5th Grade Math student-ebook-course-11

* 3. Formulate There are about three hundred twenty little O’s of cereal
(15) in an ounce. About how many little O’s are there in a one-pound box?

Write an equation and solve the problem (1 pound = 16 ounces).

* 4. There are 31 days in August. How many days are left in August after
(11) August 3? Write an equation and solve the problem.

* 5. Compare: 3 − 1 1 − 3
(14)

* 6. Represent Subtract 5 from 2. Use words to write the answer.
(14)

* 7. The stock’s value dropped from $28.00 to $25.50. Use a negative
(14) number to show the change in the stock’s value.

* 8. Predict What are the next three numbers in the following sequence?
(10)
. . ., 6, 4, 2, 0, , , , . . .

* 9. What is the temperature reading on this 0
(10, 14) thermometer? Write the answer twice, once �5

with digits and an abbreviation and once with
words.

�10

°F

10. $10 − 10¢
(1)

11. How much money is 1 of $3.50?
(6) 2

12. Connect To which hundred is 587 closest?
(9)
587

400 500 600 700

13. 9 + 87 + 654 + 3210 14. 574 × 76
(1) (2)
4320 4320
15. 99 316(62).49336 493
(2)

Analyze Find each unknown number. Check your work.

17. 1200 ÷ w = 300 18. 63w = 63
(4) (4)
76  76  1
19. m 1m 20. w + $65 = $1000
(4) (3)

21. 3 + n + 12 + 27 = 50
(3)

80 Saxon Math Course 1

22. There are 10 millimeters in 1 centimeter. How many millimeters long is
(7) this paper clip?

cm 1 2 3 4 5

23. (8 + 9 + 16) ÷ 3
(5)

24. What is the place value of the 5 in 12,345,678?
(12)

25. Which digit occupi4e3s92t0he ten-billions place in 123,346564,79839,000?

(12)

26. Represent Use the numbers 19, 21, and 40 to write two addition facts
(1) and two subtraction facts.

* 27. Arrange these num7mb6ers i1n order from least to greatest:
(14) 0, −1, 2, −3

28. Of the seventeen students in Angela’s class, eight play in the school
(6) band. What fraction of the total number of students in Angela’s class
are in the band?

* 29. Analyze Reggie sold buttons with a picture of his school’s mascot
(15) for 75¢ each. If Reggie sold seven buttons, how much money did he
receive?

* 30. What number is neither positive nor negative?
(14)

Early Finishers Franklin D. Roosevelt, the thirty-second president of the United States, was
born on January 30, 1882 in Hyde Park, New York. His presidency began
Real-World March 4, 1933 and ended April 12, 1945.
Application
a. How old was Franklin D. Roosevelt when he took office?

b. How many years did he serve as president?

Lesson 15 81

LESSON Rounding Whole Numbers
Estimating
16

Power Up Building Power

facts Power Up D
mental
a. Number Sense: 3 × 30 plus 3 × 2
math b. Number Sense: 4 × 20 plus 4 × 3
c. Number Sense: 150 + 20
d. Number Sense: 75 + 9
e. Number Sense: 800 − 50
f. Number Sense: 8000 − 500
g. Measurement: How many yards are in 6 feet?
h. Calculation: Start with 1. Add 2; multiply by 3; subtract 4; then divide

by 5. What is the answer?

problem Find the next four numbers in this sequence: 2, 3, 5, 8, 9, 11, 14, 15, . . .
solving

New Concepts Increasing Knowledge

rounding When we round a whole number, we are finding another whole number,
whole usually ending in zero, that is close to the number we are rounding. The
number line can help us visualize rounding.
numbers
667

600 610 620 630 640 650 660 670 680 690 700

In order to round 667 to the nearest ten, we recognize that 667 is closer
to 670 than it is to 660. In order to round 667 to the nearest hundred, we
recognize that 667 is closer to 700 than to 600.

Example 1

Round 6789 to the nearest thousand.

Thinking Skill Solution
The number we are rounding is between 6000 and 7000. It is closer to 7000.
Verify

Draw a number
line to verify
the answer is
correct.

82 Saxon Math Course 1

Example 2

Round 550 to the nearest hundred.

Solution

The number we are to round is halfway between 500 and 600. When the
number we are rounding is halfway between two round numbers, we round
up. So 550 rounds to 600.

estimating Rounding can help us estimate the answer to a problem. Estimating is a
quick way to “get close” to the answer. It can also help us decide whether an
answer is reasonable. In some situations an estimate is sufficient to solve a
problem because an exact answer is not needed. To estimate, we round the
numbers before we add, subtract, multiply, or divide.

Example 3
Estimate the sum of 467 and 312.

Solution

Estimating is a skill we can learn to do in our head. First we round each
number. Since both numbers are in the hundreds, we will round each number
to the nearest hundred.

467 rounds to 500

312 rounds to 300

To estimate the sum, we add the rounded numbers.

500
+ 300

800

We estimate the sum of 467 and 312 to be 800.

Example 4

Stephanie stopped at the store to pick up a few items she needs. She
has a $10 bill and a couple of quarters. She needs to buy milk for $2.29,
her favorite cereal for $4.78, and orange juice for $2.42. Does Stephanie
have enough money to buy what she needs?

Solution

An estimate is probably good enough to solve the problem. Milk and juice
are less than $2.50 each, so they total less than $5. Cereal is less than $5.00,
so all three items are less than $10. If tax is not charged on food, she has
enough money. Even if tax is charged she probably has enough.

Lesson 16 83

Math Example 5

Language According to this graph, about how many more people lived in Ashton in
2000 than in 1980?
Words such
as about and

approximately Population (in thousands) Population of Ashton 1970–2000

indicate that an 10

estimate, not an 8
exact answer, is
6
needed.

4

2

0 1980 1990 2000
1970

Solution

We often need to use estimation skills when reading graphs. The numbers
along the left side of the graph (the vertical axis) indicate the population in
thousands. The bar for the year 2000 is about halfway between the 6000 and
8000 levels, so the population was about 7000. In 1980 the population was
about 4000. This problem has a subtraction pattern. We subtract and find
that about 3000 more people lived in Ashton in 2000 than in 1980.

Practice Set Round each of these numbers to the nearest ten:

a. 57 b. 63 c. 45

Round each of these numbers to the nearest hundred:

d. 282 e. 350 f. 426

Round each of these numbers to the nearest thousand:

g. 4387 h. 7500 i. 6750

Estimate Use rounded numbers to estimate each answer.

j. 397 + 206 k. 703 − 598

l. 29 × 31 m. 29 591

Use the graph in example 5 to answer problems n and o.

n. Estimate About how many fewer people lived in Ashton in 1980 than in
1990?

o. Predict The graph shows an upward trend in the population of Ashton.
If the population grows the same amount from 2000 to 2010 as it did
from 1990 to 2000, what would be a reasonable projection for the
population in 2010?

84 Saxon Math Course 1

Written Practice Strengthening Concepts

* 1. What is the difference between the product of 20 and 5 and the sum of
(12) 20 and 5?

* 2. Walter Raleigh began exploring the coastline of North America in 1584.
(13) Lewis and Clark began exploring the interior of North America in 1803.

How many years after Raleigh did Lewis and Clark begin exploring
North America? Write an equation and solve the problem.

* 3. Explain Jacob separated his 140 trading cards into 5 equal groups. He
(15) placed four of the groups into binders. The remaining cards he placed

in a box. How many cards did he put in the box? Write an equation and
solve the problem. Explain why your answer is reasonable.

4. Which digit in 159,342,876 is in the hundred-thousands place?

(12)

5. In the 2004 U.S. presidential election, 121,068,715 votes were tallied for
(12) president. Use words to write that number of votes.

6. What number is halfway between 5 and 11 on the number line?

(9)

* 7. Round 56,789 to the nearest thousand.

(16)

* 8. Round 550 to the nearest hundred.
(16)

* 9. Estimate the product of 295 and 406 by rounding each number to the
(16) nearest hundred before multiplying.

10. 45 + 5643 + 287 7308 11. 40,312 − 14,908
(1) (1) 100 5367
7308 731028
12. 12 7308 100 513106207 5367 13. 100 5367
(2) 12 (2)

14. (5 + 11) ÷ 2
(5)

15. How much money is 1 of $5? 1
(6) m12 uch is 12 of $5? 14
1 1 21 4
16. 4
(6) H2 ow mon4ey

17. $0.25 × 10 18. 325(324 − 323)
(2) (5)

19. Compare: 1 + (2 + 3) (1 + 2) + 3
(9)

* 20. Explain Wind chill describes the effect of temperature and wind
(14) combining to make it feel colder outside. At 3 p.m. in Minneapolis,

Minnesota, the wind chill was −10° Fahrenheit. At 11 p.m. the wind chill
was −3° Fahrenheit. At which time did it feel colder outside, 3 p.m. or
11 p.m.? Explain how you arrived at your answer.

* 21. Formulate Your heart beats about 72 times per minute. At that rate,
(15) how many times will it beat in one hour? (Write an equation and solve
the problem.)

Lesson 16 85

22. Estimate The distance between bases on a major league baseball
(8) diamond is 90 feet. A player who runs around the diamond runs about
how many feet?

3rd 2nd
base base

1st
base

home
Hay Eaten Daily
(in kilograms)
90 ft
Refer to the bar graph shown below to answer problems 23∙26.

Hay Eaten by Elephants
100

80

60

40

20

0 Mother Father
Baby

* 23. How many more kilograms of hay does the father elephant eat each day
(16) than the baby elephant?

* 24. Altogether, how many kilograms of hay do the three elephants eat each
(16) day?

* 25. How many kilograms of hay would the mother elephant eat in one
(16) week?

* 26. Formulate Using the information in this graph, write a comparison word
(16) problem.

Analyze Find each unknown number. Check your work.

27. 6w = 66 28. m − 60 = 37
(4) (3)

29. 60 − n = 37
(3)

* 30. Represent Each day Chico, Fuji, and Rolo drink 6, 8, and 9 glasses of
(Inv. 1) water respectively. Draw a bar graph to illustrate this information.

86 Saxon Math Course 1

LESSON The Number Line: Fractions and
Mixed Numbers
17

Power Up Building Power

facts Power Up E
mental
a. Number Sense: 5 × 30 plus 5 × 4
math b. Number Sense: 4 × 60 plus 4 × 4
c. Number Sense: 180 + 12
d. Calculation: 64 + 9
e. Number Sense: 3000 − 1000 − 100
f. Calculation: $10.00 − $7.50
g. Measurement: How many millimeters are in 2 meters?
h. Calculation: Start with 5. Multiply by 4; add 1; divide by 3; then

subtract 2. What is the answer?

problem If an 8 in.-by-8 in. pan of lasagna serves four people, how many 12 in.-by-
solving 12 in. pans of lasagna should be purchased to serve 56 people? (Hint: You
may have “leftovers.”)

Understand An eight-inch-square pan of lasagna will serve four people.
We are asked to find how many 12-inch-square pans of lasagna are needed
to feed 56 people.

