LESSON 31 Reading and Writing Name -...

Saxon Math Course 2 L31-121 Adaptations Lesson 31

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Reading and WritingDecimal Numbers (page 221)

• To read the decimal number 123.123:

We say “and” for the decimal point.

“One hundred twenty-threeand one hundred twenty-three thousandths”

We say “thousandths” to conclude naming the number.

• Decimal numbers and fractions are two ways to name parts of a whole. They are read the same way.

0.3 = 3 ___

10 Both are read “three tenths.”

0.21 = 21

____ 100

Both are read “twenty-one hundredths.”

Practice Set (page 224)

Teacher Notes:• Introduce Hint #46, “Decimal Place

Value (Digit Lines),” Hint #47, “Writing Numbers,” and Hint #48, “Tenths and Hundredths.”

• Review “Spelling Numbers” onpage 9 and “Place Value” onpage 11 in the Student Reference Guide.

• Color tiles, available in the Adaptations Manipulative Kit, can be used to demonstrate tenths and hundredths.

a. Write three hundredths as a fraction. Then write three hundredths as a decimal.

____ , .

b. Name the shaded part of this circle both as a fraction and as a decimal.

000 ____

000 , .

c. In the number 16.57349, which digit is in the

thousandths place?

d. The number 36.4375 has how many decimal

places?

Use words:

e. 25.134 and

f. 100.01 and

Use digits:

g. one hundred two and three tenths .

h. one hundred twenty-five ten-thousandths .

i. three hundred and seventy-five thousandths .

j. What little word tells you that a number will have both whole-number and fraction parts? a


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1. $32.54× $26.47

$89.89× $26.47

2. average

326288349

× 401

3.

months$

12

____ 100

100

____ ?

4. 1607 Jamestown+ 1492 Columbus

5. Divide the perimeter of the s

by to find the length of a side. Then

m the length of a side by

to find the p of the

h .

6. 80% =

a. used

b. left

7. 481,462

a. nearest hundred-thousand

b. nearest thousand

8. 49,623 20,162

9. a. fraction

b. decimal

c. percent

10. hundredths place

9.87654

Written Practice (page 225)

Use work area. b. a.

a.

b.

b. .a. c.


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14. a.

b. √_____

2025 =

Take half of the exponent of

each prime factor. Multiply.

15. Follow the instructions in the textbook to complete the drawing.

11. a. 3 ___

10 0.3

b. 3 ____

100 0.3

13. a. 5 __

0 =

15 ___

24

b. 7 ___

12 =

00 ___

24

c. 0

__ 6

= 4 ___

24

16. not redtotal

000

____ 000

=

Written Practice (continued) (page 226)

12. a. (3, 3), (3, –3), (–3, –3) and (–3, 3)

17. 4 __

3 ( 3 __

4 x ) =

4 __

3 ∙ 1 __

4 Given

( 4 __ 3

∙ 3 __

4 ) x =

4 __

3 ∙

1 __

4 a. A Property

1x = 4

__ 3 ∙

1 __

4 b. C

x = 4

__ 3

∙ 1 __

4 c. I

x = 1

__ 3

4 __

3 ∙

1 __

4 =

1 __

3

a. ∙

b.

b. units c. units2

a.

b.

c.

a.

b. ∠

Use work area.

a.

b.


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18. 9n = 6 ∙ 12 19. 90° + 45° + b = 180° 20. 1

__ 2

= 0

__ 0

+ 2 __

3 =

0 __

0

21. 1 __

2 – ( 3 __

4 ∙

2 __

3 ) = 22. 3

5 __

6 =

0 __

0

– 3 1

__ 3

= 0 __

0

23. 5 __

8 ∙ 2

2 __

5 ∙

4 __

9

0 0 ___

00 ∙ 00

___ 00

∙ 00

___ 00

=

24. 2 2 __

3 ÷ 1

3 __

4

0 0 ___

00 ×

00 ___

00 =

25. 1 7

__ 8

÷ 3

0 0 ___

00 ×

00 ___

00 =

26. 3 1 __

2 =

0 __

0

+ 1 5

__ 6 =

0 __

0

27. 5 1

__ 4

= 0 __

0

– 1 5

__ 8

= 0 __

0

28. a = 3 b = 4

b

__ a +

a __

b =

29. k = 10n

the 6th term in words

10, 100, 1000, …

30. a full circle = 360°

a. 1 _ 2 of a circle

b. 1 _ 4 of a circle

c. 1 _ 8 of a circle


c. b. a.

n = b =

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• The metric system is a system of measurement used throughout most of the world.

• Use decimal numbers with the metric system.

• Units of length in the metric system are the meter, kilometer, centimeter, and millimeter.


a. The closet door is about 2 meters tall. b. A 1-gallon plastic jug can hold about how many How many centimeters is 2 meters? liters of milk? One quart is a little more than one liter.

Check the label on a gallon bottle to be exact.

mc

___ cm

1 ___

2 ___

L E S S O N

32

Teacher Notes:• Introduce Hint #49, “Reading Metric

Rulers,” and Hint #50, “Gram/Kilogram Manipulatives.”

• Review “Equivalence Table for Units” and “Formula for Celsius to Fahrenheit Conversion” on pages 1 and 2 in the Student Reference Guide.

