LESSON 6 Factors Divisibilityera6anderson.weebly.com/.../6/...2_lessons_006-010.pdfLesson 6 43 h. The number 7000 is divisible by which single-digit numbers? i. List all the common

L E S S O N

6 Factors Divisibility

Power Up Building Power

facts Power Up B

mental math

a. Calculation: $5.00 ! $2.50

b. Decimals: $1.50 " 10

c. Calculation: $1.00 # $0.45

d. Calculation: 450 ! 35

e. Number Sense: 675 # 50

f. Number Sense: 750 $ 10

g. Probability: What is the probability of rolling a number greater than 6 on a number cube?

h. Calculation: 9 " 5, # 1, $ 4, ! 1, $ 4, " 5, ! 1, $ 4 1

problem solving

The sum of five different single-digit natural numbers is 30. The product of the same five numbers is 2520. Two of the numbers are 1 and 8. What are the other three numbers?

New Concepts Increasing Knowledge

factors Recall that factors are the numbers multiplied to form a product.

3 " 5 % 15 both 3 and 5 are factors of 15

1 " 15 % 15 both 1 and 15 are factors of 15

Therefore, each of the numbers 1, 3, 5, and 15 can serve as a factor of 15.

Notice that 15 can be divided by 1, 3, 5, or 15 without a remainder. This leads us to another definition of factor.

The factors of a number are the whole numbers that divide the number without a remainder.

Thinking SkillGeneralize

For any given number, what is the number’s least positive factor? Its greatest positive factor?

For example, the numbers 1, 2, 5, and 10 are factors of 10 because each divides 10 without a remainder (that is, with a remainder of zero).

10

100

1! 10

5

100

2! 10

2

100

5! 10

1

100

10! 10

1 As a shorthand, we will use commas to separate operations to be performed sequentially from left to right. This is not standard mathematical notation.

40 Saxon Math Course 2

Lesson 6 41

Example 1

List the whole numbers that are factors of 12.

Solution

The factors of 12 are the whole numbers that divide 12 with no remainder.

We find the factors quickly by writing factor pairs.

12 divided by 1 is 12



Below we show the factor pairs arranged in order.

1, 2, 3, 4, 6, 12

Example 2

List the factors of 51.

Solution

As we try to think of whole numbers that divide 51 with no remainder, we may think that 51 has only two factors, 1 and 51. However, there are actually four factors of 51. Notice that 3 and 17 are also factors of 51.

3 is a factor of 51 17

3! 51

Thus, the four factors of 51 are 1, 3, 17, and 51.

From the first two examples we see that 12 and 51 have two common factors, 1 and 3. The greatest common factor (GCF) of 12 and 51 is 3, because it is the largest common factor of both numbers.

Example 3

Find the greatest common factor of 18 and 30.

Solution

We are asked to find the largest factor (divisor) of both 18 and 30. Here we list the factors of both numbers, circling the common factors.

Factors of 18: 1 , 2 , 3 , 6 , 9, 18

Factors of 30: 1 , 2 , 3 , 5, 6 , 10, 15, 30

The greatest common factor of 18 and 30 is 6.

divisibility As we saw in example 2, the number 51 can be divided by 3. The capability of a whole number to be divided by another whole number with no remainder is called divisibility. Thus, 51 is divisible by 3.

17 is a factor of 51

There are several methods for testing the divisibility of a number without actually dividing. Listed below are methods for testing whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10.

Tests for Divisibility

A number is divisible by . . . 2 if the last digit is even. 4 if the last two digits can be divided by 4. 8 if the last three digits can be divided by 8.

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

3 if the sum of the digits can be divided by 3. 6 if the number can be divided by 2 and by 3. 9 if the sum of the digits can be divided by 9.

A number ending in . . . one zero is divisible by 2. two zeros is divisible by 2 and 4. three zeros is divisible by 2, 4, and 8.

Discuss Why might we want to test the divisibility of a number without dividing? Why not just divide?

Explain Why does the divisibility test for 2 work?

Example 4

Which whole numbers from 1 to 10 are divisors of 9060?

Solution

In the sense used in this problem, a divisor is a factor. The number 1 is a divisor of any whole number. As we apply the tests for divisibility, we find that 9060 passes the tests for 2, 4, 5, and 10, but not for 8. The sum of its digits (9 ! 0 ! 6 ! 0) is 15, which can be divided by 3 but not by 9. Since 9060 is divisible by both 2 and 3, it is also divisible by 6. The only whole number from 1 to 10 we have not tried is 7, for which we have no simple test. We divide 9060 by 7 to find that 7 is not a divisor. We find that the numbers from 1 to 10 that are divisors of 9060 are 1, 2, 3, 4, 5, 6, and 10.

Practice Set List the whole numbers that are factors of each number:

a. 25 b. 23

c. List the factor pairs of 24.

Analyze List the whole numbers from 1 to 10 that are factors of each number:

d. 1260 e. 73,500 f. 3600

g. List the single-digit divisors of 1356.


Lesson 6 43

h. The number 7000 is divisible by which single-digit numbers?

i. List all the common factors of 12 and 20.

j. Find the greatest common factor (GCF) of 24 and 40.

k. Explain How did you find your answer to exercise j?

Written Practice Strengthening Concepts

1.(1)

Analyze If the product of 10 and 20 is divided by the sum of 20 and 30, what is the quotient?

* 2.(6)

a. List all the common factors of 30 and 40.

b. Find the greatest common factor of 30 and 40.

* 3.(4)

Connect Use braces, an ellipsis, and digits to illustrate the set of negative odd numbers.

