Apply GCF and LCM to Monomials Mississippi Standard: Apply the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM) to monomials with variables. You can find the Greatest Common Factor (GCF) of two or more monomials by finding the product of their common prime factors.

Find the GCF of Monomials Find the GCF of 16xy2 and 30xy. Find the prime factorization of each monomial. 16xy2  2  2  2  2  x  y  y 30xy  2  3  5 

Circle the common factors.

xy

The GCF of 16xy2 and 30xy is 2  x  y or 2xy.

You can find the Least Common Multiply (LCM) of two or more monomials by multiplying the factors, using the common factors only once.

Find the LCM of Monomials Find the LCM of 18xy2 and 10y. Find the prime factorization of each monomial. 18xy2  2  3  3  10y

2

xyy 5

y

2335xyy The LCM of

18xy2

Circle the common factors.

Multiply the factors, using the common factors only once.

and 10y is 2  3  3  5  x  y  y or 90xy2.

Exercises Find the GCF of each set of monomials. 1. 12x, 40x2

2. 18m, 45mn

3. 14n, 42n2

4. 4st, 10s

5. 5ab, 6b2

6. 14b, 56b2

7. 36a3b, 56ab2

8. 30a3b2, 24a2b

9. 32mn2, 16n, 12n3

Find the LCM of each set of monomials. 10. 20c, 12c

11. 16xy, 3x

12. 36ab, 4b

13. 16a2,

14. 7x, 12x

15. 21mn, 28n2

17. 75n2, 25n4

18. 10x, 20x2, 40xy

14ab

16. 20st, 50s2t

GEOMETRY For Exercises 19 and 20, use the squares shown. 19. What is the GCF of the sides of the squares? 20. What is the LCM of the sides of the squares?

4xy

16x 2

Compare Data Sets Mississippi Standard: Use a given mean, mode, median, and range to summarize and compare data sets including investigation of the different effects that change in data values have on these measures. When given the mean, median, mode, and range of data sets, you can often analyze and make comparisons of the data, without knowing the data values.

Summarize and Compare Data Sets BUILDINGS The mean, median, mode, and range of the ten tallest

buildings in New York and Texas are given in the table. Ten Tallest Buildings (meters) Mean Median Mode Range

New York

Texas

289 281 248, 319 133

260 255 none 76

Which state has a greater average building height of its ten tallest buildings? The mean, or average, height of the ten tallest buildings in New York is 289 meters. The mean height of the ten tallest buildings in Texas is 260 meters. Compare the data. 289  260 So, New York has a greater average building height of its ten tallest buildings than Texas. Which state has the same building height for more than one of its ten tallest buildings? The mode represents data values that appear most often. In New York, a height of 248 meters and 319 meters appear most often. In Texas, there are no building heights that appear more than once or most often. So, New York has the same building height for more than one of its ten tallest buildings. The tallest building in Texas is 305 meters. How tall is the tenth tallest building? The range of the ten tallest buildings in Texas is 76 meters, so subtract 76 from 305. 305  76  229 The tenth tallest building in Texas is 229 meters tall. When a value of a data set changes, you can often determine how the mean, median, mode, and range will be affected without recalculating all the measures.

Investigate a Change in Data Value SCHOOL The table at the right shows the math test scores of

Fifth Period Period Test Test Scores Scores (%) (%) Fifth

Mr. Gomez’s fifth period class. The mean is 81%, the median is 82%, the mode is 85%, and the range is 30%. If after handing back the tests, the student who received a 65% did extra credit to change his test score to 75%, how will this affect the mean, median, mode, and range? Since the test score is increasing, the mean will also increase.

90

82

85

70

75

91

95

70

72

72

85

85

70

78

90

65

95

88

82

75

78

80

82

85

85

The median is the middle number. Since 75% is still in the lower part of the test scores, it will not affect the median. The mode is the number that appears most often. A new score of 75% will result in three test scores of 75%. However, there will still be five test scores of 85%, so the mode will stay the same. The range is the difference between the greatest score and the least score. The greatest score will still be 95%. However, the least score will now be 70%. So, the range will be 95%  70%, or 25%, instead of 30%.

