Chapter 3 Rational Numbers. 3-1-A Explore: The Number Line Let’s graph - on a number line 1. Draw...
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Transcript of 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
Self-Assessment: Try pg. 131 # 1-12 on your own. Then, check answers with a partner.
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
Self-Assessment: Try pg. 136 # 1-7 on your own. Then, check answers with a partner.
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
Self-Assessment: Try pg. 148 # 1-10 on your own. Then, check answers with a partner.
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!
Self-Assessment: Try pg. 154 # 1-9 on your own. Then, check answers with a partner.
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.
Self-Assessment: Try pg. 179 # 1-10 on your own. Then, check answers with a partner.
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.
Self-Assessment: Try pg. 183 # 1-13 on your own. Then, check answers with a partner.
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.