* 4.(5)

Use digits to write four hundred seven million, six thousand, nine hundred sixty-two.

* 5.(6)

List the whole numbers from 1 to 10 that are divisors of 12,300.

* 6.(4)

Replace the circle with the proper comparison symbol. Then write the comparison as a complete sentence using words to write the numbers.

!7 !11

* 7.(6)

The number 3456 is divisible by which single-digit numbers?

8.(4)

Model Show this subtraction problem on a number line: 2 ! 5

* 9.(5)

Write 6400 in expanded notation.

Find the value of each variable:

10.(3)

x " $4.60 # $10.00 * 11.(3)

p ! 3850 # 4500

* 12.(3)

8z # $50.00 13.(3)

748621689

! n60

* 14.(3)

1426 ! k # 87

* 15.(3)

990p " 45

* 16.(3)

z8 " 32

Simplify:

17.(1)

122535

18.(1)

800! 50

19.(1)

$100.00" $48.37

20.(1)

46,302# 49,998

21.(1)

$45.00 ! 20

22.(1)

7 " 11 " 13 23.(1)

9! 43,271

24.(1)

48¢ # $8.49 # $14 25.(2)

1000 $ (430 $ 58)

26.(1)

140(16) 27.(1)

25¢! 24

28.(1)

$43.5010

* 29.(2)

a. Analyze Name the property illustrated by the following equation.

x " 5 % 5x

b. Summarize In your own words explain the meaning of this property.

* 30.(2)

Justify List the properties used in the first three steps to simplify the expression (8 & 7) & 5.

(8 & 7) & 5 Given expression

8 & (7 & 5) a.

8 & (5 & 7) b.

(8 & 5) & 7 c.

40 & 7 8 & 5 % 40

280 40 & 7 % 280


Lesson 7 45

L E S S O N

7 Lines, Angles and Planes


facts Power Up B

mental math

a. Positive/Negative: 5 ! 10

b. Decimals: $2.50 " 10

c. Calculation: $1.00 ! 35¢

d. Calculation: 340 # 25

e. Number Sense: 565 ! 300


g. Probability: What is the probability of rolling a number less than 3 on a number cube?

h. Calculation: Start with the number of years in a decade, " 7, # 5, $ 3, ! 1, $ 4.

problem solving

The two pulleys on the left are both in equilibrium. A pulley is in equilibrium when the total weight suspended from the left side is equal to the total weight suspended from the right side. Will the pulley on the far right be in equilibrium, or will one side be heavier than the other?

Understand We are shown three pulleys on which three kinds of weights are suspended. The first two pulleys are in equilibrium. We are asked to determine if the third pulley is in equilibrium or if one side is heavier than the other.

Plan We will use logical reasoning to help us understand the relative weights of the objects.

Solve From the first pulley we see that three cones are equal in weight to four cubes, which means that cones are heavier than cubes. The second pulley shows that four cubes weigh the same as four cylinders, which means that cubes and cylinders weigh the same. On the third pulley, we can mentally remove the bottom cubes on either side. We are left with two cylinders and two cubes on one side, and four cones on the other side. Because we know that cones are heavier than cubes or cylinders (which weigh the same), the pulley is not in equilibrium, and will pull to the right.

Check We found that the pulley was not in equilibrium. Another way to verify our solution is to compare the third pulley to the first pulley. Once we remove the bottom cubes, we are left with two cylinders and two cubes on the left side of the third pulley, which is equal to four cubes. The first pulley shows that four cubes equals three cones, so four cones will be heavier.

New Concept Increasing Knowledge

We live in a world of three dimensions called space. We can measure the length, width, and depth of objects that occupy space. We can imagine a two-dimensional world called a plane, a flat world having length and width but not depth. Occupants of a two-dimensional world could not pass over or under other objects because, without depth, “over” and “under” would not exist. A one-dimensional world, a line, has length but neither width nor depth. Occupants of a one-dimensional world could not pass over, under, or to either side of other objects. They could only move back and forth on their line.

In geometry we study figures that have one dimension, two dimensions, and three dimensions, but we begin with a point, which has no dimensions. A point is an exact location in space and is unmeasurably small. We represent points with dots and usually name them with uppercase letters. Here we show point A :

A •

A line contains an infinite number of points extending in opposite directions without end. A line has one dimension, length. A line has no thickness. We can represent a line by sketching part of a line with two arrowheads. We identify a line by naming two points on the line in either order. Here we show line AB (or line BA ):

A B

Line AB or line BA

The symbols AB·

and BA·

(read “line AB ” and “line BA ”) also can be used to refer to the line above.

A ray is a part of a line with one endpoint. We identify a ray by naming the endpoint and then one other point on the ray. Here we show ray AB 1AB

¡ 2:A B

Ray AB

A segment is a part of a line with two endpoints. We identify a segment by naming the two endpoints in either order. Here we show segment AB 1AB2:

BA

Segment AB or segment BA

A segment has a specific length. We may refer to the length of segment AB by writing mAB, which means “the measure of segment AB,” or by writing the letters AB without an overbar. Thus, both AB and mAB refer to the distance from point A to point B. We use this notation in the figure below to state that the sum of the lengths of the shorter segments equals the length of the longest segment.

Thinking SkillsGeneralize

A line segment has a specific length. Why doesn’t a ray or a line have a specific length? A B C

AB ! BC " AC

mAB ! mBC " mAC


Lesson 7 47

Example 1

Use symbols to name a line, two rays, and a segment in the figure at right.

Solution

The line is AB·

(or BA·

). The rays are AB¡

and BA¡

. The segment is AB (or BA).

