Upload
voanh
View
214
Download
1
Embed Size (px)
Citation preview
Slide 1 / 306 Slide 2 / 306
7th Grade Math
Review of 6th Grade
www.njctl.org
2015-01-14
Slide 3 / 306
Table of Contents
Fractions
Number System
Decimal Computation
Click on the topic to go to that section
Expressions
Equations and Inequalities
Ratios and Proportions
Geometry
Statistics
Slide 4 / 306
Fractions
Return to Table of Contents
Slide 5 / 306
List what you remember about fractions .
Hin
t
Slide 6 / 306
We can use prime factorization to find the greatest common factor (GCF).
1. Factor the given numbers into primes.
2. Circle the factors that are common.
3. Multiply the common factors together to find the greatest common factor.
Greatest Common Factor
Slide 7 / 306
1 Find the GCF of 18 and 44.
Pul
lP
ull
Slide 8 / 306
2 Find the GCF of 72 and 75.
Pul
lP
ull
Slide 9 / 306
3 Find the GCF of 52 and 78.P
ull
Pul
l
Slide 10 / 306
A multiple of a whole number is the product of the number and any nonzero whole number.
A multiple that is shared by two or more numbers is a common multiple.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ...
Multiples of 14: 14, 28, 42, 56, 70, 84,...
The least of the common multiples of two or more numbers is the least common multiple (LCM) . The LCM of 6 and 14 is 42.
Slide 11 / 306
There are 2 ways to find the LCM:
1. List the multiples of each number until you find the first one they have in common.
2. Write the prime factorization of each number. Multiply all factors together. Use common factors only once (in other words, use the highest exponent for a repeated factor).
Slide 12 / 306
EXAMPLE: 6 and 8
Multiples of 6: 6, 12, 18, 24, 30Multiples of 8: 8, 16, 24
LCM = 24
Prime Factorization:
2 3 2 4
2 2 2
2 3 23 LCM: 23 3 = 8 3 = 24
6 8
Slide 13 / 306
4 Find the least common multiple of 10 and 14.
A 2
B 20C 70D 140
Pul
lP
ull
Slide 14 / 306
5 Find the least common multiple of 6 and 14.
A 10B 30C 42D 150
Pul
lP
ull
Slide 15 / 306
6 Find the LCM of 24 and 60.P
ull
Pul
l
Slide 16 / 306
Which is easier to solve?
28 + 42 7(4 + 6)
Do they both have the same answer?
You can rewrite an expression by removing a common factor. This is called the Distributive Property.
Slide 17 / 306
The Distributive Property allows you to:
1. Rewrite an expression by factoring out the GCF.
2. Rewrite an expression by multiplying by the GCF.
EXAMPLE
Rewrite by factoring out the GCF:
45 + 80 28 + 635(9 + 16) 7(4 + 9)
Rewrite by multiplying by the GCF:3(12 + 7) 8(4 + 13) 36 + 21 32 + 101
Slide 18 / 306
7 In order to rewrite this expression using the Distributive Property, what GCF will you factor?
56 + 72
Pul
lP
ull
Slide 19 / 306
8 In order to rewrite this expression using the Distributive Property, what GCF will you factor?
48 + 84
Pul
lP
ull
Slide 20 / 306
9 Use the distributive property to rewrite this expression:
36 + 84
A 3(12 + 28)B 4(9 + 21)C 2(18 + 42)D 12(3 + 7)
Pul
lP
ull
Slide 21 / 306
10 Use the distributive property to rewrite this expression:
88 + 32
A 4(22 + 8)
B 8(11 + 4)
C 2(44 + 16)
D 11(8 + 3)
Pul
lP
ull
Slide 22 / 306
Adding Fractions...
1. Rewrite the fractions with a common denominator.2. Add the numerators.3. Leave the denominator the same.4. Simplify your answer.
Adding Mixed Numbers...
1. Add the fractions (see above steps).2. Add the whole numbers.3. Simplify your answer. (you may need to rename the fraction)
Link Backto List
Slide 23 / 306
11 3 10 2 10
+
Pul
lP
ull
Slide 24 / 306
12 5 8 1 8
+
Pul
lP
ull
Slide 25 / 306
13 Find the sum.
5 3 10
+ 7 5 10
Pul
lP
ull
Slide 26 / 306
14 Is the equation below true or false?
True False
1 8 12
+ 1 5 12
3 1 12
Pul
lP
ull
Don't forget to regroup to the whole number if you
end up with the numerator larger than the
denominator.
ClickFor reminder
Slide 27 / 306
A quick way to find LCDs...
List multiples of the larger denominator and stop when you find a common multiple for the smaller denominator.
Ex: and
Multiples of 5: 5, 10, 15
Ex: and
Multiples of 9: 9, 18, 27, 36
2 5
1 3
3 4
2 9
Slide 28 / 306
Common DenominatorsAnother way to find a common denominator is to multiply the two denominators together.
