the label it is important to determine the base for the percents given on the label. Regulations also require that standard daily nutritional value be used as the basis for all labels. The column heading “% Daily Value” has an asterisk that guides us to a footnote on the label that reads, “Percent Daily Values are based on a 2,000 calorie diet. Your daily values may be higher or lower depending on your calorie needs.” The label also includes the recommended daily intakes that are used as a basis for a 2,000 and 2,500 calorie diet. Nutrition Facts Serving Size 1/2 cup (114g) Servings per Container 4 Amount Per Serving Calories 260 Calories from Fat 120

Total Fat 13g Saturated Fat 5g Cholesterol 30mg Sodium 660 mg Total Carbohydrate 31g Dietary Fiber 0g Sugar 5g Protein 5g

% Daily Value*20%25%10%28%11%

0%

Vitamin A 4% Calcium 15%

Vitamin C 2% Iron 4%

*Percent Daily Values are based on a 2,000 calorie diet. Your daily values may be higher or lower depending on your calorie needs: Calories 2,000 2,500 Total Fat Sat Fat Cholesterol Sodium Total Carbohydrate Dietary Fiber

Less than Less than Less than Less than

65 g 20 g 300 mg 2,400 mg 300 g 25 g

80 g 25 g 300 mg 2,400 mg 375 g 30 g



50

Analyzing Nutritional Labels, page 2

Are the recommended daily intakes proportional to the daily calories? First examine the recommended daily intake values for Total Fat. If the values are proportional,

calories 2,500calories 2,000

g 80g 65=

Before we continue we need to realize that the recommended values have been rounded and if cross products are reasonably close, we will say the relationships are proportional. To verify that the Total Fat recommended is or is not proportional to the number of calories per day, check the cross products.

65 × 2,500 = 2,000 × 80

162,500 = 160,000

If we round the values to the nearest ten-thousand, we can say that the recommended daily intake of Total Fat is proportional. If Total Fat is proportional to Calories, than we can determine the recommended Total Fat intake for diets based on any number of calories. EXAMPLE 1 Find the recommended intake value of Total Fat for a 1,600 calorie diet. Round value to the nearest

multiple of five. x Find cross products. 21

=

Multiply 1,600 × 66. 2 = Divide by 2,000.

140=x

g 52000,2000,000,140000,2

66600,1000,000,

g 65600,

=

=×

x

xx

Rounded to the nearest five grams, the recommended Total Fat for a 1,600 calorie diet is 50 grams.

1. Determine if the following recommended daily intake values are proportional (based on rounded amounts). Show calculations to justify your answers.

Saturated Fat ____________ Cholesterol ____________

Sodium ____________ Total Carbohydrate ____________

Dietary Fiber ____________



51


2. Find the recommended daily intake values for a 1,600 calorie diet for the other recommended values that are proportional.

3. Find the recommended daily intake values for a 1,200 calorie diet for the recommended values that are proportional.

4. Find the recommended daily intake values for a 1,000 calorie diet for the recommended values that are proportional.

5. Is the footnote giving the daily recommended daily intake value necessary? Could those values have been determined by other information given on the label? Justify your response with a specific example.

Calories from Fat Another important nutritional fact is the percent of calories that are derived from fat. The label gives both the total calories for a defined serving and the calories from fat. The percent can be calculated from these values. It is recommended that less than 30% of the calories that we consume in a day come from fat. In examining our sample label, we see that 120 calories of the 260 calories are from fat. What percent is this?

%15.46260

000,12000,12260

260120

100

=

=

=

=

R

R

R

R

This product has over 45% of its calories from fat. The product label also gives the number of calories per gram for fat, carbohydrates, and protein. Check the value given for the fat calories per gram to see if it is consistent with the other information given. 9 calories per gram × 13 g = 117 calories from fat Again, label values are generally rounded.



52


6. Find the number of calories from carbohydrates. 7. Find the number of calories from protein. 8. Discuss why label values use liberal rounding conventions. In what situations would a more

conservative rounding approach be advisable? Analyze two actual nutrition labels. 9. Label 1: Product description __________________________

Serving size: __________ Percent of calories from fat __________ Total calories from carbohydrates __________ Total Calories from protein __________ 10. Label 2: Product description __________________________

Serving size: __________ Percent of calories from fat __________ Total calories from carbohydrates __________ Total Calories from protein __________




53


Analyzing Nutrition Labels Out-of-Class Assignment Name_________________________ 1. Plan a 500 calorie meal and give the following details. Turn in labels or copies of labels that were used.

Menu

Serving Size

Total Calories

Fat Calories

Carbohydrate Calories

Protein Calories

Menu Totals

2. Select a favorite recipe and from nutrition labels complete the following table. Turn in labels or copies of

labels that were used. Recipe for: _______________________

Ingredient

Amount

Total Calories

Fat Calories

Carbohydrate Calories

Protein Calories

Recipe Totals 3. In the recipe you selected, define a serving and calculate the total milligrams and the percent of the

recommended daily intake of sodium and cholesterol per serving. Serving:

Sodium: _____mg _____% of recommended daily intake Cholesterol: _____mg _____% of recommended daily intake



54

WHAT PERCENT TAX DO I REALLY PAY? Outcome 1: Determine the percent of tax withheld for a given taxable income. True or false? As your income increases and you move to a higher tax bracket, your total taxable income is taxed at the higher rate. Before you answer that question, examine the withholding tax for the income in the table below based on a recent IRS Table for Percentage Method of Withholding for a single taxpayer paid monthly (IRS Publication 15, Circular E).

Calculate the withholding tax for each amount of monthly income subject to withholding. Then, find the percent that the withholding amount is of the income subject to withholding for one month. Finally, assuming that the monthly income subject to withholding is the same for each month of a year, find the annual income and the annual withholding. Monthly Income Subject to

Withholding

Withholding

Percent of Income

Annual Income

Annual Withholding 1. $221 2. $250 3. $750 4. $1,500 5. $1,750 6. $2,242 7. $3,000 8. $3,500 9. $4,000 10. $4,788 11. $5,000 12. $7,000 13. $9,000 14. $10.804 15. $12,000



55

What Percent Tax do I Really Pay?, page 2

16. $15,000 17. $23,333 18. $39,000 19. $50,000 20. $100,000 Now, true or false? As your income increases and you move to a higher tax bracket, your total taxable income is taxed at a higher rate. Explain why you think your answer is correct. Outcome 2: Determine the percent of income tax for a given annual taxable income. At the end of a year taxpayers are to pay income tax owed for the year based on the total taxable income for the year. The Tax Schedule X for single taxpayers for a recent year is provided from an IRS publication.