Example 2

In the figure below, AB is 3 cm and AC is 7 cm. Find BC.

A B C

Solution

BC represents the length of segment BC . We are given that AB is 3 cm and AC is 7 cm. From the figure above, we see that AB ! BC " AC . Therefore, we find that BC is 4 cm.

Formulate Write an equation using numbers and variables to illustrate the example. Then show the solution to your equation.

A plane is a flat surface that extends without end. It has two dimensions, length and width. A desktop occupies a part of a plane.

Two lines in the same plane either cross once or do not cross at all. If two lines cross, we say that they intersect at one point.

A DM

C B

Line AB intersects line CD at point M.

If two lines in a plane do not intersect, they remain the same distance apart and are called parallel lines.

Reading MathThe symbol # means is parallel to. The symbol $ means is perpendicular to.

R

Q

T

S

In this figure, line QR is parallel to line ST. This statement can be written with symbols, as we show here:

QR· 7 ST

·

Lines that intersect and form “square corners” are perpendicular lines. The small square in the figure below indicates a “square corner.”

M

QP

N

In this figure, line MN is perpendicular to line PQ. This statement can be written with symbols, as we show here:

MN·

!PQ·

Lines in a plane that are neither parallel nor perpendicular are oblique. In our figure showing intersecting lines, lines AB and CD are oblique.

BA

An angle is formed by two rays that have a common endpoint. The angle at right is formed by the two rays MD

¡ and MB

¡. The common endpoint

is M. Point M is the vertex of the angle. Ray MD and ray MB are the sides of the angle. Angles may be named by listing the following points in order: a point on one ray, the vertex, and then a point on the other ray. So our angle may be named either angle DMB or angle BMD .

When there is no chance of confusion, an angle may be named by only one point, the vertex. At right we have angle A .

An angle may also be named by placing a small letter or number near the vertex and between the rays (in the interior of the angle). Here we see angle 1.

The symbol ! is often used instead of the word angle. Thus, the three angles just named could be referred to as:

!DMB read as “angle DMB”

! A read as “angle A”

!1 read as “angle 1”

Angles are classified by their size. An angle formed by perpendicular rays is a right angle and is commonly marked with a small square at the vertex. An angle smaller than a right angle is an acute angle. An angle that forms a straight line is a straight angle. An angle smaller than a straight angle but larger than a right angle is an obtuse angle.

Right Acute Straight Obtuse

Example 3

a. Which line is parallel to line AB?

b. Which line is perpendicular to line AB?

Solution

a. Line CD 1or DC· 2 is parallel to line AB.

b. Line BD 1or DB· 2 is perpendicular to line AB.

Conclude How many right angles are formed by two perpendicular lines?

A

C

B

D

B

D

MAngle DMB

or angle BMD

AAngle A

Angle 1

1


Lesson 7 49

Example 4

There are several angles in this figure.

a. Name the straight angle.

b. Name the obtuse angle.

c. Name two right angles.

d. Name two acute angles.

Solution

a. ! AMD (or ! DMA) b. ! AMC (or !CMA)

c. 1. ! AMB (or ! BMA) d. 1. ! BMC (or ! CMB)

2. ! BMD (or ! DMB) 2. ! CMD (or !DMC)

On earth we refer to objects aligned with the force of gravity as vertical and objects aligned with the horizon as horizontal.

Example 5

A power pole with two cross pieces can be represented by three segments.

a. Name a vertical segment.

b. Name a horizontal segment.

c. Name a segment perpendicular to CD .

Solution

a. AB 1or BA2 b. CD 1or DC2 or EF 1or FE2 c. AB 1or BA2

The wall, floor, and ceiling surfaces of your classroom are portions of planes. Planes may be parallel, like opposite walls in a classroom, or they may intersect, like adjoining walls.

We may draw parallelograms to represent planes. Below we sketch how the planes of the floor and two walls appear to intersect.

WALLWALL

FLOOR

Although the walls and floor have boundaries, the planes of which they are a part do not have boundaries.

A

CE

DF

B

A M D

B C

Example 6

Sketch two intersecting planes. Which word below best describes the location where the planes intersect?

A Point B Line C Segment

Solution

We draw parallelograms through each other to illustrate the planes. In our sketch the intersection appears to be a segment. However, the actual planes extend without boundary, so the intersection continues without end and is a line.

Lines in the same plane that do not intersect are parallel. Lines in different planes that do not intersect are skew lines.

Connect Can you identify where planes intersect in your classroom?

Practice Set a. Name a point on this figure that is not on ray BC:

A B C D

b. In this figure XZ is 10 cm, and YZ is 6 cm. Find XY.

X Y Z

c. Draw two parallel lines.

d. Draw two perpendicular lines.

e. Draw two lines that intersect but are not perpendicular. What word describes the relationship of these lines?

f. Draw a right angle.

g. Draw an acute angle.

h. Draw an obtuse angle.

i. Two intersecting segments are drawn on the board. One segment is vertical and the other is horizontal. Are the segments parallel or perpend icular?

j. Describe a physical example of parallel planes.

k. Describe a physical example of intersecting planes.

l. Lines intersect at a point and planes intersect in a .

m. Connect If a power pole represents one line and a paint stripe in the middle of the road represents another line, then the two lines are

A parallel B intersecting C skewed


Lesson 7 51

n. Model Sketch a part of the classroom where three planes intersect, such as two adjacent walls and the ceiling.


* 1.(3)

If the product of two one-digit whole numbers is 35, what is the sum of the same two numbers?

