### Comprehensive PSSA Review Powerpoint

```Created by: Miss Jessie Minor
Purpose: PSSA Review for 7th Grade
(Can be used as enrichment or remediation for most middle school levels)
Contents: Concept explanations & practice problems.
Sources: Common Core Standards from PDE website.
Reinforcement:
www.studyisland.com
www.ixl.com
www.mathmaster.org
and PSSA Coach workbook
EXPERIMENTAL PROBABILITY!
IN ORDER TO CALCULATE
EXPERIMENTAL PROBABILITY OF AN
EVENT USE THE FOLLOWING
DEFINITION:
P(Event)=
Number of times the event occurred
Number of total trials
COACH LESSON 30
2
EXPERIMENTAL PROBABILITY!
A student flipped a coin 50 times. The coin landed on heads 28 times.
Find the experimental probability of having the coin land on heads
P(heads) = 28 = .56 = 56%
50
It is experimental because the outcome will change every
time we flip the coin.
3
PRACTICE EXPERIMENTAL PROBABILITY!
4
THEORETICAL PROBABILITY!
The outcome is exact!
When we roll a die, the total possible outcomes
are 1, 2, 3, 4, 5, and 6. The set of possible
outcomes is known as the sample space.
PRACTICE THEORETICAL PROBABILITY!
Find the prime numbers– since 2, 3, and 5 are the
only prime numbers in the same space…
P(prime numbers)= 3/5 = 60%
COACH LESSON 29
5
RATE/ UNIT PRICE/ SALES TAX!
RATE: comparison of two numbers
Example: 40 feet per second or 40 ft/ 1 sec
UNIT PRICE: price divided by the units
Example: 10 apples for \$4.50
Unit price: \$4.50 ÷ 10 = \$0.45 per apple
SALES TAX: change sales tax from a percent to a decimal, then multiply it by
the dollar amount; add that amount to the total to find the total price
Example 1: \$1,200 at 6% sales tax = 6 ÷ 100 = 0.06 x 1,200 = 72
1200
+ 72
\$1272
COACH LESSON 4
6
PRACTICE SALES TAX!
Example 2: Rachel bought 3 DVDs. Using the 6%
sales tax rate, calculate the amount of tax she paid if
each DVD costs \$7.99?
7
DISTANCE FORMULA!
Distance formula: distance = rate x time
OR
D = rt
Example 1: A car travels at 40 miles per hour for 4 hours. How far did it travel?
d=rt
d=40 miles /hr x 4 hrs
d = 160 miles.
We can also use this formula to find time and rate.
We just have to manipulate the equation.
Example 2: A car travels 160 miles for 4 hours. How fast was it going?
d = rt
160 miles = r (4 hours)
160 miles ÷ 4 hrs = r
40 miles/hr = r
COACH LESSON 23
8
PRACTICE THE DISTANCE FORMULA!
DISTANCE
=
RATE X
TIME
WITH THIS FORMULA WE CAN FIND ANY OF THE THREE QUANTITIES, RATE,
TIME, OR DISTANCE, IF AT LEAST TWO OF THE QUANTITIES ARE GIVEN.
If the time and rate are given, we can find the distance:
EXAMPLE: How far did Ed travel in 7 hours if he was going 60 miles per/hour?
d = rt
d = 60miles/hr x 7 hrs
d = 420 miles
Or if the distance and rate are given, we can find the time:
d = rt
420miles = 60 miles/hr x t
(420 miles ÷ 60 miles/hr) = 7 hours
9
PRACTICE USING THE DISTANCE FORMULA!
Michael enters a 120-mile bicycle race. He bikes 24 miles an hour. What is
Michael's finishing time, in hours, for the race?
A
B
C
D
2
5
0.2
0.5
10
RATIOS & PROPORTIONS!
Ratio: comparison of two numbers.
Example: Johnny scored 8 baskets in 4 games.
The ratio is 8 = 2
4 1
Proportion: 2 ratios separated by an equal sign .
If Johnny score 8 baskets in 4 games how many baskets will he score in 12 games?
1. Set up the proportion
4 games
12 games
2. Cross multiply & Divide
4x = 8 ( 12 )
4x = 96
x = 96
4
COACH LESSON 7
11
FRACTIONS!
