chapter 3 rational numbers. 3-1-a explore: the number line let’s graph - on a number line 1. draw...

Chapter 3Rational Numbers

3-1-A Explore: The Number Line

Let’s graph - on a number line

1. Draw a number line. Place a zero on the right side an a -1 on the left. Divide the line into fourths.

2. Starting from the right, label the line with -1/4, -2/4, and -3/4.

3. Draw a dot on the number line on the -3/4 mark.

You have already graphed integers and positive fractions on a number line. Today, you will graph

negative fractions.

-1

0-½ -¼-¾

Remember!The denominator of the fraction determines the number of sections to be marked on the number line between two integers!

Graph the pair of numbers on a number line. Then write which number is less.

Remember the steps!

1. Draw a number line. Place a zero on the right side an a -2 on the left. Divide the line into the appropriate parts.

2. Starting from the right, label the line with the fractions.

3. Draw a dot on the number line to mark the values.

Self-Assessment: Try pg. 127 # 1-8 on your own. Then, check answers with a partner.

3-1-B Terminating & Repeating Decimals

The table shows the winning speeds for a 10-year period at the Daytona 500.

1. What fraction of the speeds are between 130 and 145 miles per hour?

2. Express this fraction using words and then as a decimal.

3. What fraction of the speeds are between 145 and 165 miles per hour? Express this fraction using words and decimals.

Year Winner Speed (mph)

1999 J. Gordon 148.295

2000 D. Jarrett 155.669

2001 M. Waltrip 161.783

2002 W. Burton 142.971

2003 M. Waltrip 133.870

2004 D. Earnhardt Jr. 156.345

2005 J. Gordon 135.173

2006 J. Johnson 142.667

2007 K. Harvick 149.335

2008 R. Newman 152.672

•7/20• Think: 35/100 so

0.35•5 ¾• Think: 75/100 so

5.75•3/25• Think: 12/100 so

0.12•-6 ½• Think: 50/100 so -

6.5

TIPYou should use MENTAL MATH whenever possible when writing fractions as decimals. Think about if

the denominator is a factor of 10, 100, or 1,000.

Fractions to Decimals: Mental Math!

A goal for today is to change fractions to decimals. Try

these!

Fractions to Decimals: Division

Any fraction can be written as a decimal by dividing its numerator by its denominator!

You should get 0.375!

You should get -0.025. Remember to keep the

negative sign!

You should get-0.875 2.125

7.45

Not all fractions are TERMINATING DECIMALS.

Remember, a TERMINATING DECIMAL is a decimal with digits that end.

REPEATING DECIMALS have a pattern in their digit (s) that repeats forever!

Consider 1/3. When you divide 1 by 3, you get 0.3333...

Use BAR NOTATION to indicate a that a number pattern repeats indefinitely. A bar is written over only the digit (s) that

repeat.

PRACTICE:Write each as a decimal.1. 7/92. 2/33. -3/114. 8 1/3-------------------------------------------

---Use the table to find what fraction of the fish in an aquarium are goldfish. Write in simplest form.

Determine the fraction of the aquarium made up by each fish. Write the answer in simplest form!

a) mollyb) guppyc) angelfish

Fish Amount

Guppy 0.25

Angelfish 0.4

Goldfish 0.15

Molly 0.2


3-1-C Compare & Order Rational Numbers

The batting average of a softball player is found by comparing the number of

hits to the number of times at bat. Melissa had 50 hits in 175 at bats. Harmony

had 42 hits in 160 at bats.

1. Write the two batting averages as fractions.

2. Which girl had the better batting average? Explain.

3. Describe two methods you could use to compare the batting averages.

RATIONAL NUMBERS:numbers that can be expressed as a ratio of two integers expressed as a fraction (in which the denominator is not zero). Includes common fractions, terminating and repeating decimals, percents, and all integers.

Rational Numbers

Integers

Whole Numbers

0.8

20%

2.2

½

1 2/3

-1.44

-3 -1

21

Today, your goal is to be able to compare and order RATIONAL NUMBERS (fractions, mixed numbers, and decimals).

Graph each rational number on a number line. Mark off equal size increments of 1/6 between -2 and -1.

You won’t always be comparing rational numbers that have common denominators. A COMMON

DENOMINATOR is a common multiple of the denominators of two or more fractions.

The LEAST COMMON DENOMINATOR or LCD is the LCM of the denominators. The LCD is used to

compare fractions!

What is the least common

denominator?What does that

make your numerators?

In Mr. Reed’s math class, 20% of the students own Sperry shoes. In Mrs. Crowe’s math class, 5 out of 29 students own Sperry. In which math class does a greater fraction of students own Sperry?

Express each number as a decimal and then compare.20% = 0.2

5/29 = .1724

Since 0.2 > 0.1724, 20% > 5/29

Therefore, a greater fraction of students in Mr. Reed’s class own Sperry shoes.

In a second period class, 37.5% of students like to bowl. In a fifth period class, 12 out of 29 students like to bowl. In which class does a greater fraction of the students like to bowl?

Remember to

line up the

decimal points

and compare

using place

value!

