Download At the Beach

Review of 7th Grade Math Concepts
Fractions
Addition with fractions and mixed numbers
Find a common denominator and then rewrite the fractions using this
denominator.
Add the numerators and keep the same denominator.
Simplify if necessary
Subtraction with fractions and mixed numbers
Find a common denominator and then rewrite the fractions using this
denominator.
When borrowing, the denominator tells how many you borrowed and the
numerator tells how many you already have. So, add them to find the new
amount.
Subtract the numerators and keep the same denominator.
Simplify if necessary.
Change a mixed number to an improper fraction
Multiply the denominator and whole number.
Add the numerator.
Write over the denominator.
Change an improper fraction to a mixed number
Divide the numerator by the denominator.
The quotient is the whole number part.
The remainder is the numerator and you keep the same denominator.
Multiplication with fractions and mixed numbers
Rewrite any mixed numbers as improper fractions.
Cross cancel if possible
Multiply the numerators and the denominators.
Simplify if necessary.
Reciprocal
Definition: Two numbers whose product is 1.
To find the reciprocal of a number, exchange the numerator and the
denominator. If it is a mixed number first write it as an improper fraction.
5
5 * 2 10


12 12 * 2 24
3 3*3 9


8 8 * 3 24
19

24
+
5
-
4
1
6 1
7
4
4
6
6
6
5
5

6
6
2
1
4 4
6
3
2 4 * 3  2 14


3
3
3
23
3
5
4
4
6 2 6 * 2 12 4
* 


7 3 7 * 3 21 7
Number
2
5
Reciprocal
5
2
3 1 = 13
4
4
4
13
Division with fractions and mixed numbers
2
1 8 16
2 3  
Rewrite any mixed numbers as improper fractions.
3
5 3 5
Keep the first fraction, change the division sign to a multiplication sign, and
8 5
8 * 5 40 5
* 


use the reciprocal of the next fraction.
3 16 3 *16 48 6
Follow multiplication rules for fractions.
Change a fraction to a decimal
Divide the numerator by the denominator.
Keep dividing until the decimal portion repeats or terminates.
Change a fraction to a percent
Make the denominator be 100 or change the fraction to a decimal and then
change it to a percent.
7
 .875
8
terminates
5
 .45
11
repeats
4 4 * 20 80
*

 80%
5 5 * 20 100
7
 .875  87.5%
8
1
Decimals
Addition with decimals
Line up the decimal points.
Add zeros to make sure both numbers have the same amount of decimal
places.
Add and then move down the decimal point.
Subtraction with decimals
Line up the decimals points.
Add zeros to make sure both numbers have the same amount of decimal
places.
Subtract and then move down the decimal point.
Multiplication with decimals
Multiply as if they were whole numbers.
Place the decimal point in the product so that it has the same number of
decimals places as the sum of the decimals places in the factors.
4.56 + 3.9 =
12 – 3.456 = 12.000
- 3.456
8.544
6.26 * 4.3 =
6.26
4.3
1878
25040
26.918
*
Division with decimals
Move the decimal point the same number of places to the right in both the
divisor and dividend until the divisor is a whole number.
Divide as you divide with a whole number.
Move up the decimal point.
Change a decimal to a fraction
Read the decimal and write it as a fraction.
Simplify if needed.
Change a decimal to a percent
Move the decimal point two places to the right.
4.56
+ 3.90
8.46
11.124 ÷ 2.7
2.7 11.124
4.12
27 111.24
4.7 is four and seven tenths
7
4
10
2.35 = 235%
.24 = 24%
.9 = 90%
.007 = .7%
Percents
Change a percent to a decimal
Move the decimal point two places to the left.
Change a percent to a fraction
Write the percent with a denominator of 100.
Finding the percent of a number mentally
Find 1% by moving the decimal point 2 places to the left.
Find 10% by moving the decimal point 1 place to the left.
Find 50% by dividing the number by 2.
Use those percents to find other percents.
Steps to use when solving a proportion
Set the cross products equal to each other.
Multiply
Undo multiplication by using division
Solving percent problems using equations
Remember:
IS means equal to
OF means multiply
Change a percent to a decimal before using it in the equation.
432% = 4.32
65% = .65
60% = .6
.9% = .009
35
7
7
35% 

