Transcript EXPERIMENTAL PROBABILITY - Central Cambria School District

COMPREHENSIVE REVIEW FOR
MIDDLE SCHOOL MATHEMATICS
2013
COMPREHENSIVE REVIEW FOR
MIDDLE SCHOOL MATHEMATICS
Purpose: Mathematics Review for 7th Grade
(Can be used as enrichment or remediation for most middle school levels)
Contents: Concept explanations & practice problems.
Sources: PA Standards-PDE website.
Additional Reinforcement:
www.studyisland.com
www.ixl.com (links provided throughout)
www.mathmaster.org (links provided throughout)
and PSSA Coach workbook
Created by: Jessie Minor
EXPERIMENTAL PROBABILITY!
IN ORDER TO CALCULATE EXPERIMENTAL
PROBABILITY OF AN EVENT USE THE
FOLLOWING DEFINITION:
P(Event)=
3
Number of times the event occurred
Number of total trials
Coach Lesson 30
EXPERIMENTAL PROBABILITY!
Example:
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.
4
Experimental Probability IXL
PRACTICE EXPERIMENTAL PROBABILITY!
A spinner is divided into five equal sections numbered 1 through 5.
Predict how many times out of 240 spins the spinner is most likely to
stop on an odd number.
F.
G.
H.
I.
80
96
144
192
Marilyn has a bag of coins. The bag contains 25 wheat pennies, 15
Canadian pennies, 5 steel pennies, and 5 Lincoln pennies. She picks a
coin at random from the bag. What is the probability that she picked a
wheat penny?
F.
G.
H.
I.
5
10%
25%
30%
50%
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 of the sample space above– since
2, 3, and 5 are the only prime numbers in the same
space…
60
P(prime numbers)= 3/5 = ______%
6
Coach Lesson 29
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
7
Unit Prices IXL
COACH LESSON 4
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?
$7.99 x 3 = $23.97
$23.97 x 0.06 = $1.4382
Sales Tax = $1.44
8
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
9
COACH LESSON 23
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
10
PRACTICE USING THE DISTANCE FORMULA!
Gilda’s family goes on a vacation. They travel 125 miles in the
first 2.5 hours. If Gilda’s family continues to travel at this rate,
how may miles will they travel in 6 hours?
Distance = rate x time
300 miles
Michael enters a 120-mile bicycle race. He bikes 24 miles an
hour. What is Michael's finishing time, in hours, for the race?
d = rt
A
B
C
D
11
2
5
0.2
0.5
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
8 baskets =
4 games
2. Cross multiply & Divide
4x = 8 ( 12 )
4x = 96
x = 96
4
x= 24 baskets
12
x baskets
12 games
Ratios Word Problems IXL
COACH LESSON 7
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
9
6
2 3
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
13
Least Common Denominator IXL
COACH LESSON 1
PRACTICE FRACTIONS!
14
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
15
=
24
20
REDUCE!!!
1
1/5
Multiplying Fractions IXL
Dividing Mixed Numbers IXL
COACH LESSON 2
PRACTICE MULTIPLYING FRACTIONS!
3
4
X
5
8
16
5
6
1
X 7
49
13
1
91
5
9
X
4
9
4
5
Multiplying & Dividing Mixed Numbers!
When multiplying or dividing mixed numbers, always change
them to improper fractions, then multiply.
Example 1:
7
4
Example 2:
17
1¾ x 1½ =
x
3 = 21
2 8
2
5
8
12 x 2 ½ =
12 x 5 = 60 =30
1
2 2
Dividing Mixed Numbers IXL
Dividing Mixed Numbers!
When dividing any form of a fraction, change the division to
multiplication, then reciprocate the 2nd fraction.
Example:
18
1¾ ÷ 1½
=
7
4
÷ 3
2
7
4
x
2 =
3
14 =11/6
12
Dividing Fractions IXL
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
2
12
2
2
24: 2 2 2 3
32: 2 2 2 2 2
19
32
2 6
2
2 3
2x2x2x3x2x2 = 96
2
2
2
2
16
2
8
2
2
4
2
22 2
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,
20
so 8 is the GCF of 24 and 32.
