lesson 51 scientific notation for name -...

Saxon Math Course 2 L51-203 Adaptations Lesson 51

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Scientific Notation for Large Numbers (page 363)

• Numbers used in science are often very large or very small.

• Scientific notation is a way to express numbers as a product of a decimal and a power of 10.

Example: 9,461,000,000,000 = 9.461 × 1012

• The power of 10 shows where the decimal point is located when the number is written in standard form.

• To write a large number in standard form:

1. Shift the decimal point to the right the number of places shown by the positive exponent.

2. Use zero as a placeholder.

Example: Write 4.26 × 106 in standard form.

4.26 × 106 4260000 4,260,000 6 places

• To write a large number in scientific notation:

1. Place the decimal point to the right of the first nonzero digit.

2. Use the power of 10 to show the real location of the decimal point.

3. Omit terminal zeros.

Example: Write 405,700,000 in scientific notation.

405,700,000 4.05700000 4.057 × 108

8 places

Practice Set (page 365)

Write each number in scientific notation.

a. 15,000,000 × b. 400,000,000,000 ×

c. 5,090,000 × d. two hundred fifty billion ×

e. two point four times ten to the fifth ×

Write each number in standard form.

f. 3.4 × 106 g. 1 × 105

Compare:

h. 1.5 × 105 1.5 × 106 i. one million 1 × 106

j. Use words to show how 9.3 × 107 is read. nine point

Teacher Note:• Refer students to “Converting

TO Scientific Notation” and “Converting FROM Scientific Notation” on page 21 in the Student Reference Guide.


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1.

1 2 3 4 5

50

Test

Test Scores

60

70

80

90

100S

core

Bob’s scoreClass average

3. sides

______ inches

___ 9

___ ?

4. cans

_____ $

1 ___

___

? 3.36

3.36

cans ________

$ saved

6 ___

1 __

?

5. convinced+ unconvinced

total

6. a. twelve million

b. 17,600

7. Shift.

a. 1.2 × 104 =

b. 5 × 106 =

8. a. 1

__ 8 =

b. 87 1

__ 2 % =

9. a. 1 kg ≈ 2.2 lb

b. 176 lb ∙ kg

_______ lb

=

10. Fraction Decimal Percent

a. b. 40%

c. d. 4%

Written Practice (page 366)

2. average

7080

0+ 00

a.

b.

a.

b.

a.

b.

a. ×

b. ×

Use work area.

a. kg

_______ lb

b.


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11. percent of 360° 12. a.

15. Estimate the perimeter.

x

y

16. Estimate the area.

x

y

13. positive, single-digit, even numbers

a. sample space

b. probability that Luis is correct

14.

17. 24

___ x =

60 ___

40 18.

6 ___

4.2 =

n __

7

19. 5m = 8.4 20. 6.5 – y = 5.06

6.500 6.500

Written Practice (continued) (page 367)

25%

50%

d.b.

a.

c.1212 %

Sales Tax

$1 $0.06

$2 $0.

$3 $0.

$4 $0.

$5 $0.

ZAWD

YB XC

a.

b.

c.

d.

a. { , , , }

b.

x = n =

m = y =

b. a. ∠

b.


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28.

29. area of △ABC 30. See the Student Reference Guide.

21. 52 + 33 + √___

64 = 22. 16 cm ∙ 10 mm _______

1 cm =

23. 8 days 3 hr 15 min– 5 days 18 hr 50 min

days hr min

24. 3 yd 2 ft 5 in.+ 1 yd 9 in.

yd ft in. Simplify.

25. 6 2

__ 3

+ ( 5 1 __

4 – 3

7 __

8 ) =

5

1 __

4 =

___

– 3 7 __

8 =

___

6

2 __

3 =

___

+ =

___

27. Show two ways to evaluate x(x + y) when x = 0.5 and y = 0.6.

0.5( ) 0.5( )

( ) or

the D Property of

m


26. 3 1 __

3 ∙ ( 2

2 __

3 ÷ 1

1 __

2 ) =

y

x

Use work area. m∠A = m∠B = m∠C =


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Order of Operations (page 369)

• When more than one operation occurs in the same expression, perform the operation in the order listed below.

• Another good way to remember the order of operations is with the sentence “Please excuse my dear Aunt Sally.” Each initial letter stands for an order-of-operations word.

ParenthesesExponentsMultiplicationDivisionAdditionSubtraction

• A division bar may serve as a symbol of inclusion. Simplify above and below the bar before dividing.

Example: 32 + 3 ∙ 5

_________ 2 =

9 + 15 _______

2 =

24 ___

2 = 12

• Hint: Put parentheses around multiplication and division before solving the problem.

Example: 3 – 3 ÷ 3 + 3 ∙ 3 3 – (3 ÷ 3) + (3 ∙ 3) = 3 – 1 + 9 = 11


Simplify:

a. 5 + 5 ∙ 5 – 5 ÷ 5 = b. 50 – 8 ∙ 5 + 6 ÷ 3 =

c. 24 – 8 – 6 ∙ 2 ÷ 4 = d. 23 + 32 ÷ 2 ∙ 5

______________ 3

=

Evaluate:

e. ab – bc f. ab + a __

c

if a = 5, b = 3, and c = 4 if a = 6, b = 4, and c = 2

g. x – xy

if x = 2

__ 3 and y =

3 __

4

Teacher Note:• Refer students to “Order of

Operations” on page 22 in the Student Reference Guide.Order of Operations

1. Simplify within parentheses (or other symbols of inclusion) from innermost to outermost, before simplifying outside of the parentheses.

