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Saxon Math Intermediate 5 331 Adaptations Lesson 51

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5151 Teacher Notes:

• Introduce Hint #38 “Multiplication by Two Digits.”

• Review Hint #13 “Multiplication (Carrying on Fingers).”

• Review “Multiplication Table” on page 5 and “Properties of Operations” on page 22 in the Student Reference Guide.

• For additional practice, students may complete Targeted Practice 51.

• Multiplying by Two-DigitNumbers

New ConceptNew Concept

• Use a four-step process to multiply by two-digit numbers.

Example

43× 12

1. Multiply by the ones digit 43× 12

86(ignore the tens digit):

2. Cross out the ones digit 4 3× 12

86Xwhen you are finished with it:

3. Indent the next line using X as a placeholder. 4 3× 12

8 6

+ 4 3 X

Then multiply by the tens digit.

4. Add the two answers: 4 3 × 1 2

8 6

+ 4 3 X5 1 6

page 325

X

X

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Saxon Math Intermediate 5 332 Adaptations Lesson 51

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• The Distributive Property lets us find the answer to thesame problem 2 different ways.

Example

25 × (10 + 2)

Choice 1: Find the sum; then multiply.

25 × 12 = 300

Choice 2: Multiply each addend; then add the products.

(25 × 10) + (25 × 2) = 250 + 50 = 300

Lesson PracticeMultiply.

a. 3 2× 1 2

+00000X

b. $0. 6 2× 2 3

+000000X

c. 4 8× 6 4

+00000X

d. 2 4 6× 2 2

+000000X

e. $1. 4 7× 3 4

+000000X

f. 8 7× 6 3

+00000X

g. Musoke wants to multiply 12 by (20 + 3). Show her two choices for multiplying.Find each answer.

Distributive Property

Choice 1: 1 2× 1 21 2

+00000X

Choice 2: (12 × ) + (12 × )

+ =

h. Use your textbook to complete this problem.

Math Language

Distributive Property: allows multiplication problems to be rewritten:a(b + c) = ab + ac.

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Saxon Math Intermediate 5 333 Adaptations Lesson 51

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1.

Day Number of Visitors

Wednesday 47

Thursday 76

Friday 68

Saturday

Science Fair

day 1 day 2

day 3

320

–

2. Which stamp did he use 3 of?

( 3 × ) + = stamp stamp

× 3 + 3

3. 3 __ 4 of 60 60 raisins

___ raisins

___ raisins

___ raisins

34Arthur ate

1.4

Arthur did noteat

.

___ raisins

) _____

6 0

is of

__ = ? ___ 60

0 __ 4 = percent

4. standard form

(1 × 1000) + (1 × 1)

5. 1760

words:

Use work area.

6. Shade all but 1 _ 6 .

percent not shaded See page 19 in the

Student Reference Guide.

6 __ 6 =

Written PracticeWritten Practice page 329

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Saxon Math Intermediate 5 334 Adaptations Lesson 51

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Written Practice,Written Practice, continued continued page 330

7. sixty-two thousand, four hundred ninety

9.

inch 1 2 3

10.

average

4 8

) ___

11. 1 __ 2 of 10 1 __

3 of 12

12. parentheses first

(1 + 2 + 3 + 4 + 5) ÷ 5 =

13. 4 3× 1 2

+00000X

14. $0. 7 2× 3 1

+000000X

8. 2 8 3 5

A 2000 B 2700

C 2800 D 2900

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Saxon Math Intermediate 5 335 Adaptations Lesson 51

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15. 2 4 8× 2 4

+000000X

16. $1. 9 6× 5 3

+000000X

17. 876236244795

+ 8473

18. $10.00– $ 9.92 $

19. 600× 50

20. short division

$ .

8 ) __________

$6 . 0 0

23. 3

1 __ 4

+ 2 2 __ 4

24. parentheses first

5 5 __ 8 – 3 3 __

8 – 1 1 __

8 =

5 5 __

8

– 3 3 __ 8

– 1 1 __

8

25. 1 ft = in.

( × ) + =

× 1 2

+00000X

+ 9 inches

page 330

21. short division

$ .4 )

_____________ $4 1 . 3 6

22. 9x = 4275

x =

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Saxon Math Intermediate 5 336 Adaptations Lesson 51

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Written Practice,Written Practice, continued continued

26. Distributive Property

15 × (20 + 4) =

Choice 1: 1 5×

+00000X

Choice 2: (15 × ) + (15 × )

+ =

Use work area.

27. a. two speeds used to find the range of the data: ,

b. median speed: cheetahzebrawart hog

lion

) ______

RRR c. average of all animals

The animal with the average speed that is the closest to the average of all

animals is .Use work area.

28. expanded notation

205,000

( × ) + ( × )

Use work area.

29. Is 11 1 _ 4 inches closer to 11 inches or 12 inches?

2__4

= 1__2

30. 24 mi.

Kenley Bernardo

24 24

18 ×

Use work area.

page 331

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Saxon Math Intermediate 5 337 Adaptations Lesson 52

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5252 Teacher Notes:

• Review Hint #4 “Place Value (Digit Lines),” and Hint #5 “Writing Numbers.”

• Review “Spelling Numbers” and “Place Value” on pages 12 and 13 in the Student Reference Guide.

• Review Reference Chart “Spelling Numbers.”

• For additional practice, students may complete Targeted Practice 52.

• Naming Numbers ThroughHundred Billions

New ConceptNew Concept

To name a number using words:

1. Insert commas.

Start at the right and put a comma after every three digits.