* 2.(2)

Analyze Name the property illustrated by this equation:

! 5 · 1 " ! 5

* 3.(6)

List the factor pairs of 50.

* 4.(4)

Use digits and symbols to write “Two minus five equals negative three.”

5.(5)

Use only digits and commas to write 90 million.

6.(6)

List the single-digit factors of 924.

7.(4)

Arrange these numbers in order from least to greatest:

#10, 5, #7, 8, 0, #2

* 8.(4)

Generalize Use words to describe the following sequence. Then find the next three numbers in the sequence.

…, 49, 64, 81, 100, …

* 9.(7)

To build a fence, Megan dug holes in the ground to hold the posts upright. Then she attached rails to connect the posts. Which fence parts were vertical, the posts or the rails?

* 10.(6)

a. List the common factors of 24 and 32.

b. Find the greatest common factor of 24 and 32.

11.(4)

Connect The temperature at noon was 3$C. The temperature at 5:00 p.m. was #4$C. Did the temperature rise or fall between noon and 5:00 p.m.? By how many degrees?

Find the value of each variable.

* 12.(3)

6 · 6 · z " 1224 13.(3)

$100.00 # k " $17.54

14.(3)

w # 98 " 432 15.(3)

20x " $36.00

* 16.(3)

w20

! 200 * 17.(3)

300x ! 30

18.(6)

Explain Does the quotient of 4554 % 9 have a remainder? How can you tell without dividing?

Simplify:

19.(1)

36,475! 55,984

20.(1)

476" 38

21.(1)

$80.00 ! $72.45 22.(1)

$68.00 " 40

* 23.(2)

Justify Show the steps and the properties that make this multiplication easier to perform mentally: 8 · 7 · 5

24.(2, 4)

Compare: 4000 " (200 " 10) (4000 " 200) " 10

25.(1)

Evaluate each expression for a # 200 and b # 400: a. ab b. a ! b c. ba

* 26.(7)

Refer to the figure at right to answer a and b. a. Which angle is an acute angle?

b. Which angle is a straight angle?

* 27.(7)

What type of angle is formed by perpendicular lines?

Refer to the figure below to answer problems 28 and 29.

X Y Z

* 28.(7)

Name three segments in this figure.

* 29.(7)

Conclude If you knew mXY and mYZ, describe how you would find m XZ.

* 30.(7)

Model Sketch two intersecting planes.

Early FinishersReal-World Application

Lindy and seven friends played miniature golf at a new course. Par (average score for a good player) for the course is 63. The players scores were recorded as numbers above or below par. For example a score of 62 was recorded as –1 (one under par). The recorded scores were:

1 !1 3 !3 5 0 !2 3

a. What were the seven scores?

b. Arrange the scores in order from least to greatest.

c. How many of the scores are par or under par? List the scores.

A M C

B


Lesson 8 53

L E S S O N

8 Fractions and Percents Inch Ruler


facts Power Up A

mental math


b. Decimals: $0.25 " 10


d. Number Sense: 325 # 50


f. Number Sense: 200 " 10

g. Measurement: Convert 2 hours into minutes

h. Calculation: Start with a score, # 1, $ 3, " 5, # 1, $ 4, # 1, $ 2, " 6, # 3,$ 3.

problem solving

The number 325 contains the digits 2, 3, and 5. These three digits can be ordered in other ways to make different numbers. Each order is called a permutation of the three digits. The smallest permutation of 2, 3, and 5 is 235. How many permutations of the three digits are possible? Which number is the greatest permutation of 2, 3, and 5?

Understand We are told that digits can be arranged in different permutations. We are asked to find how many permutations of the digits 2, 3, and 5 are possible, and to find the greatest permutation of the three digits.

Plan We will make an organized list by working from least to greatest: first we will list the permutations that begin with 2, then with 3, then with 5.

Solve First, we write each permutation that begins with 2. Then we write each permutation that begins with 3. Finally, we write each permutation that begins with 5.

235, 253

325, 352

523, 532

There are six possible permutations of the three digits. The greatest permutation is 532.

Check We found the number of possible permutations and the greatest permutation of the three digits. We kept an organized list to ensure we did not accidentally forget any permutations.


fractions and percents

Fractions and percents are commonly used to name parts of a whole or parts of a group.

At right we use a whole circle to represent 1. The circle is divided into four equal parts with one part shaded. One fourth of the circle is shaded, and 34 of the circle is not shaded.

Since the whole circle also represents 100% of the circle, we may divide 100% by 4 to find the percent of the circle that is shaded.

100% ! 4 " 25%

We find that 25% of the circle is shaded, so 75% of the circle is not shaded.

A common fraction is written with two numbers and a division bar. The number below the bar is the denominator and shows how many equal parts are in the whole. The number above the bar is the numerator and shows how many of the parts have been selected.

numerator denominator

14

division bar

A percent describes a whole as though there were 100 parts, even though the whole may not actually contain 100 parts. Thus the “denominator” of a percent is always 100.

25 percent means 25100

Math LanguageInstead of writing the denominator 100, we can use the word percent or the percent symbol, %.

A mixed number such as 2 34 includes an integer and a fraction. The shaded

circles below show that 2 34 means 2 ! 3

4. To read a mixed number, we first say the integer, then we say “and”; then we say the fraction.

Two and three fourths

It is possible to have percents greater than 100%. When we write 2 34 as a

percent, we write 275%.

Connect How do we write 3 14 as a percent?

Example 1

Name the shaded part of the circle as a fraction and as a percent.


Lesson 8 55

Solution

Two of the five equal parts are shaded, so the fraction that is shaded is 25.