2. Simplify powers and roots.

3. Multiply and divide in order from left to right.

4. Add and subtract in order from left to right.


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1. ( × × ) ÷ ( + + ) = 2. 40 in. ∙ cm _________

in. =

3. 202.020 202.002

4. CD$

25

___

___ ?

5. a. pageshr

___ 1

? ___ pages

b. 330 202

c. No. We did not need to know

.

6. 75% =

7. a. 2,756,300,000 miles = ×

b. 4,539,600,000 miles

words:

miles

8. Shift.

a. 1.6 × 1010 dollars =

b. 2.4 × 108 dollars

words:

dollars

9. a. 3

__ 8 =

b. 6.5% =

10. 3. ___

27


did not

60 passengers

disembarked4

4 disembark

passengers

passengers

passengers

passengers pages

Use work area.

Use work area. Use work area.

b. a.

a.

b.

40 in. 100 cm


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11. Fraction Decimal Percent

a. b. 250%

c. d. 25%

12. 9 ) ___

70

13. 14.

15. area of triangle = 1 _ 2 bh 16. 8

__ f

= 56

____ 105

17. 12

___ 15

= w ___

2.5 18. p + 6.8 = 20

20.0 20.0

19. q – 3.6 = 6.4

6.4 20.0

20. 53 – 102 – √___

25 =


1.00.9

B

AD

C 12

8

6 6

Use work area. b. a.

f =

w =

q =

a.

b.

a.

b.

c.

p =


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21. Put parentheses around multiplication and division.

4 + 4 ∙ 4 – 4 ÷ 4 =

22. 4.8 – 0.24

_________ (0.2)(0.6)

=

4.84 0.24

0.2 0.6

23. 5 hr 45 min 30 s+ 2 hr 53 min 55 s

hr min s

24. 6 3

__ 4 + ( 5

1 __

3 ∙ 2

1 __

2 ) 6

3 __

4 =

___

+ =

___

25. 5 1

__ 2 – ( 3

3 __

4 ÷ 2 ) = 5

1 __

2 =

___

– =

___

26. estimated and exact answers

8.5758.5758.575

8.575

27. Shift.

0.8 × 1.25 × 106 =

28. Evaluate: ab + a __

b

if a = 4 and b = 0.5

29. 1.4 m ∙ 100 cm

_______ 1 m

= 30. football

basketball

baseball

total



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Ratio Word Problems (page 375)

• To solve ratio word problems:

1. Make and complete a ratio box.

Write the given numbers in the box.Write a letter in the box that answers the question asked.

2. Write a proportion using the numbers in the ratio box.

3. Solve the proportion.

Cross multiply.Divide by known factor.

Example: The ratio of salamanders to frogs was 5 to 7. If there were 20 salamanders, how many frogs were there?

salamanders

____________ frogs

5

__ 7

= 20

___ F

5 ∙ F = 7 ∙ 20

5F = 140

F = 140

____ 5

F = 28 frogs


Solve each of these ratio word problems. Begin by completing the ratio box.

a. The girl-boy ratio was 9 to 7. If 63 girls attended, how many boys attended?

boys

b. The ratio of sparrows to bluejays at the bird sanctuary was 5 to 3. If there were 15 bluejays in the sanctuary, how many sparrows were there?

sparrows

c. The ratio of tagged fish to untagged fish was 2 to 9. Ninety fish were tagged. How many fishwere untagged?

untagged fish

d. Calculate the ratio of boys to girls in your classroom. Then calculate the ratio of girls to boys.

boys girls boysgirls

xxx

____ x

girlsboys

xxx

____ x

Teacher Notes:• Review Hint #27, “Rate.”

• Review “Ratio” and “Proportion (Rate) Problems” on page 19 in the Student Reference Guide.

Ratio Actual Count

5 20

7 F

salamanders

frogs

Ratio Actual Count

9 637 B

girlsboys

Ratio Actual Count

5 Ssparrowsbluejays

Ratio Actual Count

2U

taggeduntagged


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2. percent in Jan. – Jun.

4. a. average

$3.95$3.95$3.953.95

$3.95

1.

3. boys in Apr. – Jun.

5. did not attend

a.

b. xx

___ 12

of 840

6. a. one trillion =

b. 475,000 =

7. Shift.

a. 7 × 102 =

b. 2.5 × 106 2.5 × 105


partpartwhole

a. ×

b. ×

8. a. 35 yd ∙ xxxft ______

xxxyd =

b. 2000 cm ∙ xxxcm _______

xxxcm =

9. 54 = ∙ ∙ ∙

36 = ∙ ∙ ∙

LCM = ∙ ∙ ∙ ∙

10. xxxmi

______ xxxhr

∙ xxxkm

______ xxxmi

≈ xxxkm

______ xxxhr

a.

b.

b. The average would

because this book costs

than the average.

per

Use work area.

a.

b.

a.

b.

a.

b.


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11. 12. a. 4 __

5 =

b. 0.06 =

13. m ___

cm

1 ___

xx

xx ___

? 14.

15. 16. 18

____ 100

= 90

___ p

17. 6

__ 9 =

t ___

1.5 18. 8 = 7.25 + m

19. 1.5 = 10n 20. √___

81 + 92 – 25 =

Fraction Decimal Percent

a. b. 150%

c. d. 15%

Use work area.


Ratio Actual Count

trumpetersdrummers

165

lilac rose

a.

b.

a.

b.

p =

t =

8.00 8.00

m =

n =


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21. parentheses for × and ÷

16 ÷ 4 ÷ 2 + 3 × 4 =

22.