Plan We will draw a diagram to help us visualize the problem. Then we
will write an equation to find the number of 12-inch-square pans of lasagna
needed.

Solve First, we find the size of each serving by “cutting” the 8-inch-square
pan into four pieces. Then we see how many pieces of the same size can be
made from the 12-inch-square pan:

8 in. 4 in. 4 in. 12 in. 4 in. 4 in. 4 in.

4 in. 4 in.
8 in. 4 in.

4 in. 12 in.

Each person eats one 4 in.
4-inch-square piece.

One 12-inch-square pan of lasagna can be cut
into nine 4-inch-square pieces.

One 12-inch-square pan of lasagna can serve 9 people. We use this

information to write an equation: N × 9 = 56. We divide to find that
N = 6 R2. Six pans of lasagna would only provide 54 pieces, so we must
buy seven pans of lasagna in order to serve 56 people.

Lesson 17 87

Check We found that we need to buy seven 12-inch-square pans of
lasagna to serve 56 people. It would take 14 8-inch-square pans of lasagna
to feed 56 people, and one 12-inch-square lasagna feeds about twice as
many people as one 8-inch-square pan, so our answer is reasonable.

New Concept Increasing Knowledge

Math Language On this number line the tick 1marks show1 the location of the integers:

Recall that 44

integers are the –3 –2 –1 0 1 2 3

set of counting There are points on the number line between the integers that can be named
with fractions or mixed numbers. A mixed number is a whole number plus
numbers, their Ha afrlafwctaioynb.eHtwalefwenay−b1eatwndee−n2120is�a−nd1121. is 12. Halfway between 1 and 2 is 112.

opposites, and 121

zero. 121

–3 –2 1 –2 –1 1 –1 – 1 0 1 1 121 2 2 1 3
2 2 2 2 2

We count from zero.

The distance between consecutive integers on a number line may be divided
into halves, thirds, fourths, fifths, or any other number of equal divisions. To
determine which fraction or mixed number is represented by a point on the
number line, we follow the steps described in the next example.

Example 1
Point A represents what mixed number on this number line?

A

0123

Solution

We see that point A represents a number greater than 2 but less than 3. So
point A represents a mixed number, which is a whole number plus a fraction.
To find the fraction, we first notice that the segment from 2 to 3 has been
divided into five smaller segments. The distance from 2 to point A crosses
three of the five segments. Thus, point A represents the mixed number 2 35.
Analyze Why did we focus on the number of segments on the number line
and not the number of vertical tick marks?

88 Saxon Math Course 1

Activity

Inch Ruler to Sixteenths

Materials needed:
• inch ruler made in Lesson 7

In Lesson 7 we made an inch ruler divided into fourths. In this activity we will
divide the ruler into eighths and sixteenths. First we will review what we did in
Lesson 7.
We used a ruler to make one-inch divisions on a strip of tagboard.

1234

Then we estimated the halfway point between inch marks and drew new
marks. The new marks were half-inch divisions. Then we estimated the
halfway point between the half-inch marks and made quarter-inch divisions.

1234

We made the half-inch marks a little shorter than the inch marks and the
quarter-inch marks a little shorter than the half-inch marks.
Now divide your ruler into eighths of an inch by estimating the halfway
point between the quarter-inch marks. Make these eighth-inch marks a little
shorter than the quarter-inch marks.

1234

Finally, divide your ruler into sixteenths by estimating the halfway point
between the eighth-inch marks. Make these marks the shortest marks on
the ruler.

1234

Example 2
Estimate the length of this line segment in inches. Then use your ruler to
find its length to the nearest sixteenth of an inch.

Solution
The line segment is about 3 inches long. The ruler has been divided into
sixteenths. We align the zero mark (or end of the ruler) with one end of the
line segment. Then we find the mark on the ruler closest to the other end
of the line segment and read this mark. As shown on the next page, we will
enlarge a portion of a ruler to show how each mark is read.

Lesson 17 89

Thinking Skill 1 1 3 5 3 7 9 5 11 13 7 15
16 8 16 1 16 8 16 16 8 16 3 16 8 16
Connect 1 341 1
inch 1 2 42 2 2 3 4
How is a number 1
line like a ruler? 4

1 21116 13 5 3
8 16 16 8

1 1 1 1 121278 2 7 7 7
2 2 8 8 8

1 1 1 1Es1241timate We find that the line segment is about 2 7 inches long. This is the 7
2 8 8
nearest sixteenth because the end of the segment aligns more closely to the
2 1 1
2
7 7 mark (which equals11361164 ) than it does to th11e64 13 mark or to the 15 mark. 15
8 8 16 16 16
1 13 1
16 8 16 � 1121156 4
Practice Set a12. 121: 1 5
Generalize Continue this sequence to 176, 13 4 16 3
15 116116, 18, 136, 14, 15186, 38, 8
15 16 121, 61…36p
16

15 3 b. What nu7mber is1halfway between −2 and −3?
4 16
8 16 2

c. Represent What number is halfway between 2 and 5? Draw a number

1 11 31 line to sho3w1the number15that is halfw5ay3between 2 3and7 5. 1 7 1
3 3 116 5 3 7 1165 8 3 71 156d1128836. 1P68312o41i7n6t116A repr21es14e167n6ts w18 hat21m1i6xe8d 1n36umber 8 16 2 16 2
8 16 416 8 16 146 8 126 7 5 3
C1o6nnect on this41 number 16 8

1 line? 13 15 3 7 1
31513 7 16 31513 7 8 1 153 7 16 4 1 3 A7 16 81 16 2
51 13513 1168468 163 1216648 16 1 21468 16 5 15 3 7 3 17
68476 11 1 116166848 216 8 16 8 2 8 16 7 2 161 2
16 126 2 1 16 4 16
8 16 2

0 1 1216, 18, 136, 14, 156, 38, 176, 12, p

Use your ruler to find th1e16l,en18g, 1t3h6,o14f ,e1a56c,h38o,f17t6h,e12s,e line segments to the nearest
sixteenth of an inch: p

e.

f.

12, p 116, 18, 136, 14, 111566,, 1838,, 113766,, 1412,, 15p6g,.38, 176,S12t,repngthe1n16in, g18,C13o6n, c14e, p156ts, 38, 176, 12,

Written Practice p

6, ,3818, ,111716636,, ,181214,, ,131p6156, ,14183, ,1516376, ,38142, ,17165p6, ,1238, , p171(611162,).,1218,Ws,ixp1h36ah,tu14ins,dt1rh5e6ed, s38u, m176o, f12t,wpelve thousand, five hundred and ten thousand,
ten?

90 Saxon Math Course 1

2. In 1903 the Wright brothers made the first powered airplane flight. In
(13) 1969 Americans first landed on the moon. How many years was it from

the first powered airplane flight to the first moon landing? (Write an
equation and solve the problem.)

* 3. Linda can run about 6 yards in one second. About how far can she run
(15) in 12 seconds? Write an equation and solve the problem.

* 4. A coin collector has a collection of two dozen rare coins. If the value of
(15) each coin is $1000, what is the value of the entire collection? Write an

equation and solve the problem.

* 5. Estimate Find the sum of 5280 and 1760 by rounding each number to
(16) the nearest thousand before adding.

438(620). 480 7. 6 � 66 � 6
3 33
(5)

8. Represent The letters a, b, and c represent three different numbers.
(1) The sum of a and b is c.

a+b=c

13 1 sRu1e56batrrraacntgioentheequlea83tttieornsst.oHfionrtm: Toa1n7b6oethseurready12doiutioanrraenqgueadtio4tnh38e0anledttetwrsoin the
8 16 4

6, 14, 156, 38, 176, 12, p correct order, choose numbers for a, b, and c that make a + b = c true.

Then try those numbers in place of the letters in your three equations.

52805280
494. R4e4write 2 ÷ 3 with a division bar, but do not divide.

(2)

10. Explain A square has sides 10 cm long. Describe how to find its
(8) perimeter.

* 11. Connect Use a ruler to find the leng4th80of the line seg5m2e8n0t below to the 6�
(17) nearest sixteenth of an inch. 3 44 3

Find each unknown number. Check your work.

12. $3 − y = $1.75 13. m − 20 = 30
(3) (3)

14. 12n = 0 15. 16 + 14 = 14 + w
(4) (3)

16. Compare: 19 × 21 20 × 20
(9)
18. 5280
17. 100 − (50 − 25) (2) 44
(5)

19. 365 + 4576 + 50,287
(1)

20. What number is missing in the following sequence?
(10)

5, 10, , 20, 25, …

21. Which digit in 987,654,321 is in the hundred-millions place?

(12)

22. 250,000 ÷ 100 23. $3.75 × 10
(2) (2)

Lesson 17 91

* 24. Estimate An 8-ounce serving ohaf l2f%themgilrkamcosnatareincsa2r6b41oghraydmrsatoefsp. Arobtoeiunt,
(15) fat, and carbohydrates. About

how many grams are not from carbohydrates?
11
4 What is the sum of the first six posit2ive odd numbers?
25. Analyze
(10)

26. Explain How can you find 1 of 52? 1
(6) 4 2

1 27. A quarter is 1 of a d1ollar. 1
4 (6) 4 2
a. How many quar2ters are in one dollar?

11
4 b. How many quarters are i2n three dollars?
mark is halfway between 1
* 28. 1 the 2 1 and
(17)
On an inch ruler, w4hich 41-inch 2mark
the 21-inch mark?

* 29. Point A represents what mixed number on the number line below?

(17) 1
2A

1
12
2

1 3456
2

* 30. A segment that is 1 of an inch long is how many sixte1enths of an inch
(17) 2
long?
2

Early Finishers Read the problem below. Then decide whether you can use an estimate to
answer the question or if you need to compute an exact answer. Explain how
Real-World you find your answer.
Application
Manny has one gallon of paint. The label states that it covers about 350−450
square feet. Manny wants to paint the living room, dining room and family
room of his home. The lateral surface area of all of the rooms totals about
800 square feet. How much more paint should Manny buy?

92 Saxon Math Course 1

LESSON Average
Line Graphs
18

Power Up Building Power

facts Power Up B
mental
a. Number Sense: 4 × 23 equals 4 × 20 plus 4 × 3. Find 4 × 23.
math b. Number Sense: 4 × 32
c. Number Sense: 3 × 42
d. Number Sense: 3 × 24
e. Geometry: A hexagon has sides that measure 15 ft. What is the

perimeter of the hexagon?
f. Measurement: How many days are in 2 weeks?
g. Measurement: How many hours are in 2 days?
h. Calculation: Start with a half dozen. Add 2; multiply by 3; divide by 4;

then subtract 5.What is the answer?

problem In his pocket Alex had seven coins totaling exactly one dollar. Name
solving a possible combination of coins in his pocket. How many different
combinations of coins are possible?

New Concepts Increasing Knowledge

average Here we show three stacks of books; the stacks contain 8 books, 7 books,
and 3 books respectively. Altogether there are 18 books, but the number of
books in each stack is not equal.

If we move some of the books from the taller stacks to the shortest stack,
we can make the three stacks the same height. Then there will be 6 books
in each stack.