PRACTICE LCM AND GCF!
What is the least common multiple of 3, 6, and 27?
A
B
C
D
3
27
54
81
What is the greatest common factor of 12, 16, and 20?
A
B
C
D
21
2
4
6
12
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
22
24
36
48
72
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
23
Absolute Value IXL
PRACTICE ABSOLUTE VALUE!
If x=-24 and y=6, what is the value of the expression
|x + y|?
A
B
C
D
24
18
30
-18
-30
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.
25
Distributive Property IXL
COACH LESSON 20
Associative & Commutative Property!






Associative

Commutative
Always has parentheses
A ( B X C) = B (C X A)
FOR MULTIPLICATION

A + (B + C) = B + (C + A)
FOR ADDITION



AXB=BXA
FOR MULTIPLICATION
A+B=B+A
FOR ADDITION
Properties for Multiplication IXL
26
Commutative Property for Addition IXL
Stem and Leaf Plots, Box – and – Whisker Plots
We use stem and leaf plots to organize
scores or large groups of numbers.
To arrange the numbers into a stem and leaf plot, the tens place
goes in the stem column and the ones place goes in the leaf column.
Example: We will arrange the following numbers in a stem & leaf
plot: 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
135
002
001
011
6
2
1
2
6
4
1
3
8
56689
2335778
58
Stem-and-Leaf-Plots IXL
27
COACH LESSON 24
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.
Stem
2
3
4
5
Leaf
135
002
001
78
011
668
2456689
1123357
2358
Upper quartile- 47
Lower quartile- 32
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.
28
COACH LESSON 27, 25
Box-and-Whisker Plot!
Lower
extreme
First
Second
quartile or quartile
lower
or median
quartile
Inter quartile
Range
29
Third
quartile or
upper
quartile
Upper
extreme
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
Stem
2
3
4
30
Leaf
3 5 6 7 7 7 9
0 1 5 9
0 5
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
31
Leaf
Range: 26
0
5 9
Mode: 14
1
4 4 4 6 6 9
Median: 17.5
2
1 2 2
3
0 1 1
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 in the statement.
This is why we use order of operation instead of the
Distributive Property.
32
COACH LESSON 5
PRACTICE ORDER OF OPERATIONS!
Karen is solving this problem:
(3² + 4²)² = ?
Which step is correct in the process of solving the problem?
A
(3² + 4⁴)
B
(9² + 16²)²
C
(7²)²
D
(9 + 16)²
More Practice!
1.) 3 + 2(4 x 3)
3.) (22 + 14) – 6
33
3 + 2(12)
3+ 24
27
36 – 6
30
2.) 12 - 15 - 3
4.) 64 – 8 + 8
-3 -3
-6
56 + 8
64
PRACTICE ORDER OF OPERATIONS!
Simplify the expression below.
(6² - 2⁴) · √16
A
16
B
64
C
80
D
108
Order of Operations Math Masters
1.) 2³ = 2 x 2 x 2 = 8
4.) √144 = 12
2.) 3⁴ = 3 x 3 x 3 x 3 = 81
5.) √64 =
3.) 4² = 4 x 4 =
34
8
16
Order of Operations IXL
FINDING THE MISSING ANGLE 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
Angle 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
Angle a = 130°
35
Measuring Angles IXL
Practice finding the measure of <A in the triangle ABC below!
A
30°
C
m<A + 90 + 30 = 180
m<A =
36
60 °
B
A square has 4 angles which each measure 90 degrees.
D
A
45
45
C
37
45
What is the total
measure of the
interior angles
of a square?
45
B
360 °
Pythagorean Theorem!
To find the missing hypotenuse of a right triangle, we use
the formula…
C²
C²
C²
C²
Height =
6 in
=
=
=
=
A²
+
B²
(6)² in + (8)² in
36 in² + 64 in²
100 in²
√C²= √100 in²
Base = 8 inches
38
C =
10 in²
Pythagorean Theorem MathMasters
AREA OF A TRIANGLE!