23. 12 2

__ 3

+ ( 5 5 __

6 ÷ 2

1 __

3 ) = 12

2 __

3 =

x ___

xx

+ 12 2 __

3 =

x ___

xx

24. 8 3 __

5 – ( 1

1 __

2 ∙ 3

1 __

5 ) = 8

3 __

5

– 8 3 __

5

25. 26. estimate

6.85

4 1 ___

16

27. Evaluate: ab

___ bc

if a = 6, b = 0.9. and c = 5

28. 2 1 _ 2 dozen eggs = eggs

flatseggs

1 ___

xx

? ___

xx

29. 30.

3 yd 1 ft 7 1

__ 2

in.

+ 3 yd 2 ft 6 1

__ 2

in.

3 yd 1 ft xx

___ 72

in.

partpartwhole

ab

c

m∠b =

m∠c =

m∠a =

10.6010.6010.60

1 0.60


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c. What is Jenna’s hourly rate of pay? d. How much would Jenna earn working 20 hours?

a. The trip took how many hours to complete? b. During the trip the car used how many gallons of gas?

Rate Word Problems (page 380)

• We solve rate word problems the same way we solved ratio word problems:

1. Make and complete a rate box.

Write the given numbers in the box.Write a letter in the box that answers the question asked.

2. Write a proportion using the same numbers in the rate box.

3. Solve the proportion.Cross multiply.Divide by known factor.


Use a rate box to help you solve these rate problems.

On a 600-mile trip, Dixon’s car averaged 50 miles per hour and 30 miles per gallon.

Teacher Notes:• Review Hint #27, “Rate.”

• Review “Ratio” and “Proportion (Rate) Problems” on page 19 in the Student Reference Guide.

Rate Actual Count

miles 50 600

hours 1 A

Rate Actual Count

$ 68.80 C

hours 1

Rate Actual Count

$ D

hours 1 20

Rate Actual Count

pounds 1

$ E

Rate Actual Count

miles 30 600

gallons 1 B

Jenna earned $68.80 working 8 hours.

The price of one type of cheese is $2.60 per pound.

e. What is the cost of a 2.5-pound package of cheese?

f. How could we find the cost of a half-pound package of cheese?

D the cost per pound by .


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1. 1776 Declaration of Independence+ 1750

Jefferson died

1743 Jefferson bornyears lived

2. average

190 195197201

203

3. 4. lb

__ $

1 ___

___

?

5. (LCM ÷ GCF)

LCM of 4 and 6 GCF of 4 and 6

6. 80% =

7. Shift.

a. 405,000 =

b. 0.04 × 105 =

8. a. 106 ∙ 102 = 10 (ax · ay = ax+y)

b. 106 ÷ 102 = 10

( ax

__ ay

= ax–y )

9. a. 5280 ft ∙ _____

yd ___

ft =

b. 300 cm ∙

_____ mm

____ cm

=

10. 3.1415926

Ratio Actual Count

women 5

men M


a.

b.

a. ×

b. Use work area.

a.

b.


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11. a. mi

___ hr

___ 1

? ___

b. mi

___ hr

___ 1

___

?

13. 26

__ 22

A 23 B 24

C 13 D 3

14.

15. 16. a. I Property of

b. D Property

c. A Property of

d. I Property of

17.

19. 6.2 = x + 4.1

6.2 6.2

20. 1.2 = y – 0.21

1.20 6.20


18. The average score is likely to be b the

median score. The mean “balances” low scores with

h scores. The scores above the median are

not far enough a the median to allow the

balance point for all the s to be at or above

the m .

12. percent of 360°

Use work area.

Use work area.

a.

b.

x = y =

c.

d.

a.

b. b.

a.


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21. 24

___ r =

36 ___

27 22.

w _____

0.16 = 6.25

23. 112 + 13 – √_____

121 = 24. parentheses for × and ÷

24 – 4 × 5 ÷ 2 + 5 =

25. (2.5)2

______ 2(2.5)

= 26. 1 weeks 5 days 14 hr+ 2 weeks 6 days 10 hr

weeks days hr

27. 3 5 ___

10 + ( 9

1 __

2 – 6

2 __

3 ) =

9 1 __

2 =

___

– 6 2 __

3 =

___

3 5 ___

10 =

___

+ =

___

29. 30. Refer to the graph you drew in #29.


Simplify.

A (–1, 3)B (–4, 3)C (–4, –1)

X (1, 3)Y (4, 3)Z (4, –1)

y

x0–6 –5 –4 –3 –2 –1 21 3 4 5 6–1

–4–3–2

–5–6

654

23

1

28. 7 1 __

3 ∙ ( 6 ÷ 3

2 __

3 ) =

Use work area.

r = w =

a.

b.

c. ∠


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Average and Rate Problems with Multiple Steps (page 386)

• To find a missing sum of an average:

average × number of items = missing sum averagenumbers of items )

____________ missing sum

average × number of items

missing sum

• To find a missing number from an average:

average × numbers of items = sum

sum – sum of known numbers = missing number averagenumbers of items )

________ sum

average × number of items

sum

• To find a missing number that makes a new average:

average A × number of items = sum A

average B × total number of items = sum B

sum B – sum A = missing number

Example: After 4 tests, Annette’s average score was 89. What score does Annette need on her fifth test to bring her average up to 90?

average A × number of items = sum A 89 × 4 = 356

average B × total number of items = sum B 90 × 5 = 450

sum B – sum A = missing number 450 – 356 = 94

• To solve multi-step rate problems, break the problem into smaller parts and work each part separately.