Exercises BASEBALL For Exercises 1–4, use the information in the

Number of Runs Batted In by Season Leader, 1997–2006

table on the number of runs batted in by the season leader for the National League and American League. 1. Which league had a greater average number of runs

National League

American League

142.9 144 147 32

147.7 146 145 28

Mean Median Mode Range

batted in by its season leaders from 1997–2006? 2. From 1997–2006, the greatest number of runs batted

in by the season leader for the National League was 160 runs. What was the fewest number of runs batted in by the season leader?

Source: The World Almanac

3. From 1997–2006, the fewest number of runs batted in by the season leader

for the American League was 137 runs. What was the greatest number of runs batted in by the season leader? 4. Write a statement comparing the middle number of runs batted in by the

National League and American League season leaders from 1997–2006. ONLINE For Exercises 5 and 6, use the list below that shows the number of hours Ms. Wright’s students spent online last week. The mean of the data is 6, the median is 7, the mode is 7, and the range is 12.

7

4

7

9

3

1

5

8

10

0

9

4

0

11

3

1

9

0

7

11

12

7

6

8

4

10

5. If the student who spent 12 hours online thought about it more and

changes his or her number of hours to 9 hours, how will this affect the mean, median, mode, and range? 6. If Ms. Wright recorded a student who said 1 hour as 11 hours on the list,

how will this affect the mean, median, mode, and range when she corrects the data value to 1 hour and recalculates the measures?

Multiply and Divide with Scientific Notation Mississippi Standard: Multiply and divide numbers written in scientific notation. You can use scientific notation to simplify computations with very large and/or very small numbers. To multiply numbers in scientific notation, regroup to multiply the factors and multiply the powers of ten. Then simplify. To multiply the powers of ten, use the Product of Powers .

Product of Powers

Words

To multiply powers with the same base, add their exponents.

Symbols

Arithmetic 32



35



32  5

Algebra or

37

xa

 xb  x a  b

Multiplication with Scientific Notation Evaluate the expression (1.3  102)(2.5  101). (1.3  102)(2.5  101)  (1.3  2.5)(102  101)  (3.25)(102  101)  3.25 

Commutative and Associative Properties Multiply 1.3 by 2.5.

102  1

Product of Powers

 3.25  103

Add the exponents.

 3.25  1,000

103  1,000

 3,250

Move the decimal point 3 places.

Evaluate the expression (4.2  103)(1.6  104). (4.2  103)(1.6  104)  (4.2  1.6)(103  104)  (6.72)(103  104)  6.72 

Commutative and Associative Properties Multiply 4.2 by 1.6.

103  4

Product of Powers

 6.72  107

Add the exponents.

 6.72  10,000,000

107  10,000,000

 67,200,000

Move the decimal point 7 places.

To divide numbers in scientific notation, regroup to divide the factors and divide the powers of ten. Then simplify. To divide the powers of ten, use the Quotient of Powers . Quotient of Powers

Words

To divide powers with the same base, subtract their exponents.

Symbols

Arithmetic 48  43



48  3

603 Prerequisite Skills

or

Algebra 45

xa   xa  b, x xb

0

Division with Scientific Notation 9.45  106 2.1  10

Evaluate the expression  3 .



 

Associative Property

10  4.5  3

 10 

Divide 9.45 by 2.1.

 4.5  106  3

Quotient of Powers

 4.5  103

Subtract the exponents.

 4.5  1,000

103  1,000

 4,500

Move the decimal point 3 places.