Lesson 18 93

By making the stacks equal, we have found the average number of books in
the three stacks. Notice that the average number of books in each stack is
greater than the number in the smallest stack and less than the number in the
largest stack. One way we can find an average is by making equal groups.

Example 1

In four classrooms there were 28 students, 27 students, 26 students,
and 31 students respectively. What was the average number of students
per classroom?

Solution

The average number of students per classroom is how many students there
would be in each room if we made the numbers equal. So we will take the
total number of students and make four equal groups. To find the total
number of students, we add the numbers in each classroom.

28 students
27 students
26 students
+ 31 students
112 students in all

We make four equal groups by dividing the total number of students by four.

28 students
4 112 students
If the groups were equal, there would be 28 students in each classroom.
The average number of students per classroom would be 28.

Notice that an average problem is a “combining” problem and an “equal
groups” problem. First we found the total number of students in all
the classrooms (“combining” problem). Then we found the number of
students that would be in each group if the groups were equal (“equal
groups” problem).

Example 2 4 1132 students

Use counters to model the average of 3, 7, and 8.

Solution

Thinking Skill This question does not tell us whether the numbers 3, 7, and 8 refer to
books or students or coins or quiz scores. Still, we can find the average of
Connect these numbers by combining and then making equal groups. We can model
the problem with counters. Since there are three numbers, there will be
What are some three groups of counters with 3, 7, and 8 counters in the groups. To find the
other real- average we make the groups equal. One way to do this is to first combine all
life situations the counters in the three groups. That gives us a total of 18 counters.
where we might
need to find 3 + 7 + 8 = 18
the average of a
set of numbers? Then we divide the total into three equal groups.

18 ÷ 3 = 6

94 Saxon Math Course 1

That gives us three groups of six counters. We find that the average of 3, 7,
and 8 is 6.

Example 3
What number is halfway between 27 and 81?

Solution

The number halfway between two numbers is also the average of the two
numbers. For example, the average of 7 and 9 is 8, and 8 is halfway between
7 and 9. So the average of 27 and 81 will be the number halfway between 27
and 81. We add 27 and 81 and divide by 2.

Average of 27 and 81 = 27  81
2

27  81 = 108
2 2

= 54

The number halfway between 27 and 81 is 54.

line graphs The average we have talked about in this lesson is also called the mean. We
will learn more about average and mean in later lessons.

Line graphs display numerical information as points connected by line
segments. Whereas bar graphs often display comparisons, line graphs often
show how a measurement changes over time.

Example 4

Reading Math This line graph shows Margie’s height in inches from her eighthHeight (in inches)
birthday to her fourteenth birthday. During which year did Margie grow
On this line graph, the most?
the horizontal axis
is divided into Margie’s Height
equal segments 68
that represent
years. The vertical 64
axis is divided into
equal segments 60
that represent
inches. The labels 56
on the axes tell us
how many years 52
and how many
inches. 48

44
8 9 10 11 12 13 14
Age (in years)

Solution

From Margie’s eighth birthday to her ninth birthday, she grew about two
inches. She also grew about two inches from her ninth to her tenth birthday.
From her tenth to her eleventh birthday, Margie grew about five inches.
Notice that this is the steepest part of the growth line. So the year Margie
grew the most was the year she was ten.

Lesson 18 95

Practice Set Your teacher might ask you to keep a line graph of your math test scores.
The Lesson Activity 2 Test Scores Line Graph can be used for this purpose.

a. There were 26 books on the first shelf, 36 books on the second shelf,
and 43 books on the third shelf. Velma rearranged the books so that
there were the same number of books on each shelf. After Velma
rearranged the books, how many were on the first shelf ?

b. What is the average of 96, 44, 68, and 100?

c. What number is halfway between 28 and 82?

d. What number is halfway between 86 and 102?

e. Find the average of 3, 6, 9, 12, and 15.

Use the information in the graph in example 4 to answer these questions:

f. How many inches did Margie grow from her eighth to her twelfth
birthday?

g. During which year did Margie grow the least?

h. Predict Based on the information in the graph, would you predict that
Margie will grow to be 68 inches tall?

Written Practice Strengthening Concepts

1. Jumbo ate two thousand, sixty-eight peanuts in the morning and three
(11) thousand, nine hundred forty in the afternoon. How many peanuts did

Jumbo eat in all? What kind of pattern did you use?

2. Jimmy counted his permanent teeth. He had 11 on the top and
(11) 12 on the bottom. An adult has 32 permanent teeth. How many

more of Jimmy’s teeth need to grow in? What kind of pattern did
you use?

* 3. Olivia bought one dozen colored pencils for an art project. Each pencil
(15) cost 53¢ each. How much did Olivia spend on pencils? What kind of

pattern did you use?

* 4. Estimate Find the difference of 5035 and 1987 by rounding each
(16) number to the nearest thousand before subtracting.

* 5. Find the average of 9, 7, and 8.
(18)

* 6. What number is halfway between 59 and 81?
(18)

* 7. What number is 6 less than 2?

(14)

8. $0.35 × 100 9. 10,010 ÷ 10 10. 34,180 ÷ 17
(2)
(2) (2)

96 Saxon Math Course 1

11. $3.64 + $94.28 + 87¢ 12. 41,375 − 13,576
(1) (1)

13. 125 × 16 14. 4 ∙ 3 ∙ 2 ∙ 1 ∙ 0
(2) (5)

Analyze Find each unknown number. Check your work. 1
2
15. w − 84 = 48 16. 234 = 6
(3) (4) n

17. (1 + 2) × 3 = (1 × 2) + m

(3, 5)

18. Model Draw a rectangle 5 cm long and 3 cm wide. What is its
(8) perimeter?

19. What is the sum of the first six positive even numbers?

(10)

20. Generalize Describe the rule of the following sequence. Then find the
(10) missing term.

1, 2, 4, , 16, 32, 64, …

21. Compare: 500 × 1 500 ÷ 1
(9)

234 22. Generalize What number is 1 of 1110?
n (6) 2

23. What is the place value of the 7 in 987,654,321?

(12)

Refer to the line graph shown below to answer problems 24–26.

Heart Rates During Various Activities
200

Heartbeats per Minute 140

80

20 Walking Running
Resting

* 24. Estimate Running increases a resting person’s heart rate by about how
(18) many heartbeats per minute?

* 25. Estimate About how many times would a person’s heart beat during a
(18) 10-minute run?

* 26. Formulate Using the information in the line graph, write a word problem
(18) about comparing. Then answer the problem.

* 27. Explain In three classrooms there are 24, 27, and 33 students
(18) respectively. How many students will be in each classroom if some
students are moved from one classroom to the other classrooms so that
the number of students in each classroom is equal? How do you know
that your answer is reasonable?

Lesson 18 97

28. A dime is 1 of a dollar. 2 1
(6) 10 4

a. How many dimes are in a dollar?

b. How many dimes are in three dollars?

1 * 29. Mode1l10 Use a ruler to dra2w41 a rectangle that is 2 1 inches long and 1 3 1 3
10 (17) 4 4 4
inches wide.

* 30. Formulate Word problems about finding an average include which two
(18) types of problems? (Select two)

combining separating comparing equal groups

Early Finishers The cost of one cubic foot of natural gas in my town is $0.67. My meter
reading on July 31 was 1518. On August 31 it was 1603. The meter measures
Real-World natural gas in cubic feet.
Application
a. How many cubic feet of gas did I use in the month of August (between
the two meter readings)?

b. What was my total gas cost for the month?

98 Saxon Math Course 1

LESSON Factors
Prime Numbers
19

Power Up Building Power

facts Power Up D
mental
a. Number Sense: 3 × 64
math b. Number Sense: 3 × 46
c. Number Sense: 120 + 18
d. Calculation: 34 + 40 + 9
e. Calculation: 34 + 50 − 1
f. Calculation: $20.00 − $12.50
g. Measurement: How many decades are in 2 centuries?
h. Calculation: Start with 100. Divide by 2; subtract 1; divide by 7; then

add 3. What is the answer?

problem Doubles tennis is played on a rectangular-
solving shaped court that is 78 feet long and
36 feet wide. In the drawing of the doubles
tennis court, how many rectangles can you
find?

New Concepts Increasing Knowledge

factors Recall from Lesson 2 that a factor is one of the numbers multiplied to form a
product.

2 × 3 = 6 Both 2 and 3 are factors.

1 1 × 6 = 6 Both 1 and 6 are factors.

Thinking Ski4ll 1

We see that each of the n4umbers 1, 2, 3, and 6 are factors of 6. Notice that

How are the when we divide 6 by 1, 2, 3, or 6, the resulting quotient has no remainder

two definitions (that is, it has a remainder of zero). We say that 6 is “divisible by” 1, 2, 3,

of factor and 6.

6 connected? 6 33 6 3 22 11 32

6 1 6 226 6 1 6 2 6 336 6 66 6 2 6 3 6 6

6666
0000

This leads us to another definition of factor.

66

0 The f0actors of a given number are

the whole numbers that divide the

given number without a remainder.

Lesson 19 99

We can illustrate the factors of 6 by arranging 6 tiles to form rectangles.
With 6 tiles we can make a 1-by-6 rectangle. We can also make a 2-by-3
rectangle.

12
6
3

The number of tiles along the sides of these two rectangles (1, 6, 2, 3) are the
four factors of 6.

Example 1
What are the factors of 10?

Solution

The factors of 10 are all the numbers that divide 10 evenly (with no
remainder). They are 1, 2, 5, and 10.

10 5 10 5 10 2 5 2 15 21 2
1 10 2 10 1 10 2 10 151010 2 10 5 10 10 102 10 51100 10 5 10

We can illustrate the factors of 10 with two rectangular arrays of tiles.

12
10
5

The number of tiles along the sides of the two rectangles (1, 10, 2, 5) are the
factors of 10.

Example 2

How many different whole numbers are factors of 12?

Solution

Twelve can be divided evenly by 1, 2, 3, 4, 6, and 12. The question asked
“How many?” Counting factors, we find that 12 has 6 different whole-number
factors.
Twelve tiles can be arranged to form three different shapes of rectangles. The
lengths of the sides illustrate that the six factors of 12 are 1, 12, 2, 6, 3, and 4.

1
12

2 3
6 4

Model Draw tiles to illustrate the factors of 18. How many different shapes
of rectangles can you make?

100 Saxon Math Course 1

prime Here we list the first ten counting numbers and their factors. Which of the
numbers numbers have exactly two factors?

Number Factors

11

2 1, 2

3 1, 3

4 1, 2, 4

5 1, 5

6 1, 2, 3, 6

7 1, 7

8 1, 2, 4, 8

9 1, 3, 9

10 1, 2, 5, 10

Counting numbers that have exactly two factors are prime numbers. The
first four prime numbers are 2, 3, 5, and 7. The only factors of a prime
number are the number itself and 1. The number 1 is not a prime number,
because it has only one factor, itself.