Since the whole circle (100%) is divided into five equal parts, each part is 20%.

100% ! 5 " 20%

Two parts are shaded. So 2 # 20%, or 40%, is shaded.

Example 2

Which of the following could describe the part of this rectangle that is shaded?

A 12

B 40% C 60%

Solution

There is a shaded and an unshaded part of this rectangle, but the parts are not equal. More than 12 of the rectangle is shaded, so the answer is not A. Half of a whole is 50%.

100% ! 2 " 50%

Since more than 50% of the rectangle is shaded, the correct choice must be C 60%.

Between the points on a number line that represent integers are many points that represent fractions and mixed numbers. To identify the fraction or mixed number associated with a point on a number line, it is first necessary to discover the number of segments into which each length has been divided.

Example 3

Point A represents what mixed number on this number line?

7 8

A

9

Solution

We see that point A represents a number greater than 8 but less than 9. It represents 8 plus a fraction. To find the fraction, we first notice that the segment from 8 to 9 has been divided into five smaller segments. The distance from 8 to point A crosses two of the five segments. Thus, point A represents the mixed number 8 25.

Note: It is important to focus on the number of segments and not on the number of vertical tick marks. The four tick marks divide the space between 8 and 9 into five segments, just as four cuts divide a strip of paper into five pieces.

inch ruler A ruler is a practical application of a number line. The units on a ruler are of a standard length and are often divided successively in half. That is, inches are divided in half to show half inches. Then half inches are divided in half to show quarter inches. The divisions may continue in order to show eighths, sixteenths, thirty-seconds, and even sixty-fourths of an inch. In this book we will practice measuring and drawing segments to the nearest sixteenth of an inch.

Here we show a magnified view of an inch ruler with divisions to one sixteenth of an inch.

1

†b¢

F

§d¶

A

®f™

H

¨hÆ

inch

Bear in mind that all measurements are approximate. The quality of a measurement depends upon many conditions, including the care taken in performing the measurement and the precision of the measuring instrument. The finer the gradations are on the instrument, the more precise the measurement can be.

For example, if we measure segments AB and CD below with an undivided inch ruler, we would describe both segments as being about 3 inches long.

inch 321 4

C D

A

We can say that the measure of each segment is 3 inches ! 12 inch (“three inches plus or minus one half inch”). This means each segment is within 12 inch of being 3 inches long. In fact, for any measuring instrument, the greatest possible error due to the instrument is one half of the unit that marks the instrument.

We can improve the precision of measurement and reduce the possible error by using an instrument with smaller units. Below we use a ruler divided into quarter inches. We see that AB is about 31

4 inches and CD is about 23

4 inches. These measures are precise to the nearest quarter inch. The greatest possible error due to the measuring instrument is one eighth of an inch, which is half of the unit used for the measure.

inch 321 4

C D

A


Lesson 8 57

Example 4

Use an inch ruler to find AB, BC, and AC to the nearest sixteenth of an inch.

A B C

Solution

From point A we find AB and AC. AB is about 78 inches, and AC is about 21

2 inches.

A B C

inch 321

We move the zero mark on the ruler to point B to measure BC. We find BC is about 15

8 inches.

A B C

inch 21

Just as we have used a number line to order integers, we may use a number line to help us order fractions.

Example 5

Arrange these fractions in order from least to greatest:

12, 1

4, 58, 7

16

Solution

The illustrated inch ruler can help us order these fractions.

14, 7

16, 12, 5

8

Practice Set a. What fraction of this circle is not shaded?

b. What percent of this circle is not shaded?

c. Half of a whole is what percent of the whole?

Model Draw and shade circles to illustrate each fraction, mixed number, or percent:

d. 23 e. 75% f. 23

4

Evaluate Points g and h represent what mixed numbers on these number lines?

3 4

g

5

12 13

h

14

i. Find XZ to the nearest sixteenth of an inch.

YX Z

j. Jack’s ruler is divided into eighths of an inch. Assuming the ruler is used correctly, what is the greatest possible measurement error that can be made with Jack’s ruler? Express your answer as a fraction of an inch.

k. Arrange these fractions in order from least to greatest:

14, 1

2, 18, 1

16


* 1.(4, 8)

Represent Use digits and a comparison symbol to write “One and three fourths is greater than one and three fifths.”

* 2.(8)

Refer to practice problem i above. Use a ruler to find XY and YZ.

3.(1)

What is the quotient when the product of 20 and 20 is divided by the sum of 10 and 10?

* 4.(6)

Analyze List the single-digit divisors of 1680.

* 5.(8)

Evaluate Point A represents what mixed number on this number line?

3 4

A

5

6.(2, 4)

a. Replace the circle with the proper comparison symbol.

3 ! 2 2 ! 3

b. Analyze What property of addition is illustrated by this comparison?

7.(5)

Use words to write 32500000000.

* 8.(8)

a. What fraction of the circle is shaded?

b. What fraction of the circle is not shaded?

* 9.(8)

a. Analyze What percent of the rectangle is shaded?

b. What percent of the rectangle is not shaded?

* 10.(8)

What is the name of the part of a fraction that indicates the number of equal parts in the whole?


Lesson 8 59


11.(3)

a ! $4.70 " $2.35 12.(3)

b # $25.48 " $60.00

13.(3)

8c " $60.00 14.(3)

10,000 ! d " 5420

* 15.(3)

e15

! 15 * 16.(3)

196f

! 14

17.(2, 3)

Justify Give a reason for each of the first two steps taken to solve the equation 9 # (n # 8) " 20.