Teacher Notes:• Introduce Hint #52, “Complex

Average.”

• Refer students to “Complex Average” on page 25 in the Student Reference Guide.

• Students may use a calculator for multi-step average problems.

known numberknown numberknown number

+ known numbersum of known numbers

sum– sum of known numbers

missing number

a. Tisha scored an average of 18 points in each of her first five basketball games. Altogether, how many points did she score in her first five games?

points

The sum is missing.

average × number of items = missing sum

b. The average of four numbers is 45. If three of the numbers are 24, 36, and 52, what is the fourth number?

A number is missing from the average.

average × number of items = sum

sum – sum of known numbers = missing numbers

18× 18

average

missing sum

45× 18

average

sum

2436

× 52 sum of known numbers


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c. After five games of bowling, Ralph’s averagescore was 91. After six games, his average was 89. What was his score in the sixth game?

A number is missing that makes a new average.

average A × number of items = sum A

average B × total number of items = sum B

sum B – sum A = missing number

d. Vin’s babysitting earnings are shown in the table in your textbook. If he wants to earn an average of at least $20 per month over 4 months, what is the minimum he can earn in June?

A number is missing from the average.

average × number of items = sum

sum – sum of known numbers = missing number

1. 2. A number is missing from the

average.

3. quarts$

12

___ x

1 __

?

4.


$20× $18

average

sum

$18$22

× $15 sum of known numbers

e. Ray is buying cheese to make macaroni and cheese. Cheddar cheese is on sale at $3.60 for 10 oz. American cheese costs $3.04 for 1 _ 2 lb. Ray needs at least 1 1 _ 2 lb cheese for the macaroni and cheese. Which type of cheese is the better buy?

(16 oz = 1 lb)

f. Carla drove 180 miles from her home to Houston in 3 hours. On her return home the traffic was slow and the trip took 4 hours. Find Carla’s average rate of speed (mph) for her drive to Houston, for her return trip, and for the round trip to the nearest mile per hour.

to Houston return trip round trip

180 mi

_______ 180 hr

= mph 180 mi

_______ 180 hr

= mph 360 mi

_______ 360 ihr

= mph

, ,

Ratio AC

13 Rwhite black 85

187678

81

$2.89 $2.89

inch 321 4

A B C

5. a. 3 ___

10 of 30 b.

3 ___

10 =

b. a.

91× 18

average A

sum A

89× 18

average B

sum B

89× 18

sum B sum A

Cheddar American

$lb

3.60

_____ x

? __

1

1 __

2 lb = oz

$ __

lb

3.04 _____

x

? __

1

Practice Set (continued) (page 389)


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6. Shift.

a. 675 million =

b. 1.86 × 105 =

7. a. 108 ∙ 102 = 10

b. 108 ÷ 102 = 10

8. a. 24 ft ∙ xxin

____ xxft

=

b. 60 mi

______ 1 hr

∙ xxxhr

______ xxmin

=

9. The product of two hundredths and twenty-five thousandths is five ten-thousandths.

10. a. $hr

7 ___

xx

? ___

xx b.

$ __

hr

xx ___

7

? ___

xx

11.

12. 13.

14. (6 × ) + 2 = 1 ft = ? in.

68 in.

15. a. lapsmin

5 __

4

? ___

xx

b. lapsmin

5 __

4

xx ___

?


$35.00 $35.00

FRACTION DECIMAL PERCENT

1 _ 5 a. b. %

c. 0.1 d. %e. f. 75%

Use work area.

a.

b.

16. Show two ways to evaluate b(a + b)

when a = 1 _ 4 and b = 1 _ 2 .

1

__ 2 ( 1 __

2 + )

1 __

2 ( 1 __

2 + )

( 1 __ 2 ) or

xx ___

x +

xx ___

x

Use work area.

17. 30

___ 70

= 21

___ x 18.

1000 _____

w = 2.5

x = w =

a. ×

b.

a.

b.

a.

b. 0.02 ×

a.

b.

a.

b. ,

c. ∠

a.

b.

c.


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21. 10 yd ∙ 36 in. ______

1 yd =

22.

Simplify.

23. 122 – 43 – 24 – √____

144 =


50 + 30 ÷ 5 ∙ 2 – 6 =

25. 6 2

__ 3

× 5 1

__ 4 × 2

1 ___

10 =

26. Work left to right.

3 1 __

3 ÷ 3 ÷ 2

1 __

2 =

27. 3.47 + (6 – 1.359) = 28. over, over, up

$1.50 ÷ 0.075 =

29. 30. Sum of angles in triangle is 180°.


6 – ( 7 1

__ 3

– 4 4

__ 5

) =

8 yd 2 ft 7 in.+ 3 yd 2 ft 5 in.

3 yd 1 ft 5 in.

6.000 5.000

3.470 5.000

Given

a. C

b. I Property

c. I Property

(4)(3.7)(0.25)

(4)(0.25)(3.7)

(1)(3.7)

3.7

2 5 ___

12 =

xx ___

xx

6 5

__ 6 =

xx ___

xx

+ 4 7 __

8 =

xx ___

xx

7 1

__ 3

= xx

___ xx

– 4 4

__ 5

= xx

___ xx

6 1 __

3 =

xx ___

xx

– 4 4 __

5 =

xx ___

xx

Use work area.

m∠a =

m∠c =

m∠b =


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1

4

3

2

1 2 3 4

Input Output x y

0 0

1 2

2 4

3 6

4 8

Plotting Functions (page 393)

• A function is a mathematical rule that tells the relationship between two sets of numbers.