Calculate the income tax owed for the year for the following monthly incomes subject to be taxed. Then, find the percent the income tax is of the taxable income. Monthly Income Subject to

Withholding Annual Taxable

Income Amount of Income Tax

Percent of Tax

1. $221 2. $250 3. $750 4. $1,500 5. $1,750 6. $2,242 7. $3,000 8. $3,500 9. $4,000 10. $4,788



56

What Percent Tax do I Really Pay?, page 3

Compare the tax withheld for a year with the tax owed for the year. Summarize your findings. Suppose two persons both had an annual taxable income of $18,000. One person had a seasonal variation in the wages earned and earned $750 in taxable income a month for three months and $1,750 in taxable income a month for nine months. Calculate the amount of tax withheld during the year. The second person earned $1,500 in taxable income each month and the annual taxable income, the annual withholding, and the annual taxes owed have been found in exercise 4 on each table. Compare the withholding and tax owed for both persons. In your own words, explain the difference between the tax withheld from taxable income on a regular basis with the annual income tax owed for the year. Summarize what you learned in this activity:



57

ESTIMATING MEASURES Outcome: Estimate linear and circular measure in inches. Materials: Empty tennis ball can, tennis ball, tape measure. 1. Examine the tennis ball can and estimate in inches the following measures. Height: ________ Circumference: ________

2. Use a tape measure to find a close approximation in inches for the measure of the tennis can. Height: ________ Circumference: ________

3. Compare your estimates with the measured results. Were the results as you expected? Explain. 4. Measure the diameter of a tennis ball. Diameter of tennis ball: ________ 5. Explain the procedure you used to find the diameter. 6. Relate the height of the tennis can to the diameter of the tennis ball. 7. Relate the circumference of the tennis can to the diameter of the tennis ball. 8. Write a rule or procedure for estimating the circumference of a circle when the diameter is known. 9. Write a rule or procedure for estimating the diameter of a circle when the circumference is known.



58

Estimating Measures, page 2

Estimating Measures Out-of-Class Assignment Name_________________________ Find five circular objects that the circumference can be easily measured with a tape measure. Trace the circumference of each object on paper. Make the traced circle dark enough to be seen from the back side of the paper or cut out the circle. Fold each circle in half. Measure and record the diameter of each object. Estimate the circumference of each object, measure each circumference, and compare the measured results with your estimates. Description Measured Estimated Measured of Object Diameter Circumference Circumference Object 1: ______ ______ ______ ______ Object 2: ______ ______ ______ ______ Object 3: ______ ______ ______ ______ Object 4: ______ ______ ______ ______ Object 5: ______ ______ ______ ______ Find five different circular objects: Measure and record the circumference of each object and estimate each diameter. Measure each diameter and compare the results with your estimates.

Description Measured Estimated Measured of Object Circumference Diameter Diameter Object 1: ______ ______ ______ ______ Object 2: ______ ______ ______ ______ Object 3: ______ ______ ______ ______ Object 4: ______ ______ ______ ______ Object 5: ______ ______ ______ ______ Summarize what you learned in this activity:



59

WHAT IS PI, π? Objective: Discover the relationship between the circumference and diameter of a circle. 1. Find the calculator value of π .

EXEsome calculators: other calculators: ππ

(If the label is above a key, asπ

, access the π function by first pressing a shift, inverse, or 2nd

function key.) 2. What is π? (a) Find 10 circular objects of different sizes and trace the circles onto a sheet of paper. (b) Use a flexible rule, like a tape measure, to measure the circumference (C) of each circular object. If no flexible rule is available, use string and a rigid rule. (c) Cut out the 10 circles of different sizes. (d) Fold the cut-out circles in half and measure the diameter (d).

(e) Calculate Cd

. Use decimal equivalents rounded to the nearest thousandth and record your

results on the chart in Exercise 3.

(f) Find the difference between π and the calculated value, Cd

. (Use the calculator value of π).

error = −π Cd

(g) Find the percent of error (rounded to the nearest tenth of a percent).

π

error ofamount 100

error ofpercent =

3. Complete the chart with the results from Exercise 2.

Circular Object Number 1 2 3 4 5 6 7 8 9 10

b. C d. d

e. Cd

f. error g. % error

4. Use your findings to argue that the following statement is true or false: The quotient of the circumference and diameter of any size circle is a fixed value or π. 5. What is a reasonable percent error for this investigation? Check the measurements and calculations for measurements with an unreasonable percent error. Summarize what you learned in this activity:



60

CIRCLE, BAR, AND LINE GRAPHS Outcome 1: Interpret data from various types of graphs. 1. Circle Graphs (a) Find three examples of circle graphs in the newspaper or periodical articles. Tape, glue, or staple newspaper articles or a copy to a standard sheet of paper (one article per sheet). Give the name of the newspaper or periodical and the date. (b) Briefly explain what the graph shows. (c) Make up three problems that use the information on the graph to find other (not given) information. (d) Solve the problems from part c. 2. Bar Graphs (a) Find three examples of circle graphs in the newspaper or periodical articles. Tape, glue, or staple newspaper articles or a copy to a standard sheet of paper (one article per sheet). Give the name of the newspaper or periodical and the date. (b) Briefly explain what the graph shows. (c) Make up three problems that use the information on the graph to find other (not given) information. (d) Solve the problems from part c. 3. Line Graphs (a) Find three examples of circle graphs in the newspaper or periodical articles. Tape, glue, or staple newspaper articles or a copy to a standard sheet of paper (one article per sheet). Give the name of the newspaper or periodical and the date. (b) Briefly explain what the graph shows. (c) Make up three problems that use the information on the graph to find other (not given) information. (d) Solve the problems from part c. Outcome 2: Plan, design, and conduct an investigation in which the findings are reported in writing and graphically.

4. Design a project and gather appropriate data. Make a report of your findings and display data graphically. Use as many different types of graphs as practical.




61

CRITI QUING GRAPHS Outcome 1: Critiquing graphs found in recent publications. Collect a variety of graphs from several sources (newspapers, magazines, internet, etc.). Select two graphs to critique: one you evaluate to be a good graphical representation of the information presented and the other to be a poor graphical representation of the information presented. Answer the following questions about your selected graphs. Sample of good graphical representation (Attach graph or copy of graph.)

Sample of poor graphical representation. (Attach graph or copy of graph.)

1. What type of graph is used to display the information?

2. Briefly describe the purpose of the graph. 3. Do the titles and notes explain the data

satisfactorily? Explain why or why not. 4. What additional information would have been

helpful? 5. What suggestions would you make for a better

presentation of this data? 6. What questions would you ask the researcher about

how the data were collected?

1. What type of graph is used to display the information?

2. Briefly describe the purpose of the graph. 3. Do the titles and notes explain the data

satisfactorily? Explain why or why not. 4. What additional information would have been

helpful? 5. What suggestions would you make for a better

presentation of this data? 6. What questions would you ask the researcher about

how the data were collected? Summarize what you learned in this activity:



62

ENGINEERING NOTATION Outcome: Change a number from ordinary notation to engineering notation. In Chapter 5 we examined some metric prefixes for very large and very small units of measure. In Chapter 10 we examined power-of-10 exponents and scientific notation. We will now examine power-of-10 exponents that are multiples of 3.

