Download Now you would construct your box and whisker plot.

2
3
This reference book was created using the Ohio
Department of Education Mathematics Standards
for 7th grade and the Lebanon City Schools 7th
Grade Pre-Algebra Pacing Guide. I have done
extensive studying and researched many articles
to provide the most accurate information. I have
looked at techniques that will enable students to
obtain the greatest benefit in Mathematics
Education. This reference book is meant to be a
guide and tool for students and parents to use
when they need extra information and resources
on topics that are being studied in class.
Review Topics
Decimal Place Value……………………………………………
Adding and Subtracting Decimals…………………………...
Multiplying and Dividing Decimals……………………………
Repeating/Terminating Decimals…………………………….
Rounding………………………………………………………….
Equivalent Fractions……………………………………………. .
Adding and Subtracting Fractions……………………………
Multiplying and Dividing Fractions……………………………
Long Division………………………………………………………
6
7
7
8
9
9
10
11
12
Number Sense
Absolute Value…………………………………………………..
Exponents………………………………………………………...
Scientific Notation………………………………………………
Adding and Subtracting Integers……………………………
Multiplying and Dividing Integers…………………………….
Order of Operations…………………………………………….
One and Two Step Equations…………………………………
Percent and Percent Proportions…………………………….
Percent of Increase/Decrease………………………………..
Sales Tax and Discounts………………………………………..
15
15
16
18
19
20
21
23
26
Patterns, Functions and Algebra
Sequences……………………………………………………….
Finding Patterns…………………………………………………
Input/Output Tables…………………………………………….
Coordinate Plane……………………………………………….
Slope and y-intercept………………………………………....
Graphing Equations…………………………………………….
Solve Inequalities………………………………………………..
Simplifying Expressions (combine like terms)……………….
29
30
31
33
35
37
40
42
5
Measurement and Geometry
Customary Unit Conversion…………………………………..
Metric Unit Conversion…………………………………………
Geometric Shapes……………………………………………..
Perimeter…………………………………………………………
Area……………………………………………………………….
Surface Area…………………………………………………….
Volume……………………………………………………………
Scale Factor/Proportions………………………………………
Types of Triangles………………………………………………..
Pythagoreans Theorem………………………………………..
Translation, Rotation, Reflection, Dilation…………………..
44
45
47
47
48
50
53
55
56
57
59
Data Analysis and Probability
Measures of Central Tendency………………………………
Scatter plot………………………………………………………..
Histogram…………………………………………………………
Box and Whiskers………………………………………………..
Stem and Leaf Plot……………………………………………..
Bar Graph…………………………………………………………
Circle Graph………………………………………………………
Probability………………………………………………………...
Theoretical Probability………………………………………….
Experimental Probability………………………………………
Tree Diagram…………………………………………………….
Simple Probability……………………………………………….
Compound Probability…………………………………………
62
63
64
65
67
69
70
71
72
72
74
75
76
Glossary of Terms……………………………………………………….
78
Technology Links……………………………………………………….
87
Homework Help………………………………………………………..
96
Ohio State Standards…………………………………………………
99
Real Number Properties Review…………………………………….
101
Multiplication Chart…………………………………………………….
102
6
7
Decimal Place Value
.
Tenths
Hundredths
Thousandths
Ten
thousandths
Do you notice that the decimal point separates whole numbers from
decimals? Decimals are parts of a whole number and are another way of
expressing a fraction. In other words, 0.7 can also be expressed as
.
When reading a decimal aloud, you would say:
1. The whole number
2. (And ) for the decimal point
3. The whole number of the decimal with the place value of the last digit.
Example:
21,300.25
Twenty-one thousand three hundred AND twenty-five hundredths
8
Adding and Subtracting Decimals
When adding and subtracting decimals you MUST line up the
decimal points in the problem. Use place holding zeros when necessary.
For example:
12 – 4.08 =
AND
4.26 + 13.77=
12.00
- 4.08
7.92
13.77
+ 4.26
18.03
Be sure to line up the decimals!
Multiplying and Dividing Decimals
When multiplying decimals, don’t consider the decimals until after you
have multiplied. Then, count the number of digits BEHIND the decimal(s)
in the problem. This is how many space you will move the decimal to the
LEFT in your answer.
For example:
.004 x .02=
.004
x.02
.00008
4x2=8
There are 5 digits behind the decimals in the problem
8.0 → move decimal 5 places left → .00008
If the number in a division box has a decimal, but the number outside of
the division box does not have a decimal, place the decimal point in the
quotient (the answer) directly above the decimal point in the division box.
Divide normally.
.002
5 0.010
9
If the number outside of the division box has a decimal, but the number
inside of the division box does not, move the decimal on the outside
number however many places needed to make it a whole number. Then
to the right of the number in the division box (a whole number with an
"understood decimal" at the end) add as many zeros to match the
number of places the decimal was moved on the outside number. Place
the decimal point in the quotient directly above the new decimal place
in the division box.
20
0.05 1  0.05 1.00  5 100
If both the numbers inside and outside of the division box have decimals,
count how many places are needed to move the decimal point outside
of the division box (the divisor) to make it a whole number. Move the
decimal point in the number inside of the division box (the dividend) the
same number of places. Place the decimal point in the quotient (the
answer) directly above the new decimal point.
.2
.05 .01  5 1  5 1.0
Repeating Decimal
A decimal that repeats in a specific pattern over and over. It never
terminates or stops.
Example:
1
= .3333
3
The line over the decimal indicates it is repeating.
Terminating Decimal
A decimal that terminates, means it stops at a certain point.
Example: ¾ = .75
10
Rounding
When rounding, determine which place value you need to round to.
Underline that digit. Look to the right. This digit will determine if your
underlined digit stays the same or goes up one digit. Make the changes
(if necessary) to the digit then fill in the rest of the places with zeros.
4 and below= stay the same
5 and above= move up one digit
Example: Round to the nearest hundreds place.
2,542
→
2,500
The number to the right of the hundreds digit is a 4. You would keep the
hundreds digit the same. Then the 4 and 2 need to be replaced with
zeros.
Round to the nearest thousands place.
67,902
→
68,000
Equivalent Fractions
Equivalent fractions are fractions that have the same value.
1 2