• Gram and kilogram weights can be found in the Adaptations Manipulative Kit.

Saxon Math Course 2 L32–125 Adaptations Lesson 32

Metric System (page 228)

• Use the chart below to convert metric length to U.S. Customary length.

• Units of liquid capacity in the metric system are liters and milliliters.

• Units of weight in the metric system are grams and kilograms.

Examples of Metric Prefixes

Prefix Unit Relationship

kilo-hecto-deka-

deci-centi-milli-

kilometer (km)hectometer (hm)dekameter (dkm)

meter (m)decimeter (dm)centimeter (cm)millimeter (mm)

1000 meters100 meters10 meters

0.1 meter0.01 meter0.001 meter

• Temperature in the metric system is measured in degrees Celsius (°C).

• The metric system is based on powers of 10. A prefix tells what power of 10 the base unit is multiplied by. For instance, the prefix “kilo” tells to multiply a unit by 1000. So 1 kilometer is 1000 meters.

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1. average

42483584

9418

2. hr

____ min

1 ___

? ____

206 3.

440 _____

1760 =

4. a. sides

cm

4 ___

1 __

?

Now find the perimeter of the pentagon.

b. D the perimeter of the square by to find

the length of a side. Then m the length of a

side by to find the perimeter of the

p .

6. 313

489



c. A metric ton is 1000 kilograms, so a metric ton is about how many pounds? A kg mass weighs about

2.2 pounds.

kglb

1 ___

1000 _____

d. A temperature increase of 10° on the Celsius scale is equivalent to an increase of how many degrees on the Fahrenheit scale? (See Example 4 in textbook.)

°C

___ °F

100

____ 180

10

___

e. After running 800 meters of a 3-kilometer race, Michelle still had how many meters to run?

kmm

1 ___

3 __

? – 800

f. A 30-cm ruler broke into two pieces. One piece was 120 mm long. How long was the other piece?

cmmm

1 ___

30 ___

? – 120

g. About how many inches long was the ruler in exercise f. before it broke? Round off.

in.cm

1 ___

? ___

30

5. 3,197,270

a. nearest million

b. nearesthundred-thousand

7. a. about love and chivalry b. not about love and chivalry

8. not shaded a. fraction

b. decimalc. percent

Practice Set (continued) (page 231)

a.

b. a.

a.

b.

a.

b. .

c.

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12. a. 2500 in expanded notation

b.

c. ) ______

2500 =

14. Complete the triangle. Measure the angles to check that they measure 45°.

15. a. area of smaller square b. area of larger square c. total area

16. perimeter 17. 10 ∙ 6 = 4w 18. 180° – s = 65°

180°× 165°

19. 1 __

4 =

___

3

__ 8

=

___

+ 1 __

2 =

___

20. 5 __

6 =

___

– 3

__ 4 =

___

13. liters$

35

___ 1 ___


9. 3.025 in words

and

Take half of the exponent of each prime factor. Multiply.

10. seven and forty hundredths meters

11. $15 ∙ 2 ____

2 1

__ 2 · 2

w =

Use work area.

Use work area.

c.

a. ( × ) + ( × )

b. ∙

a.

b.

c.

.

s =

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21. 5 ___

16 =

___

– 3 ___

20 =

___

22. 8 __

9 ∙ 1

1 __

5 ∙ 10

___ ∙

___ ∙

___ =

23. 6 1 __

6 =

___

– 2 1

__ 2 =

___

24. 4 5 __

8 =

___

+ 1 1 __

2 =

___

25. 2 __

3 + ( 2 __

3 ÷

1 __

2 )

26. 25

___ 36

∙ 9 ___

10 ∙

8 ___

15 = 27. estimated and exact answers

5 2 __

5 ÷

9 ___

10

___ ×

___ =

28. estimated and exact answers

7 3

__ 4

=

___

+ 1 7

__ 8

=

___

30. a. chord but not a diameterb. central right anglec. inscribed angle



29. (–5, 3), (–5, –2), and (2, –2)

a. coordinates of the fourth vertex

b. area

a. ( , )

b.

a.

b. ∠

c. ∠


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Comparing Decimals Rounding Decimals (page 235)

• To compare decimal numbers, write the numbers vertically with the decimal points aligned. Then compare each place value.

Examples: 0.12 0.012 0.12 0.12 > 0.012 0.012

0.4 0.400 0.4 0.4 = 0.400 0.400

• Terminal zeros to the right of the decimal point have no value:

1.3 = 1.30 = 1.300 = 1.3000

• To round a decimal number:

1. Underline the place value you are rounding to.

2. Circle the digit to the right.

3. If the circled digit is 5 or more, add 1 to the underlined number.If the circled digit is less than 5, do not change the underlined number.

4. Everything after the underline becomes zero.

5. Remove all terminal zeros after the decimal point.

Examples: 3.14 1 59 3.14000 3.14

43 9 6.4315 4400.0000 4400


a. 10.30 10.3 10.3010.3

b. 5.06 5.60 5.065.60

c. 1.1 1.099 1.11.099

Teacher Note:• Review Hint #12, “Comparing

Numbers,” and Hint #42, “Estimating or Rounding.”

d. Round 3.14159 to the nearest ten-thousandth.

.

e. Round 365.2418 to the nearest hundred.

f. Round 57.432 to the nearest whole number.

g. Simplify 10.2000 by removing extraneous zeros.