Metric Prefixes Prefix Abbreviation Meaning Power-of-10 Yotta- Zetta- Exa- Peta- Tera- Giga- Mega- kilo

Y X E P T G M k

one septillion times one sextillion times one quintillion times one quadrillion times one trillion times one billion times one million times one thousand times

2410× 2110× 1810× 1510×

1210× 910× 610× 310×

Standard Unit × 010milli- micro- nano- pico- femto- atto- zepto- yocto-

m µ n p f a z y

one thousandth of one million times one billionth of one trillionth of one quadrillionth of one quintillionth of one sextillionth of one septillionth of

310−× 610−×

910−× 1210−×

1510−× 1810−× 2110−×

2410−×

These prefixes correlate with a modification of scientific notation called engineering notation. Engineering notation, like scientific notation, has a first factor or coefficient and a power-of-10 factor. The first factor is greater than 1 and less than 1,000. That is, the first factor has one, two, or three digits to the left of the decimal. The power-of-10 factor has an exponent that is a multiple of 3. To change an ordinary number to engineering notation: 1. Indicate with a caret (^) where the decimal should be positioned. a. If there are 1, 2, or 3 digits to the left of the decimal, the decimal will not shift and the power-of-10 factor

will be 10 . 0

b. If there are more than three digits to the left of the decimal point, insert commas as appropriate to separate the place-value periods, and place the caret (the new position of the decimal) at the leftmost comma.

c. If there are no nonzero digits to the left of the decimal, count from the decimal to the right in groups of three places until you have at least one, but no more than three, nonzero digits to the left of the caret (new position of the decimal).



63


2. Determine the exponent of the power-of-10 factor and its sign by counting the number of places from the new

position of the decimal to the old position. The result will be a multiple of 3. Example Change to engineering notation: (a) 2,400,000 (b) 2,400 (c) 24 (d) 0.24 (e) 0.024 (f) 0.0000024 (a) 2,400,000 Place the caret at the leftmost comma. 000,4002∧ 2.6 × 106 N → O = +6 (b) 2,400 Place the caret at the leftmost comma 4002∧ 2.4 × 10 N → O = +3 (c) 24 Place the caret at the decimal point. The ∧24 power-of-10 exponent can only be 0 or a multiple of 3. 24 × 106 N → O = 0 (d) 2,400,000 Place the caret after a place value with an exponent

that is a multiple of three. There can be no more than 3 nonzero digits on the left of the caret.

∧240.0

240 × 103 N → O = −3 (e) 2,400,000 Place the caret after a place value with an exponent


^0.024

24 × 10−3 N → O = −3 (f) 2,400,000 Place the caret after a place value with an exponent


4∧000002.0

2.4 × 10−6 N → O = −6 Measures that are expressed in metric units can be written in engineering notation by showing the

power-of-ten factor or by translating the power-of-10 factor to the appropriate metric prefix. Example Write the engineering notation using metric prefixes. (a) 4,382,000 Σ (b) 0.00001 ∆ (c) 5,870 Wµ (d) 3,500 MHz (a) 4,382,000 Σ = 4.382 × 106 Ω 106 = M 4.382 M= Ω (b) 0.00001 ∆ = 10 × 10−6 ∆ 10−6 = µ = 1- µ∆



64


(c) 5,870µW = 5.87 × 103 µW` µ = 10−6 = 5.87 × 103 × 10−6 W Combine power-of-10 factors. = 5.87 × 10−3 W 10−3 = m = 5.87 mW (d) 3,500 MHz = 3.5 × 103 MHz M = 106 = 3.5 × 103 × 106 Hz Combine power-of-10 factors. = 3.5 × 109 Hz 109 = G = 3.5 GHz Exercises Write each number in engineering notation. 1. 3,800,000 2. 5,600 3. 78 4. 52,000 5. 80,000,000 6. 5,830,000 7. 1,736,500,000 8. 41,980,000 9. 0.78 10. 0.33 11. 0.0000011 12. 0.000000008 13. 0.0009832 14. 0.0000719 15. 0.0120307 16. 0.000675 17. 0.00000092 18. 0.004 19. 8,400,000 20. 5,900 21. 41 22. 31,000 23. 17,000,000 24. 129,000,000 25. 3,084,000,000 26. 0.0982 27. 0.0000018 28. 0.000989 29. 0.007 30. 0.12 31. 0.00000000035 32. 5,200,000,000 Change each unit to engineering notation using metric prefixes. 33. 428,000 Σ 34. 5,700,000 V 35. 3,520,000,000 W 36. 79,000,000 Hz 37. 0.000081 s 38. 0.0973 ∆ 39. 0.00000000541 s 40. 0.89 s 41. 0.00058 MΣ 42. 0.00077 sµ 43. 2,980,000 ps 44. 7,810,000 mW 45. 42,300 kV 46. 1,572,000 kW 47. 5,096,000 Σ 48. 0.000000008 ∆ 49. 182,000 Hz 50. 1,600 Σ 51. 5,200,000 V 52. 97,000,000 W 53. 0.00049 s 54. 0.00052 ∆ 55. 0.588 ∆ 56. 10.53 s 57. 246.7 V 58. 5,082 W 59. 42,000 mW 60. 7,800 kΣ 61. 5,729 Wµ 62. 25,000 ps 63. 4,800 GHz 64. 4,000 ns 65. 0.000001 Ms 66. 0.0047 GΣ 67. 0.0000829 W 68. 0.000000023 fs Summarize what you learned in this activity:



65

ESTIMATING RADICALS Squares and Square Roots Outcome 1: Distinguish between squares and square roots and rational and irrational numbers. 1. Complete the following table of numbers and their squares.

n 1 2 3 4 5 6 7 8 9 10 n2

n 11 12 13 14 15 16 17 18 19 20 n2

Table 1 The numbers represented by n are called perfect squares. The principal square root of each 2

n2 is n. 2. Using your calculator, complete the following table, rounding answers to the nearest thousandth. some calculators: n other calculators: n =

n 1 2 3 4 5 6 7 8 9 10

n

n 11 12 13 14 15 16 17 18 19 20 n

n 21 22 23 24 25 n

Table 2 The square root of a perfect square is a rational number. The square root of a whole number that is not a perfect square is an irrational number. The decimal equivalents of irrational numbers will never terminate or repeat. We normally use approximations that are found by rounding. 3. Using the answers in Table 2, position the principal square roots of each value of n on the

following number line.

21



66

Squares and Square Roots, page 2

Outcome 2: Determine between which two whole numbers an irrational number will fall.

1 1 4 2= =, 2 and 3 are between 1 and 4. 2 and 3 are between 1 and 4 or between 1 and 2.

4 2 9= =, 3 5, 6, and 7 are between 4 and 9. 5 , 6 , and 7 are between 4 and 9 or between 2 and 3.