Ex: and 3 x 5 = 15
2 5
1 3
1 3
x 5
x 5 5 15
2 5
6 15
x 3
x 3
Slide 29 / 306
15 2 5 1 3
+
Pul
lP
ull
Slide 30 / 306
16 3 10 2 5
+
Pul
lP
ull
Slide 31 / 306
17 5 8 3 5
+
Pul
lP
ull
Slide 32 / 306
18
A
5 3 4
+ 2 7 12
=
7 1612
B 8 4 12
C
7 5 8
D
8 1 3
Pul
lP
ull
Slide 33 / 306
19
A
2 3 8
+ 5 5 12
=
7 1924
7 8 20
B
7 8 12
C
8 7 12
D
Pul
lP
ull
Slide 34 / 306
20
5 2 10
5 5 12
A
3 1 4
+ 2 1 6
=
B
5 1 2
C
6 5 12
D
Pul
lP
ull
Slide 35 / 306
Subtracting Fractions...
1. Rewrite the fractions with a common denominator.2. Subtract the numerators.3. Leave the denominator the same.4. Simplify your answer.
Subtracting Mixed Numbers...
1. Subtract the fractions (see above steps..). (you may need to borrow from the whole number)2. Subtract the whole numbers.3. Simplify your answer. (you may need to simplify the fraction)
Link Backto List
Slide 36 / 306
21 7 8 4 8
Pul
lP
ull
Slide 37 / 306
22 6 7
4 5 P
ull
Pul
l
Slide 38 / 306
23 2 3
1 5 P
ull
Pul
l
Slide 39 / 306
24 Is the equation below true or false?
True False
4 5 9
3 9
3 2 9
Pul
lP
ull
Slide 40 / 306
25 Is the equation below true or false?
True False
2 7 9
1 9
1 2 3
1
Pul
lP
ull
Slide 41 / 306
26 Find the difference.
4 7 8 2 3
8
Pul
lP
ull
Slide 42 / 306
27 6 7 3 5
Pul
lP
ull
Slide 43 / 306
A Regrouping Review
When you regroup for subtracting, you take one of your whole numbers and change it into a fraction with the same denominator as the fraction in the mixed number.
3 3 5
= 2 5 5
3 5
= 2 8 5
Don't forget to add the fraction you regrouped from your whole number to the fraction already given in the problem.
Slide 44 / 306
5 1 4
3 7 12
5 3 12
3 7 12
4 1212
3 7 12
3 12
4 1512
3 7 12
1 8 12
1 2 3
Slide 45 / 306
28 Do you need to regroup in order to complete this problem?
Yes or No
3 1 2
1 4
Pul
lP
ull
Slide 46 / 306
29Do you need to regroup in order to complete this problem?
Yes or No
7 2 3
3 46
Pul
lP
ull
Slide 47 / 306
30 What does 17 become when regrouping? 3 10 P
ull
Pul
l
Slide 48 / 306
31 What does 21 become when regrouping? 5 8 P
ull
Pul
l
Slide 49 / 306
32
2 1 12
A
1 2224
B
4 1 6 2 1
4=
1 1112
C
1 1 12
D
Pul
lP
ull
Slide 50 / 306
33
A
3 1321
B
6 2 7 3 2
3=
3 8 21 2 2
3C
2 1321
D
Pul
lP
ull
Slide 51 / 306
34
A
6 1 6
B
15 8 1012
=
7 5 6 7 1
6C
6 2 12
D
Pul
lP
ull
Slide 52 / 306
Multiplying Fractions...
1. Multiply the numerators.2. Multiply the denominators.3. Simplify your answer.
Multiplying Mixed Numbers...
1. Rewrite the Mixed Number(s) as an improper fraction. (write whole numbers / 1)2. Multiply the fractions.3. Simplify your answer.
Link Backto List
Slide 53 / 306
35
1 5
x 2 3
= Pul
lP
ull
Slide 54 / 306
36
2 3
x 3 7
= Pul
lP
ull
Slide 55 / 306
37
= 4 9
3 8( )
Pul
lP
ull
Slide 56 / 306
38
True
False
x 1 2
=5 5 1
x 1 2
Pul
lP
ull
Slide 57 / 306
39
A
x 4 73
B
C
3 5 7
D
1221
12 7
1 5 7
Pul
lP
ull
Slide 58 / 306
40
True
False
x =2 1 4 3 1
8 6 3 8
Pul
lP
ull
Slide 59 / 306
41
15 1 4
A
18 1 8
B
20 3 8
C
19 1 8
D
5 8( )5 2
5(3 ) Pul
lP
ull
Slide 60 / 306
Dividing Fractions...
1. Leave the first fraction the same.2. Multiply the first fraction by the reciprocal of the second fraction.3. Simplify your answer.
Dividing Mixed Numbers...