Therefore, to determine whether a number is prime, we may ask ourselves
the question, “Is this number divisible by any number other than the number
itself and 1?” If the number is divisible by any other number, the number is
not prime.

Example 3

The first four prime numbers are 2, 3, 5, and 7. What are the next four
prime numbers?

Solution

We will consider the next several numbers and eliminate those that are not
prime.

8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

All even numbers have 2 as a factor. So no even numbers greater than two
are prime numbers. We can eliminate the even numbers from the list.

8 , 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Since 9 is divisible by 3, and 15 is divisible by 3 and by 5, we can eliminate 9
and 15 from the list.

8 , 9 , 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Each of the remaining four numbers on the list is divisible only by itself and
by 1. Thus the next four prime numbers after 7 are 11, 13, 17, and 19.

Lesson 19 101

Thinking Skill Activity

Conclude Prime Numbers

Why don’t List the counting numbers from 1 to 100 (or use Lesson Activity 3 Hundred
the directions Number Chart). Then follow these directions:
include “Circle
the number 4” Step 1: Draw a line through the number 1. The number 1 is not a prime
and “Circle the number.
number 6”?
Step 2: Circle the prime number 2. Draw a line through all the other multiples
of 2 (4, 6, 8, etc.).

Step 3: Circle the prime number 3. Draw a line through all the other multiples
of 3 (6, 9, 12, etc.).

Step 4: Circle the prime number 5. Draw a line through all the other multiples
of 5 (10, 15, 20, etc.).

Step 5: Circle the prime number 7. Draw a line through all the other multiples
of 7 (14, 21, 28, etc.).

Step 6: Circle all remaining numbers on your list (the numbers that do not
have a line drawn through them).

When you have finished, all the prime numbers from 1 to 100 will be circled
on your list.

Practice Set List the factors of the following numbers:

a. 14 b. 15

c. 16 d. 17

Justify Which number in each group is a prime number? Explain how you
found your answer.

e. 21, 23, 25

f. 31, 32, 33

g. 43, 44, 45

Classify Which number in each group is not a prime number?
h. 41, 42, 43

i. 31, 41, 51

j. 23, 33, 43

Prime numbers can be multiplied to make whole numbers that are not prime.
For example, 2 ∙ 2 ∙ 3 equals 12 and 3 ∙ 5 equals 15. (Neither 12 nor 15 are
prime.) Show which prime numbers we multiply to make these products:

k. 16 l. 18

102 Saxon Math Course 1

Written Practice Strengthening Concepts

Math Language 1. If two hundred fifty-two is the dividend and six is the quotient, what is
(4) the divisor?
The dividend is
the number that * 2. In 1863, President Abraham Lincoln gave his Gettysburg Address,
is to be divided. (11, 15) which began “Fourscore and seven years ago … .” A score equals

twenty. What year was Lincoln referring to in his speech? Explain how
you found your answer.

* 3. The temperature in Barrow, Alaska was –46°F on January 22, 2002.
(14) It was 69°F on July 15, 2002. How many degrees warmer was it on

July 15 than on January 22?

* 4. If 203 turnips are to be shared equally among seven rabbits, how many
(15) should each receive? Write an equation and solve the problem.

* 5. What is the average of 1, 2, 4, and 9?
(18)

6. Predict What is the next number in the following sequence?
(10)

1, 4, 9, 16, 25, , ...

7. A regular hexagon has six sides of equal length. If each side of a
(8) hexagon is 25 mm, what is the perimeter?

8. One centimeter equals ten millimeters. How many millimeters long is the
(7) line segment below?

cm 1 2 3 4 5

* 9. What are the whole-number factors of 20?
(19)

* 10. How many different whole numbers are factors of 15?
(19)

* 11. Classify Which of the numbers below is a prime number?
(19)
A 25 B 27 C 29

12. 250,000 ÷ 100 13. 1234 ÷ 60
(2)
(12)
15. $3.45 × 10
* 14. 6  18  9 (2)
(15) 3

Find each unknown number. Check your work.

16. $10.00 − w = $1.93 17. w  4
(3) (4) 3

18. Represent The letters a, b, and c represent three different numbers.
(2) The product of a and b is c.

ab = c

Rearrange the letters to form another multiplication equation and two
division equations.

1

2 Lesson 19 103

w  4
3

* 19. Arrange these numbers in order from least to greatest:
(17)
3, −2, 1, 21, 0

20. Compare: 123 ÷ 1 123 − 1
(9)

21. Which digit in 135,792,468,000 is in the ten-millions place? 1
2
(12)

* 22. Connect Round 123,456,789 to the 1n38earest million.
(16)

23. How much money is 1 of $11.00?
(6) 2

24. If a square has a perimeter of 48 inches, how long is each side 1of83 the
(8) square?

25. (51 + 49) × (51 − 49)1 3 w  4
(5) 8 3

* 26. Classify Which of the numbers below is a prime number?
(19) A 2
B 22 C 222

* 27. Prime numbers can be multiplied to make whole numbers that are not
(19) prime. To make 18, we perform the multiplication 2 ∙ 3 ∙ 3. Show which
prime numbers we multiply to make 20.

* 28. The dictionaries are placed in three stacks. There are 6 dictionaries
(18) in one stack and 12 dictionaries in each of21the other two stacks. How
many dictionaries will be in each stack if some dictionaries are moved

from the taller stacks to the shortest stack so that there are the same

number of dictionaries in each stack?

* 29. Model Draw a square with sides that are 1 3 inches long.
(17) 8

* 30. Explain How can you use the concepts of “even” and “odd” numbers
(10, 19) to determine whether a number is divisible by 2.

Early Finishers The school newspaper reported that middle school students own an average
of 47 CDs. Yolanda asked seven of her friends how many CDs they owned
Real-World and got these results.
Application
32, 49, 21, 59, 37, 44, 52

a. What is the average number of CDs owned by Yolanda’s seven friends?

b. How does this average compare to the average reported in the school
newspaper?

104 Saxon Math Course 1

LESSON Greatest Common Factor (GCF)

20

Power Up Building Power

facts Power Up C
mental
a. Number Sense: 6 × 23
math b. Number Sense: 6 × 32
c. Number Sense: 640 + 1200
d. Calculation: 63 + 20 + 9
e. Calculation: 63 + 30 − 1
f. Number Sense: $100.00 − $75.00
g. Measurement: How many minutes are in 2 hours?
h. Calculation: Start with 10. Multiply by 10; subtract 1; divide by 9; then

add 1. What is the answer?

problem A card with a triangle on it in the position
solving shown is rotated 90° clockwise three times.
Sketch the pattern and draw the triangles in
the correct positions.

New Concept Increasing Knowledge

The factors of 8 are

1, 2, 4, and 8

The factors of 12 are

1, 2, 3, 4, 6, and 12

Math Language We see that 8 and 12 have some of the same factors. They have three factors

Recall that a in common. These common factors are 1, 2, and 4. Their greatest common

factor is a whole factor—the largest factor that they both have—is 4. Greatest common factor
number that is often abbreviated GCF.
divides another

whole Example 1
number

without a Find the greatest common factor of 12 and 18.

remainder. Solution

The factors of 12 are: 1, 2, 3, 4, 6, and 12.

The factors of 18 are: 1, 2, 3, 6, 9, and 18.

We see that 12 and 18 share four common factors: 1, 2, 3, and 6. The
greatest of these is 6.

Lesson 20 105

Example 2
Find the GCF of 6, 9, and 15.

Solution

Visit www. The factors of 6 are: 1, 2, 3, and 6.
SaxonPublishers.
com/ActivitiesC1 The factors of 9 are: 1, 3, and 9.
for a graphing
calculator activity. The factors of 15 are: 1, 3, 5, and 15.

Practice Set The GCF of 6, 9, and 15 is 3.

Note: The search for the greatest common factor of two or more numbers
is a search for the largest number that evenly divides each of them.
In this problem we can quickly determine that 3 is the largest number
that evenly divides 6, 9, and 15. A complete listing of the factors might
be helpful but is not required.

Generalize What is the GCF of any two prime numbers? Explain your answer.

Find the greatest common factor (GCF) of the following:

a. 10 and 15 b. 18 and 27

c. 18 and 24 d. 12, 18, and 24

e. 15 and 25 f. 20, 30, and 40

g. 12 and 15 h. 20, 40, and 60

i. Analyze Write a list of three numbers whose GCF is 7.

Written Practice Strengthening Concepts

1. What is the difference between the product of 12 and 8 and the sum of
(12) 12 and 8?

2. Saturn’s average distance from the Sun is one billion, four
(12) hundred twenty-nine million kilometers. Use digits to write that

distance.

3. Which digit in 497,325,186 is in the ten-millions place?

(12)

* 4. Estimate Jill has exactly 427 beads, but when Dwayne asked her how
(16) many beads she has, Jill rounded the amount to the nearest hundred.

How many beads did Jill say she had?

* 5. The morning temperature was −3°C. By afternoon it had warmed to
(14) 8°C. How many degrees had the temperature risen?

106 Saxon Math Course 1

* 6. In three basketball games Allen scored 31, 52, and 40 points. What was
(18) the average number of points Allen scored per game?

* 7. Find the greatest common factor of 12 and 20.

(20)

* 8. Find the GCF of 9, 15, and 21.

(20)

9. Connect How much money is 1 of $3.24? 28  42
4 14
(6)

4110. 5432 ÷ 10 11. 28  42
(2) (5) 14

12. 56,042 + 49,985 13. 37,080 ÷ 12
(1) (2)

14. $6.47 × 10 15. 5 × 4 × 3 × 2 × 1
(2) (5)

Analyze Find each unknown number. Check your work.

16. w − 76 = 528 17. 14,009 − w = 9670
(3) (3)

18. 6w = 90 19. q − 365 = 365
(4) (3)

20. 365 − p = 365
(3)

21. Generalize Find the missing number in the following sequence:

(10)

, 10, 16, 22, 28, . . .

22. Compare: 50 −1 49 + 1
(9)

23. Predict The first positive odd number is 1. What is the tenth positive
(10) odd number?

24. Explain The perimeter of a square is 100 cm. Describe how to find the
(8) length of each side.

* 25. Estimate Estimate the length of this key to the nearest inch. Then
(17) use a ruler to find the length of the key to the nearest sixteenth of
an inch.

11
88

26. A “bit” is 1 of a dollar.
(6) 8
a. How many bits are in a dollar?

b. How many bits are in three dollars?

* 27. In four boxes there are 12, 24, 36, and 48 golf balls respectively. If the
(18) golf balls are rearranged so that there are the same number of golf balls
in each of the four boxes, how many golf balls will be in each box?

Lesson 20 107

* 28. Classify Which of the numbers below is a prime number?
(19)
A5 B 15 C 25

* 29. Explain List the whole-number factors of 24. How did you find your
(19) answer?

* 30. Ten billion is how much less than one trillion?
(12)

Early Finishers Tino’s parents like mathematics, especially prime numbers. So they made a
plan for his allowance. They will number the weeks of the year from 1 to 52.
Real-World On week 1, they will pay Tino $1. On weeks that are prime numbers, they will
Application pay him $3. On weeks that are composite numbers, they will pay him $5.

a. How much money will Tino receive for one year? Show your work.

b. Would Tino receive more or less if his parents paid him $5 on “prime
weeks” and $3 on “composite weeks”? Show your work.