87654321 87,654,321

2. Use words to name the number. Say the place value every time you come to a comma. 87,654,321 eighty-seven million, six hundred

fifty-four thousand, three hundred twenty-one

• Use the chart below to help name the commas.

hund

red

s

tens

ones

mill

ions

com

ma

__

Millions Units (Ones) Thousands

__ __ __ __ __ __ __ __thou

sand

s co

mm

a

hund

red

s

tens

ones

bill

ions

com

ma

__

Billions

__ __

hund

red

s

tens

ones

hund

red

s

tens

ones

Lesson PracticeName the value of the place held by the zero in each number.

a. 345,052 b. 20,315,682

c. 1,057,628 d. 405,176,284

page 332

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Saxon Math Intermediate 5 338 Adaptations Lesson 52

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e. In 675,283,419,000, which digit is in the ten-billions place?

f. In which of the following numbers does the 7 have a value of seventy thousand?

A 370,123,429 B 1,372,486 C 4,703,241 D 7,000,469

g. Write the value of the 1 in 321,987,654.

Use words to name each number. (First insert commas, counting from the right.)

h. 21462300

twenty-

i. 19650000000

nineteen

Use digits to write each number for problems j–l.

j. nineteen million, two hundred twenty-five thousand, five hundred

, ,

k. seven hundred fifty billion, three hundred million

, , ,

l. two hundred six million, seven hundred twelve thousand, nine hundred thirty-four

, ,

m. Write 7,500,000 in expanded notation.

( × ) + ( × )

Lesson Practice, continued

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Saxon Math Intermediate 5 339 Adaptations Lesson 52

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1. 5 dozen

gave away 24

2. 1 __ 2 of Marco’s weight

3. $3.60

$0.00

$10.00

$00.00

4. 002007001607

centuries _________ years 1 __ 0 = ? __

0

6. Finish drawing this rectangle.

Shade all but 3 __ 8 .

2 in.

1 in

.

See page 19 in the Student Reference Guide.

percent not shaded

1 __ 8 = percent

7. 250,000

words:

Use work area.

5. standard form

(1 × 100) + (4 × 10) + (8 × 1)

Written PracticeWritten Practice page 336

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Saxon Math Intermediate 5 340 Adaptations Lesson 52

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Written Practice,Written Practice, continued continued

8.

4

9

2

average

49

2 ) ____ RR

I found the av number of books in 3 stacks.

Use work area.

9. hundred millions

789,456,321

10. 1236 11. 102,345,678

12. 5 7× 2 2

+00000X

13. $0. 8 3× 4 7

+000000X

14. 1 6 7× 8 9

+000000X

15. $1. 9 6× 4 6

+000000X

page 336

16. 843734295765

+ 9841

17. $26.38– $19.57

18. 3041– w

2975

w =

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Saxon Math Intermediate 5 341 Adaptations Lesson 52

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19. short division

4328 _____ 4 =

20. short division

5670 _____ 10

=

21. short division

$78.40 _______ 4 =

22. 3 ___ 10

2

+ 1 4 ___ 10

23. 2 3 __ 4 5 3 __

4

– –

24. $1.43

$ .

$ .

+ $ .

$10.00

– $ .

25. Which arrow could be pointing to 3 9 __ 10 on the number line below?

5

A B C D

0

page 337

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Saxon Math Intermediate 5 342 Adaptations Lesson 52

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Written Practice,Written Practice, continued continued

26. Distributive Property

25 × (20 + 4) =

Choice 1: 25 × Choice 2: (25 × ) + (25 × )

Use work area.

27. 2, 9, 2, 5, 4, 5, 1, 5, 4, 5, 5, 4, 12, 4

a. 0 5 10 15

b. median: mode: range:

c. outliers: and

d. data cluster: and Use work area.

28. expanded notation

three million, two hundred thousand

( × ) + ( × )Use work area.

page 337

29. difference in temperatures 30. (10 × ) –

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Saxon Math Intermediate 5 343 Adaptations Lesson 53

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5353 Teacher Notes:

• Introduce Hint #39 “Area and Perimeter Vocabulary.”

• Refer students to “Perimeter, Area, Volume,” “Length and Width,” and “Circle” on pages 17 and 14 in the Student Reference Guide.

• Display reference chart “Circle.”

• Perimeter

• Measures of a Circle

New ConceptNew Concept

• Perimeter is the distance around a figure. To find perimeter add all sides. Formula for the perimeter of a rectangle: P = 2l + 2w

• Regular polygons All sides are equal. Formula for the perimeter of a regular polygon: P = number of sides × side length

circumference

diameter

radius

• Circumference distance around the circle (perimeter)

• Diameter distance across the whole circle through its center

Diameter = 2 × radius (d = 2r)

• Radius distance from its center to its edge

Radius = 1 _ 2 of diameter (r = 1 _ 2 d)

ActivityActivity page 342

Measuring Circles

• Use your textbook to complete this activity.

• Perimeter

Math Language

Length – the longer side

Width – the shorter side

page 339

• Measures of a Circle

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Saxon Math Intermediate 5 344 Adaptations Lesson 53

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Lesson Practice

a. What is the length of this rectangle? 5 in.

3 in. b. What is the width of the rectangle?

c. What is the perimeter of the rectangle? (Perimeter Add all sides.)

d. What is the perimeter of this right triangle? 3 cm

4 cm

5 cm (Perimeter Add all sides.)

e. What is the perimeter of this square? ______

4 ft (Perimeter Add all sides.)

f. Do we use units, square units, or cubic units to measure the perimeter?

u

What do we call the perimeter of a circle?

g. What do we call the distance across a circle through its middle?

h. If the radius of a circle is 6 inches, what is the diameter of the circle?

d = 2r

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Saxon Math Intermediate 5 345 Adaptations Lesson 53

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1. (3 × ) – 15 =

× 3

– 15

2. average

13

+ 9 ) ___

00

I a the

number of and

divided by .

3. 1 __ 3 of 30

) _____

3 0

30 students

____ students

____ students

____ students

13 walked home

23

did notwalk home

See page 19 in the Student Reference Guide.

00 ___ 3 = percent

4. average

35

7

) ___

00

I all the

together and

divided by .