1. Rewrite the Mixed Number(s) as an improper fraction(s). (write whole numbers / 1)2. Divide the fractions.3. Simplify your answer.
Slide 61 / 306
To divide fractions, multiply the first fraction by the reciprocal of the second fraction. Make sure you simplify your answer!
Some people use the saying "Keep Change Flip" to help them remember the process.
3 5
x 8 7
= 3 x 8 5 x 7
= 2435
3 5
7 8
=
1 5
x 2 1
= 1 x 2 5 x 1
= 2 5
1 5
1 2
=
Slide 62 / 306
42
True
False
8 10
= 5 4
x 8 10
4 5 P
ull
Pul
l
Slide 63 / 306
43
True
False
2 7
= 3 4 2 7
8
Pul
lP
ull
Slide 64 / 306
44
1A
3940
B
C
8 10
= 4 5
4042
Pul
lP
ull
Slide 65 / 306
45
Pul
lP
ull
Slide 66 / 306
To divide fractions with whole or mixed numbers, write the numbers as an improper fractions. Then divide the two fractions by using the rule (multiply the first fraction by the reciprocal of the second).
Make sure you write your answer in simplest form.
5 3
x 2 7
= 1021
2 3
=1 1 2
3 5 3
7 2
=
6 1
x 2 3
= 12 3
=6 1 2
1 6 1
3 2
= = 4
Slide 67 / 306
46
= 1 2 2 2
31
Pul
lP
ull
Slide 68 / 306
47
= 1 2 2 2
31
Pul
lP
ull
Slide 69 / 306
48
= 1 2 52
Pul
lP
ull
Slide 70 / 306
Decimal Computation
Return to Table of Contents
Slide 71 / 306
List what you remember about decimals.
Slide 72 / 306Some division terms to remember....
· The number to be divided into is known as the dividend
· The number which divides the other number is known as the divisor
· The answer to a division problem is called the quotient
divisor 5 20 dividend
4 quotient
20 ÷ 5 = 4
20__5
= 4
Slide 73 / 306
When we are dividing, we are breaking apart into equal groups
EXAMPLE 1
Find 132 3
Step 1 : Can 3 go into 1, no so can 3 go into 13, yes
4
- 12 1
3 x 4 = 1213 - 12 = 1Compare 1 < 3
3 132
3 x 4 = 1212 - 12 = 0Compare 0 < 3
- 12 0
2
Step 2 : Bring down the 2. Can 3 go into 12, yes
4
Slide 74 / 306
EXAMPLE 2(change pages to see each step)
Step 1: Can 15 go into 3, no so can 15 go into 35, yes
2
-30 5
15 x 2 = 3035 - 30 = 5Compare 5 < 15
15 357
Slide 75 / 306
2
-30 5
15 35715 x 3 = 4557 - 45 =12Compare 12 < 15
7 - 45 12
Step 2 : Bring down the 7. Can 25 go into 207, yes
3
EXAMPLE 2(change pages to see each step)
Slide 76 / 306
2
-30 5
15 357.0
7 - 45 120 - 120 0
3
Step 3: You need to add a decimal and a zero since the division is not complete. Bring the zero down and continue the long division.
15 x 8 = 120120 - 120 = 0Compare 0 < 15
.8
EXAMPLE 2(change pages to see each step)
Slide 77 / 306
49 Compute.
Pul
lP
ull
Slide 78 / 306
50 Compute.
Pul
lP
ull
Slide 79 / 306
51 Compute.
Pul
lP
ull
Slide 80 / 306
If you know how to add whole numbers then you can add decimals. Just follow these few steps.
Step 1: Put the numbers in a vertical column, aligning the decimal points.
Step 2: Add each column of digits, starting on the right and working to the left.
Step 3: Place the decimal point in the answer directly below the decimal points that you lined up in Step 1.
Slide 81 / 306
C
52 Add the following:
0.6 + 0.55 =
A 6.1
B 0.115click
C 1.15
D 0.16
Slide 82 / 306
53 Find the sum
1.025 + 0.03 + 14.0001 =
15.0551click
Slide 83 / 306
54 Find the sum:
5 + 100.145 + 57.8962 + 2.312 = 165.3532click
Slide 84 / 306
What do we do if there aren't enough decimal places when we subtract?
4.3 - 2.05
Don't forget...Line Them Up!
4.32.05
What goes here?
4.3 02.05
2.25
2 1
Slide 85 / 306
55
5 - 0.238 =4.762click
Slide 86 / 306
56
12.809 - 4 =8.809click
Slide 87 / 306
57
4.1 - 0.094 = 4.006click
Slide 88 / 30658
17 - 13.008 = 3.992click
Slide 89 / 306
If you know how to multiply whole numbers then you can multiply decimals. Just follow these few steps.
Step 1: Ignore the decimal points.
Step 2: Multiply the numbers using the same rules as whole numbers.
Step 3: Count the total number of digits to the right of the decimal points in both numbers. Put that many digits to the right of the decimal point in your answer.