108 Saxon Math Course 1

INVESTIGATION 2

Focus on
Investigating Fractions with Manipulatives

In this investigation you will make a set of fraction manipulatives to help you
answer questions in this investigation and in future problem sets.

Activity

Using Fraction Manipulatives

Materials needed:
• Investigation Activities 4–8
• scissors
• envelope or zip-top bag to store fraction pieces

Preparation:

To make your own fraction manipulatives, cut out the fraction circles on the
Investigation Activities. Then cut each fraction circle into its parts.

Thinking Skill Model Use your fraction manipulatives to help you with these exercises:

Connect 1. What percent of a circle is 1 of a circle?
2
What percent
is one whole 2. What fraction is half of 12?
circle?

3. What fraction is half of 14?

4. Fit three 1 pieces together to form 3 of a circle. Three fourths of a circle
4 4
is what percent of a circle?

5. Fit four 1 pieces together to form 4 of a circle. Four eighths of a circle is
8 8
what percent of a circle?

6. Fit three 1 pieces together to form 3 of a circle. Three sixths of a circle is
6 6
what percent of a circle?

7. Show that 84, 36, and 2 each make one half of a circle. ( We say that 48,
36, and 2 4 12.)
4 all reduce to

8. The fraction 2 equals which single fraction piece?
8

9. The fraction 6 equals how many 41s?
8

10. The fraction 2 equals which single fraction piece?
6

11. The fraction 4 equals how many 31s?
6

12. The sum 1  1  1 is 38. If you add 3 and 28, what is the sum?
8 8 8 8

Investigation 2 109

12 3 3 6

13. Connect Form a whole circle using six of the 1 pieces. Then remove
16. What fraction of the circle is left? 6 represents
(subtract) What equation
1
6 your model?

14. rDeemmoovninsgtra13.teWshuabttrfraacctitniogn13isfrloemft?1 by forming a circle of 3 and then
3

15. Use four 41s to demonstrate the subtraction 1 14. Then write the answer.

16. oEnigehct i18rcslefo(r1mon83 e), cthirecnlew. Ihf a83tofrfaacctiiorcnleofisthreemciorcvleedrefrmomains?

17. What percent of a circle is 1 of a circle?
3

18. What percent of a circle is 1 of a circle?
6

Fraction manipulatives can help us compare fractions. Since 1 of a circle is
can see that 2
larger than 1 of a circle, we
3

1 7 1
2 3

Model For problems 19 and 20, use your fraction manipulatives to
construct models of the fractions. Use the models to help you write the
correct comparison for each problem.

Readin12g7M13ath 1239.Co43mpare: 2  33 1 1 3 7 3
3 48 3 5 5 10
 means is less
than; 20. Compare: 2 3
= means is 3 8
eqmuaelaton54s; is
greater than. Represent We can also draw pictures to help us compare fractions.

21114. 173 1
Draw two re4ctangles of the4same size. Sha2de 3
1 What fraction of one rectangle and
5 the rectangle that has
of the other rectangle. represents

the larger amount shaded?

22. Draw and shade rectangles to illustrate this comparison:

3 7 3
5 10

Problems 23–29 involve improper fractions. Improper fractions are

fractions that are equal to or greater than 1. In a fraction equal to 1 the
3
numerator equals the denominator (as in 3 ). In a fraction greater than 1

the numerator is greater than the denominator (as in 4 ).
3

Work in groups of two or three students for the remaining problems.

23. Show that tfhoeurimopf trhoepe14rpfrieaccetisonto45meqakuealas the mixed number 1 1 by
combining whole circle. 4

24. The improper fraction 7 equals what mixed number?
4

25. The improper fraction 3 equals what mixed number?
2

110 Saxon Math Course 1

1 2 12, 14, 13, 1
12 12 6

26. Form 1 1 1ci21rcles1using on1ly 14s. How man1y 1 pieces are n1e12eded to make1
1 12? 2 4 12, 144 , 13, 1
1217. 42 4 6 3
12 12 11
1 1 4 1 11 6 1 ma1n21y 1
31 1 2 34 4 6 E4xplain How 1 pieces are3needed to mak3e two whole circ6les?
28. 3
3 3 THhoewimdopry1o12opuekr nfroawctiyoonur341a1e2122nqsuwaelsr is corre1ct?
what m4ix12,e14d, 1 1
4 1 2

n13,u61mber?

1 2 1 29. Co12,n14v,e13r,t 11 to a mixed number.
12 12 8 66

30. Evaluate An analog clo1ck can serve a2s a 21, 411,1311,261 1
12, 14, 13,ohvth16iansen.udHhaaslonrwmedfasemrrmkaennoaycrfkf81etw1o1f2oefforlff1tt2w2haseo1clf2afirtarchelscei.in;rAcae1lt6te1a12,cahoon’’cocdlfloost1chco2kkesthee
10 2
fractions of a circle? 93

84
76 5

12, 14, 13, 1
6

Hint: Try holding each fraction piece at arm’s length in the direction of 1
the clock (as an artist might1e21xtend a thumb14 toward a subje14ct.) 2
1

After you have completed the exercises, gather and store your fraction

manipulatives fo1r 1later use. 1 1 1 1 1
4 4 2 3
2

extensions a. Justify CUosme mexuatmatpivleesPar1on12pde1n1r2toynceaxnambepal14epsp1lt2io2edsutoppthoertao14drdditi21iso,pn14ro,av13ne,d16 1 1
2
that the

b. subtraction of two fractions. math or12e, s14t,im13,a16tion and use th1at12

12

Analyze Ch12oose between1m2 ental

method to answer each of the feoalltoewni.nHg1o2q2wuemsuticohnso.fEthxep12lpa, ii14nz,zya13o,wu16racshnooict e.
1. One-fourth of was
1

a pizz1a2

eaten? 1

2. Each of 3 students ate 1 of a new box o161f cereal. What amoun1t2 of
8
cereal in the box was eaten?

3. More th1an half of the s1t1udents in the class are girls. What fraction
of the s8tudents in the c6lass are boys?
num1ber21l,in14e, .13, 1
c. Estimate Copy this 6 11

86

0 12, 14, 13, 1 1 2 31
6
8

Then estimate the placement of the following fractions on the number
line. Use your fraction pi21e,c14e,s13t,o16help you if you need to.
3 3 93 33 103 3 33 3 81433 3 3 9 433 33 7 39 3 43481p302359 3 313340 9 2 1831303459 592 1 10 7 592 110 1284177,p3014,21318, 148p177302 1
3 4 4 54 24 32 4 3 42 4 2 4 2 5 42 7 25 3 4 5 8 3 7 83 6 8

1 1 11 1 1 1 1 1 11 1 11 1 1 11 1 1
2 2 22 2 2 4 2 2 24 2 24 2 2 4 24 4 4

Investigation 2 111

LESSON Divisibility

21

Power Up Building Power

facts Power Up D
mental
a. Number Sense: 4 × 42
math
b. Number Sense: 3 × 76

c. Number Sense: 64 + 19

d. Number Sense: 450 + 37

e. Calculation: $10.00 − $6.50 678
6
f. Fractional Parts: 1 of 24 100 4030 1, 12, 0, 2, 1
2 4

g. Measurement: How many months are in a year?

h. Calculation: Start with 25, × 2, − 1, ÷ 7, + 1, ÷ 2

problem Here is part of a randomly-ordered multiplication table. 48 30 42
solving What is the missing product? 32 ? 28
56 35 49

New Concept Increasing Knowledge

Thinking Skill There are ways of discovering whether some numbers are factors of other
numbers without actually dividing. For instance, even numbers can be
Discuss divided by 2. Therefore, 2 is a factor of every even counting number. Since
even numbers are “able” to be divided by 2, we say that even numbers are
Without dividing “divisible” by 2.
by 2, how can
you tell that a Tests for divisibility can help us find the factors of a number. Here we list
number is even? divisibility tests for the numbers 2, 3, 5, 9, and 10.

Last-Digit Tests

Inspect the last digit of the number. A number
is divisible by …

2 if the last digit is even.
5 if the last digit is 0 or 5.
10 if the last digit is 0.

112 Saxon Math Course 1

Sum-of-Digits Tests

Add the digits of the number and inspect the
total. A number is divisible by …

3 if the sum of the digits is divisible by 3.
9 if the sum of the digits is divisible by 9.

Example 1 1556
Which of these numbers is divisible by 2?
365 1179

Solution

To determine whether a number is divisible by 2, we inspect the last digit of
the number. If the last digit is an even number, then the number is divisible
by 2. The last digits of these three numbers are 5, 9, and 6. Since 5 and 9 are
not even numbers, neither 365 nor 1179 is divisible by 2. Since 6 is an even
number, 1556 is divisible by 2. It is not necessary to perform the division to
answer the question. By inspecting the last digit of each number, we see that
the number that is divisible by 2 is 1556.

Example 2 1556
Which of these numbers is divisible by 3?
365 1179

Solution

To determine whether a number is divisible by 3, we add the digits of the
number and then inspect the sum. If the sum of the digits is divisible by 3,
then the number is also divisible by 3.

The digits of 365 are 3, 6, and 5. The sum of these is 14.

3 + 6 + 5 = 14
We try to divide 14 by 3 and find that there is a remainder of 2. Since 14 is
not divisible by 3, we know that 365 is not divisible by 3 either.

The digits of 1179 are 1, 1, 7, and 9. The sum of these digits is 18.

1 + 1 + 7 + 9 = 18
We divide 18 by 3 and get no remainder. We see that 18 is divisible by 3, so
1179 is also divisible by 3.

The sum of the digits of 1556 is 17.

1 + 5 + 5 + 6 = 17
Since 17 is not divisible by 3, the number 1556 is not divisible by 3.

By using the divisibility test for 3, we find that the number that is divisible
by 3 is 1179.

Discuss Is 9536 divisible by 2? by 3? How do you know?

Lesson 21 113

Example 3

Which of the numbers 2, 3, 5, 9, and 10 are factors of 135?

Solution

First we will use the last-digit tests. The last digit of 135 is 5, so 135 is
divisible by 5 but not by 2 or by 10. Next we use the sum-of-digits tests.
The sum of the digits in 135 is 9 (1 + 3 + 5 = 9). Since 9 is divisible by both
3 and 9, we know that 135 is also divisible by 3 and 9. So 3, 5, and 9 are
factors of 135.

Predict Will the product of any two prime factors of 135 given above also be
a factor of 135?

Practice Set a. Which of these numbers is divisible by 2?
123 234 345

b. Which of these numbers is divisible by 3?