• We have made function tables with input and output numbers before.


For this function table:

a. Write the rule in words.

To find the number of h , multiply the number of

d by 24.

b. Find the missing number in the table.

c. Express the rule as an equation. h =

d. Plot four points that satisfy this function:the number of bicycle wheels (y) is twice the number of bicycles (x).2x = y

• Put any number into a function (replace x with any number), and get an output number (the number for y).

Example: Here is a function table for y = 2x.

• To show all the points on a function, draw a line through the points already plotted. The arrowhead shows that the line continues (because you could put numbers larger that 4 into the function).

• We can graph or plot this function with a set of unconnected points on a coordinate plane. The input number is the x-coordinate and the output number is the y-coordinate for each point.

• This plot only shows a few of the points in the function.

• The line continues in both directions because a function also includes negative numbers. This line represents all the pairs of numbers for the function y = 2x.

Input Output days (d) hours (h)

1 24

2 48

5 120

10


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e. Make a function table of 4 pairs for y = 3x. Then plot the pairs and draw a line through the points.

f. Write the rule for this function as an equation. y =

Name another pair not named on the graph. (–3, )

g. The point on the graph in f that satisfies the function is

( , ). One way we can tell is the point

( , ) is on the line. Another way is to substitute

y = 4x with numbers and we get 4 = ( ).

1. a. sample space {H1, , , , ,

, T1, , , , , }

b. heads, prime numbertotal

x ___

xx =

x ___

xx =

decimal

2.

length ______ width

4 ___

xx =

xx ___

w


Use work area.

3. Read carefully.

$2 + [ 1 _ 2 hr

____ $

1 ___

xx

xx ___

? ] =

4. average

Ratio Actual Count

length 4 12

width 3 w

12 ft


x y

–1 –30 0

1

2

a. b.

85 84

60 84


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8. Shift.

a. six hundred ten thousand

b. 1.5 × 104 =

9. Round the answer.

m

___ yd

1 ___

xx

? ___

xx

10. a. 1 __

6 =

b. 1

__ 6

=

11. pennies = 1 dollar

pennies = 1 million dollars

12. 11 million 1.1 × 106

13. even, two-digit numberUse the times table.

14. Round off and change to feet. 15. Round off and change to feet.


b. a. × b. a.

5. Brand X oz$

xx

___ x

1 __

?

¢ per ounce

Brand Y oz$

xx

___ x

1 __

?

¢ per ounce

Brand is a better buy.

6.

7. a. Name two pairs of vertical angles.

∠Q and ∠T

∠R and ∠Q

part

part

whole

b. Name two angles that are supplemental to ∠RPS.

∠R and ∠S

Use work area.

QR

P

T S

Use work area. c.

a.

b.

×

47 in.12

35 in.12


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21.

Simplify.

22.

23. 18

___ 19

∙ 19

___ 18

= 24. 2 3

__ 4 + ( 5

1 __

6 – 1

1 __

4 ) =

25. 3 3 __

4 ∙ 2

1 __

2 ÷ 3

1 __

8 = 26. 3

3 __

4 ÷ 2

1 __

2 ∙ 3

1 __

8 =


16. °C

___ °F

100

____ x

xx ___

? 17.

3 ___

2.5 =

48 ___

c 18. k – 0.75 = 0.75

19. 152 – 53 – √_____

100 = 20. parentheses for × and ÷

6 + 12 ÷ 3 ∙ 2 – 3 ∙ 4 =

c = k =

0.75 0.75

5 yd 2 ft 3 in.+ 2 yd 2 ft 9 in.

2 yd 2 ft 9 in.

5 yd 2 ft 3 in.+ 2 yd 2 ft 9 in.

2 yd 2 ft 9 in.

5 1

__ 6

= x ___

xx

– 1 1

__ 4

= x ___

xx

2 3

__ 4

= x ___

xx

– 1 1

__ 4

= x ___

xx


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c.

30. a. b.

c. We put an a at the end

of the line to show that is c .

27. Look at the times table.

1, 4, 9, 16, 25, …

The first five numbers in the s

are the s of the first five counting

n . So the 99th number in the

sequence is .

28. Measure the shortest and longest sides of this triangle to the nearest millimeter.

The relationship of the two measurements is:

The longest side is

t the length

of the shortest side.

29. radius = 1 _ 2 diameter

m ___

cm

1 ___

xx

xx ___

?

r d1 2

2

3


Use work area. Use work area.

30

Use work area.

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Teacher Note:• Review “Exponential Powers”

on page 8 and “Converting TO Scientific Notation” on page 21 in the Student Reference Guide.

L E S S O N

57 Negative Exponents Scientific Notation for

Small Numbers (page 400)

• Look for a pattern in these examples:

• You subtracted the exponents as you divided.

• Now continue the pattern:

• A number with a negative exponent is the reciprocal of the same number with a positive exponent.

10−3 = 0.001 = 1 ____ 1000 10 3 = 1000 = 1000 ____ 1

10−3 and 10 3 are reciprocals.