9 3 16 4= , = 10, 11, 12, 13, 14, and 15 are between 1 and 16. 10 , 11, 12 , 13 , 14 , and 15 are between 1 and 16 or between 4 and 5. 4. Using the table and the fact that n2 n= , decide between which two whole numbers the principal square roots of the following numbers will fall. EXAMPLE: Since 30 is between 25 and 36, 30 is between 25 and 36 or between 5 and 6. (a) 40 (b) 46 (c) 105 (d) 380 (e) 250 (f) 300 (g) 150 (h) 214 (i) 375 (j) 180 Summarize what you learned in this activity:



67

RATIONAL EQUATIONS Patient Charting Efficiency Outcome: Use rational equations in career applications. Accurate, up-to-date, and complete patient data has always been an important part of medical treatment; whether in a doctor’s office or in a hospital setting. It is vitally important in the Intensive Care Unit (ICU) of a hospital where patients’ lives may hang by a thread. Teaching hospitals accept and train a certain number of student nursing interns from affiliated medical schools. Suppose you and your team of personnel consultants have been hired by a teaching hospital to help them decide which of two nursing interns should be assigned to the ICU. After observing the two interns, Pat and Rosa, your team found them to be about equal in terms of procedure expertise, bedside manner, nursing knowledge/nomenclature, work ethic, attitude, academic background, experience, and hygiene. Your team decides to test them on one of the most important and time-consuming tasks: charting.

This hospital does not yet have computerized charting, and currently uses 1 night-shift nurse named Janet to visit and chart its ICU patients. Janet spends 60 - 90 minutes (70 min. average) to chart two ICU patients, and 90 - 120 minutes (100 min average) to chart her maximum of three ICU patients.

Pat and Janet work side-by-side one night to care for 2 patients. Your team found that it took them 48 minutes, working together, to chart these patients.

The next night Janet and Rosa worked side-by-side to care for 3 patients. (An additional patient was admitted which required about the same amount of care as each of the other two patients.) Your team found it took Janet and Rosa together 66 minutes to chart these patients.

1. Inspect the data given and state your opinion on which nursing intern seems to be the faster charter.

2. Estimate how long you think it would have taken Pat to chart the two patients alone. Write and solve an algebraic equation to find Pat’s time to chart these two patients when working alone.

3. Estimate how long you think it would have taken Rosa to chart the three patients when working alone. Write and solve an algebraic equation to find Rosa’s time to chart the 3 patients alone.

4. Estimate which intern is the faster charter based on your findings in problems 2 and 3. 5. Verify your estimate by writing and solving an equation to decide who should be hired in order to

get the faster charter.

Summarize what you learned from this activity:



68

QUADRATIC EQUATIONS Calculating Vehicular Speed from Skid Marks and Road Conditions Outcome: Use quadratic equations in career applications. The Good Decisions Through Teamwork project on pages 612–613 of Chapter 15 introduces two formulas that are used by law enforcement officers, insurance investigators, and private investigators to determine the speed of a motor vehicle based on skid marks and road conditions. The first formula is the formula for calculating the drag factor based on road conditions. The drag factor, or coefficient of friction, is a number between 0.03 (poor stopping conditions) and 1.3 ideal stopping conditions). The drag factor varies according to road surface type (asphalt, concrete, gravel, dirt, or grass), age of road, weather conditions (precipitation, wind, air temperature, air pressure, relative humidity), road surface conditions (wet, dry, ice, hail, oil, sand, dirt/gravel or other debris on road), grade of road, etc. Officers determine the drag factor of specific road conditions by conducting drag factor tests under similar conditions. Each test consists of braking to a stop from a specified speed (usually 25 - 35 mph), then measuring the length of the skid marks. At least two tests are conducted and the longest skid distance is used in the formula if the two distances are within 5%. The formula for calculating the drag factor (f) is f = s2/30D, where s is the test speed and D is the length of the skid marks. Usually the weight of the vehicle is negligible in determining speed, but weight can be a factor under extremely poor stopping conditions such as black ice, oil, slick, or oil mixed with water (which happens at the beginning of a rain occurring after a long dry period). Once a drag factor is determined, the estimated speed that the vehicle was traveling is calculated using a second formula. Under most stopping conditions the distance of the skid marks D is related to the speed of the car s, the drag factor f, and the percentage of braking before stop n (written as a decimal), by the formula: D = s2/30 fn. EXAMPLE Suppose tests indicate the drag factor on a wet, 15 year old asphalt highway is 0.5. Estimate the

speed of a car, traveling in a 45 mph speed limit zone, which left 388 ft. skid marks showing continuous braking before coming to a stop.

D = s2/30 fn 388 = s2/30(0.5)(1.00) Continuous braking: n = 100% = 1.00 388 = s2/15 5,820 = s2 ±76 = s The car was traveling approximately 76 mph.

1. Your car was run off an icy road by an out of control car which left continuos skid marks 572 feet long before it stopped. The police report found the drag factor to be 0.13. Approximate the speed of the car that ran you off the road.



69

Calculating Vehicular Speed from Skid Marks and Road Conditions, page 2

2. Solve the formula D = s2/30 fn for s.

3. Use the rearranged version of the formula to estimate your speed if you started braking when you entered the deep sand shoulder of the road (which has a drag factor of 0.35 under these conditions) and continuously braked to a stop leaving 129 feet of skid marks.

4. Research the drag factors of dry concrete and deep sand. Which of these two surfaces takes longer

to stop on, with other factors being equal? Explain why runaway truck ramps along mountain highways are often made of deep sand. What other surface would be appropriate? Why?

5. Research the difference between glare ice and black ice. Which of these road surface conditions

takes longer to stop on, with other factors being equal? Contact your State Highway Patrol Office or local police department if you want to verify your findings. Summarize what you learned in this activity:



70

WHAT IS THE NATURAL EXPONENTIAL e? Outcome: Discover the effect of large values of n in the expression (1 + 1/n)n.

When a quantity grows or decays exponentially, that means that the rate of increase or decrease is affected by an exponent. Several formulas dealing with exponential growth or decay use the natural exponential . This irrational e

number, like π , is a specific value. The value is the limit of the expression n

n

+

11 as n gets larger and larger. To

see this relationship, we will evaluate this expression for several values of . n

1. Evaluate the expression n

n

+

11 for the following values of n.

n

n

n

+

11

1 2121)11(1

111 ==+=

+

10 101010

)1.1()1.01(1011 =+=

+ = 2.59374246

100 1,000

10,000 100,000

1,000,000 10,000,000

100,000,000

2. Find the calculator value of e. To determine the calculator value of e, find the value of e using your calculator. 1

1exmost calculators:

1some calculators: ex

=

(If the label is above the key, as ex

, access the e function by first pressing a shift, inverse, or x

second function key.)

3. How does the calculator value of e compare to your calculated value when n is 100,000,000? Summarize what you learned in this activity:



71

COMPOUNDED AMOUNT AND COMPOUND INTEREST Outcome: Compare the two compound amount formulas for compound interest, A = p(1 + r/n)nt, and A = pent

The formula for finding the compound amount is: A = nt

nr

p

+1 .

In the formula A is the accumulated amount, p is the principal, t is time in years, r is the decimal equivalent of the rate per year, and n is the number of compounding periods per year. We will examine the effect of the number of compounding periods per year by evaluating the expression for several different periods for a principal of $1 at a rate of 10% per year for 1 year. Complete the chart.