7% 
100 20
100
Number: 56
1% = .56 10% = 5.6 50% = 28
15% = 10% + 5% = 5.6 + 2.8 = 8.4
30% = 3 * 10% = 3 * 5.6 = 16.8
4 10

9 n
4 * n = 10 * 9
4n = 90
÷ 4 = ÷4
n = 22.5
What ____ is ____% of _____?
What number is 18% of 117?
n = .18 * 117
n = 21.06
2
Solving percent problems using proportions
Use the proportion at the right. To decide what goes where in the
proportion remember the following:
The % number has a percent sign with it.
The OF number is ALWAYS after the word OF
The is number can be before the word IS or after the word IS
Follow the steps to solve a proportion.
is
%

of 100
36 is 30% of what number?
36 30

n 100
36 * 100 = 30n
3600 = 30n
÷ 30 ÷ 30
120 = n
Integers
Absolute Value
The absolute value of a number is its distance from
zero on a number line.
Adding Integers with the same sign
1. Find the absolute value of each integer
2. Add the absolute values.
3. Keep the sign of the integers for your answer
Adding Integers with different signs
1. Find the absolute value of each integer
2. Subtract the smaller absolute value from the larger
absolute value.
3. Use the sign of the integer with the greater absolute
value for you answer. (BIG GUY RULES)
Subtracting Integers
1. Keep the first integer.
2. Change the subtraction sign to an addition sign.
3. Change the second integer to its opposite.
4. Add using the addition rules for integers with the
same sign or different signs.
Multiplying and Dividing Integers
1. Multiply or divide using the absolute value of each
integer (or ignoring the signs)
2. Count the number of negative signs in the original
problem. If there are:
- an odd number of negative signs the answer is
negative
- an even number of negative signs the answer is
positive
|5| = 5
|-5| = 5
- 5 + -7 = -12
-20 + -11 = -31
-5 + 6 = 1
-5 + 2 = -3
6 + -10 = -4
-3 + 11 = 8
-5 - -6 becomes -5 + 6
5 – 10 becomes 5 + -10
-5 – 2 becomes -5 + -2
8 – (-8) becomes 8 + 8
-8 – (-8) becomes -8 + 8
which equals 1
which equals -5
which equals -7
which equals 16
which equals 0
3 * -8 * -2 = 48
5 * -6 = -30
-36 ÷ -4 = 9
-50 ÷ 10 = -5
Number Sense and Operations
Expression
A mathematical phrase made up variables and/or
numbers and operations.
A dentist earns twice as much as she did last year. If
her salary last year was p, write an expression for her
current salary.
3x -11
½x
y+8
2p
3
Equation
A mathematical statement that two expressions are
equal.
Two-step equations
Whatever you do to one side of the equation you
MUST do to the other side.
Undo any addition or subtraction first
Undo any multiplication or division
Unit Rates
To find a unit rate make the second quantity be one.
Ratios
Use a ratio and a proportion to solve a problem.
Ratio: - the relationship between two quantities;
expressed as a fraction, or quotient of one number
divided by the other. The ratio of 3 to 2 can be
written 3 / 2 or 3 : 2.
Standard Form to Scientific Notation
To write a number in scientific notation, count how
many places you must move the decimal point to get a
number greater than or equal to 1 and less than 10.
The number of decimal places you move the decimal
point is the power of 10.
Scientific Notation to Standard Form
To write a number in standard form move the decimal
point to the right the number of places indicated by the
power of 10.
Order of Operations
The rules for evaluating expressions using order of
operations:
P – Parentheses – simplify inside the parentheses
E – Exponents
M D – multiplication and division from left to right
A S – addition and subtraction from left to right
Multipyling by ½
Remember that multiplying by ½ is the same as
multiplying by 0.5 or dividing by 2.
Division Symbols
All of these symbols mean to divide.
10
2
10/2
x – 10 = 6
3y = 9
x - 4 = 14
4y + 10 = 74
4
-10 = - 10
+4= +4
4y = 64
x = 18
÷4 = ÷ 4
4
y = 16
*4 = *4
x = 72
Rate: 504 miles in 12 hrs.
Unit Rate: Divide each quantity by 12 to get
42 miles in 1 hr.
Ratio of boys to girls is 6:5. If there are 392 students,
how many of them are girls?
In this sample there are 11 total students, 6 boys and 5
girls. Write a proportion to solve this problem:
5 = n
11 392
5 * 392 = 11n
1960 = 11n
÷ 11 = ÷ 11
178.2 = n
There are 178 girls.
245,000 = 2.45 * 105
1,890,000 = 1.89 * 106
45.68 = 4.568 * 101
6,000,000,000 = 6 * 109
6.75 * 104 = 67,500
9 * 106 = 9,000,000
3.4 * 107 = 34,000,000
6 + 5(24 ÷ 3 + 2) – 4
6 + 5(8 + 2) – 4
6 + 5(10) – 4
6 + 50 – 4
56 – 4
52
243 ÷ (-3)2 – 5 + 6
243 ÷ 9 – 5 + 6
27 – 5 + 6
22 + 6
28
36 * ½ = 36 * 0.5 = 36 ÷ 2
18 = 18
= 18
10 = 5
2
10/2 = 5
10 ÷ 2 = 5
10 ÷ 2
4
Data Analysis, Statistics and Probability
Venn Diagram
A Venn Diagram is used to display data. This
diagram shows the following information about
30 students that were surveyed.