.

h. Estimate the sum of 8.65, 21.7, and 11.038 by rounding each decimal number to the nearest whole

number before adding. 08.65

21.7

11.038


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1. We m 5 times 12 inches to find the number

of inches in 5 f . Then we a

8 inches to find the t number of inches in

5 feet, 8 inches.

(5 × 12) + 8 = 62 inches

3. 176,581+ 189,862

4. Complete the function table.

5. a. perimeter of hexagon cm

each side of hexagon cm

perimeter of octagon

b. Divide the p of the h by

to find the length of a side. Multiply the

length of a side by to find the perimeter of

the o .

7. 8. a. nearest hundredth

15.73591

b. 15.73591

3.14

6.

2. average

424338475152

+ 49

Use work area.

Use work area.

Use work area.

a.

b.

c.

.


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9. a. 150.035 b. 0.0015

words: words:

and

11. a. 0.128 0.14 0.1280.14

b. 0.03 0.0015 0.030.0015

12.

a. centimeters

b. millimeters

13. Draw ray OD so that angle COD measures 60°. 14. whole number but not a counting number

n ∙ =

15.

∙ ∙ ∙ ∙ =

16. 8m = 4 ∙ 18

17. 135° + a = 180°

180° 135°

18. 3 __

4 =

00 ___

00

5

__ 8

= 00

___ 00

+ 1

__ 2

= 00

___ 00

10. a. one hundred twenty-five thousandths

b. one hundred and twenty- five thousandths

Use work area.

a. .

b. .

b. a.

Use work area.

m =

a =

a.

b.


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19. 3

__ 4 =

00 ___

00

– 1 __

6 =

0 ___

00

20. 4 1 __

2 =

00 ___

00

– 4 3 __

8 =

0 ___

00

21. 3

__ 8

∙ 2 2

__ 5

∙ 3 1

__ 3

xxx

____ x

∙ xxx

____ x

∙ xxx

____ x

=

22. 2 7 ___

10 ÷ 5

2 __

5

x ____ xxx ×

x ____

xxx =

23. 5 ÷ 4 1 __

6

x ____ xxx ×

x ____

xxx =

24. 6 1 __

2 =

00 ___

00

– 2 5 __

6 =

0 ___

00

25. 3 __

4 + ( 1 __

2 ÷

2 __

3 ) 26. There are 4 aces in a deck.

not aces/cards in deck

00

___ 52

=

27. a. 54 = 54 + y

y =

b. I Property of a

28. 5 ÷ 4 1 __

6

The quotient will be than 1

because a larger number is d

by a s number.

29. The mixed numbers are g than

and 5, so the sum is g

than . The numbers are l

than 9 and , so the s is

less than 15.

30. Complete the triangle. Label the 30° angle and the 60° angle.

Use work area. Use work area.



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Decimal Numbers on the Number Line (page 241)

• If each centimeter on a centimeter ruler is divided into 10 equal parts, then each part is 0.1-centimeter (1-millimeter) long.

Find the length of this segment

a. in millimeters. 23 mm

b. in centimeters. 2.3 cm

• If a number line is divided so there are 100 equal parts between whole numbers, then each mark is one hundredth (0.01) from the next mark.

Find the number on the number line indicated by each arrow.


a. Find the length of this segment in b. Find the length of the segment to the nearest

centimeters. . millimeter.

c. The scale above can be precise within what fraction of a millimeter?

d. Seventy-five centimeters is how many e. Carmen’s wardrobe closet is 1.57 meters tall. meters? 1 cm = 0.01 m How many centimeters tall is it? 1 m = 100 cm

.

f. What point on a number g. What decimal number names the point marked by A on this line is halfway between number line? 2.6 and 2.7?

.

.

h. Estimate the length of this segment in centimeters. Then i. 4.6 4.45

use a centimeter ruler to measure its length.

j. 2.5 cm 25 mm

1 cm = 10 mm

4.6 4.45


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1. average 2. parts per cubic meter

3. $12.55 $13.95

4. 5.

perimeter of square

each side of square

7. First round the length to and

the width to . Then

m the l and the

w .

8. nearest thousandth

7.49362

9. a. 200.02 b. 0.001625

words: words:

and

10. a. one hundred seventy-five millionths

b. three thousand, thirty and three hundredths


188212

203

1698 1024

6. 40% =

a.

b.

parts per

Wright brothers’ flight years later landed on the moon

1903 1966

Use work area. .

a. . b. .

$12

Use work area.

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11. a. 6.174 6.17401

b. 14.276 1.4276

12.

a. centimeters

b. millimeters

13. What decimal number?

15. a. halfway between 7 and 8

b. halfway between 0.7 and 0.8

16. 15 ∙ 20 = 12y 17. 180° = 74° + c

180° 174°

18.

19. 20.


a. .

b.

a. .

b. .

5 __

6 =

x ___

xx

2 __

3 =

x ___

xx

+ 1 __

2 =

x ___

xx

5 ___

36 =

x ___

xx

– 1 ___

24 =

x ___

xx

5 1 __

6 =

x ___

xx

– 1 2 __

3 =

x ___

xx

6.1746.17401

14.27611.4276

b. a.

14.