Simplify:

a. 5−2 = b. 3 0 = c. 10−4 =

Write each number in scientific notation.

d. 0.00000025 e. 0.000000001 f. 0.000105

× × ×


103

103 �10

1� 10

1� 10

1

101

� 101

� 101

� 100 � 1

102

103 �10

1� 10

1

101

� 101

� 10� 10�1 �

110

�1

101

101

103 �10

1

101

� 10 � 10� 10�2 �

1100

�1

102

105

103 �10

1� 10

1� 10

1� 10 � 10

101

� 101

� 101

� 102 � 100

104

103 �10

1� 10

1� 10

1� 10

101

� 101

� 101

� 101 � 10

2 0 = 1

3−2 = 1 __

32 =

1 __

9

If a number a is not zero, then

a0 = 1

a−n = 1/an

• To write a small number in standard form:

1. Shift the decimal point to the left the number of places shown by the negative exponent.

2. Use zero as a placeholder.

Example: Write 6.32 × 10−7 in standardform.

.000000632 0.000000632

7 places to the left

Example: Compare: zero 1 × 10−3

1 × 10−3 = 0.001

zero < 1 × 10−3

• To write a small number in scientific notation:

1. Place the decimal to the right of the firstnonzero digit.

2. Use the power of ten to show the reallocation of the decimal.

Example: Write 0.0000033 in scientific notation.

0000003.3 3.3 × 10−6

6 places to the right

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1. 2. A number is missing that makes a new average.

88× 50

90× 50

3. 2($0.39 + 3 × $0.25) = 4. $pt

__ ?

__ 1

5. a. 2 __

5 of min =

b. 2 __

5 =

c. There are min in 1 hr.

6. a. 186,000 =

b. 0.00004 =


Write each number in standard form. j. I shifted the decimal point:

g. 4.5 × 10−7 . in g places to the left

h. 1 × 10−3 . in h places to the left

i. 1.25 × 10−5 . in i places to the left

Compare:

k. 1 × 10−3 1 × 10 2 l. 2.5 × 10−2 2.5 × 10−3

m. Use digits to write “three point five times ten to the negative eight.”

. × 10


Use work area.

a. ×

b. ×

7. Shift.

a. 3.25 × 10 1 =

b. 1.5 × 10−6 =

8. a. 2−3 =

b. 5 0 =

c. 10−2 =a.

b.

a.

b.

c.


7

Ratio Actual Count

5walkers

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9. a. 2000 mL ∙ mL _____

mL =

b. 10 L ∙ ___ qt

__ L =

10. composite numbers on

a number cube:

11. $tickets

__ ? __

1

13. Compare:

a. 2.5 × 10−2 2.5 ÷ 10 2

b. one millionth 1 × 10−6

c. 3 0 2 0

14.

15. 16. Evaluate 4acif a = 5 and c = 0.5.

17. estimate

$19.89 3.987

18. y = 3x ÷ 5 19. 20 2 + 10 3 – √___

36 =

xy

12.

.

Use work area.

Use work area.

a.

b.

c. b. a.

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20. 48 ÷ 12 ÷ 2 + 2(3) = 21.

3 yd 2 ft 1 in.– 1 yd 2 ft 3 in.

yd ft in.

22.

4 gal 3 qt 1 pt 6 oz+ 1 gal 2 qt 1 pt 5 oz

gal qt pt oz

Simplify.

23. 48 oz ∙ 1 pt

______ 16 oz

=

24. 5 1

__ 3 ∙ ( 7 ÷ 1

3 __

4 ) = 25.

5 1

__ 6 = _

3 5 __

8 = _

+ 2 7 ___

12 = _

26.

1 ___

20 = _

– 1 ___

36 = _

27. Shift.

(4.6 × 10−2) + 0.46 =

28. 10 – (2.3 – 0.575)

2.300.000

10.000+ 10.000

29. 0.24 × 0.15 × 0.05 = 30. 10 ÷ (0.14 ÷ 70) =



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Teacher Note:• Students will require scissors and

paper to complete the activity in the Student Edition.

L E S S O N

58 Symmetry (page 406)

• A figure on paper has reflective symmetry if it can be divided in half so that the halves are mirror images of each other.

• A line of symmetry divides a figure so that the halves are mirror images. The figures show the three lines of symmetry in an equilateral triangle.

• A figure has rotational symmetry if it re-appears in its original position more than once in a full turn.

• The square shown below has four lines of symmetry and rotational symmetry. The square is in its original position every quarter turn. (Turn your paper to see this.)

• Do the Activity on page 407.



• Figures in a coordinate plane can also have reflective and rotational symmetry. In the figure below, the y-axis is the line of symmetry.

(–2, 2)

(–3, 0)

(2, 2)

(3, 0)

y

x

a. Show the two lines of symmetry on this rectangle.

c. Which of these letters have rotational symmetry? Which have reflective symmetry?

V W X Y

All four have symmetry.

Only has symmetry.

b. The y-axis is a line of symmetry for a triangle. The coordinates of two of its vertices are (0, 1) and (3, 4). What are the coordinates of thethird vertex?

( , )

d. Circle the name of the polygon that doesnot necessarily have rotational symmetry.Sketch an example of the polygon without rotational symmetry.

rectangle square

triangle parallelogram

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1. kilometerstimes

__ 1

? __ 2. $

1 _ 2 hours __

1

? __

3. n ∙ 17 = 340

n =

n + 17 =

4. part

part

whole

5. First find the missing sum.

Then make a new average.

120+ 5

118124

142

6.

7. 8. a. 1.5 × 10 7 =

b. 2.5 × 10−4 =

c. 10−1 =

d. 10 0 =

9. galqt

1

__ 0

2 __

?