Slide 90 / 306
23.2x 4.04
928
92800 0000
93.728
}
There are a total of three digits to the right of the decimal points.
There must be three digits to the right of the decimal point in the answer.
EXAMPLE
Slide 91 / 306
59 Multiply 0.42 x 0.032 0.1344click
Slide 92 / 306
60 Multiply 3.452 x 2.1 7.2492click
Slide 93 / 306
4.7383661 Multiply 53.24 x 0.089
Slide 94 / 306
DividendDivisor
Step 1: Change the divisor to a whole number by multiplying by a power of 10.
Step 2: Multiply the dividend by the same power of 10.
Step 3: Use long division.
Step 4: Bring the decimal point up into the quotient.
Divide by Decimals
Quotient
Slide 95 / 306
15.6 6.24
Multiply by 10, so that 15.6 becomes 1566.24 must also be multiplied by 10
156 62.4
.234 23.4
Multiply by 1000, so that .234 becomes 23423.4 must also be multiplied by 1000
234 23400
Try rewriting these problems so you are ready to divide!
Slide 96 / 306
62 Divide
0.78 ÷ 0.02 = 39click
Slide 97 / 306
63
10 divided by 0.25 = 40click
Slide 98 / 306
64
12.03 ÷ 0.04 = 300.75click
Slide 99 / 306
There are two types of decimals - terminating and repeating.
A terminating decimal is a decimal that ends.All of the examples we have completed so far are terminating.
A repeating decimal is a decimal that continues forever with one or more digits repeating in a pattern.
To denote a repeating decimal, a line is drawn above the numbers that repeat. However, with a calculator, the last digit is rounded.
Slide 100 / 306
Examples:
6600 2342 2200 14200 13200 10000 8800 12000 11000 10000 8800 12000 11000
63 48 45 39 36 32 27 51 45 60 54 6
Slide 101 / 306
65
click
Slide 102 / 306
Slide 103 / 306
67
click
Slide 104 / 306
Statistics
Return to Table of Contents
Slide 105 / 306
List what you remember about statistics.
Slide 106 / 306
Measures of Center Vocabulary:
· Mean - The sum of the data values divided by the number of items; average
· Median - The middle data value when the values are written in numerical order
· Mode - The data value that occurs the most often
Slide 107 / 306 Slide 108 / 306
Slide 109 / 306 Slide 110 / 306
Slide 111 / 306Measures of Variation Vocabulary:
Minimum - The smallest value in a set of data
Maximum - The largest value in a set of data
Range - The difference between the greatest data value and the least data value
Quartiles - are the values that divide the data in four equal parts.
Lower (1st) Quartile (Q1) - The median of the lower half of the data
Upper (3rd) Quartile (Q3) - The median of the upper half of the data.
Interquartile Range - The difference of the upper quartile and the lower quartile. (Q3 - Q1)
Outliers - Numbers that are significantly larger or much smaller than the rest of the data
Slide 112 / 306
Slide 113 / 306 Slide 114 / 306Quartiles
There are three quartiles for every set of data.
LowerHalf
UpperHalf
10, 14, 17, 18, 21, 25, 27, 28
Q1 Q2 Q3
The lower quartile (Q1) is the median of the lower half of the data which is 15.5.
The upper quartile (Q3) is the median of the upper half of the data which is 26.
The second quartile (Q2) is the median of the entire data set which is 19.5.
The interquartile range is Q3 - Q1 which is equal to 10.5.
Slide 121 / 306 Slide 122 / 306
Slide 123 / 306 Slide 124 / 306
The mean absolute deviation of a set of data is the average distance between each data value and the mean.
Steps
1. Find the mean.2. Find the distance between each data value and the mean. That is, find the absolute value of the difference between each data value and the mean.3. Find the average of those differences.
*HINT: Use a table to help you organize your data.
Slide 125 / 306
Let's continue with the "Phone Usage" example.Step 1 - We already found the mean of the data is 56.Step 2 - Now create a table to find the differences.
48 8
52 4
54 2
55 1
58 2
59 3
60 4
62 6
Data Value
Absolute Value of the Difference|Data Value - Mean|
Slide 126 / 306
Step 3 - Find the average of those differences.
8 + 4 + 2 + 1 + 2 + 3 + 4 + 6 = 3.75 8
The mean absolute deviation is 3.75.
The average distance between each data value and the mean is 3.75 minutes.
This means that the number of minutes each friend talks on the phone varies 3.75 minutes from the mean of 56 minutes.