.

b. ( , ) c.

y = c =


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21. 1 ___

10 ∙ 2

2 __

3 ∙ 3

3 __

4 22. 5

1 __

4 ÷ 1

2 __

3

x ____ xxx ×

x ____ xxx =

23. 3 1

__ 5 ÷ 4

x ____ xxx ×

x ____ xxx =

24. 6 7 __

8 =

x ___

xx

+ 4 1

__ 4 =

x ___

xx

25. 1 __

8 + ( 5 __

6 ∙

3 __

4 ) =

26. 1 mm = 0.1 cm 1 cm = 10 mm

24 mm = cm 3.6 cm = mm

Now find the difference

a. in cm

b. in mm

27. equivalent to 22 ∙ 23

Add the exponents.

A 25 B 26 C 12 D 24

28. least to greatest

0.365 0.3575 0.36

29. x = 5 y = 10

y __

x – x =

30. a. red

_____ total

xxx

____ x =

___ b.

white _____

total

xxx ____

x =

___

c. blue

_____ total

xxx

____ x d.

green ______

total

xxx ____

x =


a.

b.

, ,

Use work area.


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Adding, Subtracting, Multiplying, and Dividing Decimal Numbers (page 247)

• The chart below summarizes the rules for adding, subtracting, multiplying and dividing decimal numbers.

• Notice that there are two kinds of decimal division problems: dividing by a whole number and dividing by a decimal number. For the next few lessons, we will divide only by a whole number.

• Following are examples using the Decimals Chart:

3.6 + 0.36 + 36 5 – 4.32 0.23 × 0.4 0.0144 ÷ 8 A and E A, E, and F B C and F

• When multiplying or dividing a decimal by 10, 100, or 1000, there is a shortcut.

Shift the decimal point one place for each zero.

When multiplying, shift the decimal to the right.

When dividing, shift the decimal to the left.

Examples: shift shift 3.257 × 100 = 325.7 3.257 ÷ 10 = 0.3257


a. 1.2 + 3.45 + 23.6 b. 4.5 + 0.51 + 6 + 12.4 c. What is the perimeter of this rectangle?

d. Drew ran .50 meters in 8.46 seconds. e. 16.7 – 1.936 f. 12 – 0.875Mathea ran 50 meters in 8.52 seconds. Drew ran how many seconds faster than Mathea?

Teacher Notes:• Introduce Hint #51, “Decimal

Arithmetic Reminders Chart.”

• Refer students to “Multiplication and Division of Decimal Numbers by 10, 100, 1000” and “Decimal Arithmetic Reminders Chart” on pages 6 and 7 in the Student Reference Guide.

+ – × ÷ by whole ÷ by decimal

A Line up the decimal points.

B Multiply. Then count decimal places.

C Decimal point is up. D Decimal point is over, over, up.

E 1. Place a decimal point to the right of a whole number.

F 2. Fill empty places with zeros.

Decimals Chart

3.600.36

+ 36.0039.96

5.00– 4.32

0.68

0.00188 )

_______ 0.0144

0.23× 00.4

0.092

2 places1 places3 places

1.201.20

+ 1.20

4.501.201.20

+ 1.20

8.52 8.46

16.736– 11.236

12.736– 11.236


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g. h. (0.06)2 i. Six inches is how many centimeters? (1 in. = 2.54 cm)

in.

___ cm

1 ___

xx

6 __

?

j. 0.3 × 0.2 × 0.1 = k. shift l. What is the area of (0.04)(10) = this rectangle?

m. 14.4 ÷ 6 n. 0.048 ÷ 8 o. 3.4 ÷ 5 p. 0.3 ÷ 6short division

q. A loop of string 0.6 meter long is arranged to form a square. sides

_____ m

4 ___

xx

1 __

?

How long is each side of the square?

1. A all the bills together and

d by .

2.

3. mo

___ $o

x ______

15.60

1 __

?

Now compare.

4. Faster means less time.

(1 × ) + 22 = s 1 min = ?s

Now find Li Jinyi’s time.

5. perimeter of square

each side of square

6. a. 1 ___

11 of

110 ____

1 =

b.

4.2 0.24 0.06

0.06

) ______

1 4 .4 ) ________

0. 0 4 8 ) ______

3. 4 0 ) ______

0. 3 0

7. a. area of 10 × 10

b. area of 6 × 6

c. area of figure

8. not shaded

c.

a.

b. a. c.

b. .


Use work area.

2 1

__ 2

= x ___

xx

1 3

__ 4

= x ___

xx

a.

b.

110– 110


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10. a. 100.075

and

b. twenty-five hundred-thousandths

.

9.

11.

12. a. 10.38 gal

$2.28 9 ___

10

b. It is p to

pay 9 __ 10 of one cent.


14. This figure illustrates the multiplication of which two decimal numbers? What is their product?

15.

16. 15x = 9 · 10 17. f + 4.6 = 5.83 5.83 4.63

18. 8y = 46.4


a. ( , ) b.

Use work area.

a. . b.

Use work area. .