5. 123,456,789,000

A 2 billion B 20 billion

C 200 billion D 2000 billion

6. factors of 8: 1, , , 8

factors of 12:

1, , , , , 12

, ,

Written PracticeWritten Practice page 342

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Saxon Math Intermediate 5 346 Adaptations Lesson 53

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Written Practice,Written Practice, continued continued

7. 1890 1820

decades years 1 __ = ? __

8. digits

nineteen million, four hundred ninety thousand

, ,

11. 300

×

12. 800

×

13. 5t = 500

t =

14. $5.6 4× 7 8

+000000X

15. 8 6 5× 7 4

+000000X

9. 4 2 __ 3

−

6

+

10. 2 2 __ 3

+

4 2 __ 3

–

page 343

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Saxon Math Intermediate 5 347 Adaptations Lesson 53

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16. 9 8 3× 7 6

+000000X

17. $63.14– $42.87 $

18. 3106– 875

19. $68.09 $43.56 $27.18+ $14.97 $

20. short division

$31.65 _______ 5 =

21. short division

4218 _____ 6 =

22. short division

5361 ÷ 10 =

23. 1 2 3 6

A 1230 B 1240

C 1200 D 1300

24. What is the length of this rectangle?

3 cm

2 cm

25. P = 2l + 2w

3 cm

2 cm

page 343

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Saxon Math Intermediate 5 348 Adaptations Lesson 53

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Written Practice,Written Practice, continued continued

26. Distributive Property

35 (20 + 1) =

Choice 1: 35 × Choice 2: (35 × ) + (35 × )

Use work area.

27. expanded notation

2,050,000

( × ) + ( × )

Use work area.

28. Draw an equilateral triangle.

See page 15 in the Student Reference Guide.

Use work area.

29. Is 8 5 _ 8 inches closer to 8 inches or 9 inches?

4 __ 8 = 1 __

2

Use work area.

30.

Mississippi

New Jersey

Illinois

Michigan

State Highest Elevation(in feet)

Key: 200 feet

a. highest elevation of about 2000 feet

b. elevations

greatest to least

2000, , ,

c. elevation nearest sea level least

b. Use work area.

c.

a.

page 344

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Saxon Math Intermediate 5 349 Adaptations Lesson 54

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5454 Teacher Notes:

• Introduce Hint #40 “Long Division, Part 1” and Hint #41 “Long Division: Canceling Matching Zeros.”

• For additional practice, students may complete Targeted Practice 54.

• Dividing by Multiples of 10

New ConceptNew Concept

• Use long division with two-digit divisors:

1. Divide, multiply, subtract, and bring down.2. Use zero as a placeholder.3. Place a digit above each digit.4. Make sure any remainder is smaller than the divisor.

Example

01 5 R 430 )

_____ 4 5 4

3 0 1 5 4 1 5 0 4

DivideMultiplySubtractBring down

• To help divide by a two-digit number, mentally remove the last digit from each number.

• To check division: 1 5 R 430 )

_____ 4 5 4

1. Multiply the answer (quotient) by the divisor. 15× 30 450

2. Add the remainder. (This sum should equal the dividend). 15

× 30 450+ 4 454

page 345

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Saxon Math Intermediate 5 350 Adaptations Lesson 54

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• If the dividend and the divisor both end in 0, cancel matching zeros.

1. Rewrite the problem using a division bar.

2790 _____ 40

2. Cancel matching zeros.

3. Use short division.

4. Only if the answer is to be written with a remainder, add the zero back to the end of the remainder.

R40 )

______ 2 7 9 0

0 6 9 R 34 )

______ 2 7 39

so

0 0 6 9 R 3040 )

_______ 2 7 39 0

• Don’t cancel matching zeros if it is a money problem.

Lesson PracticeDivide.

a. $ .30 )

__________ $4 . 2 0 b.

R60 )

_________ 7 2 5 c.

$ .40 )

__________ $4 . 8 0 d.

$ .20 )

__________ $3 . 2 0

e. Cancel matching zeros. f. 10 ) _________

3 4 5 Use short division.

Add the zero back to the remainder.

R50 )

_________ 6 1 0 610 ____

50

R5 )

_____ 6 1

g. Show how to check this division answer. h. Estimate. Is the answer correct?

2 3 R5

40 ) _____

9 2 5

$4.60 _____ 18

) _____

123

Check: 23

× +

Correct or not?

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Saxon Math Intermediate 5 351 Adaptations Lesson 54

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Written PracticeWritten Practice page 347

1. 0000$3.18

0000$0.00

0000$5.25

0000$0.00

My answer is reasonable because if I round all of the numbers I would add

+ , which equals . Then I would

subtract – , which equals .

My answer is close to the estimate.

2. 1236 3. 2 _ 3 of a yard 36 in.

____ in.

____ in.

____ in.13

23

is __ of

2 __ 3 = ? ___ 36 )

____ 23

4. 987,654,321

A 700 B 7,000,000 C 700,000 D 7000

5. Shade 1 _ 2 . Shade 2 _ 4 .

2 _ 4 = percent

6. one dollar = cents

a. 1 _ 4 of a dollar

b. 2 _ 4 of a dollar

a.

b.

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Saxon Math Intermediate 5 352 Adaptations Lesson 54

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Written Practice,Written Practice, continued continued

7. 3,150,000,000

words:

Use work area.

8. factors of 9: 1, , 9

factors of 12:

1, , , , , 12

,

9. long division

R30 )

_________ 4 5 4

10. long division

R40 )

__________ $5 . 6 0

12. 500

×

13. 5 6 3× 4 6

+000000X

14. $4 . 3 2× 6 8

+000000X $

15. 25 1 __ 4

+

16. 36 2 __ 3

−

17. short division

2947 ÷ 8 =

page 348

11. Cancel matching zeros.

Add the zero back to the remainder.

R50 )

_________ 7 6 0

760 ____ 50

R5 )

_____ 7 6

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Saxon Math Intermediate 5 353 Adaptations Lesson 54

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18. 7564 ÷ (90 ÷ 10) =

) ____

23 ) ____________

7 5 6 4

19. 12,345− 6,789

20. $3.65$2.47$4.83

+ $2.793.65

21.