L

qt

1 __

? __

10. Round the answer.

3.45

multiple-choice5

not multiple-choice5

60 questions

___ questions

___ questions

___ questions

___ questions

___ questions

C

AO

B

a.

b.

b.

c.

a.

d.

a. , ,

b. ,

c.

d.

b.

c.

a.

2 gal 8 liters

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11. 20, 15, 10, , , , … 12.

13. a. To find y, m x by . 14.

15. 16. a. lines of symmetry

b. A regular pentagon have

r symmetry

because it looks the same

when I turn the page.

17. a. mi

hr

__ 1 __

?

b. mi

hr

__ 1 __

?

18. 1.5

___ 2

= 7.5

___ w

19. 1.7 – y = 0.17

1.7.00

20. 10 3 – 10 2 + 10 1 – 10 0 =


, , Use work area.

x y

0 0 3 12 6 24 2

D BC

A

w =

a.

b.

a. ( , )

b.

b.

c. y =

y =

a.

b.

c.

Use work area.

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21. 6 + 3(2) – 4 – (5+3) = 22.

1 gal 2 qt 1 pt+ 1 gal 2 qt 1 pt

gal qt pt Simplify.

23.

1 day 3 hr 15 min– 8 hr 30 min

24. 2 mi ∙ 5280 ft _______

1 mi =

25. 10 – ( 5 3 __

4 – 1

5 __

6 ) =

5 3

__ 4 = _

– 1 5 __

6 = _

1000– 1000

26. ( 2 1 __

5 + 5

1 __

2 ) ÷ 2

1 __

5 =

2 1

__ 5

= _

+ 5 1 __

2 = _

27. 3 3 __

4 ∙ ( 6 ÷ 4

1 __

2 ) = 28. Evaluate: b2 – 4ac

if a = 3.6, b = 6, and c = 2.5

29. least to greatestChange 3 _ 2 and – 1 _ 2 to decimals.

‒1, 3

__ 2

, 2.5, 0, – 1 __

2 , 2

30. Lindsey could double both n

before dividing, making the equivalent

d problem 70 ÷ .

She could also double both of these

n making 140 ÷ 10.a. , , , , ,

b. , , Use work area.

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Teacher Note:• Refer students to “Adding Two

Signed Numbers” on page 26 in the Student Reference Guide.

L E S S O N

59 Adding Integers on the Number Line (page 413)

• Integers: all counting numbers, their opposites, and zero (does not include decimals or fractions)

The dots on this number line mark the integers from –5 through +5.

• Signed numbers: all integers except zero

• Zero: neither positive nor negative(The sum of two opposites is always zero.)

+3 + –3 = 0

• Absolute value: a number’s distance from zeroWriting a vertical bar on each side of a number shows absolute value.

�3� = 3 The absolute value of 3 equals 3.

�–3� = 3 The absolute value of –3 equals 3.

�–2� + �2� = 4

• To add integers on the number line:

Always begin the problem at zero.Go right for positive.Go left for negative.

Example: (–2) + (+5) + (–4)

1. Start at zero2. Count left 2 units. (–2)3. Count right 5 units. (+5)4. Count left 4 units. (–4)The answer is –1.


Sketch on a number line to show each addition problem.

a. (–2) + (–3) b. (+4) + (+2)

c. (–5) + (+2) d. (+5) + (–2)

e. (–4) + (+4) f. (–3) + (+6) + (–1)

Find each absolute value:

g. �– 1 _ 4 � = h. �11� = i. �–0.05� =

Simplify:

j. �–3� + �3� = k. �3 – 3� = l. �5 – 3� =

m. 4362 ft above 0 ft sea level 126 ft below

n. $525

4

3 1 1025 4

3 1 1025 4

3 1 1025 4

0 1 2 3 4 5 6

311 0 2 54

2 1014 3 2 3 4


=

=

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1. 2 of 5-by-7 prints = 6 of wallet-size prints =

50× 50

3. average maximum for first seven days 4. 7° F 0° F –9° F

5. Ratio Actual Count

7redgreen

6.

won

failed towin

20 games

games

games

games

games

7. �‒3� �3� 8. a. 4,000,000,000,000 =

b. 3.67 × 10 9 miles =

9. a. 1 × 10−6 =

b. 1 mm 1 × 10−3 m

m m

10. 300 mm ∙ m ______

mm =

=

=

a.

b.

a. ×

b.

2. Maximum Temperature Readingsfor Tri-City Area, August 1–7

20 1 3 4 5 6 7

80

0

82

84

86

88

90

92

Tem

per

atur

e (F

)

Date

Average maximumtemperature for August

a.

b.

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11. 12. a. (+2) + (–5)

b. (–2) + (+5)

13. a. To find y, add to . 14.

15. 16. 4.4 = 8w

17. 0.8

___ 1

= x ___

1.5 18. n +

11 ___

30 =

17 ___

30

17

___ 30

= _

11

___ 20

= _

19. 0.364

______ m

= 7 20. Think reciprocal.

2−1 + 2−1 =


33 1 10 224

3 311 02 2 4

x y

0 12

2 14

8 20

12

50

yx

20

60

15

35

30

50

yx

20

60

15

35

30

b.

c. y =

Use work area.Use work area.

w =

x = n =

m =

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21. √___

64 – 2 3 + 4 0 = 22.

3 yd 2 ft 7 1 _ 2 in.

+ 1 yd 5 1 _ 2 in.

23.