ADDING AND SUBTRACTION – FIND COMMON DENOMINATORS!
Use factor trees, find prime factors , circle ones that are the same circle the ones
by themselves. Multiply the circled numbers.
EXAMPLE:
5
12
+
8
9
12
2
6
2 3
9
3 3
12: 2 2 3
9: 3 3
3 x 3 x 2 x 2 = 36
Common denominator = 36
3 x 5 = 4 x 8 = 15 + 32 = 47
36
36
36 36
36
COACH LESSON 1
12
PRACTICE FRACTIONS!
13
MULTIPLYING & DIVIDING FRACTIONS!
Multiplying fractions : cross cancel and multiply straight across
¹4 X ¹5
¹5 ²8
=
1
2
Dividing fractions : change the sign to multiply, then reciprocate the
2nd fraction
3 ÷ 5
4
8 =
3 X 8
4
5
= 24
20
REDUCE!!!
COACH LESSON 2
14
PRACTICE MULTIPLYING FRACTIONS!
3 X 5
4
6
1 X
49
7
13
5 X
9
4
5
15
Multiplying & Dividing Mixed Numbers!
When multiplying or dividing mixed numbers, always change
them to improper fractions.
Example 1:
Example 2:
1¾ x 1½ =
7
x
3 = 21
4
2
8
12 x 2 ½ =
12 x 5 = 60 = 30
1
2 2
16
Dividing Mixed Numbers!
When dividing any form of a fraction, change the division to
multiplication, then reciprocate the 2nd fraction.
Example:
1¾ ÷ 1½
7
4
÷ 3
2
7
4
x
=
2 =
3
14 = 11/6
12
17
LEAST COMMON MULTIPLE!
LCM : Least Common Multiple : the smallest number that 2 or more numbers will divide
into
Example: Find the LCM of 24 and 32
You can multiply each number by 1,2,3,4… until you find a common multiple which is 96.
Or you can use a factor tree:
24
32
2 12
2
2
24:
32:
2 2 2 3
2 2 2 2 2
2 6
2 2 3
2 16
2
2
8
2
2
2 4
2
2
2 2 2
2x2x2x3x2x2 = 96
18
GREATEST COMMON FACTOR!
GCF~ GREATEST COMMON FACTOR : The Largest factor that will
divide two or more numbers. In this case we would multiply the
factors that are the same.
24: 2 2 2 3
32: 2 2 2 2 2
Example: 2x2x2 = 8, so 8 is the GCF of 24 and 32.
19
PRACTICE LCM AND GCF!
20
PRACTICE LCM AND GCF!
What is the greatest common factor (GCF) of 108 and 420 ?
A
B
C
D
6
9
12
18
What is the least common multiple (LCM) of 8, 12, and 18 ?
A
B
C
D
24
36
48
72
21
ABSOLUTE VALUE!
ABSOLUTE VALUE: the number itself without the sign; a number’s distance
from zero
The symbol for this is | |
Example:
The absolute value of |-5| is 5
The absolute value of |5| is 5
22
PRACTICE ABSOLUTE VALUE!
23
DISTRIBUTIVE PROPERTY
A(B + C)
=
AB + AC
Solving 2 step equations:
subtract 8
divide by 4
(We distributed A to B and then A to C)
4(x + 2) = 24
4x + 8 = 24
4x = 16
x= 4
•Remember when solving 2 step equations do addition and subtraction first
then do multiplication and division.
•This is opposite of (please excuse my dear aunt sally,) which we use on
math expressions that don’t have variables.
COACH LESSON 20
24
Associative & Commutative Property!
Associative
• Always has parentheses
• A ( B X C) = B (C X A)
• FOR MULTIPLICATION
Commutative
• AXB=BXA
• FOR MULTIPLICATION
• A+B=B+A
• A + (B + C) = B + (C + A)
http://www.mathmaster.org/video/associative-property-for-multiplication/?id=932
25
Stem and Leaf Plots, Box – and – Whisker Plots
We use stem and leaf plots to organize scores or large groups of numbers.
Example: To arrange the following numbers into a stem and leaf
plot, the tens place goes in the stem column and the ones place
goes in the leaf column.