) _____

3 6

22. What is the radius of the circle?

30 mm

r = 1 __ 2 d

23. Perimeter Add all sides.

6 cm

8 cm

10 cm

24.

21inch

25. 1896

five decades

26. Use the Distributive Property to solve 150 × (10 + 2).

Choice 1: 1 5 0× 1 2

+000000X

Choice 2: (150 × ) + (150 × )

+ =

page 348

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Saxon Math Intermediate 5 354 Adaptations Lesson 54

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Written Practice,Written Practice, continued continued

27. Position 1 2 3 4 5

Term 6 12 18 24 30

a. Rule: the

position by .

b. 20th term

20

× a. Use work area.

b.

28. 11 hours− 8 hours ($14 an hour)

hours ($21 an hour)

$14

×

$21

×

$

+ $

29.

8 cm 8 cm

6 cm

Could this triangle be scalene?

Why or why not?

of the sides of this triangle have the

same length. The sides of a scalene triangle all

have lengths.Use work area.

30. 776 ÷ 38

) ____

÷

is a reasonable estimate because 776 rounds to

, and 38 rounds to .

÷ = Use work area.

page 349

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Saxon Math Intermediate 5 355 Adaptations Lesson 55

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5555 Teacher Notes:

• Review Hint #38 “Multiplication by Two Digits.”

• Review “Multiplication Table” on page 5 in the Student Reference Guide.

• Multiplying by Three-Digit Numbers

New ConceptNew Concept

Multiplying by three-digit numbers

1. Multiply by the ones digit.

2 3 4× 1 2 3

7 0 2

2. Cross out the ones digit when you’re finished with it.

2 3 4× 1 2 3

7 0 2X

3. Indent the next line using x as a placeholder, and multiply by the tens digit:

2 3 4× 1 2 3

7 0 24 6 8 X

X

4. Cross out the tens digit when you’re finished with it.

2 3 4× 1 2 3

7 0 24 6 8 X

X

page 350

X

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Saxon Math Intermediate 5 356 Adaptations Lesson 55

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5. Indent the next line using x as a placeholder, and multiply by the hundreds digit:

2 3 4× 1 2 3

7 0 24 6 8 X

+0002 3 4 X X

XX

6. Add the three answers.

2 3 4× 1 2 3

7 0 24 6 8 X

+0002 3 4 X X2 8 7 8 2

XX

Lesson PracticeFind each product. Carry on your fingers if the number is 5 or less.

a. 3 4 6× 3 5 4

X+0000 0 0 X X

b. 4 8 7× 6 3 4

X+0000 0 0 X X

c. 4 0 3× 7 6 8

X+0000 0 0 X X

d. compatible numbers

705

678 ×

e. 739 ÷ 18

÷ 20

Since 20 is a factor of , I changed 739 to and

I changed 18 to . is a reasonable estimate

because ÷ = .

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Saxon Math Intermediate 5 357 Adaptations Lesson 55

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Written PracticeWritten Practicepage 352

1. $4.65

+ $4.

$10.00

– $04.

My answer is reasonable because if I round all of the numbers I would add

+ , which equals . Then I would

subtract – , which equals . My

answer is close to the estimate.

Use work area.

2. 3 __ 4 of 276

) _________

2 7 6

276 pages

___ pages

___ pages

___ pages

___ pages

34read

14not read

is __ of

1 __ 2 = ? ____ 276

3. n Ubangi

Loire26 shorter

n − = 26

n =

Use work area.

4. ten-millions place

98,765,432

5. 1 yard = inches

( × ) + =

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Saxon Math Intermediate 5 358 Adaptations Lesson 55

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Written Practice,Written Practice, continued continued

6. Shade all but 1 __ 3 .

See page 19 in the Student Reference Guide.

1 __ 3 = percent

8. long division

60 )

__________ $7 . 2 0

10. Cancel matching zeros.

Add the zero back to the remainder.

R80 )

_________ 9 8 0

980 ____ 950

R8 )

_____

11. Carry on your fingers.

2 3 4

× 1 2 37 0 2

4 6 8 X+0002 3 4 X X

9. Cancel matching zeros.

Add the zero back to the remainder.

R70 )

_________ 8 5 0

850 ____ 850

R7 )

_____ 8 5

page 352

7. Use digits to write:

six hundred seventy-nine million, five hundred forty-two thousand, five hundred

, ,

Use work area.

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Saxon Math Intermediate 5 359 Adaptations Lesson 55

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12. $ 3. 7 5× 2 6

+000000X

13. 6 0 4× 7 8 9

X+0000 0 0 X X

14. perimeter = number of sides

× side length

10 mm

15. Use mental math.

400 × 800 =

16. Use mental math.

60 × 500 =

17. Use mental math.

900 × 90 =

18. 300400

+ 500

19. 6000− 2000

20. Cancel matching

zeros.

400 ____ 20

=

21. 6 5 ___ 11

+ 5 4 ___ 11

22. 3 2 __ 3

− 3 0

23. 3 1 __ 3

−

7 2 __ 3

−

24. 2150 fans

$2 each paid

money paid

page 352

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Saxon Math Intermediate 5 360 Adaptations Lesson 55

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Written Practice,Written Practice, continued continued

25. 3000 seats in stadium

227 empty seats

filled seats

27. Draw an isosceles triangle. See page 15 in the Student Reference Guide.

Use work area.

28. 1 dollar = dimes

5 ) ______

29. Is 5 7 _ 8 inches closer to 5 inches or 6 inches?

4 __ 8 = 1 __

2

30. 689 ÷ 19

÷

is a reasonable estimate because 689 is close to ,

19 is close to , and ÷ = 35.

Use work area.

page 353

26. expanded notation

1,200,000

( × ) + ( × )

Use work area.