Slide 127 / 306 Slide 128 / 306
Slide 129 / 306
FREQUENCY
8
6
4
2
030- 40- 50- 60- 70- 80- 90-39 49 59 69 79 89 99
GRADE
Grade Tally Frequency30-39 I 140-49 050-59 060-69 I 170-79 IIII 480-89 IIII III 890-99 III 3
TEST SCORES95 85 9377 97 7184 63 8739 88 8971 79 8382 85
SAMPLES:
Data
TEST SCORES87 53 9585 89 5986 82 8740 90 7248 68 5764 85
FREQUENCY
8
6
4
2
040- 50- 60- 70- 80- 90-49 59 69 79 89 99
GRADE
Grade Tally Frequency40-49 II 250-59 III 360-69 II 270-79 I 180-89 IIII II 790-99 II 2
FrequencyTable
Histogram
Slide 130 / 306
A box and whisker plot is a data display that organizes data into four groups
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10 80 90 100 110 120 130 140 150
The median divides the data into an upper and lower half
The median of the lower half is the lower quartile.
The median of the upper half is the upper quartile.
The least data value is the minimum.
The greatest data value is the maximum.
Slide 131 / 306
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10 80 90 100 110 120 130 140 150
median
25% 25%25%25%
The entire box represents 50% of the data. 25% of the data lie in the box on each side of the median
Each whisker represents 25% of the data
Slide 132 / 306
Slide 133 / 306 Slide 134 / 306
Slide 135 / 306 Slide 136 / 306
Slide 137 / 306 Slide 138 / 306
A dot plot (line plot) is a number line with marks that show the frequency of data. A dot plot helps you see where data cluster.
Example:
35 40 45 5030
xxxxxx
xxx
xxx
xxxx
xx
xxx
xxxxx
Test Scores
The count of "x" marks above each score represents the number of students who received that score.
Slide 139 / 306 Slide 140 / 306
Slide 141 / 306 Slide 142 / 306
Slide 143 / 306
Number System
Return to Table of Contents
Slide 144 / 306
List what you remember about the number system.
Slide 145 / 306
{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...}
Definition of Integer:
The set of natural numbers, their opposites, and zero.
Define Integer
Examples of Integers:
Slide 146 / 306
-1 0-2-3-4-5 1 2 3 4 5
Integers on the number line
NegativeIntegers
PositiveIntegers
Numbers to the left of zero are less than zero
Numbers to the right of zero are greater than zero
Zero is neitherpositive or negative
`
Zero
Slide 147 / 306 Slide 148 / 306
Slide 149 / 306 Slide 150 / 306
Slide 151 / 306
Absolute Value of Integers
The absolute value is the distance a number is from zero on the number line, regardless of direction.
Distance and absolute value are always non-negative (positive or zero).
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
What is the distance from 0 to 5?
Slide 152 / 306
Slide 153 / 306 Slide 154 / 306
Slide 155 / 306 Slide 156 / 306
To compare integers, plot points on the number line.
The numbers farther to the right are greater.
The numbers farther to the left are smaller.
Use the Number Line
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Slide 157 / 306 Slide 158 / 306
Slide 159 / 306 Slide 160 / 306
Slide 161 / 306 Slide 162 / 306
Comparing Rational NumbersSometimes you will be given fractions and decimals that you need to compare.
It is usually easier to convert all fractions to decimals in order to compare them on a number line.
To convert a fraction to a decimal, divide the numerator by the denominator.
4 3.00-28 020 -20 0
0.75
Slide 163 / 306 Slide 164 / 306
Slide 165 / 306 Slide 166 / 306
The coordinate plane is divided into four sections called quadrants.
The quadrants are formed by two intersecting number lines called axes.
The horizontal line is the x-axis.
The vertical line is the y-axis.
The point of intersection is called the origin. (0,0)
0 x - axisy - axis
origin
(+, -)
(-, +)
(-, -)
(+, +)
Slide 167 / 306
To graph an ordered pair, such as (3,2):· start at the origin (0,0)· move left or right on the x-axis depending on the first number· then move up or down from there depending on the second
number · plot the point
(3,2)
Slide 168 / 306
Slide 169 / 306 Slide 170 / 306
Slide 171 / 306 Slide 172 / 306
Slide 173 / 306 Slide 174 / 306
Expressions
Return to Table of Contents
Slide 175 / 306
List what you remember about expressions.
Slide 176 / 306
Exponents
Exponents, or Powers, are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition.
These are all equivalent:
24 Exponential Form2∙2∙2∙2 Expanded Form16 Standard Form
In this example 2 is raised to the 4th power. That means that 2 is multiplied by itself 4 times.
Slide 177 / 306
Powers of IntegersBases and Exponents
When "raising a number to a power",
The number we start with is called the base, the number we raise it to is called the exponent.
The entire expression is called a power.
You read this as "two to the fourth power."
24
Slide 178 / 306
Slide 179 / 306 Slide 180 / 306
Slide 181 / 306 Slide 182 / 306
Slide 183 / 306
What does "Order of Operations" mean?
The Order of Operations is an agreed upon set of rules that tells us in which "order" to solve a problem.