3 like basketball only
4 like basketball and soccer
5 like soccer only
3 like basketball and baseball
2 like soccer and baseball
4 like only baseball
7 like all three
2 don't like any of the three
It also shows that 18 students like soccer, 16
like baseball, and 17 like basketball
Mean, Median, Mode and Range
Mean (average) – divide the sum of the values
by the number of values.
Median – Arrange the values in order from least
to greatest. The median is the one in the
middle.
Mode – The most common data value. If no
value occurs more than once there is no mode.
If two values occur more than once and equally
there are two modes.
Range – the difference between the highest and
lowest values.
Stem-and-Leaf Diagrams
Shows how often data values occur.
STEM: - the tens digit
LEAF: - the ones digit.
Example: 43 has a stem of 4 and a leaf of 3
Values: 4, 2, 5, 3, 4, 6, 4
Mean: 28 ÷ 7 = 4
Median: 2, 3, 4, 4, 4, 5, 6 (4 is the middle)
Mode: 4 occurs the most often
Range: 6 – 2 = 4
Values: 5, 3, 6, 3, 4, 6
Mean: 27 ÷ 6 = 4.5
Median: 3, 3, 4, 5, 6, 6 There are two middle values so add
them together and divide by 2.
9 ÷ 2 = 4.5
Mode: 3 and 6
Range: 6 – 3 = 3
Stem
4
5
6
Leaf
23556
012
566
The values represented are 42, 43, 45, 45, 46, 50, 51, 52, 65,
66 and 66.
Probability
The probability of an event compares the
number of ways the event can occur to the
number of possible outcomes. It is expressed as
a fraction, decimal, percent or ratio:
Example: There are 4 white marbles, 3 blue marbles, 6 green
marbles and 2 red marbles in a bag. What is the probability
you pick a blue marble from the bag?
number of ways event can happen = 3
number of possible outcomes
15
Probability (event)
= number of ways event can happen
number of possible outcomes
5
Tree Diagram
A tree diagram is a way to show all the different
outcomes for a given situation.
Example: How many ways can a burglar describe the length
and color of a suspect’s hair? Make a tree diagram to show
this:
Color
Black
S
T
A
R
T
Organized List
You can make an organized list to count the
number of possible descriptions.
Length
Short
Medium
Long
Brown
Short
Medium
Long
Blonde
Short
Medium
Long
So, there are 9 different possible descriptions.
Hair Color
Hair Length
Black
Short
Black
Medium
Black
Long
Brown
Short
Brown
Medium
Brown
Long
Blonde
Short
Blonde
Medium
Blonde
Long
So, there are 9 different possible descriptions.
Geometry
Congruent Figures
If two figures are congruent, then their
corresponding parts are congruent. Let's look
at the corresponding parts of triangles ABC
and DFE.
Here, angle A corresponds to angle D, angle
B corresponds to angle F, and angle C
corresponds to angle E. Side AB corresponds
to side DF, side BC corresponds to side FE,
and side CA corresponds to side ED.
6
Similar Figures
If two objects have the same shape, they are
called "similar." When two figures are similar,
the ratios of the lengths of their corresponding
sides are equal.
AB = AC
6 = 10
DE
DF
3
5
You can use this information to find the
length of a missing side by writing a
proportion.
Angles formed by intersecting lines
Vertical angles – angles on opposite sides of the
intersection of two lines. <2 and <3
1 2
Corresponding angles - The angles are in the same
3 4
position from one parallel line to the other parallel
line. <1 and <5; <2 and <6; <4 and <8; <3 and <7
5
6
Alternate Interior angles - The angles are on opposite
sides of the transversal and on the inside of the lines
7 8
it intersects. <3 and <6; <4 and <5
Alternate Exterior angles - The angles are on
