3-1-A Explore: The Number Line Previously, you have graphed
integers and positive fractions on a number line. Today, you will
graph negative fractions. 0 - - -
Slide 3
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 identify 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. When
signaled, compare your work with your partners. Are there any
differences?
Slide 4
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. YearWinner Speed
(mph) 1999J. Gordon148.295 2000D. Jarrett155.669 2001M.
Waltrip161.783 2002W. Burton142.971 2003M. Waltrip133.870 2004 D.
Earnhardt Jr. 156.345 2005J. Gordon135.173 2006J. Johnson142.667
2007K. Harvick149.335 2008R. Newman152.672
Slide 5
Look at the following example: 7/20 We can easily change this
to have a denominator of 100 by multiplying the numerator and
denominator by 5. This would make the fraction 35/100 or 0.35. Now
you try! 5 Think: 75/100so5.75 3/25 Think: 12/100so0.12 -6 Think:
50/100so-6.5 Mental Math! Converting Fractions to Decimals Because
our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals. If
the denominator of a fraction is a power or multiple of ten, then
you can use place value to write the fraction as a decimal.
Slide 6
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 Think: The top number goes
in the house.
Slide 7
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.
Slide 8
FishAmount Guppy0.25 Angelfish0.4 Goldfish0.15 Molly0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled,
compare your work with your partners. Are there any
differences?
Slide 9
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?
Be ready to explain how you found your answer. 3.Describe two
different methods you could use to compare the batting
averages.
Slide 10
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 2/3 -1.44 -3-1 2 1
Slide 11
What is the least common denominator? What does that make your
numerators? Be careful! Negative numbers may look bigger because
they have a larger absolute value. However, the larger the
negative, the smaller the numbers actual value! You wont 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!
Slide 12
In Mr. Reeds math class, 20% of the students own Sperry shoes.
In Mrs. Crowes 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. Reeds class own Sperry shoes. Now you
try! 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?
Slide 13
Remember to line up the decimal points and compare using place
value! 3.44 3.1415926 3.14 3.4444444444 Self-Assessment: Try pg.
136 # 1-7 on your own. When signaled, compare your work with your
partners. Are there any differences?
Slide 14
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 Running3 High Top2
Slide 15
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 7 th grade, you
should already know fraction addition & subtraction rules! But
your CHALLENGE is to complete some of the problems without those
rules
Slide 16
Add and Subtract Like Fractions To add or subtract like
fractions, add or subtract the numerators and write the result over
the denominator. Key Concepts Review
Slide 17
Add and Subtract Unlike Fractions To 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
Slide 18
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 down With 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!
Slide 19
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!
Slide 20
Practice adding and subtracting with fraction tiles.
Slide 21
Questions Answers Practice Without Tiles!
Slide 22
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled,
compare your work with your partners. Are there any
differences?
Slide 23
3-2-D Add & Subtract Mixed Numbers BabyBirth Weight
Adelaide Stephen Micah Nora 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.
Slide 24
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.
Slide 25
Many times, it is not possible to subtract the whole numbers
and fractions separately. In this case, two different strategies
could be used: 1.Convert mixed numbers to improper fractions OR
2.Borrow from the whole number and add 1 to the fraction. IMPROPER
FRACTION: Has a numerator that is greater than or equal to the
denominator.
Slide 26
Real World Problems! Self-Assessment: Try pg. 154 # 1-9 on your
own. When signaled, compare your work with your partners. Are there
any differences?
Slide 27
Fraction Discovery #3 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!
Slide 28
3-3-BMultiply Fractions For the problem, create a sketch or
model to solve. Represent these two situations with equations. Are
the equations the same or different? Two-thirds of the students
chose pizza at lunch. One-half of those students chose pepperoni
pizza.
Slide 29
Key Concepts 1 2 15 When multiplying with mixed numbers, you
MUST change the mixed numbers to improper fractions BEFORE you
multiply.
Slide 30
Evaluate each verbal expression: a)One-half of five-eighths
b)Four-sevenths of two-thirds c)Nine-tenths of one-fourth
d)One-third of eleven-sixteenths
Slide 31
Fraction Discovery #4 With a partner, complete Fraction
Discover #4 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!