9.45 106 9.45  106    3 2.1 10 2.1  103 6

Exercises Multiply or divide. Express using exponents. 1. 51  54

25 2

4. 3

2. 65  64

3. 102  103

76 7

109 10

5. 5

6.  6

Evaluate each expression. Express the result in scientific notation and standard form. 7. (2.6  105)(1.9  102)

8. (5.3  104)(0.9  103)

9. (3.7  102)(1.2  102)

10. (3.3  103)(2.1  102)

11. (8.5  103)(1.1  101)

12. (3.9  102)(2.3  106)

13. (6.45  105)(1.2  103)

14. (4.18  104)(0.9  105)

15.  3

8.37  108 2.7  10

16.  2

17.  5

9.72  109 1.8  10

18.  3

8.32  107 1.3  10

20.  6

4.2  108 1.68  10

22.  8

19.  5 21.  2

8.04  105 6.7  10 4.64  104 2.9  10

6.3  1010 1.5  10

9.0  1011 2.4  10

23. BASEBALL The table shows the 2007 salaries of

six Major League Baseball players. About how many times greater is Alex Rodriguez’s salary than Juan Castro’s salary? 24. ASTRONOMY The Sun burns about

4.4  106 tons of hydrogen per second. How much hydrogen does the Sun burn in one year? (Hint: one year  3.16  107 seconds) 25. OCEANS The area of the Pacific Ocean is

6.0  107 square miles. The area of the Atlantic Ocean is 2.96  107 square miles. About how many times greater is the area of the Pacific Ocean than the Atlantic Ocean?

2007 Major League Baseball Salaries Player

Team

Juan Castro Coco Crisp Nomar Garciaparra Chipper Jones Kazuo Matsui Alex Rodriguez

Cincinnati Reds Boston Red Sox Los Angeles Dodgers Atlanta Braves Colorado Rockies New York Yankees

Source: USA Today

Salary (dollars) 9.25  105 3.83  106 8.52  106 1.23  107 1.5  106 2.27  107

The Density Property Mississippi Standard: Develop a logical argument to demonstrate the ‘denseness’ of rational numbers. Examine the number line below. Find another integer that lies between the integers 2 and 3. 0

-2 -1

1

2

3

4

The integers 1, 0, 1, and 2 all lie between 2 and 3 on the number line. On the number line above, find a number that lies between 1 and 2. Their 1 2

average, 1, is one number that lies between 1 and 2.

Find a Number Between Two Given Numbers 1 3

1 2

Find a number that lies between  and  on the number line below. 0

1 6

1 3

2 3

1 2

5 6

1

One number would be their average.









1 1 1 1 2 3          2 3 2 2 6 6



1 5   

1 1 Rewrite  and  with a common denominator. 3 2 Add the numerators.

2 6 5 12

 

Multiply.

1 3

5 12

1 2

The rational number, , lies between  and . Find a number that lies between 7 and 6.5. One number would be their average. 1 1 [7  (6.5)]  (13.5) 2 2

 6.75

Add 7 and 6.5. Multiply.

The rational number, 6.75, lies between 7 and 6.5.

The process above of finding another number between any two given numbers can be continued indefinitely. This suggests the density property .

Density Property for Rational Numbers

Words

Between every pair of distinct rational numbers, there are infinitely many rational numbers.

You can use the density property to solve real-world problems.

Apply the Density Property BAKING Genevieve’s grandmother gave her a family recipe for apple pie.

Her grandmother does not use an exact amount of sugar, but told 1 4

1 2

Genevieve to use somewhere between 1 and 1 cups of sugar. If Genevieve wants to use an exact amount of sugar that is somewhere 1 4

1 2

between 1 cups and 1 cups, how much sugar can she use? One possible amount is their average.





 1 3  2 2 4 1 11   2 4



1 1 1 1 1 2  1  1   1  1 2 4 2 2 4 4

11 8

1 2 Rename  as . 2 4 Add the whole numbers and add the fractions. 3 Rewrite 2 as an improper fraction. 4

3 8

  or 1

Simplify.

3 8

So, Genevieve can use 1 cups of sugar.