Compounding Period

n A =

n

n.

+

1011

Annually

1 A = n

n.

+

1011

Semi-annually

2 A =

2

210

1

+

.1 = 1.1025

Quarterly 4 Monthly Weekly Daily

Hourly Every minute Every second

Calculator pattern: ( x y1 + .1 ÷ n ) n = The formula for finding the accumulated amount by compounding continuously (every instant) is A p , where p is the principal, r is the decimal equivalent of the rate per year, and t is the number of years.

ent=

1. Find the accumulated amount, compounded continuously, of $1 at 10% per year for 1 year. Calculator: .1ex = (multiply by 1 mentally.)

2. Compare the accumulated amounts for a $1 investment at 10% per year for 1 year using the two formulas. 3. If you were earning interest on an investment, under what conditions would you prefer using the

formula, A = nt

nr

+1p ? If you were making the calculations for or estimating interest on an

investment earning daily interest, which formula would you choose and why? Summarize what you learned in this activity:



72

DEVELOPING TABLE VALUES Outcome: Use formulas to expand tables. Tables often do not include the entries needed to make calculations. Tables are generally developed from a formula, and if this formula is known, the table can be extended to include additional values by using the formula to calculate values. A common table that is found in many personal finance and business mathematics references is a table for the future value of an annuity. Future value is another term for the accumulated amount of an investment or loan. An annuity is a series of equal periodic payments put into an interest-bearing account for a specific number of periods. The future value of an annuity would then project the accumulated amount that an account would have if regular, equal deposits (payments or investments) were made for a certain period of time at a set rate of interest. As with most financial tables, the entries are based on a specific regular payment, usually $1, $100, or $1,000. The given table is based on a payment of $1, and is based on a financial distinction of annuities called ordinary annuity. For an ordinary annuity, payment is made at the end of the period. Another type of annuity is an annuity due. For an annuity due, payment is made at the beginning of the period.

Before we expand the table, look an at example that uses the table. To find the future value of an ordinary annuity using a $1.00 Ordinary Annuity Future Value Table: 1. Select the periods row corresponding to the number of interest periods. 2. Select the rate-per-period column corresponding to the period interest rate. 3. Locate the value in the cell where the periods row intersects the rate-per-period column. 4. Multiply the annuity payment by the table value from step 3.

Future value = annuity payment × table value

EXAMPLE 1: Use the table to find the future value of a semiannual ordinary annuity of $6,000 for 5 years at 12% annual interest compounded semiannually.

5 years × 2 periods per year = 10 periods

yearper periods 2rateinterest annual 12%

= 6% period interest rate

Table value for 10 periods at 6% is 13.181. future value of annuity = annuity payment × table value

= $6,000 × 13.181 = $79,086

The future value of the ordinary annuity is $79,086.



73

Developing Table Values, page 2

Future Value of $1.00 Ordinary Annuity Rate per period

Periods 2% 3% 4% 5% 6% 7% 8% 9% 10% 12% 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2 2.020 2.030 2.040 2.050 2.060 2.070 2.080 2.090 2.100 2.120 3 3.060 3.091 3.122 3.153 3.184 3.215 3.246 3.278 3.310 3.374 4 4.122 4.184 4.246 4.310 4.375 4.440 4.506 4.573 4.641 4.779 5 5.204 5.309 5.416 5.526 5.637 5.751 5.867 5.985 6.105 6.353 6 6.308 6.468 6.633 6.802 6.975 7.153 7.336 7.523 7.716 8.115 7 7.434 7.662 7.898 8.142 8.394 8.654 8.923 9.200 9.487 10.089 8 8.583 8.892 9.214 9.549 9.897 10.260 10.637 11.028 11.436 12.300 9 9.755 10.159 10.583 11.027 11.491 11.978 12.488 13.021 13.579 14.776

10 10.950 11.464 12.006 12.578 13.181 13.816 14.487 15.193 15.937 17.549

11 12.169 12.808 13.486 14.207 14.972 15.784 16.645 17.560 18.531 20.655 12 13.412 14.192 15.026 15.917 16.870 17.888 18.977 20.141 21.384 24.133 13 14.680 15.618 16.627 17.713 18.882 20.141 21.495 22.523 24.523 28.029 14 15.974 17.086 18.292 19.599 21.015 22.550 24.215 26.019 27.975 32.393 15 17.293 18.599 20.024 21.579 23.276 25.129 27.152 29.361 31.772 37.280 16 18.639 20.157 21.825 23.657 25.673 27.888 30.324 33.003 35.950 42.753 17 20.012 21.762 23.698 25.840 28.213 30.840 33.750 36.974 40.545 48.884 18 21.142 23.414 25.645 28.132 30.906 33.999 37.450 41.301 45.559 55.750 19 22.841 25.117 27.671 30.539 33.760 37.379 41.446 46.018 51.159 63.440 20 24.297 26.870 29.778 33.066 36.786 40.995 45.762 51.160 52.275 72.052

Table values show the future value, or accumulated amount of the investment and interest, of a $1 investment for a given number

of periods at a given rate per period. Table values can be generated using the formula: FV of $1 per period =

−+

R

R N 1)(1,

when FV is the future value, R is the rate per period, and N is the number of periods. The table used to calculate the future value of an ordinary annuity is an example of a table which can be expanded to include additional columns or rows by using the formula to calculate each entry. The table does not include any

columns for percents that are mixed numbers. Let’s add a column for %21

8 . The formula is given at the bottom of

the table.

FV of the $1 per period = R

R N 1)1( −+ where FV is the future value, R is the rate per period, and N is the number of

periods.



74


EXAMPLE 2: Find the table value (FV of $1 per period) for %21

8 per period for 1 period.

R = %8 or 0.085 and N = 1 21

FV = R

R N 1)1( −+ Substitute values.

FV = 085.0

1)085.01( 1 −+ Add within parentheses.

FV = 085.0

1)085.1( 1 − Raise to power.

FV = 085.0

1085.1 − Simplify numerator.

FV = 085.0085.0

Divide.

FV = 1 (Notice this result is consistent with other table values for 1 period.) 1. Continue the process for each entry for periods 2 through 20 and record the results of your work in the space

provided.

Periods %21

8

Periods %21

8

1 1 2

We can add additional rows to the table if a different number of periods is needed. Let’s add a row for 21

periods. EXAMPLE 3: Find the first entry in the Future Value of a $1 Ordinary Annuity for 21 periods at 2$ per period. R = 0.02 and N = 21

FV = R

R N 1)1( −+ Substitute values.

FV = R

1)002.01( 21 −+ Add within parentheses

FV = 02.0

1)02.1( 21 − Raise to power.

FV = 02.0

1515666344.1 − Simplify numerator and divide.

FV = 25.78331719 or 25.783 Notice this value is reasonable given the table value for 20 periods at 2% is 24.297.