2 4
We often use equivalent fractions when reducing fractions to their lowest
terms.
We do this by finding the greatest common factor of the numerator and
denominator. We then divide the numerator and denominator by the
GCF to get the equivalent fraction.
27
27  9 3
 GCF  9  

36
36  9 4
GCF = Greatest Common Factor
11
Adding and Subtracting Fractions
When the denominators are the same, you may add and subtract across
the numerator with out making any changes. The denominator always
stays the same.
1 2 3
 
5 5 5
Example:
or
7 3 4
 
8 8 8
When the denominators are different, you must find a common
denominator. This is done by finding the least common multiple (LCM) of
the numerator and denominator. Then you must use the LCM as the
denominator for both fractions. Make equivalent fractions for each
fraction in the problem.
5= 5, 10, 15, 20
3 1
 
Example:
Multiplies of 3= 3,6, 9, 12, 15, 18
5 3
LCM = 15
Find equivalent fractions,
3 3 3 9
1 1 5 5



and 
5 5  3 15
3 3  5 15
So…
9 5 14
 
15 15 15
The same rules apply for subtracting fractions
5 1
 
6 2
Example :
Multiples of
2= 2, 4, 6
6= 6,12,18
5
 stays
6
LCM = 6
and
1 1 3 3


2 23 6
5 3 2
22 1
   reduces

6 6 6
62 3
12
Multiplying and Dividing Fractions
When multiplying fractions, multiply across the numerators and
denominators. Put your answer in lowest terms (simplest form/reduce)
when necessary.
3 4 12
 
7 5 35
3 2 6
66 1
  

9 3 18 18  6 3
or
When dividing fractions, follow the
K= Keep
C=Change
rule.
F=Flip
In a division problem keep the first fraction the same, change the division
sign to a multiplication sign, flip or find the reciprocal of the second
fraction.
3 4
 