Exercises Identify a number that lies between points A and B on each number line. 1. 0

1

A

B

2

3

3. 0

0.5

1

1.5

2. 4

5

A

B

2

2.5

A

B

-4 -3 -2 -1

4. 1 10

2 10

A

B

3 10

4 10

0

1

5 10

6 10

Identify a number that lies between each pair of numbers. 3 4

6. 10 and 9

5. 6 and 7

1 2

8. 2 and 1 11. 4 and 3

2 3

7.  and 1

3 4

1 3

10. 5 and 4

9. 4 and 4 12. 8.25 and 8.75

13. 15.5 and 16

14. SCHOOL For reading class, Dylan is recording the number of hours he reads

1 2

each week. This week, Dylan needs to read between 1 and 2 hours. What 1 2

is a possible time between 1 and 2 hours that Dylan can read? 15. CROSS COUNTRY For cross-country practice, the coach told the runners

1 2

3 4

they needed to run between 5 and 5 miles. Give a possible distance between 1 2

3 4

5 and 5 miles that a runner can run.

16. Demonstrate the density property for rational numbers with several

examples of your own.

Algebraic Properties Mississippi Standard: Apply algebraic properties in problem-solving. Review the properties in the table below. These properties can be applied when problem-solving. Properties

Commutative Property sum or product.

The order in which numbers are added or multiplied does not change the 6776 3883

Examples

abba abba

The way in which numbers are grouped when added or multiplied does not Associative Property change the sum or product. (2  7)  4  2  (7  4) (3  4)  5  3  (4  5)

Examples

(a  b)  c  a  (b  c) (a  b)  c  a  (b  c)

To multiply a sum by a number, multiply each addend by the number Distributive Property outside the parentheses. 2(7  4)  2  7  2  4 (5  6)3  5  3  6  3

Examples

Additive Identity

a(b  c)  ab  ac (b  c)a  ba  ca

The sum of any number and 0 is the number. 707

Examples

a0a

The product of any number and 0 is 0.

Zero Property of Multiplication

900

Examples Multiplicative Identity

a00

The product of any number and 1 is the number. 313

Examples

a1a

Identify Properties Name the property shown by each statement. (7  3x)  2x  7  (3x  2x)

0  5a  0

Associative Property

Zero Property of Multiplication

Use Properties to Simplify Expressions Simplify each expression. Justify each step. 4  (x  13) 4  (x  13)  4  (13  x)

Commutative Property

 (4  13)  x

Associative Property

 17  x

Add 4 and 13.

6(x  7) 6(x  7)  6(x)  6(7)  6x  42

Distributive Property Multiply.

Apply Properties to Problem Solving MUSEUMS Three friends are going to the science museum. The cost of

admission is $x each. It will cost an additional $4 to view a movie on the 3-D screen. Write and simplify an expression that represents the total cost for the three friends. The cost of admission plus the movie can be represented by (x + 4). Multiply this cost by the number of friends, 3(x + 4). 3(x  4)  3(x)  3(4)  3x  12

Distributive Property Multiply.

So, the total cost for the three friends is $3x  $12. MUSEUMS Refer to Example 5. A fourth friend will meet the group of friends at the museum but will not go to the movie. Write and simplify an expression that represents the total cost for the four friends. The cost for the fourth friend is $x. Add this to $3x  $12.

3x  12  x  3x  x  12  4x  12

Commutative Property Add.

So, the total cost for the four friends is $4x  $12.

Exercises Name the property shown by each statement. 1. 3n  m  m  3n

2. 0  18d  18d

3. (7y  8)  10y  7y  (8  10y)

4. 20xy  1  20xy

5. 3(6a  7b)  3  6a  3  7b

6. 82  0  0

Simplify each expression. Justify each step. 7. 1  (6  x)

8. 5(6a)

9. 11  (6  n)

10. 5(x  8)

11. 15(4w)

12. 9(x  2)

13. 9  2y  11  5y

14. 4(x  7)  2x

15. 11n  7(2  3n)

16. ANIMALS A zebra can run up to 40 miles per hour. An elephant can run up to

x miles per hour. Write and simplify an expression to find how many more miles a zebra will run in six hours than an elephant. 17. CELL PHONES Seven friends have similar cell phone plans. The price of each

plan is $x. Three of the seven friends pay an extra $4 per month for unlimited text messaging. Write and simplify an expression that represents the total cost of the seven plans.