40, 30, 43, 48, 26, 50, 55, 40, 34, 42, 47, 47, 52, 25, 32, 38, 41, 36,
32, 21, 35, 43, 51, 58, 26, 30, 41, 45, 23, 36, 41, 51, 53, 39, 28
Stem
2
3
4
5
Leaf
135668
0022456689
001112335778
0112358
COACH LESSON 24
26
Stem
2
3
4
5
Leaf
135668
0022456689
001112335778
Upper quartile- 47
0112358
Lower quartile- 32
MODE—The number that occurs the most often—The mode of these scores– is 41.
RANGE—The difference between the least and greatest number—is 37.
MEDIAN—The middle number of the set when the numbers are arranged in order—
it is 40.
MEAN– Another name for average is mean.
FIRST QUARTILE OR LOWER QUARTILE —The middle number of the lower half of
scores—is 32.
THIRD QUARTILE OR UPPER QUARTILE—The middle number of the upper half of
scores—is 47.
COACH LESSON 27, 25
27
Box-and-Whisker Plot!
Lower
extreme
First quartile
or lower
quartile
Second
quartile or
median
Third quartile
or upper
quartile
Upper
extreme
Inter
quartile
Range
28
PRACTICE STEM & LEAF/ BOX & WHISKERS!
Make a stem and leaf plot from the following numbers. Then make a box
and whiskers diagram.
25, 27, 27, 40, 45, 27, 29, 30, 26, 23, 31, 35, 39
29
PRACTICE STEM & LEAF/ BOX & WHISKERS!
Below are the number of points John has scored while playing the last 14
basketball games. Finish arranging John’s points in the stem and leaf plot and
then find the range, mode, and median.
Points: 5, 14, 21, 16, 19, 14, 9, 16, 14, 22, 22, 31, 30, 31
Stem
Leaf
Range:
0
Mode:
1
Median:
2
3
30
Order of Operations!
3(4 + 4)
÷
3 - 2
3 (8)
÷
3 - 2
24
÷
3 - 2
8
-
2
=6
Note that there are not any variables is the statement.
This is why we use order of operation instead of the Distributive Property.
COACH LESSON 5
31
PRACTICE ORDER OF OPERATIONS!
1.) 3 + 2(4 x 3)
2.) 12 - 15 - 3
3.) (22 + 14) – 6
4.) 64 – 8 + 8
32
PRACTICE ORDER OF OPERATIONS!
http://www.mathmaster.org/video/exponent-properties-involving-products/?id=1889
1.) 2³
2.) 3⁴
3.) 4²
=
2x2x2 =
4.)
144
=
5.)
64
=
= 3x3x3x3 =
=
4x4 =
33
FINDING THE MISSING SIDE OF A TRIANGLE!
a
50°
65°
b c
Finding b:
Since the sum of the degrees of a triangle is
180 degrees,
we subtract the sum of 65 + 50 = 115
from 180
180 - 115 = 65
…so
b = 65°
Finding c:
If b = 65 to find c we know that a straight line is
180 degrees
so if we subtract 180 – 65 = 115°
…so
Angle c = 115°
Finding a:
To find a we do the same thing.
180 – 50 = 130
…so
a = 130°
34
PRACTICE FINDING THE MEASURE OF <A IN THE
TRIANGLE ABC BELOW!
A
30
C
B
m<A + 90 + 30 = 180
m<A =
35
A square has 4 angles which each measure 90 degrees.
D
A
45
45
C
45
45
B
36
Pythagorean Theorem
To find the missing hypotenuse of a right triangle, we
us the formula…
c²
C²
C²
C²
Height = 6 in
Base = 8 inches
C
=
=
=
=
A² + B²
6²in + 8²in
36 in² + 64 in²
100 sq in
=
in²
= 10 in²
http://www.mathmaster.org/video/pythagorean-theorem/?id=1922
37
AREA OF A TRIANGLE!
A = base x height
2
Area = base x height
2
A = 10in x 8 in
2
A = 80 in²
2
Height= 8 in
Base= 10 in
A =
40 in²
Definition of height is a line from the opposite vertex perpendicular to the base.
COACH LESSON 12
38
PRACTICE FINDING THE AREA OF A TRIANGLE!