× =

a.

b.

x = f = y=


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19. w – 3.4 = 12 12.0 13.4

20. 3.65.65.65

33.65

21. 1 1 __

2 =

x ___

xx

2 2 __

3 =

x ___

xx

+ 3 3

__ 4 =

x ___

xx

22. 1 1 __

2 ∙ 2

2 __

3 ∙ 3

3 __

4 23. 1

1 __

6 – ( 1 __

2 +

1 __

3 ) =

1 __

2 =

x ___

xx

+ 1

__ 3 =

x ___

xx

24. 3 1 ___

12 =

x ___

xx

– 1 3

__ 4

= x ___

xx

25. shift

1.2 ÷ 10 =

26. (0.3) (0.4) (0.5) =

27. 5 __

6 ∙

2 __

2 =

10 ___

12

I Property of multiplication

28. a. Inches Centimeters

1 2.54 2 3 4

b. Rule: For every i there are

cm.

29. a. 6.452 4.912

b. 4.2 0.9

30. Complete the triangle. Label the 45° angles.



Use work area.Use work area.

– 0000

× 0000

estimate

calculate

estimate

calculate

w =


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Ratio Sample Space (page 255)

• Ratio is a way to describe a relationship between two numbers.

• The ratio of 3 to 4 can be written several ways:

with the word to 3 to 4

as a fraction 3 _ 4

as a decimal number 0.75

with a colon 3:4

• Write ratios as reduced fractions but not as mixed numbers.

Example: In a class of 28 students, there are 12 boys. What is the boy-girl ratio?

First find the number of girls in the class.12 boys

+ 1? girls28 total

12 boys+ 16 girls

28 total

Write the ratio as a reduced fraction. boys ____ girls 12 __ 16 = 3 _ 4

• The sample space is a list of all possible outcomes in a probability experiment.

• The sample space can be shown in different ways.

Example: A coin is tossed once, and then it is tossed again. Each time a coin is tossed, it can land either heads (H) or tails (T).

A tree diagram shows the process of the probability experiment.

1st toss 2nd toss

H

H

T

T

H

T

Outcomes

HH

HT

TH

TT

The outcomes from the tree diagram can be summarized in braces or a table.

{HH, HT, TH, TT}

• The fundamental counting principle says that you multiply the number of outcomes of each part of an experiment to find the total number of outcomes.

• Tossing a coin has 2 outcomes (heads or tails). So tossing a coin twice has 2 × 2 = 4 outcomes.


Teacher Notes:• Review Hint #29, “Probability.”

• Refer students to “Ratio” on page 22 and “Number Cube Chart” on page 25 in the Student Reference Guide.

• Review “Probability, Chance, Odds” on page 25 in the Student Reference Guide.

a. biglittle

___ =

___ b. boys

+ 14 girls.30 total

boys

_____ girls

___ =

___

Outcomes for 2 Coin Tosses

HH HT

TH T T


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c.

c. 3 won+ lost

8 total

won

____ lost

___

d. 5 red

+ 3 bluetotal

blue

_____ total

___

e. The name for the list of all possible outcomes

of a probability experiment is s

s .

f. { , , , }H T

g.

h. tails, less than 3total possible

___ =

i. Which sample space below is the better way to show the possible outcomes of rolling two number cubes? Why? Use the better sample space to find the probability of rolling a sum of 7.

Sample space 1: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Sample space 2:

The better way is sample space # because

it shows more than one way to roll most sums.

It shows all combinations.

combinations of 7total possible

___ =


1. cities with morecities with less

___ = 2. average

66.320.118.345.850.8

16.5

3. pagesdays

___

1

? __

7

4. 59.48 56.24

5. 40% =

a. never played rugby

b. had playednever played

___ =


a.

b. to

to


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c.

11. heads, eventotal possible

_____ =

6. One way to find BC in m is

to first convert AB to mm and AC to

mm. Then s 40 mm from

mm.

7. length = width + 5 cm

a. area

b. perimeter

8. estimate

1014 1390 1936

10. a. twelve million

b. twelve millionths

12. 13.

14. a. 85% = 85

____ 100

= b. 144

____ 600

= ∙ ∙ ∙ ∙ ∙ ________________ ∙ ∙ ∙ ∙ ∙ =

• List two factors of each given number. • Continue to factor until each factor is

a prime number. • Write the prime factors in order. • Cancel matching factors.


9. a. Round to three decimal places. a. 6.857142

b. 6.8571420 1.9870

15. estimate 6 3

__ 4 hr

$10.90

She earned a little

than .

Use work area.

Use work area.

.

× =

a.

b.

b. .

a.

b. a. .

b. a.


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c.

16.

19. estimated and exact

4.274.27

4.27

20. estimated and exact

4.200 4.270

21. Convert.

3

1 __

2 =

___

1 1 __

3 =

___

+ 2 1 __

4 =

___

22. 3 1 __

2 ∙ 1

1 __

3 ∙ 2

1 __

4 =

25. Multiply the reciprocal.

2 3 __

4 ÷ 4

1 __

2 =

26. Multiply the reciprocal.

5 – ( 2 __ 3 ÷

1 __

2 ) =

27. 1.4 ÷ 8 = 28. (0.2)(0.3)(0.4) =

29. a. shift 12.25 × 10 =

b. shift 12.25 ÷ 10 =


17. 8y = 122 18. w

__ 4 = 1.2

30. area = 25 units2

each side = units

23. 3 5 __

6 – ( 2 __

3 –

1 __

2 ) =

2 __

3 =

___

– 1

__ 2 =

___

24. 8 5 ___

12 =

___

– 3 2 __

3 =

___

b. ∠

c. ∠ a. ∠

( , ),( , ),( , ),( , )

w =y =

a.

b. Use work area.


