Download Chapter 3.notebook

Chapter 3.notebook
January 20, 2016
What is a
rational number?
Write 4 as a
5
decimal.
You must be specific
about three things!
0.8
State two
rational numbers between
­5.4 and ­5.5
Express A as a mixed fraction.
A
5
­5.41
6
5 8/10
­5.42
Write two equivalent fractions!
­8
9
­16/18
Oct 31­3:53 PM
What is a
rational number?
Write 4 as a
5
decimal.
State two
rational numbers between
­5.4 and ­5.5
You must be specific
about three things!
Express A as a mixed fraction.
A
5
6
Write two equivalent fractions!
­8
9
Oct 31­3:53 PM
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Chapter 3.notebook
January 20, 2016
Rational
Numbers
any # that can
be written as fraction
any number including decimal that repeats
→ any
#
that ends
Sep 10­10:59 AM
Express each fraction as a decimal, then sort as a repeating or terminating
decimal.
­5
9
6
27
27
33
­8
5
20
­10
18
12
Oct 31­5:25 PM
2
Chapter 3.notebook
The numerator is LARGER
than the denominator.
7
3
January 20, 2016
Improper vs. Mixed Fractions This is a Improper Fraction
Mixed Fraction Integer + Fraction
2 31
Oct 31­5:27 PM
Arrange the numbers from least to greatest.
Change the numbers to decimals!
­0.375, 0.555..., ­2.5, ­1.25, 0.7, 2.666...
Least...
­2.5
­1.25
­0.375
0.555...
0.7
2.666...
...Greatest
Oct 27­10:29 PM
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Chapter 3.notebook
January 20, 2016
Which rational number is larger??
(Decimals may be used on this side.)
(NO Decimals please!!.)
­13
16
­12
15
2
3
­0.8125
­0.8
3
4
8
12
9
12
16
24
18
24
Oct 31­5:43 PM
1) Identify wether the number is rational or non­rational
2
3
1.234567.....
1.66
­2.25
2)Express each fraction as a decimal
a) 4
5
b) 9
c) 3 6
11
­
3)Express decimal as a fraction mixed fraction
a) 0.1
b) 4.75
c) ­3.222
Feb 2­7:51 PM
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Chapter 3.notebook
January 20, 2016
Sept. 14
3.2 Adding Rational Numbers
Write each mixed number as an improper fraction:
1)
3
3
5
2)
-5
5
6
Copy this DOWN and Hand this IN
3) Put the fractions in order from least to greatest
,
­ 12 , ­ 45 , ­ 11
15
2
32
,1
20
Oct 29­9:43 AM
Adding Fractions
When adding fractions you need a COMMON DENOMINATOR:
­5
6
+
1)
8 8
=
2) ­8 + ­4
7
7
=
Oct 29­10:22 AM
5
Chapter 3.notebook
January 20, 2016
Determine the sum of each of the following
1) ( (=
­3 + ­3
7
7
2) a) 2.7 + 1.8 b) ­3.7 + 4.5 c) 2.7 + (­8.7)
Nov 1­9:01 PM
What happens if the denominators are different?
Find a Common Denominator
by determining the LCM.
L owest
C ommon
M ultiple
Oct 30­3:39 PM
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Chapter 3.notebook
January 20, 2016
Multiples
Find a common denominator:
4 + 8
5
3
+
=
5
3
5 x 1 = 5
5 x 2 = 10
5 x 3 = 15
5 x 4 = 20
LCM
3 x 1 = 3
3 x 2 = 6
3 x 3 = 9
3 x 4 = 12
3 x 5 = 15
Oct 30­3:43 PM
What about mixed numbers?
1
3
3
5
2 + 2
Step 1: Write each mixed number as an improper fraction.
+
Step 2: Find a common denominator, and then add numerators.
+
Oct 30­4:02 PM
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Chapter 3.notebook
January 20, 2016
Practice!
1)
1
2
2
3
1
4
(-1 ) +(-2 )
5 + (-3 )
7
8
2)
Oct 30­4:28 PM
1) Identify whether the number is rational or non­rational
i) 3
4
­3.286754....