Slide 184 / 306
The P stands for Parentheses : Usually represented by ( ). Other grouping symbols are [ ] and { }. Examples: (5 + 6); [5 + 6]; {5 + 6}/2
The E stands for Exponents : The small raised number next to the larger number. Exponents mean to the ___ power (2nd, 3rd, 4th, etc.) Example: 2 3 means 2 to the third power or 2(2)(2)
The M/D stands for Multiplication or Division : From
left to right. Example: 4(3) or 12 ÷ 3
The A/S stands for Addition or Subtraction : From left to right. Example: 4 + 3 or 4 - 3
What does P E M/D A/S stand for?
Slide 185 / 306
Watch Out!
When you have a problem that looks like a fraction but has an operation in the numerator, denominator, or both, you must solve everything in the numerator or denominator before dividing.
453(7-2)
453(5)
4515
3
Slide 186 / 306
Slide 187 / 306 Slide 188 / 306
Slide 189 / 306 Slide 190 / 306
Slide 191 / 306
[ 6 + ( 2 8 ) + ( 42 - 9 ) ÷ 7 ] 3
Let's try another problem. What happens if there is more than one set of grouping symbols?
[ 6 + ( 2 8 ) + ( 42 - 9 ) ÷ 7 ] 3
When there are more than 1 set of grouping symbols, start inside and work out following the Order of Operations.
[ 6 + ( 16 ) + ( 16 - 9 ) ÷ 7 ] 3[ 6 + ( 16 ) + ( 7 ) ÷ 7 ] 3
[ 6 + ( 16 ) + 1 ] 3[ 22 + 1 ] 3
[ 23 ] 369
Slide 192 / 306
Slide 193 / 306 Slide 194 / 306
Slide 195 / 306 Slide 196 / 306
Slide 197 / 306
What is a Constant?A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative.
Example: 4x + 2
In this expression 2 is a constant.click to reveal
Example: 11m - 7
In this expression -7 is a constant.click to reveal
Slide 198 / 306
What is a Variable?
A variable is any letter or symbol that represents a changeable or unknown value.
Example: 4x + 2
In this expression x is a variable.
click to reveal
Slide 199 / 306
What is a Coefficient?
A coefficient is the number multiplied by the variable. It is located in front of the variable.
Example: 4x + 2
In this expression 4 is a coefficient.click to reveal
Slide 200 / 306
If a variable contains no visible coefficient, the coefficient is 1.
Example 1: x + 4 is the same as 1x + 4
- x + 4 is the same as
-1x + 4
Example 2:
x + 2has a coefficient of
Example 3:
Slide 201 / 306 Slide 202 / 306
Slide 203 / 306 Slide 204 / 306
Slide 211 / 306 Slide 212 / 306
Equations and Inequalities
Return to Table of Contents
Slide 213 / 306
List what you remember about equations and inequalities .
Slide 214 / 306
A solution to an equation is a number that makes the equation true.
In order to determine if a number is a solution, replace the variable with the number and evaluate the equation.
If the number makes the equation true, it is a solution.If the number makes the equation false, it is not a solution.
Determining the Solutions of Equations
Slide 215 / 306 Slide 216 / 306
Slide 217 / 306 Slide 218 / 306
Why are we moving on to Solving Equations?
First we evaluated expressions where we were given the value of the variable and had which solution made the equation true.
Now, we are told what the expression equals and we need to find the value of the variable.
When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).
This will eliminate the guess & check of testing possible solutions.
Slide 219 / 306 Slide 220 / 306
Slide 221 / 306 Slide 222 / 306
Slide 223 / 306 Slide 224 / 306
To solve equations, you must use inverse operations in order to isolate the variable on one side of the equation.
Whatever you do to one side of an equation, you MUST do to the other side!
+5+5
Slide 225 / 306 Slide 226 / 306
Slide 227 / 306 Slide 228 / 306
Slide 229 / 306 Slide 230 / 306
Slide 231 / 306 Slide 232 / 306
Slide 233 / 306
An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:
Symbol Expression Words
< A < B A is less than B
> A > B A is greater than B
< A < B A is less than orequal to B
> A > B A is greater than orequal to B
Slide 234 / 306
Slide 235 / 306 Slide 236 / 306
Slide 237 / 306 Slide 238 / 306
Remember: Equations have one solution.
Solutions to inequalities are NOT single numbers. Instead, inequalities will have more than one value for a solution.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
This would be read as, "The solution set is all numbers greater than or equal to negative 5."
Solution Sets
Slide 239 / 306
Let's name the numbers that are solutions to the given inequality.
r > 10 Which of the following are solutions? {5, 10, 15, 20}
5 > 10 is not trueSo, 5 is not a solution
10 > 10 is not trueSo, 10 is not a solution
15 > 10 is trueSo, 15 is a solution
20 > 10 is trueSo, 20 is a solution
Answer:{15, 20} are solutions to the inequality r > 10
Slide 240 / 306
Slide 241 / 306 Slide 242 / 306
Slide 243 / 306
Since inequalities have more than one solution, we show the solution two ways.