75


2. Continue the process for the period 21 row for rates of 3% through 12% and record the results of your work in the space provided.

% 2% 3% 4% 5% 6% 7% 8% 8.5% 9% 10% 11% 12% value 25.783

3. Compare each of the values in Exercises 1 and 2 with the table value above it and on each side to ensure that

it is a reasonable value. EXTENSION: Tables can be developed by using spreadsheet software such as Microsoft Excel and Corel Quattro Pro. The table is developed by using a formula that is sometimes referred to as a generating formula. This formula can be used to calculate each entry in the table by changing key data, such as the period number or the rate of interest. Generate tables using the given generating formula, number of periods, and rates of interest. 1. Future Value (FV) for a one-time, lump-sum investment of $1: FV = $1(1 + R)N for periods 1 − 30 and 35 − 100 in intervals of 5. Use the interest rates per period of 0.50%, 1%, 2%, 4%, 6%, and 10%. 2. Present values shows a lump sum amount of money (PV) that should be invested now so that the

accumulated amount will be $1 after a specified number of periods, N, at a specified rate per period, R. The generating formula is:

PV = NR)1(1$

+ for periods 1 − 30 and 35 − 100 in intervals of 5.

Use the interest rates per period of 0.50%, 1%, 2%, 4%, 6%, and 10%. 3. A sinking fund payment is amount of regular payment that should be made so that the accumulated amount

at the end of the time will be $1. The generating formula is:

SF = 1)1( −+ NR

R for periods 1 − 30 and 35 − 100 in intervals of 5.




76

INEQUALITIES Intelligence Quotient (IQ) Outcome: Use inequalities in career applications. The Good Decision Through Teamwork project on pages 665–666 of Chapter 17 introduces the concept of IQ or Intelligence Quotient. In general, an IQ score is a measure of your ability to think and reason. Since many factors determine the effectiveness of testing one’s “intelligence,” the score actually is an indication of how you compare in this ability with the majority of people in your age group. For example, a rating of 100 means that compared to the majority of other people in your age group, you have a normal rate of intelligence. Psychologists consider scores in

the range of 90 - 105 as having a normal or average IQ. The formula for determining an IQ score is: IQ = 100CAMA

⋅

where MA is the mental age of the person, and CA is their chronological age (both measured in years). Assessment accuracy may vary 5 points either way from the calculated score. EXAMPLE: An IQ test indicated that a 6.50 year-old child is able to think like a 7.25 year old child. Express the

child’s IQ score as a range of values within 5 points of the calculated score.

IQ = 100CAMA

⋅

IQ = 1006.507.25

⋅

IQ = 111.54 or 112

This test measure the child’s IQ as about 112 ± 5 points or between 107 and 117. In inequality notation: 107 < IQ < 117.

MENSA is an international organization whose mission is to offer information, support, and fraternity to intellectually gifted people. Persons of all ages who score in the top 2% on internationally recognized intelligence tests are eligible for membership. The average minimum IQ score needed is 132, but it ranges from 130 - 135 since IQ tests differ slightly in their scoring. Suppose you and your team are Mensa members who have been appointed to a MENSA committee to answer prospective members’ questions about their IQ scores.

1. The teacher of a third grade student (8.5 yr. old boy) who is achieving at the 6th grade level (11.5 mental age) asks if this student might be eligible for Mensa so they can benefit from the educational enrichment and interaction with similar children his age. Calculate the boy’s IQ given this information, and state the result as an inequality. Then give a recommendation as to whether he should be formally tested for Mensa membership.

2. The parents of a 10.0 yr. old girl called your team and said that the girl scored 128 of her Stanford

Binet IQ test. They wanted to know the range of her IQ score, and the range of her mental age. State each as an inequality.

3. A psychologist remarks at the Mensa convention that she recently tested a preschooler who was

functioning at the mental age of a twelve year old, having an IQ of 209! How old would this preschooler have to be?




77

GRAPHING EQUATIONS AND INEQUALITIES PROJECT (See graphing activities 1-7.)

Instructions for Group Project For Five-Member Groups

One group member will serve as group recorder for each activity. Rotate recorders with each activity. Each of the other four members will graph a set of equations using a computer or calculator. Activity 1 - Linear Equations in the form y = mx. Member 1: Graph equations 1 on page 83. All six graphs should appear on the axes at the same time. 6-

Be able to identify each equation to your group members. Leave the graphs on the screen until your group has answered questions 7 - 8.

Member 2: Graph equations 9 1 on page 83. All four graphs should appear on the axes at the same time. 2-

Be able to identify each equation to your group members. Leave the graphs on the screen until your group has answered question 13.

Member 3: Graph equations 14 on page 83, using the computer. All six graphs should appear on the axes at the

same time. 19-

Be able to identify each equation to your group members. Leave the graphs on the screen until your group has answered questions 20 . 21-

Member 4: Graph equations 22 on page 84. All four graphs should appear on the axes at the same 25-

time.

Be able to identify each equation to your group members. Leave the graphs on the screen until your group has answered question 26.

Recorder: Record the results of each of the four sets of graphs. Entire Group: Discuss 8, 13, 20, 21, 26, and 27 ; reach a consensus, and record your responses. 33- Activity 2: Linear Equations in the form y = mx + b Member 1: Graph 1 on page 85. Discuss 4. 3-Member 2: Graph 5 on page 85. Discuss 8. 7-Member 3: Graph 9 1 on page 85. Discuss 12. 1-Member 4: Graph 13 on page 86. Discuss 16 . 15- 18-Recorder: Record the results of each of the four sets of graphs. Entire Group: Without the use of a computer or calculator, draw a freehand sketch of the graphs of the equations 19-30. Reach a consensus on questions 4, 8, 12, and 16-18; and, record your responses.



78

Instructions for Group Project, page 2

Activity 3 - Linear Equations in the form y = k and x = k. Member 1: Graph 1 on page 87. Discuss 4. 3-Member 2: Graph 5 on page 87. Discuss 8 . 7- 9-Member 3: Graph 10 on page 87. Discuss 13. 12-Member 4: Graph 14 on page 88. Discuss 17. 16-Recorder: Record the results of each of the four sets of graphs. Entire Group: Reach a consensus on questions 4, 8, 9, 13, 17, and 18; and, record your responses.

Without the use of a computer or calculator, draw a freehand sketch of the graphs for equations 19 . 30-

Activity 4 - Linear Inequalities Many graphing computer software packages will not graph inequalities. The equation that forms the boundary for the inequality can be graphed. Test points can be used to determine which side of the boundary is to be shaded. Some graphing calculators will automatically inequalities Four members of the group will graph problems 1 on page 89. 8-Recorder: Record the results of 1 . 8- Entire Group: Discuss questions 9 1 , reach a consensus, and record your responses. 2- Activity 5 - Quadratic Equations in the form y = ax2

Member 1: Graph 1 on page 90. Discuss 4. 3-Member 2: Graph 5 on page 90. Discuss 8 . 7- 9-Member 3: Graph 10 on page 90. Discuss 13. 12-Member 4: Graph 13 - 15 on page 90. Discuss 17. Recorder: Record the results of each of the four sets of graphs. Entire Group: Reach a consensus on question 7-9 and 16-20 and record your responses. Activities 6 and 7 - Quadratic Equations in the form y = ax2 + b and y = (x + b)2

Member 1: Graph 1 for each activity. Discuss 4. 3-Member 2: Graph 5 for each activity. Discuss 8 . 7- 9-Member 3: Graph 10 for each activity. Discuss 13. 12-Member 4: Graph 14 for each activity. Discuss 17. 16-Recorder: Record the results of each of the four sets of graphs. Entire Group: Reach a consensus on question 7-9 and 16-20 and record your responses.