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Area of a Triangle Rectangular Area, Part 2 (page 264)

• A triangle has a base (b) and a height (h).

• The base is any of the three sides. The height is the perpendicular distance from the base to the opposite vertex of the triangle.

• Since a triangle has three sides and any side can be the base, a triangle may have three base-height orientations, as shown by rotating these triangles.

• Do the Activity on pages 265 and 266 of the Student Editon to understand this formula:

area of a triangle = 1 __

2 bh or

bh ___

2

• To find the area of a complex shape: Example:

1. Divide the shape into rectangular parts.

2. Find the area of each part.

3. Add the areas to find the total area.

Sometimes subtracting a “ghost” area (the area that is missing) from a rectangle is easier.There are three ways to find the area of this shape.


Find the area of each triangle. Dimensions are in centimeters. area of a triangle = 1 _ 2 bh or bh __ 2

a. b.

12

8 1010

c.6

6

d. Find the area. 14 m

5 m10 m

20 m

AB

e. Find the area by subtracting the area of the small square from the area of the large rectangle.

Teacher Notes:• Review Hint #33, “Perimeter of

Complex Shapes.” (Students must find the length of missing sides to find the area of a complex shape.)

• Review “Geometric Formulas” on page 29 in the Student Reference Guide.

• Students will need paper, a ruler, a protractor, and scissors to complete the activity in this lesson.

10 in.

4 in.

4 in.

12 in.


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f. Find the area. 6 in.

8 in.

24 in.

20 in.

A

B

g. I found my answer to f by a the areas

of two r .

h. Write two formulas for finding the area of a triangle.

A =

A =


1. + 2 played+ 2 cancelled

3 total games

playedcancelled

___

2. average

4753625646

48

3. 11.611.611.6

11.6

5. 6. perimeter

x = in.

y = in.

7. Two methods:

1. Add area of both parts.

2. S area of cut-out part from

b rectangle.

Now find the area.

8. a. 5 __

6 =

? ___

18

b. ?

__ 8

= 9 ___

24

c. 3

__ 4

= 15

___ ?

4. Subtract $1.30 from $ to find how much the

gallons of milk cost. Then d that number by

to find how much e gallon cost.

Use work area.

to

a.

b.

c.

b. a.


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9. a. decimal shaded b. decimal not shaded c. percent not shaded

10. 3184.5641

a. two decimal places

b. nearest hundred

11. a. 0.00025

b. sixty and seven hundredths

.

12. a. 2% = 2 ____

100 =

b. 720

_____ 1080

=

13. Find the length of segment BC. 14. • Intersect with oblique parallel lines.• Then shade the enclosed region.

15. a. perimeterb. areaarea of triangle = 1 _ 2 bh

16. 0.2 + 0.3 0.2 × 0.3

17. a. { , , , , ,

, , , }

b. Atotal possible

___

18. 7 ∙ 8 = 4x

= 4x


AA BB

CC

Use work area.

a.

b.

Use work area.

Use work area.

a.

b.

x =

a. .

b. .

c. a. . b.

{{{


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c.

19. 4.2 = 1.7 + y

4.2 1.7

20. m – 3.6 = 0.45

3.60+ 0.45

21. 4.5

___ w

= 3 22. 3

__ 5 ∙ 12 ∙ 4

1 __

6

23. 5

__ 6 =

___

1 3 __

4 =

___

+ 2 1 __

2 =

___

24. 5

__ 8

+ ( 1 __ 2 +

3 __

8 ) =

1

__ 2 =

___

+ 3 __

8 =

___

25. 3 9 ___

20 =

___

– 1 5 ___

12 =

___

26. a = 3 1

__ 3 b = 5

a __

b = a ÷ b

27. 22 ∙ 22 ∙ 22 = 2n 28. shift

a. 0.25 × 10 = b. 0.25 ÷ 10 =

29. Plot (3, –3). Which quadrant is the point in?

30. perimeter


b. a.

m = w =

n =

y =


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Interpreting Graphs (page 273)


Use the information from the graphs in the lesson area above to answer each question.

e. Which type of graph best displays the relationships between parts of a whole? graph

f. Which type of graph best displays change over time? graph

g. Which type of graph is best for comparing categories of data? graph

Adventure Tire Sales

Jan.

Feb.

Mar.

Represents 100 tires

Where AyishaSpends Her Day

At home12 hr

Elsewhere4 hr

At school8 hr

• In a pictograph, each picture in the chart represents a certain number of items. The legend tells how many items each picture represents.

• A bar graph shows comparisons between categories (such as classrooms or months of the year).

Number of Aluminum CansCollected by Each Homeroom

10,000

8000

6000

4000

2000

012 14 16 18

Room Number

Num

ber

of A

lum

inum

Can

s• A line graph shows how a measurement changes

over time.• A circle graph (pie graph) shows parts of a whole.

a. How many more tires were sold in February than

in January? tires

c. On which game was Paul’s score lower than his

score on the previous game? Game #

b. How many aluminum cans were collected by all

four homerooms? cans

d. What fraction of Ayisha’s day is spent somewhere

other than at home or at school?


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c.