9 # (n # 8) " 20 Given equation

9 # (8 # n) " 20 a.

(9 # 8) # n " 20 b.

17 # n " 20 9 # 8 " 17

n " 3 17 # 3 " 20

Simplify:

18.(1)

400! 500

19.(1)

79¢! 30¢

20.(1)

3625 # 431 # 687

21.(1)

6000 $ 50 22.(1)

20 % 10 % 5

23.(1)

$27.0018 24.

(1)

34566

25.(1)

Analyze If t is 1000 and v is 11, find a. t ! v b. v ! t

* 26.(4)

a. The rule of the following sequence is k " 3n ! 1. What is the tenth term of the sequence?

2, 5, 8, 11, . . .

b. Analyze What pattern do you recognize in this sequence?

27.(2, 4)

Compare: 416 ! (86 # 119) (416 ! 86) # 119

Refer to the figure at right to answer problems 28 and 29.

* 28.(7)

Name the acute, obtuse, and right angles.

29.(7)

a. Name a segment parallel to DA.

b. Name a segment perpendicular to DA.

30.(7)

Explain Referring to the figure below, what is the difference in meaning between the notations QR and QR?

Q R S

D

C

A

B

L E S S O N

9 Adding, Subtracting, and Multiplying Fractions Reciprocals


facts Power Up A

mental math


b. Decimals: $0.39 " 10





g. Measurement: Convert 6 pints into quarts

h. Calculation: Start with half a dozen, # 1, " 6, ! 2, $ 2, # 4, $ 4, ! 5, " 15

problem solving

The diameter of a circle or a circular object is the distance across the circle through its center. Find the approximate diameter of the penny shown at right.


adding fractions

On the line below, AB is 1 38 in. and BC is 1

48 in. We can find AC by measuring

or by adding 1 38 in. and 1

48 in.

A CB

1 38 in. ! 1

48 in. " 27

8 in.

Math LanguageHere, the word common means “shared bytwo or more.”

When adding fractions that have the same denominators, we add the numerators and write the sum over the common denominator.

Example 1

Find each sum:

a. 17% 2

7 % 3

7 b. 331

3% % 331

3%

IN

GOD WE TRUST

LIBERTY

1991

inch 21


Lesson 9 61

Solution

a. 17 ! 2

7 !37 ! 67 b. 331

3% " 3313% ! 662

3%

Example 2

How much money is 14 of a dollar plus 34 of a dollar?

Solution14 !

34 ! 44 ! 1 The sum is one dollar.

When the numerator and denominator of a fraction are equal (but not zero), the fraction is equal to 1. The illustration shows 4

4 of a circle, which is one

whole circle.

subtracting fractions

To subtract a fraction from a fraction with the same denominator, we write the difference of the numerators over the common denominator.

Example 3

Find each difference:

a. 3 59 ! 1 19 b. 3

5 ! 35

Solution

a. 3 59 " 1 19# 2 4

9 b. 35 "

35 #

05 # 0

multiplying fractions

The first illustration shows 12 of a circle. The second illustration shows 12 of 12 of a circle. We see that 12 of 12 is 14. We often translate the word of into a multiplication symbol. We find 12 of 12 by multiplying:

12

of 12

becomes 12 $ 1

2 # 14

To multiply fractions, we multiply the numerators to find the numerator of the product, and we multiply the denominators to find the denominator of the product. Notice that the product of two positive fractions less than 1 is less than either fraction.

Model Draw and shade a rectangle to show 14. Then show 12 of 1

4 on your rectangle. What is 12 # 1

4?

12

12 of 1

2

Thinking SkillModel

Draw a number line to show the solution to Example 1a.

44 ! 1

Example 4

Find each product:

a. 12 of 13 b. 1

2 ! 34 ! 1

5

Solution

a. 12 ! 1

3" 1

6 b. 1

2 !34 ! 1

5" 3

40

reciprocals If we invert a fraction by switching the numerator and denominator, we form the reciprocal of the fraction.

The reciprocal of 43 is 34.

The reciprocal of 34 is 43.

The reciprocal of 14 is 41, which is 4.

The reciprocal of 4 1or 41 2 is 14.

Note the following relationship between a number and its reciprocal:

The product of a number and its reciprocal is 1.

Here we show two examples of multiplying a number and its reciprocal.

43 !

34

" 1212

" 1

14 ! 4

1" 4

4" 1

Math LanguageA real number is a number that can be represented by a point on a number line.

This relationship between a number and its reciprocal applies to all real numbers except zero and is called the Inverse Property of Multiplication.

Inverse Property of Multiplication

a ! 1a " 1

if a is not 0.1

Example 5

Find the reciprocal of each number below. Then multiply the number and its reciprocal.

a. 35

b. 3

Solution

a. The reciprocal of 35 is 53. 35 ! 53 " 15

15 " 1

b. The reciprocal of 3, which is 3 “wholes” or 31, is 13. 31 ! 13 " 33 " 1

1 The exclusion of zero from being a divisor is presented in detail in Lesson 119.


Lesson 9 63

Example 6

Find the missing number: 34 n ! 1

Solution

The expression 34 n means “34 times n.” Since the product of 34 and n is 1, the

missing number must be the reciprocal of 34, which is 43.

34 ! 43 ! 12

12 ! 1 check

Generalize Give another example to show that the product of a number and its reciprocal is always 1.

Example 7

How many 34 s are in 1?

Solution

The answer is the reciprocal of 34, which is 43.

In Lesson 2 we noted that although multiplication is commutative (6 " 3 # 3 " 6), division is not commutative (6 $ 3 % 3 $ 6). Now we can say that reversing the order of division results in the reciprocal quotient.