­0.33
1.66
1.85
2)Express each fraction as a decimal(Show all work)
a) 1
5
b) 2 15
c) ­ 4 13
3)Express decimal as a mixed fraction in simplest form. (Show all work)
a) 0.4
c) 1.125
b) ­3.2
4) Determine each sum. (Show all work)
a)
6
­5
+
56
))
­2
5
b) 8
3
c) +
5
4
­3 27 + 2 14
Oct 18­2:06 PM
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Chapter 3.notebook
January 20, 2016
4) On December 18th, the temperature in Miramichi was ­21.6oC. By noon the next day, the temperature increased by 3.7 oC.
a) What was the temperature at noon on December 19 th?
b) On December 17th, the temperature was 2.1oC less than (colder than) that of December 18th. What was the temperature on the 17th?
Sep 12­10:18 PM
When subtracting Rational Numbers you must have a ... Ex)
This look similar to adding Rational Numbers
= Same Denominators
Oct 18­7:52 PM
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Chapter 3.notebook
January 20, 2016
You try ...
(Remember to write all solution in simplest form)
1) 2) 3) Oct 18­8:48 PM
When denominators are different you have to find a "common denominator"
LCM
How
By determining the LCM
Lowest Common Multiple
(of the denominators)
Oct 18­8:27 PM
10
Chapter 3.notebook
January 20, 2016
You try...
2) 1
2 5 ­ 5
1) 2
+ 2
3)
3
Oct 18­9:04 PM
STEP 1) Write each mixed number as an inproper fraction
STEP 2) Find common denominators and then subtract like before
STEP 3) Reduce all fractions Oct 18­8:56 PM
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Chapter 3.notebook
January 20, 2016
Your Turn
2)
1)
Oct 18­9:23 PM
Multiplying Rational Numbers
What rules do we use to multiply integers?
Indicate if the answer will be negative or positive. How do you know?
(­4) x 3 = negative
(­3) x (­6) = positive
2 x 8 = positive
Nov 1­6:01 PM
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Chapter 3.notebook
January 20, 2016
Copy down
When multiplying integers, we use the following rules:
(a negative #) x (a positive #) = (a negative #) a negative # x a negative # = a positive #
a positive # x a positive # = a positive #
So, when the signs are opposite, the product is negative
Copy down
and
when the signs are the same, the product is positive!
Nov 1­8:10 PM
Now, let's take a look at Fractions
.
What rules do we use to multiply fractions?
Evaluate the following expression.
How did you get your answer?
=
6 x 8
5 x 7
=
48
35
When multiplying fractions, we use this rule:
Multiply the numerator by the numerator
then
Multiply the denominator by the denominator
** Then, of course, REDUCE!! (if possible)
Nov 1­6:01 PM
13
Chapter 3.notebook
January 20, 2016
Try these out!
Use what you know about multiplying integers & fractions to evaluate the following expressions.
Don't forget to ALWAYS reduce
if possible!
(­1.5) x (­1.8)
Nov 1­6:12 PM
Jan 9­11:09 AM
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Chapter 3.notebook
January 20, 2016
Multiplying Rational Numbers in Fraction Form
Determine the product:
Look for common factors in the numerators and denominators.
11 and 44 have a common factor 11.
7 and 21 have a common factor 7.
Divide numerator and denominator by their common factors.
The signs are the same,
so the product is positive!
First, we simplify:
1
3
1
4
=
Our rule for multiplying fractions is:
numerator by numerator
denominator by denominator
So, our new expression, looks like this:
1 x 3 =
1 x 4
3
4
=
Nov 1­6:27 PM
Multiplying Rational Numbers in mixed number Form
Determine the product.
The signs are different,
so the product is negative!
Write the mixed numbers as improper fractions:
=
8 _ 7
( )
3 ( )
4
8 _ 7
( )
3 ( )
4
2
=
1
=
(2)(­7)
(3)(1)
=
­14
3
=
­4 23
Nov 1­6:31 PM
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Chapter 3.notebook
January 20, 2016
9
• Every non­zero fraction has a reciprocal.
• Fractions with a denominator of "0" are undefined.
• To find the reciprocal of a fraction, you simply flip the fraction !!
10
16
Chapter 3.notebook
January 20, 2016
Express each division question as a multiplication question !!!!
11
Dividend
Quotient
Divisor
12
17
Chapter 3.notebook
January 20, 2016
Express division as multipication by multiplying the dividend by the reciprocal of the divisor !!
=
13
math
#1
14
18
Chapter 3.notebook
January 20, 2016
#2
15
#3
16
19
Chapter 3.notebook
January 20, 2016
Jan 9­11:12 AM
Dividend Missing
(__) ÷ 4 = 3
Think: Division is the inverse of Multiplication.