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Saxon Math Intermediate 5 361 Adaptations Lesson 56

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5656 Teacher Notes:

• Review “Multiplication Table” on page 5 in the Student Reference Guide.

• For additional practice, students may complete Targeted Practice 56.

• Multiplying by Three-DigitNumbers That Include Zero

New ConceptNew Concept

• Indent using X as a placeholder.

• “Offset” when the number ends in zero.

2 4 3 0× 1 2 0

4 8 6 0+002 4 3 X X

2 9 1 6 0

• When the middle digit in the bottom number is zero, “bring down” the zero.

2 4 3× 1 0 2

4 8 6+002 4 3 0 X

2 4 7 8 6

page 354

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Saxon Math Intermediate 5 362 Adaptations Lesson 56

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2. 3 __ 6 of a minute

) ___

60

is __ of

0 __ 0 = ? ___

60

Lesson PracticeMultiply.

a. 2 3 4 0× 2 4 0

9 3 6 0+004 6 8 X X

b. $1. 2 5 0× 2 4 0

0+004 6 8 X X

c. 2 3 0 0× 1 2 0

0+004 6 8 X X

d. 3 0 4 0× 1 2 0

0+004 6 8 X X

e. 2 3 4× 2 0 4

+0000 0 0 X

f. $1. 2 5× 2 0 4

+0000 0 0 X

g. 2 3 0× 1 0 2

+0000 0 0 X

h. 3 0 4× 1 0 2

+0000 0 0 X

1. $12 $ 7 $

$30

$ $

60 seconds

___ seconds

___ seconds

___ seconds

___ seconds

___ seconds

___ seconds

of a minute36

Written PracticeWritten Practice page 356

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Saxon Math Intermediate 5 363 Adaptations Lesson 56

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onWritten Practice,Written Practice, continued continued

8.

10 mm

20 mm

P = 2l + 2w

9. Use mental math.

900 × 40 =

3. (to school + from school) × days =

×

I added the number of blocks to school and back. + = .

Then I multiplied the total number of blocks times days.

× = .

Use work area.

4. average

3629

73 ) _____

5. ten-thousands place

123,456,789

6. 5 in.

d = 2r

7. three hundred forty-five million, six hundred fourteen thousand, seven hundred eighty-four

, ,

page 356

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Saxon Math Intermediate 5 364 Adaptations Lesson 56

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Written Practice,Written Practice, continued continued

10. Use mental math.

700 × 400 =

11. 2 3 4× 3 2 0

0+0000 0 X X

12. $3. 4 5× 2 0 3

+0000 0 0 X $

13. 4 6 8× 3 8 6

X+0000 0 X X

14. w __ 5 = 6

w =

15. 4317 ÷ 6 =

16. 2703 ÷ 9 = 17. 8m = $86.08

m =

18. 79,08937,86529,453

+ 16,257

19. 43,218− 32,461

20. $100.00− $ 4.56

$100.00

page 357

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Saxon Math Intermediate 5 365 Adaptations Lesson 56

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21. 3 5 __ 6

– 1 5 __ 6

22. 4 1 __ 8

+ 6

24. Which arrow could be pointing to 1362?

13601350

A B C D

1340 1370

26. Distributive Property

150 × (200 + 3) =

Choice 1: 2 0 3 0× 1 5 0

0+0000 0 X X

Choice 2: (150 × ) + (150 × )

+ =

25. 7 1 ___ 10

words:

Use work area.

page 357

23. 1 week = days

( × ) + =

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Saxon Math Intermediate 5 366 Adaptations Lesson 56

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Written Practice,Written Practice, continued continued

27. a. Which has no remainder? Think: Divisibility.

A 543 ÷ 9 B 543 ÷ 5 C 543 ÷ 3 D 543 ÷ 2

b. Explain.

28.

a. 1 __ 4 of 100

b. 1 __ 4 = .

a.

b.

29. Is 37 3 _ 4 inches closer to 37 inches or 38 inches?

2 __ 4 = 1 __

2

30. Number of Golf Balls 1 2 3 4

Number of Dimples 392 784 1176 1568

a. Each ball has dimples.

b.

4 9× 1 2

+00000X

Use work area.

page 358

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Saxon Math Intermediate 5 367 Adaptations Lesson 57

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5757 Teacher Notes:

• Introduce Hint #42 “Probability and Chance.”

• Refer students to “Probability/ Chance” on page 10 in the Student Reference Guide.

• Use spinners to illustrate probability.

• Probability

New ConceptNew Concept

• Probability measures how likely it is for an event to happen. It is named with a number from 0 to 1 (including fractions and decimal numbers).

• Chance measures the same thing but with a percent.

• If an event is certain to happen, its probability is 1. Its chance is 100%.

• If an event is impossible, its probability is 0. Its chance is 0%.

• All other events have probabilities between 0 and 1 or chances between 0% and 100%.

12

1

0% 50% 100%

20 1

impossible unlikely

fractions less than12fractions greater than

likely certain

• Possible results of experiments that involve probability are called outcomes.

Example

Probability of picking a blue marble:

RR R

RRB B B

Y Y

number of blue marbles _____________________ total number of marbles

= 3 ___ 10

Probability of picking a marblethat is not blue:

number of marbles that are not blue ______________________________ total number of marbles

= 7 ___ 10

page 359

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Saxon Math Intermediate 5 368 Adaptations Lesson 57

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` Lesson Practice

a. What are all the possible outcomes? , , , ,

b. What is the probability that the spinner will stop on 3? 14

2

5

3 c. What is the probability of spinning a number greater

than three?

d. What is the probability of spinning an even number?

e. If the weather forecast states that the chance of rain is 40%, is it more likely to

rain or not to rain?

f. If today’s chance of rain is 20%, then what is the chance that it will not rain today?

g. Seth said that the probability of picking a

RR R

RRB B B

Y Y red marble was 1 _ 2 .