AREA = ½ (BASE X HEIGHT)
A = ½ bh
Height= 2 ft
Area = ½ bh
A = ½ (4ft)(2ft)
A = ½ 8ft
A =4 ft²
Base= 4 ft
39
FINDING THE AREA OF A PARALLELOGRAM!
h
b
40
AREA OF A RECTANGLE & A SQUARE!
Area of a RECTANGLE = Length
x
Width
Area of a SQUARE = Side
x
Side
Example:
2ft
4ft
2ft
2ft
41
PRACTICE FINDING PERIMETER!
PERIMETER IS THE OUTER DISTANCE AROUND A FIGURE.
9 FT
3FT
P = a+ b + c + …
P = 9FT + 9FT + 3FT + 3FT
P = ____ FT
42
FINDING PERIMETER AND AREA OF COMPOUND FIGURES!
To find the area of a compound figure, we simply
have to find the area of both figures, then add
them together.
6FT
AREA = LENGTH X WIDTH
A = 2FT X 6FT
A = 12FT²
2FT
3FT
7FT
AREA = LENGTH X WIDTH
A = 3FT X 5FT
A = 15 FT²
TOTAL AREA = 12FT² + 15FT² = 27FT²
43
CONGRUENT ANGLES & CONGRUENT SIDES!
Congruent angles and sides mean that they have the same
measure. Use symbols to show this!
44
Complementary angles : angles whose sum equals 90 degrees
Supplementary angles: angles whose sum equals 180 degrees
Right angle: angle measures 90 degrees ---symbol
Acute angle: angle less than 90
Obtuse angle: angle greater than 90 degrees
Congruent: when two figures are exactly the same
Similar: when two figures are the same shape but not the same size
Regular: when a figure has all equal sides
Line of symmetry: when a line can cut a figure in two symmetrical sides
COACH LESSON 17
45
Parallel lines: lines that never touch--- symbol
Perpendicular lines: lines that intersect---symbol
Skew lines: lines in different planes that never intersect
Plane: a flat, 2-Dimensional surface, formed by many points
A point (0-Dimension); A line (1-D); A plane (2-D); A solid (3-D)
Vertical angles: angles that share a point and are equal
Adjacent angles: are angles that are 180 degrees and share a side
COACH LESSON 18
46
ADJACENT ANGLES: ANGLES THAT SHARE A COMMON SIDE.
In the figure below:
ANGLES 3 AND 4 ARE ADJACENT ANGLES.
ANGLES 2 AND 3 ARE ALSO ADJACENT ANGLES.
What are some other adjacent angles?
2
3
1
4
47
REVIEW: CLASSIFY LINES!
Intersecting lines: occupy the same plane AND meet at only one
point
Perpendicular lines: two lines intersect and form right angles
(90°)
The symbol is:
Parallel lines: extend forever in both directions in the same plane
and never intersect
The symbol is:
Skew lines: a pair of lines that are not parallel but never intersect
AND occupy two different planes
48
REVIEW: CLASSIFY LINES!
Supplementary angles: sum is 180 degrees
Complementary angles: sum is 90 degrees
Straight angle: equal to 180 degrees
49
PRACTICE GEOMETRY!
What is the total number of lines of symmetry that can be drawn on the trapezoid below?
Circle One:
A .)
4
B .)
3
C .)
2
D .)
1
Which figure below correctly shows all the possible lines of symmetry for a square?
Circle One:
A.)
Figure 1
B.)
Figure 2
C.)
Figure 3
D.)
Figure 4
50
[Volume= units³ or cubed units]
4 ft
3 ft
V = 5ft x 3ft x 4ft = 60ft³
5 ft
51
Identifying similar figures!
Two figures are similar if they have exactly the same
shape, but may or may not have the same size.
The symbol is ≈
X
For example: ∆ ABC ≈ ∆ XYZ
A
Which angle is similar
to angle B?
Angle: _______
B
C
Y
Z
52
Chord: line that cuts
the circle and does
not go through the
center of the circle
Diameter: distance across
the center of the circle
way across the circle
( ½ diameter)
Segment: the area of a
circle in which a chord
creates
Sector: a pie-shaped
part of a circle made by
Circumference: distance
around the outside of
the circle
Arc: a connected section
of the circumference of
a circle
COACH LESSON 15
53
Central angles: angles in
the center of the circle
Inscribed angles: angles
on the inside of the circle
formed by two chords
COACH LESSON 15
54
PRACTICE FINDING THE CIRCUMFERENCE OF A CIRCLE!