4 5
becomes
3 5
 
4 4
Multiply across
3 5 15
 
4 4 16
ALWAYS REDUCE TO
LOWEST TERMS!
13
Long Division
1. Write the dividend and divisor in this form:
The procedure involves dividing the divisor (4) into a number for each digit of the
dividend (950).
2.The first number to be divided by the divisor (4) is the leftmost digit (9) of the dividend.
Ignoring any remainder, write the result (2), above the line over the leftmost digit of the
dividend. Multiply the divisor by that number (4 times 2) and write the result (8) under the
leftmost digit of the dividend.
3. Subtract the bottom number (8) from the number immediately above it (9). Write the
result (1), under the bottom number (8), then copy the next digit of the dividend (5) to
the right of the result of the subtraction.
4. Repeat steps 2 and 3, using the newly created bottom number (15) as the number to
be divided by the divisor (4), and write the results above and under the next digit of the
dividend.
14
5. Repeat step 4 until there are no digits remaining in the dividend. The number written
above the bar (237) is the quotient, and the result of the last subtraction is the remainder
for the entire problem (2).
The answer to the above example is expressed as 237 with remainder 2. Alternatively,
one can continue the above procedure to produce a decimal answer. We continue the
process by adding a decimal and zeroes as necessary to the right of the dividend,
treating each zero as another digit of the dividend. Thus the next step in such a
calculation would give the following:
Be patient  Follow the rules!
15
16
Absolute Value
Absolute value is the absolutely positive value of any number. It is
represented when a positive or negative integer has a bar on each side
of the digit.
6
or
8
To find the absolute value of a number, you just take the positive value of
the given number.
10  10
and
16  16
Exponents
An exponent is made up of an exponent and a base.
A base, tells you the number you will be multiplying.
A exponent, tells you how many times to multiply the base by itself.
64
6 (the base) will be multiplied by itself 4 (the exponent) times.
6 x 6 x 6 x 6 = 1296
17
Scientific Notation

Numbers expressed in scientific notation are written as the product
(multiplication) of a factor and a power of 10