1234 2345 3456

Use the divisibility tests to decide which of the numbers 2, 3, 5, 9, and 10 are
factors of the following numbers:

c. 120 d. 102

Written Practice Strengthening Concepts

Math Language 1. What is the product of the sum of 8 and 5 and the difference of
(12) 8 and 5?
The word
product is 2. Formulate In 1787 Delaware became the first state. In 1959 Hawaii
related to (13) became the fiftieth state admitted to the Union. How many years were
multiplication,
the word sum there between these two events? Write an equation and solve the
is related to problem.
addition, and the
word difference * 3. Formulate Maria figured that the bowling balls on the rack weighed a
is related to (15) total of 240 pounds. How many 16-pound bowling balls weigh a total
subtraction.
of 240 pounds? Write an equation and solve the problem.

4. An apple pie was cut into four equal slices. One slice was quickly eaten.
(6) What fraction of the pie was left?

5. There are 17 girls in a class of 30 students. What fraction of the class is
(6) made up of girls?

6. Use digits to write the fraction three hundredths.

(6)

7. How m14 uch money is 1 of $2.34?
2
(6)

8. What is the place value of the 7 in 987,654,321?

(6)

114 Saxon Math Course 1

9. Generalize Describe the rule of the following sequence. Then find the
(10) next term.

1, 4, 16, 64, , ...

10. Compare: 64 × 1 64 + 1
(9)

* 11. Which of these numbers is divisible by 9?
(21)
A 365 B 1179 C 1556

* 12. Estimate Find the sum of 396, 197, and 203 by rounding each number
(16) to the nearest hundred before adding.

* 13. What is the greatest common factor (GCF) of 12 and 16?
(20)
1 678 1,1(5212,). 04, 8,824, 014 ÷ 24
2 14. 100 4030 6
(2)
678
1 100 4030 16. 6 1, 21, 0, 2, 1 17. $4.75 × 10 1 100 403
2 (2) 4 (2) 2

Find each unknown number. Check your work.

18. $10 – w = 87¢ 19. 463 + 27 + m = 500
(3) (3)

1 * 20. Arrange thes6e78numbers in order from least to greatest:
2 6 1, 12,
(17) 0, 2, 1
4
100 4030

* 21. What is the average of 12, 16, and 23?
(18)

* 22. List the whole numbers that are factors of 28.
(19)

* 23. What whole numbers are factors of both 20 and 30?
(19)
24. Use an inch ruler to draw a line segment four inches long. Then use a
(7) centimeter ruler to find the length to the nearest centimeter.

25. (12 × 12) – (11 × 13) 11 12 1
(5) 10 2
9
* 26. Represent To divide a circle into thirds, John 8 3
(Inv. 2) first imagined the face of a clock. From the
76 4
center of the “clock,” he drew one segment 5
up to the 12. Then, starting from the center,
John drew two other segments. To which two
numbers on the “clock” did John draw the
two segments when he divided the circle into
thirds?

* 27. Model Draw and shade rectangles to illustrate this comparison:

(Inv. 2)

2 6 3 1
3 4 8

Lesson 21 115

*23 268. 34A “bit” is 1 of a dollar.
(6) 8

a. How many bits are in a dollar?

b. How many bits are in a half-dollar?

29. A regular octagon has eight sides of equal length. What is the perimeter
(8) of a regular octagon with sides 18 cm long?

* 30. Represent Describe a method for dividing a circle into eight equal
(Inv. 2) parts that involves drawing a plus sign and a times sign. Illustrate the

explanation.

Early Finishers The 22 sixth grade students at the book fair want to buy a mystery or science
fiction novel. Ten of the 15 students who want a mystery book changed their
Real-World minds and decided to look for a humorous book. How many sixth grade
Application students are NOT looking for mystery books?

Write one equation and use it to solve the problem.

116 Saxon Math Course 1

LESSON “Equal Groups” Problems with
Fractions
22

Power Up Building Power

facts Power Up C
mental
a. Number Sense: 4 × 54
math
b. Number Sense: 3 × 56
1
4 c. Number Sense: 36 + 29

1 d. Calculation: 359 − 42

proble4 m e. Calculation: $10.00 − $3.50
solving
f. Fra41ctional Parts: 1 ooff4488 1 2
2 3 3

g. Measurement: How many yards are in 9 feet?

h. Calculation: Start with 100, − 1, ÷ 9, + 1, ÷ 2, − 1, × 5

33

Truston4has 16 tickets, 4Sergio has 8 tickets, and Melina has 6 tickets. How
many tickets should Truston give to Sergio and to Melina so that they all have
the same number of tickets?

New Concept Increasing Knowledge

Here we show a collection of six objects. The

collection is divided into three equal groups. 1 1 1 of 48
1 We se41e that1there are12twoof 4o18bjects ainre13foofutrho12eboj4efc4ts8 2 14 2
4 collection. W4 e also see that4there 3 3 2
1 3
3 3
1 of 48 in 2 of the collection. 4
2 1 3
4
This collection of twelve objects is divided into 3
34ino14f 4
tfohuerceo34qllueacltigo14rno,uspos.thTehr34eerearaer34eninthereoebjoebcjtescitns of
the

3 collection.
4

Example 1

Thinking Skill Two thirds of the 12 musicians played guitars. How many of the
480 66
Infer 3 musicians played guitars? 3

Why do you Solution
divide the

musicians by 3? This is a two-step problem. First we divide the 12 musicians into three equal

groups (thirds). Each group contains 4 musicians. Then we count the number

of musicians in two of the three groups.

5280 Lesson 22 117
44

1 did not play 12 musicians
3 guitars. 4 musicians
4 musicians
2 played 4 musicians
3 guitars.

Sinc1e tishe8r.eWaerefi4ndmt14uhsaitci8anmsuinsiecaiacnh12sthopiflrad4y,8ethdegnuuitmar13bs.er of musician2s in two
third4s
3

Example 2

Cor14y has finished 3 of the 28 prob34lems on the assignment. How many
4
problems has Cory finished?

Solution

First we divide the 28 problems into four equal groups (fourths). Then we find

the number of problems in three of the four groups. Since 28 ÷ 4 is 7, there
are 7 problems in each group (in each fourth).

1 are not 28 problems
4 finished. 7 problems
7 problems
3 are 7 problems
4 finished. 7 problems

In each group there are 7 problems. So in two groups there are 14 problems,

and in three groups the1re are 21 probl2ems. We see th3at Cory has finis1hed 3
3 3 4 4 4
21 problems.

12313
3 3 4 4 4
Example 3

How much money is 3 of $3.00? 2 3
5 5 5

Solution

323 $3.00

First we divide5$3.00 into five5equal groups. T5hen

wgorfeomufiopnnsd.etWyheienademiavcoidhuengt$ro3o.uf05pm0.o$$bn03ye..605y00tinointfhienradecethhoefgtarhomeuopfiuvnet 2 of $3.00 $0.60
$0.60 in each ggrroouupp 5 $0.60
$0.60
5 $3.00 3 of $3.00 $0.60
5

Now we multiply $0.60 by 3 to find the amount of $0.60

1 2 money in thre34e groups. 1 3
3 3 4 4

$0.60

×3
$1.80

3 2 We find that 3 of $3.00 is $1.80.
5 5 5

118 Saxon Math Course 1
$0.60 in each group

5 $3.00

Example 4

W35hat number is 3 of 100? 1 of 100 3 of 100
4 4 4

Solution

We divide 100 into four equal groups. Since 100
1010 ÷ 4 is 25, the3re are 25 in eac1h group. We will
fin5d the total of th5ree of the parts5. 1 of 100 25
4

25

3 × 25 = 75 3 of 100 25
34
3 4 25 1 of 100
5 4
33
5 4 1 of 100 3 of 100
a 51 b a 35 b 4 4
2 3 100%
of 100 % 5 of
Example5 5

a. What percent of a whole circle is 1 of a circle? 3 1
5 5 5
w15314hoolfe1c0i0rcle 35343ooffa1c0i0rcle? 1
35Solut23bio. nWhat p34 erce54nt of a is 5
5 4
1 of 100 3 of 100
4 4

eAqwuhalog35lerocuirpcsle. is 100%34 . We divide 10041 %of i1n0to0 five 3 of 100 100%
100% 4
2 of 5235ao51ofbf11000%0% 20%a 51 b
5 20%a 53 b
1 TOhnreeeof35otfhtehefivfievepapratsrt52sQ1551oQR35fiRs1i0s2003%%×.
5 a. 20%3535,ow43f h1i0c0h% 1 3
b. 5 4
1 of 100 20% of 100
4
e1quals 60%. 3 5354of 100%
1 2 20%
55 5 3

24 20%
to352aio51l15lufbs1t0r0at%e each535pao53r35ofbb1l0e0m%.
Practice Set 2 of 100% 3 of 100% a 51 15b a 53 b
5 5
Model Draw a diagram

a. Tth25heroemfe1uf0os0iuc%ritahnssocfo35tuholedf 11200m%usicianas15cbould play thea53pbiano. How many of
play the piano?
2 4
3 5
b. How money is 2 of $4.50? 4
much 3 35
10 a 1 b a 3 b
c. W23hat number is 4 of 2 of 100% 3 of 100% 9 5 5
5 5 10
60?5

d. What number is 3 of 80? 9
10 10

e. Five sixths of 24 is what23 number? 4
5

f. Giov1a30nni answered 9 of the questions correctly. What percent of the
10
questions did Giovanni answer correctly?

Written Practice Strengthening Concepts 23 3

3 4 10
1. When the sum of 15 and 12 is subtracted from the product of 15 and
(12) 12, what is the d2ifference?
3 3
3 4 10

2. There were 13 original states. There are now 50 states. What fraction of
(6) the states are the original states?

23 3
3 4 10

Lesson 22 119

w  345
15

39
10 10 3 9
10 10

* 3. 3A marathon rac9e is 26 miles plus 385 yards long. A mile is 1760 yards.
10Altogether, how10many yards long is a marathon? (First use a
(11, 15)

multiplication pattern to find the number of yards in 26 miles. Then use

an addition pattern to include the 385 yards.)

* 4. Model If 2 of the 12 appl34es were eaten, h130ow many were eaten? Draw a
3
(22) diagram to illustrate the problem.

* 5. Model W23hat number is 3 of 16? Dra2w a13d0 iagram to3illustrate the 3
4 10
(22) problem. 34

* 6. 2 Model How m34 uch money is 3 of $3.50? Draw a diagram to illustrate
3 10
(22) the problem.

* 7. As Shannon rode her bike out of the low desert, the elevation changed
(14)
from –100 1fwt5to630405ft. What was the total elevation change for her
ride?

Find ewac−h1u5nk=n1wo85wn number. Check your work.
8. 345 9.
w  345
(3) (4) 15

10. 1w356¢ +34$54.78 + $34.09 11. $12.45 ÷ 3
(1) 79 (2)
14 7  9  14
12. 35 1000 335 1000 13. 3
(2) (5)

* 14. Estimate Shannon bought three dozen party favors for $1.24 each. To
(16) estimate the total cost she thought of 36 as 9 × 4, and she thought of
$1.24 as $1.25. Then she multiplied the three numbers.

a. What was Shannon’s estimate of the cost?

b. To find the cost quic3k5ly,1w0h0i0ch two n7umb93ers 1sh4ould she multiply
first?