1 qt 1 pt 6 oz

– 1 pt 12 oz 24. 2

1 __

2 hr ∙ 50 mi

______ 1 hr

=

25. ( 5 __ 9

∙ 12 ) ÷ 6 2 __

3 =


3 5 __

6 – ( 4 – 1

1 __

9 ) = 4 = _

– 1 1

__ 9

= _

3 5

__ 6 = _

– = _


( 5 5 __

8 + 6

1 __

4 ) ÷ 6

1 __

4 =

5 5 __

8 = _

+ 6 1

__ 4

= _

28. Evaluate: a – bcif a = 0.1, b = 0.2, and c = 0.3

29. average

a. 30 mi

______ hr

= mph

b. 30 mi

______ hr

= mph

c. mi _____

hr = mph

30.

votes for the winnertotal votes

_____

Candidate Votes

Vote Tally

Vasquez

Lam

Enzinwa

Use work area.

a.

b.

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L E S S O N

60 Fractional Part of a Number, Part 1

Percent of a Number, Part 1 (page 420)

• To find a fractional part of a number, use an is/of loop.

Example: Three fifths of 120 is what number?

is

of

3 __

5

? ____

120

120× 3

360

725 )

_____ 360

• If the problem has a decimal or percent, convert to a reduced fraction before using the loop.

Examples: What number is 25% of 88? What number is 0.6 of 31?

25% = 25

____ 100

= 1

__ 4 0.6 =

6 ___

10 =

3 __

5

is

of

1 __

4

? ___

88 22

4 ) ___

88

is

of

3 __

5

? ___

31 31

× 393

18.65 )

_____ 93.0


a. What number is 4

__ 5 of 71? b. Seventy-five hundredths of 14.4

is

of

4 __

5

? __ is what number?

75

____ 100

= is of

? _____

14.4

_____

c. What number is 50% of 150? d. Three percent of $39 is how

50% = is

of

? ____

150

_____ much money?

3% = is of

? ___

39

3 __

e. What number is 25% of 64? f. What is 12% of $250,000?

25% = is

of

? ___

___ 12% =

is

of __

? __

g. Find the sales tax on a $36.89 radio h. Find the total price of the radio in when the tax rate is 7 percent. problem g, including tax.

7% = is

of

7 __ _______

$36.89

$36.89.

radio

tax


Teacher Notes:• Review “Finding a Part (Fraction

or Percent) When the Whole Is Known” on page 14 in the Student Reference Guide.

• Students should use the method given on this worksheet for fractional part of number problems. Thus, students are not required to convert problems into equations.


3. a. d =

b.

4.

5. A number is missing from the average.

775

717478

78

6. a. 0.00000008 =

b. 67.5 billion =

7. 8. 23,000 ft 0 9000 ft


i. Find the total price, including 6 percent tax, for a $6.95 dinner, a 95¢ beverage, and a $2.45 dessert.

$6.95 dinner$0.95 beverage$2.45 dessert

t 6% =

is

of __

? __

$ .222 $2.222

total price


1.

7.0021.

Use work area.

=

=

9. What number is 3 _ 4 of 17?

is

of

__ ? ___

17

10. 40% of $65 is profit.

40% =

is

of

? ___

___

2. She bought: over, over, up

0.20 ) ______

10.00

There were twice as many as

she bought, so in all .

Magazines Dollars

10 2.00

part

part

whole

a.

b.

a. ×

b. ×

a.

b.

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11. a. 1

__ 3 0.33

b. �5 – 3� �3 – 5�

12.

13. a. (–3) + (–1)

b. (–3) + (+1)

14. a. b. Take half of the exponent of

each prime factor.

15. fraction of 360° 16.

17. 18.

19. p – 1 ___

30 =

1 ___

20

1 ___

20 = _

1 ___

30 = _

20. 9m = 0.117


12

16

13

a.

b.

c.

10

10

10

8

8

6 6

6

A

B

C

D

EF

10

10

10

8

8

6 6

6

A

B

C

D

EF

) ______

3600 ) ______

3600 ) ______

3600 ) ______

3600 ) ______

3600 ) ______

3600 ) ______

3600 ) ______

3600

10

10

10

8

8

6 6

6

A

B

C

D

EF

a.

b.

a. ∙ ∙

b. ∙ ∙

a.

b.

c.

a.

b.

a.

b.

p = m =

△

△

Use work area.

Use work area.

Fraction Decimal Percent

1 _ 8 a. b.

c. d. 125%

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3 2 + 4(3 + 2) – 8 ÷ 4 + √___

36 =

22. a. 3 __

4 ( 4 __

9 +

2 __

3 )

3 __

4 ∙

4 __

9 +

3 __

4 ∙

2 __

3

b. D Property

23. a. sample space

{AA, , AC, , , , CA, , }

b. outcomes with Atotal outcomes

____

25. ( 1 1 __

4 ÷

5 ___

12 ) ÷ 24 = 26. 6.5 – (0.65 – 0.065) =

0.65.

6.5.

27. 0.3 ÷ (3 ÷ 0.03) = 28. 3.5 cm ∙ m

_______ cm

=

29. The first d problem can be

multiplied by 100 ___ 100 to form the s

division problem. Since 100 ___ 100 equals

, the quotients are the

s .

$1.50

______ $0.25

= ¢

_______ ¢

30.

BC

A

a. ( , )

y

x0–6 –5 –4 –3 –2 –1 21 3 4 5 6–1

–4–3–2

–5–6

654

23

1

24. 3 3

__ 5

– ( 5 __ 6

∙ 4 ) =

3 3 __

5 = _

– _ = _

b.

Use work area.

Use work area.

b.

lesson 51 scientific notation for name -...

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