Do you agree or disagree with Seth? Why?

of the marbles are red. So the probability of getting a red marble

is , which is a fraction equal to .

h.

12

3

Which fraction best names the probability that the spinner will stop in sector 3?

A 1 __ 2 B 1 __

3 C 1 __

4 D 1 __

6

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Saxon Math Intermediate 5 369 Adaptations Lesson 57

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1. 1 ft = in.

5 ft = in.

2. centuries years 1 __ = 10 ___ ? 3. What is the perimeter of

a circle called?

Written PracticeWritten Practice page 363

4. 10 7 __ 10

words:

Use work area.

5. 2 _ 3 of an hour 60 min

____ min

____ min

____ min

23

) _____

1 2 2

is of

2 __ 3 = ? __

6. Don’t count 6 a.m. 7. quotientdivisor )

_________ dividend )

____ 2 1

×

8. 321,098,765

9. factors of 15: 1, , , 15

factors of 20: 1, , , , , 20

,

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Saxon Math Intermediate 5 370 Adaptations Lesson 57

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Written Practice,Written Practice, continued continued

10. Each side is 3 cm long. Perimeter = number of sides × length of 1 side.

11. 2 1 __ 3

+ 1 1 __ 3

3 2 __ 3

–

12. 2 2 __ 3

– 1 1 __ 3

3 1 __ 3

+

13. long division

40 ) __________

$5 . 2 0

14.

R8 )

____________ 3 1 6 1

15. divisor See page 8 in the Student

Reference Guide.

16. $43.15– $28.79

17. 4 2 3× 3 0 2

+0000 0 0 X

18. 99 36 42 75 64 98+ 17

19. $3.4 5× 3 6 0

0+0000 0 X X

page 364

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Saxon Math Intermediate 5 371 Adaptations Lesson 57

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20. 6 0 4× 5 9 8

X+0000 0 0 X X

21. 10 ___ 10

– 9 ___ 10

= 22. 4 2 _ 3 – 1 _ 3 =

23. 5 2 __ 2

–

1 1 __ 2

24. May to May 12 months

May to August

25. It is morning. What time will it be in 2 hours 20 minutes?

time now:

2 hours later:

20 minutes later:

3

121110

9

87 6 5

4

21

26. a. a millennium = years

b. half a millennium = years

c. 1 __ 2 =

a.

b.

c.

page 364

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Saxon Math Intermediate 5 372 Adaptations Lesson 57

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Written Practice,Written Practice, continued continued

27. With one roll, what is the probability of more than one dot on top?

A cube has sides.

28. average

8080

95 ) _____

29. chance

100%

– correct

incorrect

30. (10 × ) + =

× 10

5

+

page 365

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Saxon Math Intermediate 5 373 Adaptations Lesson 58

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5858 Teacher Notes:

• Review Hint #14 “Ways to Show Division.”

• For additional practice, students may complete Targeted Practice 58.• Writing Quotients with

Mixed Numbers

New ConceptNew Concept

• In division problems, sometimes we must write the remainder as a fraction.

3 R34 )

____ 1 5

3 3 __ 4 remainder __________

divisor

4 ) ____

1 5

Lesson PracticeWrite each quotient as a mixed number. (Show the remainder as a fraction.)

a. 4 ) _____

1 7 b. 20 ÷ 3 ) _____

000 c. 16 ___ 5 )

_____ 000

d. 5 )_____ 4 9 e. 21 ÷ 4 )

_____000 f. 49___

10)_____000

g. 6 ) _____

7 7 h. 43 ÷ 10 ) _____

000 i. 31 ___ 8 )

_____ 000

page 366

Written PracticeWritten Practice page 368

1. $0.35

8

$5.00

$ .

2. mixed number

) _____

2 1

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Saxon Math Intermediate 5 374 Adaptations Lesson 58

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Written Practice,Written Practice, continued continued

3. 3 _ 5 of 100 100 stamps

____ stamps

____ stamps

____ stamps

____ stamps

____ stamps

35 used

25 not used

See page 19 in the Student Reference Guide.

) _________

1 0 0 00 ___ 5 = percent

is __ of

9 __ 9 = ? __ 9

7. Cancel matching zeros.

Add the zero back to the remainder.

R30 )

_________ 6 4 0

640 ____ 000

R3 )

______

4. 1776 5. In which of these numbers does the 5 have a value of 500,000?

A 186,542,039 B 347,820,516

C 584,371,269 D 231,465,987

6. Perimeter Add all sides.

12 mm

8 mm

page 368

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Saxon Math Intermediate 5 375 Adaptations Lesson 58

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8. long division

R40 )

_________ 9 2 2

10. 1400 + m = 7200

m =

11. Offset.

$1.25 4

×

12. Cancel matching zeros.

10 )

_________ 7 0 0 700 ____

000

1 ) ______

13. 6 7 9× 4 8 9

X+0000 0 0 X X

14. 8104− 5647

15. $2.86$6.35$1.78$0.46

+ $0.62

9. Cancel matching zeros.

R50 )

_________ 8 0 0

800 ____ 000

5 )

______

w =

page 369

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Saxon Math Intermediate 5 376 Adaptations Lesson 58

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Written Practice,Written Practice, continued continued

16. 4228 _____ 7 = 17. 4635 _____

9 =

18. 5 __ 5 − 1 __ 5 = 19. 3 1 __

3 − 1 __

3 =

20. 4 6 __ 6

− 2 5 __ 6

21. mixed number

) _____

6 2

22. What is the denominator of the

fraction in 6 3 _ 4 ?

23. quotientdivisor )

_________ dividend )

_____ 000

×

page 369

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Saxon Math Intermediate 5 377 Adaptations Lesson 58

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24. 1500

five centuries

25. 12 mm

d = 2r

26. probability of red

RRB B BY

Y Y Y YY

27. right and isosceles

A B C D

page 369

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Saxon Math Intermediate 5 378 Adaptations Lesson 58

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Written Practice,Written Practice, continued continued

28. 3 _ 4 of 100

29. digits

one billion, two hundred eighty-four million, two hundred four thousand

, , ,

30. $3.48+ $0.32

$10.00

− $ .