15. 145FW0in0hd0icthhdeiggirte7iante3s31793t5c,4o21m124564m,1o9n8,f0a0c0toirs in t1he te1n-millions place?
of 32
* (12) 12 and 15.

1365.
(20)

* 17. List the whole numbers that are factors of 30.
(19)

* 18. The number 100 is divisible by which of these numbers: 2, 3, 5, 9, 10?
(21)

* 19. qMueodsetilonJsedbidanJsewb earnesdw45eor fctohrereqcutleys?tiDorn13aswcao21rdreiacgtlrya.mWthoatillpuesrtcraetnet of the
(22) the

problem.

254(09). Compare: 1 1
3 2

* 21. Classify Which of these numbers is not a prime number?
(19)
A 19 B 29 C 39

22. (3 + 3) − (3 × 3)

(5, 14)

23. Generalize Find the number halfway between 27 and 43.

(18)

120 Saxon Math Course 1

Math Language 24. What is the perimeter of the rectangle below? 4 1
(8) 5 3
Perimeter is the 15 cm
distance around 10 cm
a closed, flat
shape.

25. Use an inch ruler to find the length of the line segment below.
(7)

* 26. Analyze Corn bread and wheat bread were baked in pans of equal
(Inv. 2) size. The corn bread was cut into six equal slices. The wheat bread was

cut into five equal slices. Which was larger, a slice of corn bread or a
slice of wheat bread?

* 27. Compare these fractions. Draw and shade rectangles to illustrate the
(Inv. 2) comparison.

23 1
45 4

* 28. Model A mquaanr42ytemr oofn35athyseaarreisin41aoqf uaayretearr.oTfhaeyreeaarr?e 12 months in a to
(22) year. How Draw a diagram

illustrate the problem.

29. A “bit” is one eighth of a dollar.
(6)
a. How many bits are in a dollar?
b. How many bits are in a quarter of a dollar?

* 30. Represent The letters c, p, and t represent three different numbers.
(1) When p is subtracted from c, the answer is t.

c−p=t

Use these letters to write another subtraction equation and two addition
equations. Hint: To be sure you arranged the letters in the correct order,
choose numbers for c, p, and t that make c – p = t true. Then try those
numbers with these letters for your three equations.

Lesson 22 121

LESSON Ratio
Rate
23

Power Up Building Power

facts Power Up D
mental
a. Number Sense: 5 × 62
math
b. Number Sense: 5 × 36

c. Number Sense: 87 + 9

d. Number Sense: 1200 + 350

e. Calculation: $20.00 − $15.50

1 f. Fractional Parts: 1 of 84
4 2

g. Measurement: How many millimeters are in 3 meters?

h. Calculation: 10 × 3, + 2, ÷ 4, + 1, ÷ 3, × 4, ÷ 6

13 problem How many different bracelets can be made from
15 solving 7 white beads and 2 gray ones?

New Concepts Increasing Knowledge

ratio A ratio is a way to describe a relationship betwee41n numbers. If there are 1
13 boys and 15 girls in a classroom, then the ratio of boys to girls is 13 to 15. 2
Math Language Ratios can be written in several forms. Each of these forms is a way to write
13 and 15 are the the boy-girl ratio:
terms of the ratio.
13 to 15 13:15 13
15

Each of these forms is read the same: “Thirteen to fifteen.”

In this lesson we will focus on the fraction form of a ratio. When writing a
ratio in fraction form, we keep the following points in mind:

1. We write the terms of the ratio in the order we are asked to give them.

2. We reduce ratios in the same manner as we reduce fractions.

3. We leave ratios in fraction form. We do not write ratios as mixed
numbers.

Example 1

A team lost 3 games and won 7 games. What was the team’s win-loss
ratio?

122 Saxon Math Course 1

distance 55 miles distance 7 28 miles
time 1 hour fuel used 3
Solution 1 gallon

The question asks for the ratio in the order of wins, then losses. The team’s
$2.w89in-loss radtioistwaansce7 to 3, wdhisictahnwcee write as the fraction 37.
price numtibmeer of games won 7 number of boy
quantity 1 pound time number of games lost  3 number of girl

60 miles We leave the ratio in fraction form.
4 hour
distanDcisecuss Is the win-loss ratio the same as the loss-win rannt56uuiomm?bbeerr of games won  7
fuel used of games lost 3

Example 2

In a class of 28 students, there are 13 boys. What is the ratio of boys to
girls in the class?

7 Solution

3

To write the ratio, we need to know the number of girls. If 13 of the

28 students are boys, then 15 of the students are girls. We are asked

to write the ratio in “boys to girls” order.
2
3
number of games won 7 number of boys 13
number of games lost  3 number of girls  15

Model Use tiles of two colors to model this ratio. 2
3

rate A rate is a ratio of measures. Below are some commonly used rates. Notice
that per means “for each” and substitutes for the division sign.

Common Rates

Name Rate of Measures Example Alternate Form
Speed 55ffdduuffmdidieuuessiilieetlstslaaeululttnasansuuscnncesespeccedeedeeeddr hour2211822811gg88mmaggammlaalilillololleeiillllnoonseesnnss
dddidissttiiitistsmttmaaiittmmnaaneecnnceeecceee 5511551155hhmmohhommiiuooluleeiiruurllseesrrss
28 miles per gallon
didsitsatnacnece M515i5le15hmaohgmiuloeeirulsers didsitsatnacnece 2828mmileilses
timtime e fufeuleul suesded 1 1gaglalollnon

2 Unit price qqququppauuaprpriniaancrrciitnntecceiitteeyiiyttyy 1111$$pp2$$2popo.22.8uoo8u..9n8u8u9nd9n9nddd $2dd.didi8ssttiiit9istsmttmaaiittmmpnaaneeecnnceereccepeeounddddidissttiiitistsmttmaaiittmmnaaneecnnceeecceee
3

distqitmaupqnearupcincaretiencitteReyitaaycteh p5g1r5roo1hmbu$o1lppei2ul$.omep.r2T8usos.h9n8uead9nrpderoabftduyleipesmdletaiudsstonsiitomsfatceiftenmedteaqecnenuecaienl vgor21olv8ugedpmaitsdslhtililoitprmesatenristnmoeeacbnneelcueemm.bTehrse. rate is the number in
One number is the

rate, and the other two rn6e6u44l0660am4400thhmmebohhommdieiuooluleeriibruurllsseesyrrssamreualtbipoffdduluuiffdicdieuuesstaiileetlststaauhlulttionaanesuusncnncesesmeccedeedoeeddr adsivuirseisonthaast form the rate.
The three numbers are we show below.

qupa6ri4n0c6t4e0himtoyhmiuloeirPulserastter1n$p:2fdoud.fi8udesius9nlitetsaudaltnsnaucecnsedeceed= distance × timedistitmanece 55
time 66

Example: 165 miles = 55 miles per hour × 3 hours

In a rate problem one of the numbers is unknown. We find an unknown

60 miles produdcitsbtaynmceultiplying, and we find an unknown factor by dividing t5he
4 hour produfcutebl uystheed known factor. 6

Lesson 23 123

Example 3 distance 55 miles distance 28 miles
time 1 hour fuel used 1 gallon

On a bike trip Jeremy rode 60 miles in 4 hours. What was his average

distance 55 miles spedeidstainncmeiles pe2r8homuilre?s price $2.89 distance distance
time 1 hour Solutifounel used 1 gallon quantity 1 pound time time

price We are given the distance and time. We are asked for the speed, which is
quantity distance divided by time.

60 miles $2.89 distance distance 60 miles = 15 mfduiielseltsaunspceeedr hour
4 hour 1 pound time time 4 hour

distance 55 miles distance 28 miles
1 gallon
Example 4 time 1 hour fuel used

distance Mabro. Mutohsocwal’fsarcahreacvaenraegxepsec3t2tmo itlreasvpeel or ngaallroona65odntrtihpeuhsiignhgw1a0yg. aPlrloednsicot f
fuel used

fuel. price $2.89 distance distance

Solution quantity 1 pound time time

This is a problem about gas mileage. Notice the similar pattern.

60 hmdoiisuletrasnce = distance × fuel used
4 fuel used

We are given the rate and the fuel used. We are asked for the distance, which
is the product.

distance = 32 miles per gallon × 10 gallons

= 320 miles
Mr. Moscal can e73xpect to travel about 320 miles on 10 gallons of fuel.

7 Making a table can help us solve some rate problems. Here is the beginning
3 of a table for example 4.
7
3 Distance Traveled at 32 Miles per Gallon
FuneulmUbseerdof(ggaalmloens) wo1n number of boys
Dinsutamnbceer o(mf gilaems)es los3t2  72 3 4 5 number of girls  13
634 96 128 160 15

number of ggaammeeass.wloWosnht at is37 the ratio of dogs tonncuuammtsbbeienrraooffnbegoiigrylhssborh11o53od that has 19 cats and

PracntiucmebSereotf

number of games wloosntb.1A237ndaloyzges? What is the gnniurulm-mbbboeeyrrrooafftibogoiriynlssacl11a53ss of 30 students with
number of games

17 boys?

c. If the ratio of cars to trucks in the parking lot is 7 to 2, what is the ratio
of trucks to cars in the parking lot?

d. How long will it take a trucker to drive 400 miles at 50 miles per
hour?

e. If a four-qua23rt container of milk costs $2.48, what is the cost per
quart?

2
3

1224 Saxon Math Course 1

3

32
52 1 5

33
Written Practice Strengthening Conc53epts 2
3 5
2
35 1. How many millimeters52long is 5a ruler that is 30 cm long? 7
5 7 10
(7) 3 10

7
ddMuiarogidnreaglmclDatosasnil.luhHsaotsrwaftinme1i7s0tahhneeydpp1r1230r3o0oo53bbflle7etmhme.s3d0i1d7p0rDoabnlefminsisohndaunrinagssciglansms?enDtra8w87
2 * 2. 2
5
(22) 5a
3 2
10 3 5 78 1 1 1 1
10 8 8 8
3.
Diego walked the length o2f a football field in 100 large paces. About
3 (7) how long w2as the footbal3l fie3ld? 77

5 5 2 10 10 8

7 2 *87 4. On the open highwa1y81 the 3car traveled1 81245 miles on 7 gallons of gas.
8 3 (23) What was the car’s gas mileage for the trip in miles per gallon?

2
3

3 * 5. Model Wh7at number is 3 o2f 25? Draw a diagram to1il81lustrate the 7
10 problem. 8 5 10
(22)
33 7
5 il3lust1r0ate38
3 3 * 6. Model How much is 7 of $36.00? Draw a diagram to
5 5 mo7ney 10
(22) the problem. 10 8

2 Use your fraction 4 t53o help answer problems 7–9. 9 7
3 8 10
manipulative8s

7 (In*1v7.072). What is the sum of 3 and 4 ? 3 9
10 8 8 8
4 9 8 $3.75
4 8 9 equals8 what mixed number? $3.75 25
8 * 8. The improper fraction 8
235
(Inv. 2) 2 49
38
3 * 9. Two eighths170of a circle is 3wh4at percent of a circle? 3 4
5 $3.75 ∙ 16 8
(Inv. 2) 2 8
9 $3.75 3 $3.75
8 2 981(02). 253 3 11. 25
3 4 (2)
2 12.
3 What is the place value4 of the 6 in 36,174,591?