6 $ 3 # 2

3 " 6 ! 12

Practice Set Simplify:

a. 56 # 1

6 b. 4

5 $35

c. 35 % 1

2 %34

d. 33 #

33 # 2

3

e. 47 % 2

3 f. 5

8 $58

g. 1427% # 142

7% h. 8712% $ 121

2%

Explain Write the reciprocal of each number. Tell what your answer shows about the product of a number and its reciprocal.

i. 45

j. 87

k. 5

Find each unknown number:

l. 58

a # 1 m. 6m # 1

n. Gia’s ruler is divided into tenths of an inch. What fraction of an inch represents the greatest possible measurement error due to Gia’s ruler? Why?

o. How many 23s are in 1?

p. If a ! b equals 4, what does b ! a equal?

q. What property of multiplication is illustrated by this equation?

23 !

32

" 1


1.(1)

What is the quotient when the sum of 1, 2, and 3 is divided by the product of 1, 2, and 3?

2.(1)

Represent The sign shown is incorrect. Show two ways to correct the sign.

* 3.(4, 9)

Replace each circle with the proper comparison symbol. Then write the comparison as a complete sentence, using words to write the numbers.

a. 12

12 ! 1

2 b. "2 "4

* 4.(5)

Write twenty-six thousand in expanded notation.

* 5.(8)

a. A dime is what fraction of a dollar?

b. A dime is what percent of a dollar?

* 6.(8)

The flag to the right is a nautical flag that stands for the letter J.

a. What fraction of the flag is shaded?

b. What fraction of the flag is not shaded?

* 7.(7)

Classify Is an imaginary “line” from the Earth to the Moon a line, a ray, or a segment? Why?

* 8.(8)

Use an inch ruler to find LM, MN, and LN to the nearest sixteenth of an inch.

L M N

* 9.(6)

a. List the factors of 18.

b. List the factors of 24.

c. Which numbers are factors of both 18 and 24?

d. Which number is the GCF of 18 and 24?

* 10.(1, 9)

If n is 25, find a. n # n b. n " n

Evaluate Find the value of each variable:

11.(3)

85,000 # b $ 200,000 12.(3)

900 ! c $ 60

J

Apples0.99¢ per pound


Lesson 9 65

13.(3)

d ! $5.60 " $20.00

14.(3)

e # 12 " $30.00

15.(3)

f $ $98.03 " $12.47

16.(3)

5 ! 7 ! 5 ! 7 ! 6 ! n !1 ! 2 ! 3 ! 4 " 40

Simplify:

* 17.(9)

31115 ! 1 3

15 * 18.(9)

138 " 14

8

* 19.(9)

34 # 1

4 20.(1)

180217

21.(1)

$60.00 % $49.49 22.(1)

607 # 78

23.(9)

45 # 2

3 # 13 24.

(9)

19 " 2

9 " 49

* 25.(2)

Write the steps and properties that make this multiplication easier to perform mentally: 50 # 36 # 20

26.(9)

What property of multiplication is illustrated by this equation? 45 #

54 $ 1

* 27.(7)

Classify Lines AB and XY lie in different planes. Which word best describes their relationship?

A Intersecting B Skew C Parallel

* 28.(7)

Refer to the figure at right to answera and b.

a. Which angles are acute?

b. Which segment is perpendicular to CB?

* 29.(4, 8)

Generalize Describe the following sequence. Then find the next number in the sequence.

12, 14, 18, p

* 30.(9)

How many 25s are in 1?

A

BC

B

A

YX

L E S S O N

10 Writing Division Answers as Mixed Numbers Improper Fractions


facts Power Up A

mental math


b. Decimals: $1.25 " 10





g. Measurement: Convert 5 yd into feet

h. Calculation: 3 " 6, $ 2, " 5, # 3, $ 6, ! 3, " 4, # 1, $ 3

problem solving

In one section of a theater there are twelve rows of seats. In the first row there are 6 seats, in the second row there are 9 seats, and in the third row there are 12 seats. If the pattern continues, how many seats are in the twelfth row?


writing division

answers as mixed numbers

Alexis cut a 25-inch ribbon into four equal lengths. How long was each piece?

To find the answer to this question, we divide. However, expressing the answer with a remainder does not answer the question.

6 R 1

241

4! 25

Interpret What unit of measure does the answer 6 R 1 stand for?

The answer 6 R 1 means that each of the four pieces of ribbon was 6 inches long and that a piece remained that was 1 inch long. But that would make five pieces of ribbon!

Instead of writing the answer with a remainder, we will write the answer as a mixed number. The remainder becomes the numerator of the fraction, and we use the divisor as the denominator.


Lesson 10 67

241

4! 256

14

Thinking SkillAnalyze

What given informationshows that the answer is 6

14, rather than

6R1?

This answer means that each piece of ribbon was 6 14 inches long, which is

the correct answer to the question.

Example 1

What percent of the circle is shaded?

Solution

One third of the circle is shaded, so we divide 100% by 3.

910

91

3! 100%33

13%

We find that 33 13% of the circle is shaded.

improper fractions

A fraction is equal to 1 if the numerator and denominator are equal (and are not zero). Here we show four fractions equal to 1.

22

33

44

55

A fraction that is equal to 1 or is greater than 1 is called an improper fraction. Improper fractions can be rewritten either as whole numbers or as mixed numbers.

Example 2

Draw and shade circles to illustrate that 53 equals 123.

Solution

53 ! 12

3

Thinking SkillAnalyze

In this problem, what does the 100% represent?