A
B
C
D
4
8
16
32
*USE ∏= 3.14
HINT: Circumference= 2∏r OR ∏· D
55
PRACTICE FINDING THE AREA OF A CIRCLE!
If the diameter of a car tire is 50 cm,
what is the area of that circle?
A
B
C
D
50.14 cm²
314 cm²
7,850 cm²
1,000 cm²
*USE ∏= 3.14
HINT: Area = ∏ x r²
56
MORE PRACTICE!
A duck swims from the edge of a circular pond to a fountain in the center of
the pond. Its path is represented by the dotted line in the diagram below.
What term describes the duck's path?
A
B
C
D
chord
diameter
central angle
57
Rules:
Negative + Negative = Negative
-4 + -3 = -7
Positive + Positive = Positive
4+3=7
Negative + Positive = ?
(Keep the sign of the larger integer & subtract)
-4 + 3 = -1
58
Multiplying & Dividing Negative Numbers!
Rules:
Negative x Negative = Positive
Negative ÷ Negative = Positive
-4 x -2 = 8
-4 ÷ -2 = 2
Positive + Positive = Positive
Positive ÷ Positive = Positive
4x2=8
Negative x Positive = Negative
-4 x 2 = -8
4÷2=2
Negative ÷ Positive = Negative
-4 ÷ 2 = -2
59
Comparing & Ordering Integers!
NEGATIVE
POSITIVE
Negative integers further to the left of zero have less value.
Positive integers further to the right of zero have greater value.
Example: -3 IS GREATER THAN -6
COACH LESSON 3
60
Inequalities!
Use the following symbols for inequality number sentences:
< less than
-4 < 2
≤ less than or equal to
3≤4
>
6>3
greater than
≥ greater than or equal to
-5 ≥ -6
61
Solving One-Step Equations!
To solve for a variable in an equation, the variable must be alone on one
side of the equals sign.
Use a model or an inverse operation to solve a one step equation.
Example:
3x = 24
Step 1: Divide by 3
on both sides
of the equation
3x = 24
3
3
x =
8
COACH LESSON 21
62
Modeling Mathematical Situations!
We can translate math sentences to numbers and symbols only
Examples:
Translate: “five more than”
(5 + a quantity)
Translate: “three times a number”
(3 x n, or 3n)
When you combine both: “five more than three times a number”
5 + 3n or
3n +5
COACH LESSON 22
63
Functions!
Functions: inserting a value in for x to find y or f(x)
Example:
f(x) = 2x + 4
Then
f(x) = 2 (2) + 4
f( x) = 4 + 4
f(x) = 8
So
y=8
If x = 2
Also, a function is when we put a value in and get an answer out.
COACH LESSON 20
64
Scientific Notation!
Scientific notation -- 4.057 x 10⁶
4.057 x 10⁶
(This means to move the decimal
six places to the right.)
becomes
4,057,000
Expanded notation --- numbers written using powers of 10
Example: 4234 = (4 x 10³) + (2 x 10²) + (3 x 10¹) + (4 x 10⁰)
4000
+ 200
+
30
+
4
=
4234
Any number raised to the zero power equals 1.
10 ⁰ = 1
Any number raised to the 1st power equals that number.
8¹ = 8
65
METRIC SYSTEM & CONVERSTION!
DekaKilo-
Meter
Liter
Gram
Deci-
Hecto-
Centi-
Milli-
START where your at and move the
decimal to where you want to go.
Example 1:
4 kilometers = 4000 meters
Example 2:
36 millimeters = 3.6 centimeters
COACH LESSON 11
66
PRACTICE UNIT CONVERSIONS!
67
Weight Unit Conversions:
Use the chart and move the decimal point.
Gram = weight
Meter = distance
Liter = volume
For U.S. Customary measurement, conversions are on
charts.
68
PRACTICE WEIGHT UNIT CONVERSIONS!
The flower box in front of the main city library weighs 124
ounces. What does the flower box weigh in pounds?
69
PRACTICE MORE UNIT CONVERSIONS!