(12) $3.75
25
4 * 13. Explain Ho89w can you find 2 of a number? 3
3(22) 3 4
83
4 F4ind each unknown number. Check your work.

14. $0.35n = $35.00 15. $10.20 − m = $3.46
(4) (3)
3
2 16. Compare: 4 1
3
(17)

* 17. Analyze The length of a rectangle is 20 inches. The width of the
(8) rectangle is half its length. What is the perimeter of the rectangle?

18. Generalize Describe the rule for the following sequence. Then, write the
(10) sixth number in the sequence.

2, 4, 8, 16, . . .

19. Yesterday it snowed. The meteorologist on the radio said that it was 14º
(10) outside. What scale was the meteorologist reading? How do you know?

20. Compare: 12 ÷ 6 − 2 12 ÷ (6 − 2)
(9)

* 21. What is the greatest common factor (GCF) of 24 and 32?
(20)
22. What is the sum of the first seven positive odd numbers?
(10)

Lesson 23 125

Model Udsisetayonucrefraction55mmaniliepsulativesdtoishtaenlpceanswer2p418rombilleems s 23–25.
a. Howtimmeany 41s ar41e1inh1o?ur
* 23. fuel used 1 gall1on 1

1 (Inv. 2) b. How many 41s are in 21? 22
4 1
2

* 24. One epigrhicteh of a circle$i2s.8w9hat percednitsotafnacceircle? distance

(Inv. 2)

* 25. Writequaafnratictytion41with1apdoeunnodminator oftim8 tehat is equalttimo e21.

(Inv. 2)

* 26. There were 16 members of the 2004–2005 men’s national swim team. 5
(23) Five of them competed in the freestyle. What is the ratio of those who 6
com6p40ethmeodiuleirns the frefedusiestyltaluenstceoedthose who did not?

* 27. Classify Which prime numbers are greater than 20 but less
(19) than 30?

28. Which of the figu41res below represents a line? 1
(7) A 2

B

C

* 29. Classify Which of these numbers is divisible by both 2 and 5?
(21) A 252
B 525 C 250

* 30. If a team lost 9 games and won 5 games, then what is the team’s
(23) win-loss ratio?

Early Finishers Mrs. Akiba bought 3 large bags of veggie sticks for her students. Each bag
contains 125 veggie sticks. One sixth of Mrs. Akiba’s students did not eat
Real-World any veggie sticks. The remaining students split the veggie sticks evenly
Application and ate them. How many veggie sticks did each of the remaining students
eat?

126 Saxon Math Course 1

LESSON Adding and Subtracting Fractions
That Have Common Denominators
24

Power Up Building Power

facts Power Up C

mental a. Number Sense: 6 × 24
math b. Number Sense: 4 × 75

c. Number Sense: 47 + 39

d. Number Sense: 1500 − 250

e. Calculation: $20.00 − $14.50

f. Fractional Parts: 1 of 68
2

g. Measurement: How many yards are in 12 feet?

h. Calculation: 6 × 7, − 2, ÷ 5, × 2, − 1, ÷ 3

problem 1
solving
Tom followed the directions on the treasure map. Startin4 g at the big tree,

he walked five paces north, turned right, and walked seven more paces. He

turned right again and walked nine paces, turned left, and walked three more

paces. Finally, he turned left, and took four paces. In which direction was
t38ree? 1 5
Tom facing, and how many paces was he from the big 4 1 8
4

New Concept Increasing Knowledge 1 1
4 2

Using our fraction 1 we see that when we a38dd 2 38t21o 35 the sum is 58. 5
8 8 8
m4anipulatives,

31 51
8 81 88

8

13 5881 4118 1 1 3818 38 8282 58 5 1
48 8 2 8 2

Math Language 3 83+8282 = 55
The denominator 8 88
tells you into how
many parts the Three eighths plus two eighths equals five eighths.
whole is divided.
Likewise, if we s38ubtr38act 2 fro85m 58, then 3 are left. 51 1
2 8 8 88 8
8 3
51 8
1 8 81
8
8

1

3  2  5 1 1 83  2  5
8 8 8 8 8 8 8 8

55 – 2828=83 3
88 8

Five eighths minus two eighths equals three eighths.

1  1  1 1  1  1  3 Lesson 2124  12127
4 4 4 4 4 4 4

253
888

Notice that we add the numerators when we add fractions that have the

same denominator, and we subtract the numerators when we subtract

Thinking Skill fractio1ns that have the same doer 5nsoumbtirnaacttofrr.aTchtieo85ndsentho28amt ihn83aavtoersthoef the fractions
82dWJehunsyotidmfyoineasttohr82e do not8change when we add 8 same 3
38
denominator. 5
88

ss8385taamyethweh85en we Example 1 1 1 1 3 1 1 1 3 1 1
438 4 4 83 4 4 4 4 2 2

8

add fractions1 Add:   5  2  3    
1with the sam8e 82 8 8 5
5 2 3 8 8
8denominator? Solution 8  8  8

5  2  3  2  3 The denominators are the same. We add the numerators.
8 8  85 8 8 1
8 4
 3 1 1  1  1  1  2  1 1481 43 1  1  3 3  3  3  3 1  1 3
8 4 1 4 4 25 2 2 1 4 4 4 43 4 4 4 258 822 8
1 4 4 1 1 1 1 
 4 8 2  4 82 4 8 5 2  2 5 3
8 8 8 8

 1  3241443312314442312144132423314444133421431443144338812E2413x1a44134S2mo4223lpAOu4433ltdne3iedo2431:nh4738a121854lf14p2l8248232141u8411so33843n4141eh3443a431241lfis431421t2w8411o4312h3a41l342411v4321e8585s,41434211w82hic434212h38is1431on8712ewh82211485ole.852841  3 5  28143 38 1  1  3
 4  8 8  4 4  4
 4341 3
1 8 4
4
 1  3 413441434343 1
4 4 2

3 7814134TW178422Dhhiaisyncs821u824kc23sias?nng187w8523Sek1784wi282l4211rliEte82xa85S2411molpTAulhtd2421eiedo3d:ne11344n2om432431inato432413rsa187434r24232eth82e431s2341a85m142e24317817483.42W31443e4232828432ad4413d4318543th4413e1434n2u144334m2 e1343r2a34tor3s.43

 2   3  3 3  3 1 4321 3
 2  4 4 4 4 2 4
 5
843 7  2  5
8 8 8

4 182487285823 12

Example 4
Su43btra4378ct:
3  3 7  24 7  2  5 7  82142 5
4 4 8 8 8 8 8 8 8

Solution

The de14n2om1in2ators are the same. We subtract the numerators. 12
4 12 4
7  2 4 7  2  5 12
8 8 8 8 8 4

12 12
4 4
128 Saxon Math Course 1

Example 5

Subtract: 1  1 1
2 2 2

Solution 1  1 1
2 2 2

 1  1 1  1 1  1 I1f we 2121start21w38ith 214and subtract 12, then wha203218tis0le281412ftis201ze43ro020. 1241021  0  0 1  1  0  0
2 2 2 2 2 2 8 2 2 2 2 3 
2 1 1 4
1 2 2
2 

1 1  1  0  0
2 2 2 2
Practic3e Se84t
 43  4 3  4 F43inad. 41e338ach4184s48um 43o1rdi41fference: 3  1 b. 181�� � 81��43  8141 34818142  1  1 1  1  1
8 8 8 8 8 8 4 4  84  1 8 8 8 8 8
3 1 1 1 8 8
4  4 8  8 d.

3  1 c. 1  1  1 3 
4 4 8 8 8 4

 14  1 4  1 4  1 3 e. 84422814HyyA$7321211441213821148421AMCoo6lotonuu1423.oow0nadgluun048242142112841824112Syemeesszlt141287pcteeer41ahteenettJrn20ooyrMu,hU738421g2412affehoatiisnnnhgoure0ddtegwrui.218424wnnsshMTmieowindhsadrguieodrdttcn44293rosiCk13nhJtohteau’o1osmfedlan0aeywnow0ci738241aandn0eraur121414ri4123eotdptnssehyz-itenepswe8424t?dghan2411yo41iseDidadre?krrsgMfad$.uegw5abfas38121412211481472413o.rt0ratrrtni0aonfidncifvmie82144242121141tog41a4213738421eiaorsgarh?n8121rwknaoe?Wpa2411m414842usroWr728831hohs2811mtaboihnatelaeginlp411421ltem84ldu8201attphsstwaitetnefrt144211aoatrcee0snrtaxerhprnde.tiarishd2811dcideifidsaemdi412411ly. 8211.4211
8 8 8  8 8 8 4
1  3 2
4 2 4  4 g.
1 4
3  2 2 Writte1478n12P4284r21ac(t11i7821,c115e). 7 
4 4 41 8
23
21 2 1
4 4 1
 1  2 2
2 9
7 4 1 1  20121
8 8 2 2 
1
 8 2

1 21213821 3 
12 1 4
 2 3 2 4 4 1 1
8 8 8 2
1000 81 
1 *38(222). 38
2 8

3  1 7 7 3 81mp838114887Eraox34pnbly8181aleiny43m42a.r81dO83sneis aismonilee?83th41E34oxupslaa42inndh, os4814334wevyeon8142413483uh41fuonu42dn41r41dedyo41suixrtyanysawrdesr.. 1 3 
874 4 4 8 8 4 4
8  *8 3.
1 3 34  1 48(22) mile How1  1
8 4 88 8 1 4 4 1
3 8 of 14 

8 ec*4114(Mxh244eoo).rdoce42s41144811443ilsee78o.Un4142see7884oyfot41uhreferaxcetricoinsemsatno41ipw1878urliateti84*va(2e54ws). oto147184878rdhperl42841878p41obw12leitm84h exercises 4–8. Then 4
tha18t is solved by th81e
3  2 3 3 11 7 
434 1 4 4 4 22 8
4 
2 11  2  1 1
4 44 4 2 2
1 

4

7  4 7 7 * 6. 1  1 * 7. 1  1 1
878 1 8 8 8 27 2 2 2 2
2  (24) (24)

1 1  1 (In**21v.892).. 21IW128n h21thaet21pcelarcssernotoomf alibcriracrlye is 1 of a circle plus 1 of a circle? 1 1
2 2 2 2 4 4 4
1 1
there were 232 non2fiction books and
(23) 41 1 fiction books. What was the ratio 7of fiction to nonfiction books
1 i4n the library? 8 3
2 8

7 73 31 1
8 8 871(108). 87Ex8plain How can you find8the numb83er ha83lfw4 ay between 123 and 3241?
1
4

3 13 1
8 44 4

3 31 11 1
4 4 3 34 4 1 18 Lesson 24 8 129
1
44 44 8