79

GRAPHING EQUATIONS AND INEQUALITIES Graphing Activity 1 Outcome: Examine equations in the form y = mx. Use a graphics calculator or computer to graph the equations. Record a sketch of graphs 1 on the same axes, labeling each graph.

6-

1. y x= 2. y x= 2 3. y x= 3 4. y x= 5 5. y x= 8 6. y x= 10 7. In your own words describe the graph of y x= . 8. In your own words compare the graphs 2 to the graph of 6- y x= . Record a sketch of graphs 9 1 on the same axes, labeling each graph. 2-

9. y x= 10. y x=12

11. y x=13

12. y x=14

13. In your own words compare the graphs 10 to the graph 12-

of y x= . Record a sketch of graphs 14 on the same grid, labeling each graph. 19- 14. y x= − 15. y x= −2 16. y x= −3 17. y x= −5 18. y x= −8 19. y x= −10 20. In your own words describe the graph y x= − . 21. In your own words compare the graphs 15 to the graph 19-

of y x= − .



80

Graphing Activity 1, page 2

Record a sketch of graphs 22 on the same axes, labeling each graph. 25-

22. y x= − 23. y = −12

x

24. y x= −14

25. y x= −13

26. In your own words compare the graphs 23 to the graph 25-

of y x= − . In the equation of the line , m represents the slope (slant or steepness) of the line. y mx=Complete 27 about the graphs of the line whose equations are in the form 32- y mx= .

27. Make a general statement about the slopes of lines when m is positive (m > 0). 28. Make a general statement about the slopes of lines when m is negative (m < 0). 29. Make a general statement about the slopes of lines when m is greater than 1 (m > 1) and gets larger

and larger. 30. Make a general statement about the slopes of lines when m is a fraction between 0 and 1 (0 < m < 1)

and gets smaller and smaller (closer to 0). 31. Make a general statement about the slopes of lines when m is less than − (m < −1) and the absolute

value of m gets larger and larger. 1

32. Make a general statement about the slopes of lines when m is a fraction between 0 and 1 (−1 < m < 0)

and gets closer and closer to 0. 33. In an equation in the form y mx= , summarize the information given by the value of m? Summarize what you learned in this activity:



81

Graphing Activity 2 Outcome: Examine equations in the form y mx b= + . Use a graphics calculator or computer to graph the equations. Record a sketch of graphs 1 on the same axes, labeling each graph.

3-

1. y x= 2. y x= +1 3. y x= − 2 4. Describe the similarities and differences in the graphs 1 . 3- Record a sketch of graphs 5 - 7 on the same axes, labeling each graph. 5. y x= 4 6. y = 4x + 3 7. y = 4x − 1 8. Describe the similarities and differences in the graphs 5 . 7- Record a sketch of graphs 9 1 on the same axes, labeling each graph. 1- 9. y x= −3 10. y x= − +3 2 11. y x= − +3 1 12. Describe the similarities and differences in the graphs 9 1 . 1-



82



13. y = −23

x 14. y x= − +23

1 15. y x= − −23

2

16. Describe the similarities and differences in the graphs 13 . 15- 17. In an equation of the form y mx b= + , summarize the information given by the value of b . 18. In an equation of the form y mx b= + , summarize the information given by the value of m. Without the use of a computer or calculator, draw a freehand sketch of the graphs of the following equations.

19. y x= 20. y x= + 4 21. y = x21

22. y x= − 23. y x= − + 3 24. y x= −2

25. y = 4x − 2 26. y = 543

+x 27. y = 431

−x

28. y = −5x − 1 29. y x= −78

2 30. y x= − +32

12




83

Graphing Activity 3 Outcome: Examine equations in the form y k= and x k= . Use a graphics calculator or computer to graph the equations. Record a sketch of graphs 1 on the same axes, labeling each graph.

3-

1. 2. 3. y = 1 y = 2 y = 12

4. In your own words describe the graphs in 1 . 3- Record a sketch of graphs 5 on the same axes, labeling each graph. 7-

5. 6. y = −3 y = −2 7. y = 41

−

8. In your own words describe the graphs in 5 . 7- 9. Make a general statement about graphs of lines whose

equations are in the form y k= . Record a sketch of graphs 10 on the same axes, labeling each graph. Graphs 10 cannot be done using many computer programs or graphics calculators.

12- 12-

10. x = 1 11. x = 2 12. x = 23

13. In your own words describe the graphs in 10 - 12.



84


Record a sketch of graphs 14 - 16 on the same axes, labeling each graph. Graphs 14 - 16 cannot be done using many computer programs or graphics calculators. Use other methods.

14. x = −2 15. x = −3 16. x = 41

−

17. In your own words describe the graphs in 14 . 16- 18. Make a general statement about the graphs of the lines whose

equations are in the form x k= . Without the use of a computer or calculator, draw a freehand sketch of the graphs of the following equations.

19. 20. y = −4 y = 34

21. x = 6

22. x = −43

23. y = 3 24. x = −5

25. x = 0 26. y = 0 27. y = 52

28. x = −94

29. y = 8 30. x = 83




85

Graphing Activity 4 Outcome: Graph various inequalities and examine similarities and differences. Draw a freehand sketch of the graphs of the following inequalities. The computer or a graphics calculator can be used to check the location of the boundary line. 1. y x< 2. y x≤ 3. y x> 4. y x≥

5. y x< −3 6. y x≤ −3 7. y x> −3 8. y x≥ −3 9. In your own words describe the similarities and differences among the following pairs of graphs: 1 and 2 - 3 and 4 - 5 and 6 - 7 and 8 -

10. Make a general statement about the boundary line of the graphs of an inequality. 11. In your own words describe the similarities and differences among the following pairs of graphs: 1 and 3 - 2 and 4 - 6 and 8 - 12. Make a general statement explaining how to shade the appropriate region of the graph of an inequality. Summarize what you learned in this activity:



86

Graphing Activity 5 Outcome: Examine quadratic equations in the form y . ax= 2

Use a graphics calculator or computer to graph the equations. Record a sketch of graphs 1 on the same axes, labeling each graph.

6-

1. 2. 3. y x= 2 y x= 2 2 y x= 3 2

4. y x=12

2 5. y x=14

2 6. y x=13

2

7. In your own words describe the graph of . y x= 2

8. In your own words compare the graphs 2 to the graph of 3-

y x= 2 . 9. In your own words compare the graphs 4 to the graph of 6-

y x= 2 .