Slide 32
3-3-D Divide Fractions KEY CONCEPT: To divide a fraction,
multiply by its multiplicative inverse, or reciprocal. PAY
ATTENTION ! The divisor (or second fraction) is the ONLY fraction
that is flipped during this process. DO NOT FLIP THE FIRST
FRACTION.
Slide 33
Practice Dividing by Fractions Show your work in your notes.
Simplify when necessary.
Slide 34
Practice Dividing by Mixed Numbers To divide by a mixed number,
first rename it as an improper fraction. Self-Assessment: Try pg.
170 # 1-10 on your own. When signaled, compare your work with your
partners. Are there any differences?
Slide 35
3-4-A Multiply & Divide Monomials For each increase on the
Richter scale, an earthquakes 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 410
x 1 = 1010 1 510 x 10 = 10010 1 x 10 1 = 10 2 610 x 100 = 1,00010 1
x 10 2 = 10 3 710 x 1,000 = 10, 00010 1 x 10 3 = 10 4 810 x 10,000
= 100,00010 1 x 10 4 = 10 5
Slide 36
REMEMBER: Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE. 2 3 x 2 4 =(2 x 2 x 2)x(2 x
2 x 2 x 2)=2 7 PRODUCT OF POWERS Words : To multiply powers with
the same base, add their exponents Symbols : a m x a n = a m+n
Example : 3 2 x 3 4 = 3 2+4 = 3 6
Slide 37
Practice Multiplying Powers! 1.7 3 x 7 1 2.5 3 x 5 4 3.(0.5) 2
x (0.5) 9 4.8 x 8 5 Common Mistake: When multiplying powers, do not
multiply (evaluate) the bases that are the same! Example: 3 3 x 3 5
9 8 3 3 x 3 5 = 3 8 MONOMIAL A 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.x 5 (x 2 ) 2.(-4n 3
)(6n 2 ) 3.-3m(-8m 4 ) 4.5 2 x 2 y 4 (5 3 xy 4 ) STUCK? Remember
that the coefficients are multiplied!
Slide 38
QUOTIENT OF POWERS Words: To divide powers with the same base,
subtract their exponents. Symbols: a m a n = a m-n Example: 3 4 3 2
= 3 4-2 = 3 2 If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS using
Slide 39
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) 199910 3 200810 9 The number of fish in a
school of fish is 4 3. If the number of fish in the school
increased by 4 2 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. When signaled, compare your work with your
partners. Are there any differences?
Slide 40
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. Using what you
observed in the table, determine how 3 -1 should be defined.
PowerValue 2626 64 2525 32 2424 16 2323 8 2 4 2121 2 2020 2 -1 Take
a look at the table:
Slide 41
PRACTICE! Write each expression using a positive exponent. 6 -2
x -5 5 -6 t -4
Slide 42
When given a fraction with a positive exponent or square, you
can rewrite it using a negative exponent.
Slide 43
Perform Operations with Exponents
Slide 44
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. When signaled, compare your work with your partners. Are
there any differences?
Slide 45
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 =
425,000,000 100,000,000 When 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.
Slide 46
Check it out! Express the following numbers in standard form:
2.16 x 10 5 2.16 x 100,000 2.16 _ _ _ 2.16 0 0 0 216,000 (move the
decimal point 5 places) You try! 7.6 x 10 6 7,600,000 (move the
decimal point 6 places) 3.201 x 10 4 32,010 (move the decimal point
4 places ) WATCH OUT! Common mistake: Make sure you are counting
decimal places rather than just adding zeroes.
Slide 47
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. Practice: Express each number in standard
form: 5.8 x 10 -3 0.0058 (move the decimal 3 places left) 4.7 x 10
-5 0.000047 9 x 10 -4 0.0009 Practice: Express each number in
scientific notation. 1,457,000 1.457 x 10 6 0.00063 6.3 x 10 -4
35,000 3.5 x 10 4 0.00722 7.22 x 10 -3
Slide 48
The Atlantic Ocean has an area of 3.18 x 10 7 square miles. The
Pacific Ocean has an area of 6.4 x 10 7 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 10 3 kilometers. Mercury has an average radius of
2.44 x 10 3 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 10 2 _____ 5,000 Self-Assessment: Try
pg. 187 # 1-12 on your own. When signaled, compare your work with
your partners. Are there any differences?