opposite sides of the transversal and on the outside of
the lines it intersects. <1 and <8; <2 and <7
A nonagon
15-sided figure
Sum of the measures of the angles of a polygon
To find this use the formula:
S = (9 – 2) * 180
S = (15 – 2) * 180
S = (n – 2) * 180
S = 7 * 180
S = 13 * 180
Where n is the number of sides in the polygon.
S = 1260 degrees
S = 2340 degrees
Other Angles
Complementary angles – two angles whose sum is 90°
Supplementary angles – two angles whose sum is 180°
Obtuse angle – an angle greater than 90° and less than 180°
Acute angle – an angle less than 90°
Straight angle – an angles that measures exactly 180°
right 3 down 4 – means to move each vertex of the
Translations
A translation moves a figure right or left and up or
figure 3 places to the right on the x-axis and down 3 on
down. The size of the figure does not change.
the y-axis.
(x, y)
(x – 2, y + 3) – means to move each vertex
2 places to the left on the x-axis and up 3 on the y-axis.
Example: A triangle whose coordinates are at: A: (4,5);
Reflections
A reflection flips a figure across a line. Every point
B: (2,3) and C: (1, 4) is reflected across
on the original figure is the same distance from the
line as the matching point on the reflection.
1. the x-axis. What are its new coordinates:
When a point is reflected across the x-axis:
A: (4,-5) B: (2,-3) C: (1,-4)
- its x-coordinate stays the same
- its y-coordinate is multiplied by -1
2. the y-axis. What are its new coordinates:
When a point is reflected across the y-axis:
A: (-4,5) B: (-2,3) C: (-1,4)
- its x-coordinate is multiplied by -1
- its y-coordinate stays the same
7
Prisms
A polyhedron whose bases are congruent and
parallel. It is named by the shape of its bases.
hexagonal prism
Pyramid
A pyramid has a polygon for a base, but comes to a
point. It is named by the shape of its base.
3-D Vocabulary
face – a flat surface on a solid
edge – a segment joining two faces of a polyhedron
vertex – a corner where the edges meet
vertex
face
edge
Measurement
Converting units in the same system of measurement
You need to be able to convert between different units by using a conversion factor. Remember in your
conversion factor to make sure the unit you want to convert is in the denominator of the conversion factor. See
the examples:
Convert 15 feet to inches
Convert 144 inches to feet
15 ft * 12 inches = 180 inches
144 inches * 1 foot = 12 feet
1 foot
12 inches
Converting units in different systems of measurement
You will also need to be able to convert between different systems of measurement (i.e. metric to English). One
mile is about 1.6 kilometers. Use a conversion factor to find the answer to this type of question. Remember in
your conversion factor to make sure the unit you want to convert is in the denominator of the conversion factor.
See the examples:
How many kilometers are there in 2350 miles?
2350 miles * 1.6 km = 3860 km.
1 mile
Scale
A scale is used in drawings and the reading of
maps. It is written in the following form:
actual value
scale value
Perimeter of a Polygon
The perimeter of a shape is the distance around
the shape. To find the perimeter add up the
length of all the sides.
How many miles are there in 3440 km ?
3440 km. * 1 mile = 2150 miles
1.6 km
On a map, 1 in = 13 miles. When I measure between two
cities it is 2 ¼ inches. What is the actual distance between
the cities?
13 * 3.25 = 42.25 miles
What is the perimeter of a parallelogram with sides of 5 in,
4in, 8 in. and 4 in.?
5 + 4 + 8 + 4 = 21 inches
8