What # goes in the blank?
Any division statement can be written as an equivalent multiplication.
OR
(__) = 3 x 4
To Solve for Missing Dividend
take Divisor X Quotient
Now with Rational #s
You Try
A)
5
3
=
( )
( )
11
7
B)
____ ÷ 12.6 = 4.2 Nov 14­5:40 PM
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Chapter 3.notebook
January 20, 2016
Divisor Missing
Decimals
Think: Quotient is negative thus the BLANK must be what sign? ________
15 ÷ (__) = ­5
What # goes in the blank?
To solve for missing Divisor
take Dividend ÷ Quotient
OR
15 = ­5 x (__)
Erase to see
15 = ­5 x (__)
­5
­5
or 15 x 1 = (__)
­5
Multiply by the reciprocal
reciprocal of ­5 is 1
­5
­3 = (__)
You Try
1) ­2.5 ÷ ___ = 5
2) 1.16 ÷ ___ = 0.2
What about fractions???
Nov 14­5:40 PM
Divisor Missing &Fractions 18
­6
( )
( )
=
49
7
( ) ( ) ÷ 18
49
= ­6
7
The Quotient is Positive
Thus the divsor is _______
Divsor = Dividend ÷Quotient
Use the strategy of multilpying by the reciprocal
­6
49
( ) = ( )
7 x 18
Simplify
1
­6
49
( ) = ( )
7 x 18
7
1
3
­7
( ) =( )
3
Nov 14­7:01 PM
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Chapter 3.notebook
January 20, 2016
Your Turn
­3
15
( )
=
( )
26
2
A) b)
­12
5
( )
=
( )
21
8
Nov 14­7:54 PM
Remember from
operations
"BEDMAS".....order of
In the order that they appear
Recall
Evaluate the following
1) (­5) ­ 3[18 ÷ (­3)]2
Oct 19­4:10 PM
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Chapter 3.notebook
January 20, 2016
Evaluate the following:
It is no difference with decimals....follow BEDMAS
With decimals you may need to round your final answers
1) (­1.3) + 0.8 ÷ (­0.2) x 5
2) (­3.6) ­ 1.7 ÷ [0.6 ­ (­0.8)] 2 = (­1.3) + (­4) x 5
= (­3.6) ­ 1.7 ÷ [1.4]2 = (­3.6) ­ 1.7 ÷ 1.96 = (­1.3) + ­20
= ­21.3
(­3.6) ­ 0.867346938
=(­3.6) ­ 0.867346938
this number does not terminate
=­4.467346939
Oct 19­9:36 PM
1) 3.3 ­ [(­5) +1]3
2) (­2.7) x (3.4 ­ 8.5) ­ 6.1 ÷ 7
Oct 19­9:54 PM
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Chapter 3.notebook
=
=
=
January 20, 2016
erase
(­0.8) + ­4 x 1.5
(­0.8) + ­6
­6.8
Nov 21­3:47 PM
Remember fractions are just numbers
erase to see solutions
1)
Step 1) BRACKETS
­ find common denominator then add the #s in brackets
Common Denominator = 6
=
Reduce sum if possible
=
Step 2) Multiply next
=
Step 3) Divide next
=
­ multiply by the reciprocal of the divisor.
= x
reduce at this point to work with smaller fractions
=
Step 4) Subtract.....find common denominator
=
12
47
=
Oct 19­10:02 PM
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Chapter 3.notebook
January 20, 2016
Erase to reveal answer
1)
Remember to switch mixed to improper fractions
Make common denominators inside brackets
Complete Brackets
Multiply
Divide
Oct 19­10:38 PM
4.
Please erase to reveal answer
AS
BEDM
4
12
=
[ 112 [
=
=
(­ 14 (
x 12
=
1
4
=
1
=
( ­ 8
(
1
=
­ 32
( (
4
=
31 4
Nov 21­4:06 PM
25
Chapter 3.notebook
January 20, 2016
Page 144 & 145
­If the question deals with fractions you must work with fractions (no calculator)
­As soon as you see a decimal you can use a calculator #2 (without calculator) # 3(c,d)
#5 (a,c)
#7(a,b,c) (without calculator) #10(b,c) (without calculator) #14 (b, d)
#18(ac)
#19(b,d)
#21 (without calculator) #23a,c,d,g
26