70
Unit Multipliers:
1. Always list the conversion.
2. Identify the correct multiplier.
3. Set up the multiplication problem with units being opposite
(top & bottom)
4. Multiply & Simplify
For example: Change 240 feet to yards
a) First list the conversions:
3 feet OR 1 yard
1 yard
3 feet
b) Since we want yards multiply by
1 yard
3 feet
c) So 240 feet x 1 yard
1
3 feet
d) Then 240 feet = 80 yards
COACH LESSON 9
71
Ratios & Proportions:
A ratio is a comparison between two numbers.
Two ratios separated by an equals sign is called a proportion.
To solve a proportion, we cross multiply and divide.
Example:
4 = 2
5 = x
4x = 10
4
4
x = 10
4
x=2½
COACH LESSON 7
72
Rational & Irrational Numbers
An Irrational Number is a real number that cannot be written as a
simple fraction.
A Rational Number can be written as a simple fraction.
Irrational means not Rational.
Example: 7 is rational, because it can be written as the ratio 7/1
Example 0.333... (3 repeating) is also rational, because it can be
written as the ratio 1/3
73
Practice Irrational Numbers!
Which of these is an irrational number?
74
Rational Numbers on a Number Line!
Fraction
Decimal
Percent
Place number over
its place value and
reduce
Divide by 100
Multiply by 100
75 = 3
100
4
0.75
0.75 x 100 = 75%
125 = 1
1000 8
0.125
0.125 x 100 = 12.5%
150 = 3 = 1 ½
100 2
1.50
1.50 x 100 = 150%
COACH LESSON 4
75
Points on a Coordinate Grid!
Point of
Origin
[0, 0]
Ordered pair:
[3, 2]
3 is x value and
2 is y value
COACH LESSON 16
76
Scaling!
A scale is the ratio of the measurements of a drawing, a model, a map or a
floor plan, to the actual size of the objects or distances.
Example:
An architect’s floor plan for a museum exhibit uses a scale of 0.5 inch = 2
feet. On this drawing, a passageway between exhibits is represented by a rectangle 3.75
inches long. What is the actual length of the passageway?
To find an actual length from a scale drawing, identify and solve a proportion.
Drawing = Drawing
Actual
Actual
Let p = the actual length in feet of the passageway
Use cross
0.5 = 3.75
products to 
2
p
solve the
proportion
0.5 x p = 2 x 3.75
0.5 p = 7.5
p
= 15
COACH LESSON 14
http://www.mathmaster.org/video/scale-and-indirect-measurement/?id=1858
77
SOLVING PROBLEMS USING PATTERNS!
Example: Erin is collecting plastic bottles. On Monday she has 7 bottles, on
Tuesday she has 14 bottles, on Wednesday she has 21 bottles, and on
Thursday she has 28 bottles. If the pattern continues, how many bottles will
she have on Friday?
1) Notice the pattern:
7,14,21,28
2) Write the different operations that you can perform on 7 to get 14.
a) 7 + 7 = 14
b) 7 x 2 = 14
3) Check these operations with the next term in the pattern.
c) 14 + 7 = 21
d) 14 x 2 = 28
4) Find the next term in the pattern to determine how many bottles Erin
will have on Friday.
5) 28 + 7 = 35
COACH LESSON 19
78
Estimation!
Estimating involves finding compatible numbers that will make the
numbers easier to operate.
There are 52 weeks in a year. Leo’s salary is \$51,950. Estimate
how much money Leo makes in one week.
Divide the compatible numbers.
\$52,000 divided 52 = \$1,000
COACH LESSON 10
79
Histogram is a bar graph without the spaces between the bars.
4
3
2
1
0
a
b
c
Bar graphs have spaces to show differences in data.
4
3
2
1
0
a
b
c
COACH LESSON 26
80
Double and Triple Bar & Line Graphs are used
to show two sets of related data.
6
5
4
Series 1
3
Series 2
Series 3
2
1
0
Category 1 Category 2 Category 3 Category 4
COACH LESSON 25
81
Making Predictions!
We can use trends or patterns seen in graphs to make predictions.
6
5
4
Series 1
3
Series 2
Series 3
2
1
0
Category 1
Category 2
Category 3
Category 4
COACH LESSON 31
82
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