Make Predictions from Circle Graphs and Histograms Mississippi Standard: Use proportions, estimates, and percentages to construct, interpret, and make predictions about a population based on histograms or circle graph representations of data from a sample. You can make predictions about a given set of data displayed in a circle graph or histogram. Use percentages to make predictions about data displayed in a circle graph. Favorite Type of Television Show

Predict from a Circle Graph

4% 5% Other Drama

The circle graph shows the results of a survey of the students in the 8th grade at Oakwood Junior High. If there are 560 students at Oakwood Junior High, how many would you predict to choose reality as their favorite type of television show?

7% Fiction 40% Reality

16% Cartoon

The section of the graph representing students who chose reality is 40% of the circle. So find 40% of 560.

28% Comedy

To find 40% of 560, you can use either method. METHOD 1 Write the percent as a decimal.

40% of 560  40% 560

Write a multiplication expression.

 0.40 560

Write 40% as a decimal.

 224

Multiply.

METHOD 2 Write the percent as a fraction.

40% of 560  40% 560 40 560    100 1

 224

Write a multiplication expression. 560 1

Write 40% as a fraction. Write 560 as . Multiply.

So, about 224 students at Oakwood Junior High would choose reality as their favorite type of television show.

Predict from a Histogram

The bar at 43.0–44.9 seconds is much higher than the others and represents the most winning times. So, the winning speed of the runner in the next summer Olympics will most likely be in the 43.0–44.9 second range.

Summer Olympic Men’s 400-Meter Run Winning Times, 1896–2004 Number of Winners

The histogram shows the winning times of the men’s 400-meter run in the summer Olympics from 1896 to 2004. Predict the range of speeds that a runner finishing in first place is most likely to be in the next summer Olympics? Explain your reasoning.

12 10 8 6 4 2 0

43.0– 45.0– 47.0– 49.0– 51.0– 53.0– 44.9 46.9 48.9 50.9 52.9 54.9

Time (seconds) Source: The World Almanac

Most Popular Luxury Car Colors 5%

Exercises CARS For Exercises 1–3, use the circle graph that shows the most popular luxury car colors.

Other

9% Blue

1. If a car dealership sold 50 luxury cars in March, predict how

many were white.

26% Silver/ Gray

9% Red

2. If a car dealership sold 250 luxury cars in January through

11% Light Brown

June, predict how many were black. 3. If a parking garage has 85 luxury cars parked on a given day,

28% White

12% Black

predict how many are silver/gray.

Favorite Summertime Activity

VACATION For Exercises 4 and 5, use the circle graph that

5% 6% Other Read

shows the results of a survey of the favorite summertime activities of 7th grade students at Parson Junior High. 4. If there are 275 students at Parson Junior High, predict

how many would choose visiting an amusement park as their favorite summertime activity.

32% Amusement Park

13% Beach

5. If there are 150 students at Parson Junior High, predict

how many would choose swimming or going to camp as their favorite summertime activity.

20% Camp

HISTORY For Exercises 6 and 7, use the histogram

U.S. Presidents Age at Inauguration

that shows the age of U.S. presidents at their inauguration.

10 8 6 4 2

9

4

9

4

–6 65

–6 60

–5 55

–5 50

–4 40

9

0

4

president will most likely be in at their inauguration.

12

–4

7. Predict the 10-year age range that the next U.S.

14

45

president will most likely be in at their inauguration.

16

Number of Presidents

6. Predict the 5-year age range that the next U.S.

24% Swim

Age at Inaguration

8. Predict the range that students in Mrs. Jeng’s

second period math class will most likely score. 9. Predict the range that students in Mrs. Jeng’s third

period math class will least likely score.

Mrs. Jeng’s First Period Test Scores 14

Number of Students

SCHOOL For Exercises 8 and 9, use the histogram that shows the test scores of Mrs. Jeng’s first period math class. Mrs. Jeng teaches three math classes of the same level in the morning.

12 10 8 6 4 2 0

51–60 61–70 71–80 81–90 91–100

Score