The first is to write the inequality. The second is to graph the inequality on a number line.
In order to graph an inequality, you need to do two things:
1. Draw a circle (open or closed) on the number that is your boundary.
2. Extend the line in the proper direction.
Slide 244 / 306
Remember!
Closed circle means the solution set includes that number and is used to represent ≤ or ≥.
Open circle means that number is not included in the solution set and is used to represent < or >.
Extend your line to the right when your number is larger than the variable. # > variable variable < #
Extend your line to the left when your number is smaller than the variable. # < variable variable > #
Slide 245 / 306 Slide 246 / 306
Slide 247 / 306 Slide 248 / 306
Slide 249 / 306
Geometry
Return to Table of Contents
Slide 250 / 306
List what you remember about geometry .
Slide 251 / 306
A = length(width)A = lw
A = side(side)A = s2
The Area (A) of a rectangle is found by using the formula:
The Area (A) of a square is found by using the formula:
Slide 252 / 306
168 What is the Area (A) of the figure?
13 ft
7 ftP
ull
Pul
l
Slide 253 / 306
169 Find the area of the figure below.
8
Pul
lP
ull
Slide 254 / 306
A = base(height)A = bh
The Area (A) of a parallelogram is found by using the formula:
Note: The base & height always form a right angle!
Slide 255 / 306
170 Find the area.
10 ft 9 ft
11 ft
Pul
lP
ull
Slide 256 / 306
171 Find the area.
8 m
13 m 13 m
8 m
12 m
Pul
lP
ull
Slide 257 / 306
172 Find the area.
13 cm
12 cm
7 cm
Pul
lP
ull
Slide 258 / 306
The Area (A) of a triangle is found by using the formula:
Note: The base & height always form a right angle!
Slide 259 / 306
173 Find the area.
8 in
6 in
10 in 9 in
Pul
lP
ull
Slide 260 / 306
174 Find the area
14 m
9 m10 m 12 m
Pul
lP
ull
Slide 261 / 306
The Area (A) of a trapezoid is also found by using the formula:
Note: The base & height always form a right angle!
10 in
12 in
5 in
Slide 262 / 306
175 Find the area of the trapezoid by drawing a diagonal.
Pul
lP
ull
9 m
11 m
8.5 m
Slide 263 / 306
176 Find the area of the trapezoid using the formula.
20 cm
13 cm
12 cm
Pul
lP
ull
Slide 264 / 306
Area of Irregular Figures
1. Divide the figure into smaller figures (that you know how to find the area of)
2. Label each small figure and label the new lengths and widths of each shape
3. Find the area of each shape
4. Add the areas
5. Label your answer
Slide 265 / 306
Example:Find the area of the figure.
12 m
8 m
4 m2 m
12 m6 m
4 m2 m #1
#2
2 m
Slide 266 / 306
Pul
l
177 Find the area.
4'
3'
1'
2'
10'
8'
5'
Slide 267 / 306
178 Find the area.
12
101320
25
10 Pul
l
Slide 268 / 306
179 Find the area.
8 cm 18 cm
9 cm
Pul
l
Slide 269 / 306
Area of a Shaded Region
1. Find area of whole figure.
2. Find area of unshaded figure(s).
3. Subtract unshaded area from whole figure.
4. Label answer with units2.
Slide 270 / 306
Example
Find the area of the shaded region.
8 ft
10 ft
3 ft3 ft
Area Whole Rectangle
Area Unshaded Square
Area Shaded Region
Slide 271 / 306
180 Find the area of the shaded region.
11'
8'
3'4'
Pul
l
Slide 272 / 306
181 Find the area of the shaded region.
16"
17"
15"7"
5"
Pul
l
Slide 273 / 306
3-Dimensional SolidsCategories & Characteristics of 3-D Solids:
Prisms1. Have 2 congruent, polygon bases which are parallel to one another2. Sides are rectangular (parallelograms)3. Named by the shape of their base
Pyramids1. Have 1 polygon base with a vertex opposite it2. Sides are triangular3. Named by the shape of their base
Cylinders1. Have 2 congruent, circular bases which are parallel to one another2. Sides are curved
Cones1. Have 1 circular bases with a vertex opposite it2. Sides are curved
Slide 274 / 306
3-Dimensional Solids
Vocabulary Words for 3-D Solids:
Polyhedron A 3-D figure whose faces are all polygons (Prisms & Pyramids)
Face Flat surface of a Polyhedron
Edge Line segment formed where 2 faces meet
Vertex (Vertices) Point where 3 or more faces/edges meet
Solid a 3-D figure
Net a 2-D drawing of a 3-D figure (what a 3-D figure would look like if it were unfolded)
Slide 275 / 306
182 Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD cylinderE coneF square pyramid
Pull
Slide 276 / 306
183Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD cylinderE coneF square pyramid Pu
ll
Slide 277 / 306
184Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD pentagonal prismE coneF square pyramid Pu
ll
Slide 278 / 306
185Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD pentagonal prismE coneF square pyramid Pu
ll
Slide 279 / 306
186Name the figure.