3.443.1415926…3.143.4444444444


Add & Subtract Positive Fractions

Sean surveyed ten classmates to find out which type of tennis shoe they like to wear!

1. What fraction liked cross trainers?

2. What fraction liked high tops?

3. What fraction liked either cross trainers OR high tops?

Fractions that have the same denominator are called LIKE FRACTIONS

Fractions that do not have the same denominator are called UNLIKE FRACTIONS.

Shoe Type

Number

Cross Trainer

5

Running

3

High Top

2

You can use FRACTION TILES as a model to help

solve problems that require addition and

subtraction of fractions.

With your “elbow partner” , complete Fraction Discovery #1. In it, you will be asked to do three things:

1. Draw a model to represent the problem and use that model to find a solution (no numbers allowed)

2. Draw a model to represent the problem and AT THE SAME TIME, write an expression using numbers. Find a solution using both methods.

3. Write a numerical expression only to solve the problem.

By 7th grade, you should already know fraction addition & subtraction rules! But your CHALLENGE is to

complete some of the problems without those rules

Add and Subtract Like FractionsTo add or subtract like fractions, add or subtract the

numerators and write the result over the denominator.

Key Concepts Review

Key Concepts Review

Add and Subtract Unlike FractionsTo add or subtract like fractions with different denominators

• Rename the fractions using the least common denominator (LCD)

• Add or subtract as with like fractions• If necessary, simplify the sum or difference

Add & Subtract Negative Fractions

Can fractions be negative? YES!

Although we may not think about it much, you use negative fractions when you:• Give part of something away• Eat a part of something• Lose part of something• Pour out part of something• Go part of the way backwards• Go part of the way downWith your “elbow partner”, complete Fraction Discovery #2. Today,

you will need PINK fractions for NEGATIVE numbers and

YELLOW fractions POSITIVE.Use what you already know about INTEGER RULES and FRACTION

OPERATIONS to help you!

Key Concepts Review

When you have like

denominators, keep the

denominator and use your INTEGER RULES

to find the sum or difference in the numerator!When you have unlike

denominators, first, find a COMMON DENOMINATOR!

Then, you can just use the INTEGER RULES to find the

sum or difference in the numerator!

Practice adding and subtracting with fraction tiles.

Practice Without Tiles!Q

uest

ion

s An

swers

3-2-D Add & Subtract Mixed Numbers

Baby Birth Weight

Adelaide

Stephen

Micah

Nora

1. Write an expression to find how much more Stephen weighs than Nora.

2. Rename the fractions using the LCD.

3. Find the difference of the fractional parts and then the difference of the whole numbers.

To add or subtract mixed numbers, first add or subtract the fractions. If necessary, rename them using the

LCD. Then add or subtract the whole numbers and simplify if necessary.

Add and write in simplest form. For these problems,

you can add the whole numbers and the fractions

separately.

Subtract. Write in simplest form. For these problems, you can subtract the whole numbers and the fractions

separately.

Many times, it is

not possible to

subtract the

whole numbers

and fractions

separately. In this

case, it is often

best to convert to

IMPROPER

FRACTIONS

IMPROPER FRACTION:Has a numerator that is greater

than or equal to the denominator

Real W

orld

Pro

ble

ms!


3-3-A Explore: Fraction Discovery

With a partner, complete Fraction Discover #3

You will use rectangular models to find the answer to fraction problems.

Your challenge is to find an answer WITHOUT using rules you have learned in the past!

3-3-B Multiply Fractions

For each the first problem, create a sketch or model to solve.

Represent these two situations with equations. Are the equations the same or different?

3-3-D Divide Fractions

KEY CONCEPT:

Words: To divide a fraction, multiply by its multiplicative inverse, or reciprocal

Practice Dividing by Fractions

Practice Dividing by Mixed Numbers

To divide by a

mixed number,

first rename it

as an improper

fraction.

Estimation a

great way to

check your

solution!Mrs. Bybee bought 4 ½

gallons of ice cream to serve at

her birthday party. If a pint is 1/8 of a gallon, how many pint-

sized servings can be made?

Ms. Holloway has 8 ¼ cups of coffee. If she divides the

coffee into ¾ cup servings,

how many servings will she

have?Self-Assessment: Try pg. 170 # 1-10 on your own. Then, check answers with a partner.

3-4-A Multiply & Divide Monomials

For each increase on the Richter scale, an earthquake’s vibrations, or seismic waves, are 10 times greater! So, an earthquake of magnitude 4

has seismic waves that are 10 times greater than that of a magnitude 3 earthquake.

1. Examine the exponents of the powers in the last column. What do you observe?

2. Write a rule for determining the exponent of the product when you multiply powers with the same base.

Richter Scale

Times Greater than Magnitude 3 Earthquake

Written using Powers

4 10 x 1 = 10 101

5 10 x 10 = 100 101 x 101 = 102

6 10 x 100 = 1,000 101 x 102 = 103

7 10 x 1,000 = 10, 000 101 x 103 = 104

8 10 x 10,000 = 100,000 101 x 104 = 105

REMEMBER:Exponents are used to show repeated multiplication. Use the definition of an exponent to find a rule for multiplying

powers with the SAME BASE.