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.
39
Area of Triangles & Trapezoids IXL
COACH LESSON 12
PRACTICE FINDING THE AREA OF A TRIANGLE!
Area = ½ bh
A = ½ (2ft)(4ft)
A = ½ 8ft
Height= 4 ft
A =4 ft²
Base= 2 ft
40
FINDING THE AREA OF A PARALLELOGRAM!
h
b
41
AREA OF A RECTANGLE & A SQUARE!
Area of a RECTANGLE = Length
x
Width
Area of a SQUARE = Side
x
Side
Example:
2ft
2ft
4ft
42
2ft
A = l x w
A = s x s
A = 4ft x 2ft
A = 2ft x 2ft
A = 8ft²
A = 4ft²
Area of Rectangles Parallelograms IXL
CALCULATING PERIMETER!
PERIMETER IS THE OUTER DISTANCE AROUND A FIGURE.
9
FT
3FT
P = a+ b + c + …
P = 9FT + 9FT + 3FT + 3FT
27 FT
P = ____
43
CALCULATING 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
2FT
3FT
7FT
AREA = LENGTH X WIDTH
A = 2FT X 6FT
A = 12FT²
AREA = LENGTH X WIDTH
A = 3FT X 5FT
A = 15 FT²
TOTAL AREA = 12FT² + 15FT² = 27FT²
44
CONGRUENT ANGLES & CONGRUENT SIDES!
Congruent angles and sides mean that they have the same
measure. Use symbols to show this!
Complementary Supplementary Vertical & Adjacent Angles IXL
45
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
46
COACH LESSON 17
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
47
COACH LESSON 18
RECOGNIZING ADJACENT ANGLES!
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
Complementary Supplementary Vertical Adjacent Angles IXL
48
REVIEW: CLASSIFYING LINES!
Supplementary angles: sum is 180 degrees
Complementary angles: sum is 90 degrees
Straight angle: equal to 180 degrees
Complementary Supplementary Vertical & Adjacent Angles IXL
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
Symmetry IXL
Calculating Volume of a Quadrilateral!
[Volume= units³ or cubed units]
4 ft
5 ft
3 ft
V = 5ft x 3ft x 4ft = 60ft³
51
Volume IXL
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 ≈
For example: ∆ ABC ≈ ∆ XYZ
X
Which angle is similar to angle B?
Y
Angle: _______
A
B
52
C
Y
Z
Chord: line that
cuts the circle and
does
not
go
through the center
of the circle
Diameter:
distance
across the center of
the
circle
(double
radius)
Radius:
the distance
half 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
two radii
Circumference:
distance around the
outside of the circle
Arc:
a
connected
section
of
the
circumference
of
a
circle
53
COACH LESSON 15
Central angles: angles in
the center of the circle
formed by two radii
Inscribed angles:
angles on the inside of
the circle formed by
two chords
54
COACH LESSON 15
PRACTICE FINDING THE CIRCUMFERENCE OF A CIRCLE!
If the circumference of a circle s 16Π, what is the radius?
Hint: C= 2Πr
A
B
C
D
55
4
8
16
32
PRACTICE FINDING THE AREA OF A CIRCLE!
If the diameter of a car tire is 30 cm,
what is the area of that circle?
Round your answer.
Hint: Area = Π x r²
A
B
C
D
56
30.14 cm²
314 cm²
7,070 cm²
707 cm²
*USE ∏= 3.14
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
57
chord
radius
diameter
central angle
Adding Negative Numbers!
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
Add & Subtract Integers IXL
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
59
4÷2=2
Negative ÷ Positive = Negative
-4 ÷ 2 = -2
Multiplying & Dividing Integers IXL
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
60
COACH LESSON 3
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
One-step Linear Inequalities IXL
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
Two-step Linear Equations IXL
62
COACH LESSON 21
Modeling Mathematical Situations!