) _____

000

I the cost of the cereal and the .

Then I subtracted that from to find the total amount spent

on milk. I that amount by 2 to find the cost of each

.

Use work area.

page 370

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Saxon Math Intermediate 5 379 Adaptations Lesson 59

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5959 Teacher Notes:

• Use fraction pieces to illustrate concepts in this lesson.

• For additional practice, students may complete Fraction Activity D. • Subtracting a

Fraction from 1

New ConceptNew Concept

• If the numerator (top) and denominator (bottom) are the same, the fraction equals 1:

22

33

44

55

• When adding fractions and mixed numbers, remember to simplify any fraction names for 1 in the answer.

• To subtract a fraction from 1, rename the 1 as a fraction:

1 – 1 __ 3

3 __ 3 – 1 __

3 = 2 __ 3

• Look at the fraction that is being subtracted to decide which name for 1 to use.

page 371

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Saxon Math Intermediate 5 380 Adaptations Lesson 59

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a. Write a fraction equal to 1 that has a denominator of 3.

Compare:

b. 4 __ 4

1 c. 5 4 __ 4 6

Add:

d. 3 ___ 10

+ 7 ___ 10

= e. Use fraction manipulatives.

3 3__5

+ 2 2__5

3 3_5 + 2 2_5 = .

I added the fractions using fraction manipulatives. When I put together 3_5 and

, they made a circle. Then I added the whole circles

+ + = .

Subtract. (Rename the 1.)

f. 1 − 1__4

= 00___00

g. 1 − 2 __ 3 = 00

___ 00

h. How many fraction names for 1 are there?

A none B 16 C 247 D infinite

Lesson Practice

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Saxon Math Intermediate 5 381 Adaptations Lesson 59

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Written PracticeWritten Practicepage 374

1. 1 min = s

3 min = s

2. 5 dozen =

1 ___ 10

of that =

3. Draw a quadrilateral with horizontalparallel segments of different lengths.

Use work area.

4. factors of 8: 1, , , 8

factors of 20: 1, , ,

, , 20

, ,

5. 2 __ 5 of a minute 0 __

5 = percent

) _____

6 0

is __ of

1 __ 1 = ? __

0

7. 1 __ 4 + 3 __

4 = 8. 1 1 __

3

+ 2 2 __ 3

9. 2 5 __ 8

+ 3 __ 8

6. ) _____

1 7

It is for each bulletin board to display the same

number of sketches because 17 be divided into

equal groups.Use work area.

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Saxon Math Intermediate 5 382 Adaptations Lesson 59

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Written Practice,Written Practice, continued continued

10. Rename the 1.

1 − 1 __ 4 =

11. Rename the 1.

1 − 3 __ 8 =

12. 2 8 __ 8 − 3 __

8 = 13. 98,789

41,286+ 18,175

14. 47,150– 36,247

15. 3 6 8× 4 7 9

X+0000 0 0 X X

16. 8 9 ___ 10

words:

Use work area.

17. mixed number

15 ___ 4 =

18. long division

R

40 ) _________

6 8 7

page 374

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Saxon Math Intermediate 5 383 Adaptations Lesson 59

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onWritten Practice,Written Practice, continued continued

19. Cancel matching zeros.

Add the zero back to the remainder.

60 )

_________ 8 5 0

850 6 )

_____

20. long division

30 ) __________

$5 . 4 0

21. 5 0 7× $ 3. 6 0

0+0000 0 X X

22. Division: cancel matching zeros.

(900 − 300) ÷ 30 =

23. not equal to 3

A 2 3 __ 3 B 3 2 __

2

C 2 4 __ 4 D 2 8 __

8

24. 1 = 0__5

page 375

25. Perimeter Add all sides.

1 2 3cm

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Saxon Math Intermediate 5 384 Adaptations Lesson 59

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Written Practice,Written Practice, continued continued

26. Distributive Property

35 × (20 + 1) =

Choice 1: 35 × Choice 2: (35 × ) + (35 × )

Choice seems easier because it’s easier to multiply numbers by

than it is to multiply numbers by .

Use work area.

27. C A

C

C BB

D

a. Which two outcomes are equally likely?

b. probability of C

a.

b.

,

29. 1477

1506

+

Use work area.

30. 140

130

120

110F

28. 1, 3, 2, 1, 4, 3, 1, 2, 3, 1, 3, 2, 2, 2, 3, 4, 3, 3, 2

a.

1 2 3 4

b. median

c. mode

b.

c.

a. Use work area.

page 375

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Saxon Math Intermediate 5 385 Adaptations Lesson 60

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6060 Teacher Notes:

• Use fraction pieces to illustrate concepts in this lesson.

• For additional practice, students may complete Fraction Activity E.• Finding a Fraction to

Complete a Whole

New ConceptNew Concept

• To find a fraction to complete a whole, first rename the 1 with a matching numerator and denominator before subtracting.

Example

One third of the students are girls.

What fraction of the students are boys?

The picture below shows that the students are 3 _ 3 .

The girls are 1 _ 3 of the students.

3 _ 3 − 1 _ 3 = 2 _ 3 , so the boys must be 2 _ 3 of the students.

students

girls

boys

Lesson Practice

a. Laxmi has read one FOURTH of her book. What fraction of her book is left to read?

b. Five EIGHTHS of the gymnasts were able to do a back handspring. What fraction of the gymnasts were unable to do a back handspring?

page 377

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Saxon Math Intermediate 5 386 Adaptations Lesson 60

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Lesson Practice, continued

c. If three FIFTHS of the spectators were rooting for the home team, then what fraction of the spectators were not rooting for the home team?

Written PracticeWritten Practice page 379

1. 14 boys

girls

students

2. $2.39$2.39$4.49

+ $0.56

$10.00

$ .