A rectangular prismB cylinderC triangular
pyramidD pentagonal prismE coneF square pyramid Pu
ll
Slide 280 / 306
187 How many faces does a cube have?
Pull
Slide 281 / 306
188 How many vertices does a triangular prism have?
Pull
Slide 282 / 306
189 How many edges does a square pyramid have?
Pull
Slide 283 / 306
6 in
2 in7 in
7 in2 in
2 in6 in
A net is helpful in calculating surface area.
Simply label each section and find the area of each.
#2 #4
6 in
#1
#3
#5
#6
Surface AreaThe sum of the areas of all outside faces of a 3-D figure.
To find surface area, you must find the area of each face of the figure then add them together.
Slide 284 / 306
7 in2 in
2 in6 in
#2 #4
6 in
#1
#3
#5
#6
#1 #2 #3 #4 #5 #6
Example
Slide 285 / 306
190Find the surface area of the figure given its net.
7 yd
7 yd
7 yd
7 yd
Pul
l
Since all of the faces are the same, you can find the area of one face and multiply it by 6 to calculate the surface area of a cube.
What pattern did you notice while finding the surface area of a cube?
Slide 286 / 306
191Find the surface area of the figure given its net.
9 cm
25 cm
12 cm
Pul
l
Slide 287 / 306
Volume FormulasFormula 1
V= lwh, where l = length, w = width, h = height
Multiply the length, width, and height of the rectangular prism.
Formula 2
V=Bh, where B = area of base, h = height
Find the area of the rectangular prism's base and multiply it by the height.
Slide 288 / 306
Example
Each of the small cubes in the prism shown have a length, width and height of 1/4 inch.
The formula for volume is lwh.
Therefore the volume of one of the small cubes is:
Multiply the numerators together, then multiply the denominators. In other words, multiply across.
Forget how to multiply fractions?
Slide 289 / 306
192Find the volume of the given figure.
Pul
l
Slide 290 / 306
193Find the volume of the given figure.
Pul
l
Slide 291 / 306
194Find the volume of the given figure.
Pul
l
Slide 292 / 306
Ratios and Proportions
Return to Table of Contents
Slide 293 / 306
List what you remember about the ratios and proportions .
Slide 294 / 306
Ratio- A comparison of two numbers by division
Ratios can be written three different ways:
a to b a : b a b
Each is read, "the ratio of a to b." Each ratio should be in simplest form.
Slide 295 / 306
195 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of vanilla cupcakes to strawberry cupcakes?
A 7 : 9
B 7 27
C 7 11
D 1 : 3
Slide 296 / 306
196 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate & strawberry cupcakes to vanilla & chocolate cupcakes?
A 20 16
B 11 7
C 5 4
D 16 20
Slide 297 / 306
197 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of total cupcakes to vanilla cupcakes?
A 27 to 9
B 7 to 27
C 27 to 7
D 11 to 27
Slide 298 / 306
Equivalent ratios have the same value
3 : 2 is equivalent to 6: 4
1 to 3 is equivalent to 9 to 27
5 35 6 is equivalent to 42
Slide 299 / 306
4 125 15
x 3 Since the numerator and denominator were multiplied by the same value, the ratios are equivalent
There are two ways to determine if ratios are equivalent.1.
4 125 15
x 3
4 125 15
Since the cross products are equal, the ratios are equivalent.4 x 15 = 5 x 12 60 = 60
2. Cross Products
Slide 300 / 306
198 4 is equivalent to 8 9 18
True
False
Slide 301 / 306
199 5 is equivalent to 30 9 54
True
False
Slide 302 / 306
Rate: a ratio of two quantities measured in different units
Examples of rates: 4 participants/2 teams
5 gallons/3 rooms
8 burgers/2 tomatoes
Unit rate: Rate with a denominator of one Often expressed with the word "per"
Examples of unit rates:
34 miles/gallon
2 cookies per person
62 words/minute
Slide 303 / 306
Finding a Unit RateSix friends have pizza together. The bill is $63. What is the cost per person?
Hint: Since the question asks for cost per person, the cost should be first, or in the numerator.
$63 6 people
Since unit rates always have a denominator of one, rewrite the rate so that the denominator is one. $63 6 6 people 6 $10.50 1 person
The cost of pizza is $10.50 per person
Slide 304 / 306
200 Sixty cupcakes are at a party for twenty children. How many cupcakes per person?
Slide 305 / 306
201 John's car can travel 94.5 miles on 3 gallons of gas. How many miles per gallon can the car travel?
Slide 306 / 306
202 The snake can slither 24 feet in half a day. How many feet can the snake move in an hour?