We can translate math sentences to numbers and symbols only
Examples:
Translate: “five more than”
(5 + n)
Translate: “three times a number”
(3 x n, or 3n)
When you combine both: “five more than three times a number”
5 + 3n or
63
3n +5
COACH LESSON 22
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
A function is when we put a value in and get an answer out.
Evaluating Functions IXL
64
COACH LESSON 20
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: 4,234 = (4 x 10³) + (2 x 10²) + (3 x 10¹) + (4 x 10⁰)
4000
+ 200
+
30
+
4,234
Any number raised to the zero power equals 1.
10 ⁰ = 1
Any number raised to the 1st power equals that number.
65
8¹ = 8
4
=
METRIC SYSTEM & CONVERSTION!
Deka
Kilo
Hecto
Meter
Liter
Gram
Deci
Centi
Milli
START at the unit you currently have, then move the
decimal to the unit you’re looking for.
66
Example 1:
4 kilometers = 4000 meters
Example 2:
36 millimeters = 3.6 centimeters
COACH LESSON 11
PRACTICE UNIT CONVERSIONS!
The students in a math class measured and recorded their heights on a chart
in the classroom. Keith’s height was 1.62 meters. Which is another way to
show Keith’s height?
A
B
C
D
0.162 cm
16.20 cm
162 cm
1,620 cm
A drawing of the Greensburg Airport uses a scale of 1 centimeter = 300
meters. Runway A is drawn 12 centimeters long. How many meters is the
actual length of the runway?
F
G
H
J
67
300
360
3,000
3,600
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 PSSA charts
provided during testing time.
68
PRACTICE WEIGHT UNIT CONVERSIONS!
Which of the following is a metric unit for measuring mass?
A
B
C
D
meter
liter
pound
gram
The flower box in front of the city library weighs 124 ounces.
What does the flower box weigh in pounds?
*Hint: 1 pound = 16 ounces
A
B
C
D
69
7½
7¾
868
1984
PRACTICE MORE UNIT CONVERSIONS!
A scientist measures the mass of a rock and finds that it is 0.16 kilogram.
What is the mass of the rock in grams?
A
B
C
D
70
1.6 grams
16 grams
160 grams
1,600 grams
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
c) So 240 feet x 1 yard
1
3 feet
1 yard
3 feet
d) Then 240 feet = 80 yards
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COACH LESSON 9
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
Ratios IXL
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COACH LESSON 7
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
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Practice Irrational Numbers!
Which of these is an irrational number?
A
B
C
D
-2
√56
√64
3.14
Which of these is an irrational number?
A
√3
B
-13.5
C
7
11
D 1
√9
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Converting Rational Numbers!
Fraction
Decimal
Percent
Place number
over its place
value and
reduce
Divide by 100
Multiply by 100
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%
75 =
100
75
3
4
COACH LESSON 4
Points on a Coordinate Grid!
Quadrant II
Quadrant I
Point of
Origin
[0, 0]
Quadrant III
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Ordered pair:
[3, 2]
3 is x value
and
2 is y value
Quadrant IV
COACH LESSON 16
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
Scale & Indirect Measurement MathMaster
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COACH LESSON 14
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)
78
28 + 7 = 35
COACH LESSON 19
Estimation!
Estimating involves finding compatible numbers that will make the
numbers easier to operate.
Leo’s yearly salary is $51,950. Estimate how much money Leo
makes in one week.
$51,950 is about $52,000.
Divide the compatible numbers.
$52,000 divided by 52 = $1,000
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COACH LESSON 10
Histogram is a bar graph without the spaces between the bars.
4
2
0
Bar graphs have spaces to show differences in data.
4
2
0
a
b
c
Interpret Histograms IXL
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COACH LESSON 26
Double and Triple Bar & Line Graphs are used
to show two sets of related data.
6
5
4
Series 1
3
Series 2
2
Series 3
1
0
Category Category Category Category
1
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2
3
4
COACH LESSON 25
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 1Category 2Category 3Category 4
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COACH LESSON 31
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