3. 1900 decades 1800 years

1 ___ 10

= ? __ 0

4. What is the radius of the wheel?

24 in.

r = 1 _ 2 d

5. 4 8 7

3 2 6 +

6. Find each missing numerator.

a. 00 ___ 7 = 1

b. 4 = 3 00 ___ 4

a.

b.

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Saxon Math Intermediate 5 387 Adaptations Lesson 60

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7. 1 lb = oz

7 lb = oz

8. 1 mile

Perimeter Add all sides.

9. 1 __ 6 + 2 __

6 + 3 __

6 =

10. 3 3 __ 5

+ 1 2 __ 5

11. Rename the 1.

1 − 1 __ 8 =

12. 4 5 __ 5

– 1 2 __ 5

13. $35.24− $14.62

14. $36.72 _______ 9 = 15. mixed number

23 ___ 10

=

16. 1 _ 8 commercials

fraction not commercials Rename the 1.

1 − 1 __ 8 =

percent commercials See page 19 in the Student Reference Guide.

page 380

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Saxon Math Intermediate 5 388 Adaptations Lesson 60

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17. 3 7 4 0× 3 6 0

0+0000 0 X X

18. long division

643 ÷ 40 =

19. Division: Cancel matching zeros.

60 × (800 ÷ 40) =

20. Cancel matching zeros.

20 ) ____________

1 3 4 0

1340

2 ) ____

21. 4 __ 4 5 __

5 22. 1 = 00 ___

8

23. fraction

10

24. Count by 5s on a clock from 11:25 until 12:00.

page 380

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Saxon Math Intermediate 5 389 Adaptations Lesson 60

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25. a. probability of green

b. probability of not green

WW W W W

W

R RG

GG G

G

a.

b.

26. Think: divisibility.

A 321 ÷ 2 B 421 ÷ 3

C 521 ÷ 6 D 621 ÷ 9

27. Frequency Table

Songs per CD Frequency

9 1

10 4

11 7

12 13

13 3

14 2

a. Among Kabira’s CDs, what number of songs does a CD most frequently have?

b. How many of Kabira’s CDs have more than 10 songs?

a.

b.

page 381

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Saxon Math Intermediate 5 390 Adaptations Lesson 60

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28. Li

fe S

pan

in Y

ears

leopard rabbit pig

Average Life Span of Animals

0

18

16

14

12

10

8

6

4

2

tiger

Animal

wolf

a. The average life span of a is years longer

than the average life span of a .

b. 4 × =

b.

a. Use work area.

29. Which number sentence could be used to find e, Fabian’s age?

A (10 × 2) – 3 = e B (10 ÷ 2) + 3 = e

C (10 × 2) + 3 = e D (10 ÷ 2) – 3 = e

30. 78× 50

78× 50

My estimate is reasonable because 78 is close to and

× = .

Use work area.

page 382

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Saxon Math Intermediate 5 391 Adaptations Investigation 6

Name I N V E S T I G A T I O N©

200

8 S

axon

page 383

Focus onFocus on• Line Graphs

• To show changes in data over a period of time, make a line graph.

Here are the average temperatures in Boston for each month of a year:

Month Temp.

January 30 F

February 31 F

March 38 F

April 49 F

May 59 F

June 68 F

Month Temp.

July 74 F

August 72 F

September 65 F

October 55 F

November 45 F

December 34 F

Average Boston Temperature

• Line graphs are drawn on a grid.

• To make a line graph for the data above:

1. Label each month on the horizontal axis.

2. Label temperatures 0ºF to 80ºF on the vertical axis.

66

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Saxon Math Intermediate 5 392 Adaptations Investigation 6

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3. Place a dot at a height equal to the normal temperature for each month.

01020304050607080

Ave

rag

e Te

mp

erat

ure

(°F

)

J F M A M JMonth

J A S O N D

4. Connect the dots with line segments.

rising line temperature is increasing

falling line temperature is decreasing

steep line temperature changing quickly

6

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Saxon Math Intermediate 5 393 Adaptations Investigation 6

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01020304050607080

Ave

rag

e Te

mp

erat

ure

(°F

)

J F M A M JMonth

J A S O N D

1. In what month was the highest average temperature?

What was that temperature?

2. What is the range of the average temperatures shown in the line

graph?

3. From March through June, the average temperature increases

about how many degrees per month?

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Every two months Liz weighed Jake, her Labrador retriever, and recorded the weight in a table.

Age Birth 2 mo 4 mo 6 mo 8 mo 10 mo 12 mo

Weight (in kg) 0.5 6 12 17 21 24 27

4. Make a line graph showing Jake’s weight over this 12-month period.

5

10

15

20

25

30

Age (in months)

Wei

ght

(in

kg)

2Birth 4 6 8 10 12

5. In which time period did Jake’s weight double?

A 2 mo – 4 mo B 4 mo – 6 mo

C 6 mo – 8 mo D 4 mo – 8 mo

6. Predict Jake’s weight at 14 months.

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Saxon Math Intermediate 5 395 Adaptations Investigation 6

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Mr. Escobar invests in stocks. He has constructed a line graph to show the value of his stocks at the beginning of each year.

2003 2004 2005 20062002200120001998 19991996 1997

$2,000$4,000$6,000$8,000

$10,000$12,000$14,000$16,000

Val

ue

of

Sto

cks

Year

7. How much were Mr. Escobar’s stocks worth at the beginning of

2001?

About how much were they worth at the beginning of 2003?

8. At the beginning of which year were his stocks worth the most?

About how much were they worth then?

9. During which year did his stocks increase in value the most?

About how much was the increase?

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10. During which year did the value of his stocks decrease the

most?

About how much was the decrease?

11. Estimate the overall change in the value of his stocks from the beginning of 1996 to the beginning of 2006.

an in of about

ActivityActivity page 385

Making a Line Graph

• Use your textbook to complete this activity.

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