10. 11. y x= − 2 y = −2 2x

x

12. 13. y = −3 2 y = − 12

2x

14. y = −13

2x 15. y x= −14

2

16. In your own words describe the graph of . y x= − 2

17. In your own words compare the graphs 10 to the graph of . 12- y x= − 2

18. In your own words compare the graphs 13 to the graph of . 15- y x= − 2

19. Make a general statement about the graphs of and y x . y x= 2 = − 2

20. What information does m give in an equation in the form . y mx= 2




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Graphing Activity 6 Outcome: Examine equations in the form y a . x b= +2

Use a graphics calculator or computer to graph the equations. Record a sketch of graphs 1 on the same grid, labeling each graph.

3-

1. y = x2 + 1 2. y x= +2 3 3. y x= +2 4 4. In your own words describe the similarities and differences

among graphs 1 . 3-

Record a sketch of graphs 5 on the same grid, labeling each graph. 6-

5. y = x2 − 1 6. y x= −2 3 7. y x= −2 4 8. In your own words describe the similarities and differences

among the graphs 5 . 7- Without the use of a calculator or computer, draw a freehand sketch of the following.

9. 10. 11. 12.

y x= 2 y x= − 2 y x= 5 2 y x= −5 2

13. y x=15

2 14. y x= −15

2 15. 16. y x= +2 5 y x= −2 2

17. 18. 19. 20. y x= +2 12 y x= − +2 2 y x= − −2 3 y x= − +3 12

Summarize what you learned in the activity:



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Graphing Activity 7 Outcome: Examine equations in the form y = (x + b)2. Use a graphics calculator or computer to graph the equations. Record a sketch of graphs 1 on the same axes, labeling each graph.

3-

1. y = (x + 1)2 2. y = (x + 3)12

3. y = (x + 4)2

4. In your own words describe the similarities and differences

among graphs 1 . 3-


5. y = (x − 1)2 6. y = (x − 3)2

7. y = (x − 4)2

8. In your own words describe the similarities and differences

among graphs 5 . 7- Without the use of a calculator or computer, draw a freehand sketch of the following.

9. y = (x + 6)2 10. y = (x + 2)2 11. y = (x + 5)2 12. y = −5x2 13. y = (5x + 2)2 14. y = (3x + 4)2 15. y = (x − 2)2 16. y = (x − 5)2

17. y = (x + 6)2 18. y = (2x − 1)2 19. y = (3x − 2)2 20. y = (4x − 3)2




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GRAPHICAL REPRESENTATION Commodities Market Investing Outcome: Use graphing in consumer applications. The Good Decisions Through Teamwork project on pages 701–702 of Chapter 18 introduces commodities market investing. Suppose you and your team of investors (who live in the Mississippi River Delta region around Memphis TN) invested a total of $10,000 in locally grown cotton last year, and held the investment for ten months. According to your records, your team’s total profit P after t months of investing was approximated by the equation: P = 100t2 − 400t EXAMPLE: Find your team’s profit one month after investing. P = 100(12) − 400(1) (t = 1) P = 100(1) − 400 P = 100 − 400 P = −300 Your team’s $10,000 investment lost $300 after 1 month.

Use P = 100t2 − 400t to calculate the following, and state your interpretation of your answers in complete sentences. 1. Graph the profit equation, counting by 1’s on the horizontal “t” axis and by 500’s on the vertical

“P” axis. Why is the graph only valid between t = 0 and t = 10? 2. Use the graph to estimate the total profit after the following months: a) 3 b) 5 c) 7.5 d) when

it was sold. 3. Use the profit equation to find the exact total profit after the same amount of time: a) 3 b) 5 c) 7.5 d) when it was sold. 4. Explain any differences between your answers in problems 2 and 3. 5. Find when this investment “broke even” (Hint: When profit is equal to zero) by using a) the graph

and b) the equation. 6. Shade the region(s) in which the investment lost money and label it “loss”. Shade the region(s) in

which the investment made money and label it “profit”. 7. The lowest this investment went was $400 below the purchase price. Find the month when this

occurred. 8. If you could have held this investment for a maximum of a year at the same rate of growth, would

you have done better? Show why or why not. State the maximum possible profit for the year. 9. Calculate the annual interest rate this investment earned during the ten months of investment. 10. Suppose each member of your team invested a different amount to total $10,000. Assign an amount

that each team member invested and pro rate the amount of profit for each team member based on those investment amounts.




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SYSTEMS OF EQUATIONS Making Business Choices Outcome: Use systems of equations to make good business choices.

In the Good Decision Through Teamwork project on pages 773–774 of Chapter 20 a student was trying to make some choices regarding a new business venture. Your team will continue to advise this student. The student decides that he wants to work 20 hrs. per week, and can afford to spend $135 per week in materials. In helping your classmate with his economic forecasting, let x = the number of 12″ base centerpieces, and y = the number of 16″ base centerpieces he can make per week in his home business. 1. Write a linear equation using x and y to show his 20 hr./wk. time allotment. 2. Write a linear equation using x and y to show his $135/wk. investment allotment. 3. Solve the system of linear equations from problems 1 and 2 by the graphing method and estimate

how many of each type of centerpiece he should make each week in order to meet both his time and money allotments.

4. Solve the system to find an exact solution. This “ideal” solution represents how many of each type of centerpiece should be made weekly in order to take exactly 20 hours and cost exactly $135. If this solution isn’t the same as in problem 3, explain why.

5. If we assume that local florists will buy all he can make, how much revenue will be received from the solution found in problem 4?

6. How much did it cost him to make these centerpieces? 7. Since profit = revenue − cost, find the profit made on these centerpieces. Use this figure to

calculate the student’s hourly wage. Would he make at least as much in this 20 hr/wk part-time venture as he currently makes in his full-time minimum wage job?

8. Why is the part-time wage found in problem 7 misleading? What other material costs or labor time have not been considered?

If the 20 hr/wk and $135/wk factors are disregarded, derive the following economic forecasting formulas appropriate for different production levels. 9. Let R = revenue from florists, and write a linear equation in x and y to show the revenue received

from any number of 12″ and 16″ centerpieces. This equation is called the Revenue Formula. 10. Let C = cost of centerpiece materials, and write a linear equation in x and y to show the materials

cost of making any number of 12″ and 16″ centerpieces. This equation is called the Cost Formula. 11. Let P = profit, and write a profit formula in x and y to show the profit made on any number of 12″

and 16″ centerpieces. This equation is called the Profit Formula. 12. Using your profit formula from problem 11, which centerpiece is more profitable to make? 13. Use the profit formula to find the profit made on the sale of eight 12″, and ten 16″ centerpieces. 14. Use the cost formula to find the cost of making five 12″, and twelve 16″ centerpieces. 15. Use the revenue formula to find the revenue on four 12″, and nine 16″ centerpieces. Summarize what you learned in this activity:



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