23 x 24 = (2 x 2 x 2) x (2 x 2 x 2 x 2)=

27PRODUCT OF POWERSWords: To multiply powers with the same base, add their exponentsSymbols: am x an = am+n

Example: 32x 34 = 32+4= 36

Practice Multiplying Powers!1. 73 x 71

2. 53 x 54

3. (0.5)2 x (0.5)9

4. 8 x 85

Common Mistake:When multiplying powers, do not multiply (evaluate) the bases that are the same!

MONOMIALA number, variable,

or product of a number and one or more variables.

Monomials can also be multiplied using

the rule for the product of powers.

1.x5 (x2) 2. (-4n3)(6n2)3. -3m(-8m4)4.52x2y4

(53xy4)

QUOTIENT OF POWERSWords: To divide powers with the same base, subtract their exponentsSymbols: am ÷ an = am-n

Example: 34÷ 32 = 34-2= 32

If we get the PRODUCT OF POWERS using ADDITION, we should get the QUOTIENT OF POWERS using……

The table compares the processing speeds of a

specific type of computer in 1999 and in 2008. Find

how many times faster the computer was in 2008

than in 1999.

Year

Processing Speed

(instructions per second)

1999 103

2008 109

The number of fish in a school of fish is 43. If the number of fish in the school increased by 42 times the original number of fish, how many fish are now in

the school? Evaluate the power.


3-4-B Negative Exponents

1. Describe the pattern of the powers in the first column. Continue the pattern by writing the next two values in the table.

2. Describe the pattern of values in the second column. Then complete the second column.

3. Determine how 3-1 should be defined.

Power Value

26 64

25 32

24 16

23 8

22 4

21 2

20 ???

2-1 ???

KEY CONCEPT: NEGATIVE EXPONENET

Words: Any nonzero number to the negative

n power is the multiplicative inverse of its nth power.

PRACTICE!Write each expression using a

positive exponent.• 6-2

• x-5

• 5-6

• t-4

PRACTICE!Write each expression using a negative

exponent other than -1.

When given a fraction with a

positive exponent or square, you

can rewrite it using a

negative exponent.

Perform Operations with Exponents

Simplify x3 (x-

5)

Perform Operations with Exponents

Nanometers are often used to measure wavelengths. 1

nanometer= 0.000000001 meter. Write the decimal as a power of 10

A unit of measure called a micron equals 0.001 millimeter.

Write this number using a negative exponent.


3-4-C Scientific Notation

More than 425 million pounds of gold have been discovered in the world. If all this gold were in one place, It would form a cube seven stories on each side!

1. Write 425 million in standard form

425,000,000

2. Complete: 4.25 x _________________

100,000,000When you deal with very large numbers like 425,000,000, it can be

difficult to keep track of the zeros! You can express numbers such as this in

SCIENTIFIC NOTATION by writing the number as the product of a factor and a power of

10.

Words:A number is expressed in scientific notation when it is

written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10.

Symbols:a x 10n, where 1≤ a < 10 and n is an integer

Example:425,000,000 = 4.25 x 108

Express Large Numbers in Standard Form:2.16 x 105

2.16 x 100,000 = 216,000 (move the decimal point 5 places)

7.6 x 106

7,600,000 (move the decimal point 6 places)

3.201 x 104

32,010 (only move the decimal point 4 places)

FOCUS:

On moving the

decimal

rather than

adding the

zeros!

SMALL NUMBERS TOO!

Scientific notation can also

be used to express very

small numbers. Study the

pattern of products at the

right. Notice that

multiplying by a

NEGATIVE POWER of

10 moves the decimal point

to the LEFT the same

number of places as the

absolute value of the

exponent.

1.25 x 102 = 1251.25 x 101= 12.51.25 x 100= 1.251.25 x 10-1=0.1251.25 x 10-

2=0.01251.25 x 10-3= 0.00125EXPRESS SMALL NUMBERS

IN STANDARD FORM5.8 x 10-3 =

0.0058 (move the decimal 3 places left)4.7 x 10-5=

0.0000479 x 10-4=

0.0009

EXPRESS IN SCIENTIFIC NOTATION

1,457,0001.457 x 106

0.000636.3 x 10-4

35,0003.5 x 104

0.007227.22 x 10-3

The Atlantic Ocean has an area of 3.18 x 107 square miles. The Pacific Ocean has an area of 6.4 x 107 square miles. Which ocean has a greater area?

Since the exponents are the same and 3.18 < 6.4, the Pacific Ocean has a greater area.

Earth has an average radius of 6.38 x 103 kilometers. Mercury has an average radius of 2.44 x 103 kilometers. Which planet has the greater average radius?

Compare using <, >, or =4.13 x 10-2_____ 5.0 x 10-3

0.00701_____7.1 x 10-3

5.2 x 102_____ 5,000

Self-Assessment: Try pp. 187 # 1-12 on your own. Then, check answers with a partner.

chapter 3 rational numbers. 3-1-a explore: the number line let’s graph - on a number line 1. draw...

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