Example 3

Convert each improper fraction to either a whole number or a mixed number:

a. 53 b. 6

3

Solution

a. Since 33 equals 1, the fraction 53 is greater than 1.

53 !

33 " 2

3

! 1 " 23

! 123

b. Likewise, 63 is greater than 1.

63 !

33 "

33

! 1 " 1

! 2

Math LanguageA fraction bar indicates division. For example, 53 means 5 divided by 3.

We can find the whole number within an improper fraction by performing the division indicated by the fraction bar. If there is a remainder, it becomes the numerator of a fraction whose denominator is the same as the denominator in the original improper fraction.

a. 53 1

32

3! 5 123

b. 63

23! 6

Model How could you shade circles to show that 63 ! 2? Draw your answer.

Example 4

Rewrite 375 with a proper fraction.

Solution

The mixed number 375 means 3 " 7

5. The fraction 75 converts to 125.

75 ! 12

5

Now we combine 3 and 125.

3 " 125

! 4 25

When the answer to an arithmetic problem is an improper fraction, we may convert the improper fraction to a mixed number.


Lesson 10 69

Example 5

Simplify:

a. 45 ! 4

5 b. 52 " 3

4 c. 1 35 ! 1 3

5

Solution

a. 45 ! 45

" 85

" 1 35 b. 5

2 #34

" 158

" 1 78

c. 135 ! 13

5" 26

5" 3 1

5

Sometimes we need to convert a mixed number to an improper fraction. The illustration below shows 31

4 converted to the improper fraction 134 .

314 ! 13

4

We see that every whole circle equals 44. So three whole circles is 44 ! 4

4 ! 44,

which equals 124

. Adding 14 more totals 13

4.

Example 6

Write each mixed number as an improper fraction:

a. 3 13 b. 2 3

4 c. 12 12

Solution

a. The denominator is 3, so we use 33 for 1. Thus 313 is

33 !

33 !

33 ! 1

3" 10

3

b. The denominator is 4, so we use 44 for 1. Thus 23

4 is

44 ! 4

4 !34

" 114

c. The denominator is 2, so we use 22 for 1. If we multiply 12 by 22, we find that 12 equals 24

2 . Thus, 1212 is

12 a22b ! 12

" 242 ! 1

2" 25

2

The solution to example 6c suggests a quick way to convert a mixed number to an improper fraction.

1212 " 12 1

2 " 2 # 12 ! 12 " 24 ! 1

2 "252

"

#


Practice Set a. Alexis cut a 35-inch ribbon into four equal lengths. How long was each piece?

b. One day is what percent of one week?

Convert each improper fraction to either a whole number or a mixed number:

c. 125

d. 126

e. 2127

f. Model Draw and shade circles to illustrate that 214 ! 9

4.

Simplify:

g. 23 " 2

3 " 23 h. 7

3 # 23 i. 12

3 " 123

Convert each mixed number to an improper fraction:

j. 123

k. 356

l. 434

m. 512

n. 634 o. 10 25

p. Generalize Write 3 different improper fractions for the number 4.


* 1.(2, 9)

Represent Use the fractions 12, 13, and 16 to write an equation that illustrates the Associative Property of Multiplication.

* 2.(7)

Use the words perpendicular and parallel to complete the following sentence:

In a rectangle, opposite sides are a. and adjacent sides areb. .

3.(1)

What is the difference when the sum of 2, 3, and 4 is subtracted from the product of 2, 3, and 4?

* 4.(8)

a. What percent of the rectangle is shaded?

b. What percent of the rectangle is not shaded?

* 5.(10)

Write 323 as an improper fraction.

* 6.(4, 9)

Replace each circle with the proper comparison symbol:

a. 2 ! 2 2 " 2 b. 12 " 1

2 1

2 # 12

* 7.(8)

Connect Point M represents what mixed number on this number line?

8 9

M

10 11

Lesson 10 71

* 8.(10)

Model Draw and shade circles to show that 135 ! 8

5.

* 9.(6)

List the single-digit numbers that are divisors of 420.


10.(3)

12,500 ! x " 36,275 11.(3)

18y " 396

12.(3)

77,000 # z " 39,400 13.(3)

a8 " $1.25

14.(3)

b # $16.25 " $8.75 15.(3)

c ! $37.50 " $75.00

* 16.(8)

Arrange these fractions in order from least to greatest:

12, 38, 34, 1

16

Simplify:

17.(10)

52 "

54 18.

(9)

58 #

58 19.

(10)

1120 $

1820

20.(2)

2000 # (680 # 59) 21.(10)

100% $ 9

22.(1)

89¢ ! 57¢ ! $15.74 23.(1)

800 % 300

* 24.(10)

223 $ 22

3 * 25.(9)

23 % 2

3 % 23

* 26.(7)

Describe each figure as a line, ray, or segment. Then use a symbol and letters to name each figure.

a. M C

b. PM

c. F H

* 27.(9)

How many 59s are in 1?

* 28.(4, 8)

Generalize What are the next three numbers in this sequence?

. . ., 32, 16, 8, 4, 2, . . .

* 29.(4)

Which of these numbers is not an integer?

A #1 B 0 C 12

D 1

* 30.(4, 9)

a. Conclude If a # b " 5, what does b # a equal?

b. If wx ! 3, what does x

w equal?

c. Conclude How are wx and xw related?

Documents

LESSON 6 Factors Divisibilityera6anderson.weebly.com/.../6/...2_lessons_006-010.pdfLesson 6 43 h. The number 7000 is divisible by which single-digit numbers? i. List all the common