1.

walkerstotal athletes

x ____

xxx

2. pagesdays

345

____ x?x

____ 1

3. (5 × ) + 52 = 1 min = ? s

5. Paul’s Bowling Scores

1 6

160

0

170

180

190

200

2 3Game

Num

ber

of P

oint

s S

core

d

4 5

average Game 7 = 185

6. What is the d between

Paul’s h score and his

l score?

7. a. read b. pages to halfway

8. a. area b. perimeter

x = in.

y = in.

9. a. 7

__ 9 =

? ___

18 b.

? __

9 =

20 ___

36

c. 4

__ 5 =

24 ___

?

d. l Property

of m

10. 2986.34157

a. nearest thousand

b. three decimal places


walkers joggers total athletes

3+ 7

y

x

4. Room 16 Room 18

Number of Aluminum CansCollected by Each Homeroom

10,000

8000

6000

4000

2000

012 14 16 18

Room Number

Num

ber

of A

lum

inum

Can

s

Use work area.

a.

b.

Mira didnot read

384 pages

48 pages

Miraread

48 pages

48 pages

48 pages

48 pages

48 pages

48 pages

48 pages

a.

b.

a.

b.

c. b. .

a.


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c.

11. a. most likely

b. least likely

c. Better sample space is

# because it lists

all the p

outcomes.

12.

a. centimeters

b. millimeters

13. perimeter in centimeters

10 mm = cm

14.

The number 3.4 is between and

and is nearly h between. Point

B is too c to 3 to be 3.4. So the

best choice is .

15.

a. perpendicular to ___

AD

b. parallel to ___

AD

16. a. area of △ACD area of a triangle = 1 _ 2 bh

b. area of △CAB (△CAB is congruent to △ACD)

c. total area

17. 4.3 + a = 6.7

6.7 4.3

18. m – 3.6 = 4.7

4.7 3.6

19. 10w = 4.5 20. x ___

2.5 = 2.5


1 2 3cm

a. .

b.

10 mm

1.2 cm

a.

b.

Use work area.

2 3 4

A B C D

5

a.

b.

a = m =

w = x =

a.

b.

c.


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c.


21. 22. 1.25 ÷ 5 =

24. 5 __

8 =

___ xx

3 __

4 =

___ xx

+ 1 __

2 =

___ xx

25. 5 ÷ 3 1 __

3 =

26. 3 ___

10 – ( 1 __

2 –

1 __

5 ) =

1 __

2 =

___ xx

– 1 __

5 =

___ xx

27. 22 ∙ 24 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 =

A 4 ∙ 42 B 28 C 48 D 46

28. a. How many milliliters?

b. How much less than a liter? 1 liter = 1000 milliliters

30.

5.3700.00

00.00

29. Compare the numbers 916.3916.35916.42916.37916.5

to tell which books areout of order.

y

x

23. 5

__ 9 ∙ 6 ∙ 2

1 ___

10 =

a. b.

200 mL

100 mL

and

916.42 916.35 916.37 916.5 916.3


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Proportions (page 280)

• A proportion is a statement that two ratios are equal.

• You can cross-multiply to check that ratios are equal.If the cross-products are equal, then the ratios are equal.

5 ∙ 16 = 80 20 ∙ 4 = 80

16

___ 20

= 4

__ 5

• Solve a proportion by finding the missing number.

1. Cross-multiply.

2. Divide by known factor.

Example: 3 __

5 =

6 __

w

3 ∙ w = 5 ∙ 6

3w = 30

w = 30

___ 3

w = 10

• You have done this before: Multiply the loop. Divide by the outside number. The missing number will always be outside the loop.


Teacher Note:• Refer students to “Proportion

(Rate) Problems” on page 19 in the Student Reference Guide.

a. Multiply the loop.

Divide by the outside number.

a ___

12 =

6 __

8 a =

c. 14

___ 21

= c ___

15 c =

e. 30

____ 100

= n ___

40 n =

b. 30

___ b

= 20

___ 16

b =

d. 30

___ 25

= 24

___ d

d =

f. m ____

100 =

9 ___

12 m =


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1. 10 months × 19 months

2. Between his and

months.

3. trumpetflute

x ____

xxx = 4. average

497513436

410

5. Weeks Dollars

1 4.50

2

3

5

6. a. fewer

b. 5 or more

7. area x = mm

y = mm

8. perimeter

9.

a. decimal number

b. mixed number

10. 0.9166666

a. nearest hundredth

b. nearest hundred- thousandth


x

y

Use work area.

Use work area.

a.

b.

a. .

b.

a. .

b. .


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c.

11. $1.99 9 ___

10

9.16

12. a. one hundred and seventy-five thousandths

b. one hundred seventy-five thousandths

13.

a. acute

b. obtuse

c. straight

14. …, 100, 10, 1, 0.1, , , , …

Divide a term by to find the

next term of the sequence.

15. Multiply the loop.

Divide by the outside number.

8 ___

12 =

6 __

x

16. 16

___ y =

2 __

3

17. 21

___ 14

= n __

4 18. m + 0.36 = 0.75

19. 1.4 – w = 0.8 20. 8x = 7.2


∠

∠

∠

b. .

a. .

a.

b.

c.

Q P S

R

Use work area.

x = y =

n = m =

0.75 0.36

1.4 0.8

w = x =


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c.

