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Saxon Math Course 2 L31-121 Adaptations Lesson 31
L E S S O N
31 Name ©
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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
Saxon Math Course 2 L31-122 Adaptations Lesson 31
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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.
Saxon Math Course 2 L31-123 Adaptations Lesson 31
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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.
Saxon Math Course 2 L31-124 Adaptations Lesson 31
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c.
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
Written Practice (continued) (page 227)
c. b. a.
n = b =
Name ©
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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.
Practice Set (page 231)
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
Written Practice (page 232)
Saxon Math Course 2 L32–126 Adaptations Lesson 32
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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Written Practice (continued) (page 232)
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 ___
Saxon Math Course 2 L32–127 Adaptations Lesson 32
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
Saxon Math Course 2 L32–128 Adaptations Lesson 32
Written Practice (continued) (page 233)
29. (–5, 3), (–5, –2), and (2, –2)
a. coordinates of the fourth vertex
b. area
a. ( , )
b.
a.
b. ∠
c. ∠
Saxon Math Course 2 L33-129 Adaptations Lesson 33
L E S S O N
33 Name ©
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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
Practice Set (page 238)
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
Saxon Math Course 2 L33-130 Adaptations Lesson 33
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Written Practice (page 239)
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.
.
Saxon Math Course 2 L33-131 Adaptations Lesson 33
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Written Practice (continued) (page 239)
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.
Saxon Math Course 2 L33-132 Adaptations Lesson 33
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Written Practice (continued) (page 240)
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.
Use work area. Use work area.
Saxon Math Course 2 L34-133 Adaptations Lesson 34
L E S S O N
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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.
Practice Set (page 243)
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
Saxon Math Course 2 L34-134 Adaptations Lesson 34
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c.
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
Written Practice (page 244)
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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c.
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.
Written Practice (continued) (page 245)
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.
Saxon Math Course 2 L34-135 Adaptations Lesson 34
.
b. ( , ) c.
y = c =
Saxon Math Course 2 L34-136 Adaptations Lesson 34
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c.
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 =
Written Practice (continued) (page 245)
a.
b.
, ,
Use work area.
Saxon Math Course 2 L35-137 Adaptations Lesson 35
L E S S O N
35 Name ©
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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
Practice Set (page 251)
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
Saxon Math Course 2 L35-138 Adaptations Lesson 35
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Written Practice (page 251)
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. .
Practice Set (continued) (page 251)
Use work area.
2 1
__ 2
= x ___
xx
1 3
__ 4
= x ___
xx
a.
b.
110– 110
Saxon Math Course 2 L35-139 Adaptations Lesson 35
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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.
13. What decimal number?
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
Written Practice (continued) (page 251)
a. ( , ) b.
Use work area.
a. . b.
Use work area. .
× =
a.
b.
x = f = y=
Saxon Math Course 2 L35-140 Adaptations Lesson 35
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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.
Written Practice (continued) (page 253)
Use work area. Use work area.
Use work area.Use work area.
– 0000
× 0000
estimate
calculate
estimate
calculate
w =
Saxon Math Course 2 L36-141 Adaptations Lesson 36
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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.
Practice Set (page 260)
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
Saxon Math Course 2 L36-142 Adaptations Lesson 36
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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
___ =
Written Practice (page 261)
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
___ =
Practice Set (continued) (page 260)
a.
b. to
to
Saxon Math Course 2 L36-143 Adaptations Lesson 36
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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.
Written Practice (continued) (page 261)
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.
Saxon Math Course 2 L36-144 Adaptations Lesson 36
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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 =
Written Practice (continued) (page 262)
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.
Saxon Math Course 2 L37-145 Adaptations Lesson 37
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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.
Practice Set (page 269)
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.
Saxon Math Course 2 L37–146 Adaptations Lesson 37
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c.
Practice Set (continued) (page 270)
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 =
Written Practice (page 270)
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.
Saxon Math Course 2 L37-147 Adaptations Lesson 37
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c.
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
Written Practice (continued) (page 271)
AA BB
CC
Use work area.
a.
b.
Use work area.
Use work area.
a.
b.
x =
a. .
b. .
c. a. . b.
{{{
Saxon Math Course 2 L37-148 Adaptations Lesson 37
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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
Written Practice (continued) (page 272)
b. a.
m = w =
n =
y =
Saxon Math Course 2 L38-149 Adaptations Lesson 38
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Inc.
Interpreting Graphs (page 273)
Practice Set (page 276)
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?
Saxon Math Course 2 L38-150 Adaptations Lesson 38
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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
Written Practice (page 277)
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.
Saxon Math Course 2 L38-151 Adaptations Lesson 38
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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
Written Practice (continued) (page 278)
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.
Saxon Math Course 2 L38-152 Adaptations Lesson 38
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Written Practice (continued) (page 279)
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
Saxon Math Course 2 L39-153 Adaptations Lesson 39
L E S S O N
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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.
Practice Set (page 282)
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 =
Saxon Math Course 2 L39-154 Adaptations Lesson 39
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c.
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
Written Practice (page 282)
x
y
Use work area.
Use work area.
a.
b.
a. .
b.
a. .
b. .
Saxon Math Course 2 L39-155 Adaptations Lesson 39
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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
Written Practice (continued) (page 283)
∠
∠
∠
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 =
Saxon Math Course 2 L39-156 Adaptations Lesson 39
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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
Written Practice (continued) (page 284)
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.
, , ,
Saxon Math Course 2 L40-157 Adaptations Lesson 40
L E S S O N
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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
Practice Set (page 289)
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 .
Saxon Math Course 2 L40-158 Adaptations Lesson 40
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c.
Practice Set (continued) (page 290)
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
Written Practice (page 290)
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.
Saxon Math Course 2 L40-159 Adaptations Lesson 40
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c.
9. a. 12.054 and
b. 10 11
____ 100
and
10. area of triangle = 1 _ 2 bh
11. What decimal number?
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
Written Practice (continued) (page 290)
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
Written Practice (continued) (page 292)
0.15 0.05
15× 1.5