21. y ___

0.4 = 1.2

22. estimated and exact answer

9.609.60

9.60

23. estimated and exact answer

3.15 – (2.1 – 0.06) =

24.

25. 26. 8 1 __

3 ∙ 1

4 __

5 =

27. 5 5

__ 6 ÷ 7 = 28.

a. perimeter

b. area = 1 _ 2 bh

29. 1, 2, 3, 4, 5, 6

odd primetotal numbers

x ____

xxx =

30. least to greatest

2

__ 3

, 1

__ 2

, 5

__ 6

,

7 ___

12

2 __

3 =

x ___

12

1 __

2 =

x ___

12

5 __

6 =

x ___

12


A

BC

y =

1.2 0.4

2.1 2.1

3.15 3.15

4 5 ___

12 =

x ___

xx

+ 6 5

__ 8 =

x ___

xx

4 1 __

4 =

x ___

xx

– 1 3 __

5 =

x ___

xx

a.

b.

, , ,


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Sum of the Angle Measuresof a Triangle

Angle Pairs (page 285)

• A square has four angles and each one measures 90°.

By drawing a diagonal segment from one corner to the opposite corner, the square divides into two congruent triangles.

• Do the Activity on pages 286 and 287 to understand:A triangle has three angles and the sum of these angles is 180°.

90° + 45° + 45° = 180°

• Adjacent angles share a common side. Examples: ∠1 and ∠4, ∠x and ∠y

• Supplementary angles are two angles whose sum is 180°. Examples: ∠y and ∠z, ∠3 and ∠4

• Complementary angles are two angles whose sum is 90°. Example: ∠A and ∠B

• Vertical angles are a pair of non-adjacent angles formed by intersecting lines. Vertical angles have the same measure. Examples: ∠1 and ∠3, ∠2 and ∠4, ∠x and ∠z


a. The sides of a regular triangle are equal in length, and the angles are equal in measure. What is the measure of each angle of a regular triangle and why?

Each a measures ° because they equally share °.

Refer to rectangle ABCD to answer questions b–d.

Teacher Notes:• Refer students to “Angle Pairs” on

page 27 in the Student Reference Guide.

• Review “Angles” on page 27 in the Student Reference Guide.

• Students will need paper, a ruler, a protractor, and scissors to complete the activity in this lesson.

90º

90º90º

90º

90º45º

45º

12

43

c. What is the measure of ∠CAB and why?

∠CAB measures because it is the

third a of a triangle whose other

angles measure and .

d. Are angles ACD and CAB vertical angles? Why or why not?

They are n vertical a .

Their angles are equal in m but

they are n non-adjacent angles

formed by two i lines.

b. What is the measure of ∠ACB and why?

Answer: ; Angle ACB and ∠ACD are c .


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c.


e. Find the measure of ∠x, ∠y, and ∠z in this figure.

m∠x =

m∠y =

m∠z =

1.

a. fraction white

b. whitetotal

x ____

xxx

2. a. ( 6 × ) + 20 = 1 min = ? s

b. lapsseconds

4 ___

xx

1 __

?

3. milesgallons

xx

___ 1

? ___

xx 4. a. 212

202

b. Round off the difference.

212 183

+ ft

____ – F°

550

____ 1

? ___

xx

5. The length is twice the width.

a. perimeter

b. area

6. a. grazed

b. drank

7. Find BC in centimeters.

cmmm

1 ___

xx

? ___

xx

8. 0.083333

a. nearest thousandth

b. nearest tenth


xy

z

80+ 40

180– 140

= m∠x

180– 140

= m∠y

+ 3 redite+ 2 white

total

a. b.

a.

b.

a. b.

a.

b.

a. . b. .

a.

b.


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c.

9. a. 12.054 and

b. 10 11

____ 100

and

10. area of triangle = 1 _ 2 bh


0.7 0.8

B

0.9

12. Sum of angles in triangle is 180°.

13. Name an angle supplementary to ∠ACB that is not ∠ACD.

14. a. I Property of m

b. 120

____ 5

=

15. 8 ___

10 =

w ___

25 16.

n ___

1.5 =

6 __

9 17.

9 ___

12 =

15 ___

m

18. 4 = a + 1.8

4.0 4.0

19. 3.9 = t – 0.39

3.90 4.00

20. 1.2 m – 12 cm =

mcm

1 ___

xx

? ___

12


CD

E

A

B

Use work area.

c. a. b.

∠ b.

w = n = m =

a = t =

1.20 4.00

m


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c.

21. (0.15)(0.05) =

22.

23. 14.4 ÷ 12 =

14.4

24. 5.6 – (4 – 1.25) =

4.00– 0.00

5.60 0.00

25. 5 – (3.14 + 1.2) =

3.14 0.00

5.00 0.00

26. 6 1

__ 4

∙ 1 3 __

5 =

27. 7 ÷ 5 5 __

6

x ___

xx ×

x ___

xx =

28. 8 ___

18 =

xx ___

x

+ 12

___ 25

= xx

___ x

29. 4 2 __

5 =

xx ___

x

– 1 3 __

4 =

xx ___

x

30. Perimeter is 15.84 m.

15.84 m

each side

area


0.15 0.05

15× 1.5

LESSON 31 Reading and Writing Name -...

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