The first factor is always > than or = to 1 AND < 10

The second factor is always the number 10 with an exponent

For example:
2 x 10³ → Correct
To take a number from standard form to scientific notation, put a
decimal after the first digit. Count the number of digits (including
zeros) to the RIGHT of the decimal. The number of digits will be your
exponent for the second factor.
The speed of light is 670,000,000 miles per hour. This in scientific
notation would look like this:
6.7 x 108
1. Place the decimal point after the first digit (which makes the
factor greater than 1 but less than 10) 6.70,000,000
2. Count the number of digits to the RIGHT of the decimal point. 8
3. So… since there are 8 digits to the RIGHT of the decimal point,
the exponent of 10 (always the second factor) is 8.
670,000,000 = 6.7 x 108
To take a number from scientific notation to standard form,
8.024 x 106
1. Write out the first factor. 8.024
2. Look at the exponent. Add zeros to the first factor until you have as
many digits to the right of the decimal as your exponent. 8.024000
3. Erase the decimal point and put commas when appropriate.
8,024,000
18
Scientific Notation with Negative Exponents
When changing a decimal number (with no whole number) from standard
form to scientific notation, you must
.00002405
1. Place a decimal in the number that will make a number greater
than 1 but less than 10. Keep the original decimal.
.00002.405
2. Count the number of digits in between the two decimals. This will
be your exponent. 5
3. Erase the original decimal and all the zeros before your new
decimal. This number will become your first factor. 2.405
4. All decimal numbers (really small numbers) will have negative
exponents. Make your exponent negative.
2.405 x 105
When changing a number from scientific notation to standard form with a
negative exponent, you must
1.259 x 107
1. Write out your first factor.
1.259
2. Add as many zeros to the left of the decimal as needed so there
are as many digits as your exponent.
0000001.259
3. Put a new decimal on the left end of your number. Erase the old
decimal. This is the number in standard form.
.0000001259
19
Adding and Subtracting Integers
When adding integers…
When the signs are different:
1. Find the absolute value of each number
2. Find the difference between the numbers (subtract the absolute
values)
3. For your answer, use the sign of the number with the greater
absolute value
Example:
6  3  6  3  3
-6 + 3 =
Absolute value of -6 is 6
Absolute value of 3 is 3
6-3=3 the great absolute value is 6, it’s sign was negative
Answer
-3
When adding integers and the signs are the same,
Keep the sign and add normally.
Example:
-7 + -4 = -11
or
2 + 3= 5
When subtracting integers, use
K= Keep
1.
2.
3.
4.
C=Change
C=Change
Keep the first integer the same.
Change the subtraction sign to addition
Change the sign of the last integer
Then add using the rules of addition from above
Example:
-6 - (-9)=
-6
Keep
+
Change
9=
Change
3
20
Multiplying and Dividing Integers
When multiplying integers,
1. Change any negative integers to positive integers
2. Multiply the integers normally
3. Find the sign this way:
a. if the signs are the same then the product is positive
b. if the signs are different, then the product is negative
When dividing integers,
1. Change any negative integers to positive integers
2. Divide the integers normally
3. Find the quotient by:
a. if the signs are the same, the quotient is positive
b. if the signs are different the quotient is negative
Example:
SAME
-9 x -7 = 63
24   3  8
DIFFERENT
-9 x 7 = -63
24   8  3
Same signs = positive answer
Different Signs = negative answer
21
Order of Operations
P
E
MD
AS
Please Excuse My Dear Aunt Sally
This acronym will help you remember the steps to solving a problem using
the Order of Operations.
First……
Second…..
Third…..
Fourth….
P= Parentheses
E= Exponents
MD= Multiplication and Division
AS= Addition and Subtraction
In the third and fourth steps (MD/AS), you complete these in
order from left to right.
3 + 22(1+8) =
P
(1+8)=9
3 + 2 2 (9)=
E
2 2 =4
3 + 4(9)=
MD
4(9)=36
3 + 36=
AS
3+36=39
39
A helpful tip is to actual write out
P E MD AS next to
each problem. As you complete each step of the Order of Operations,
cross off the letter. This will allow you to keep track of what operation
needs to happen next.
22
One Step Equations
To solve one-step equations use the inverse operation.
Addition is the inverse of subtraction, so subtraction is the inverse of
addition.
Multiplication is the inverse of division, so division is the inverse of
multiplication.
Step One: Identify the inverse operation
Step Two: Get the variable (the letter) by itself
Whatever you do to one side of the equation you MUST do to the other
Addition
x  4  11
Subtraction
(inverse)
-
-4 = -4
X=7
(inverse)
x 3  9
+
+3 = +3
x = 12
Always check to make sure your answer is correct. Plug in each value for
the original variable.
X=7
7+4=11
11=11
x=12
12-3=9
9=9
Multiplication
Division
7x = 49
7x = 49
7
7
X=7
x =7
7(7)=49
49=49
(Inverse)

x
2
4
x
(Inverse) 4   2  4
4
x
x=8
x=8
8
2
4
2=2
23
Two Step Equations
Solving a two step equation is very similar to solving a one step equation.
Do the inverse operation for
addition or subtraction first.
Do the inverse operation of
multiplication or division last.
Another example:
A more advanced two step equation would follow these steps.
10 
m
2
4
24
Percents
Percent mean “per hundred”.
A percent is just a fraction. When we write the percent, we are just writing
the numerator of the fraction (the percent) over 100. 100 will ALWAYS be
the denominator of a percent fraction.
Changing fractions to percents
To change a fraction to a percent you must follow these steps.
1. Divide the numerator by the denominator (top by bottom). This
is the same process as changing a fraction to a decimal.
2. Multiply by 100.
Step 1
3
3
4
Step 2
.75
4
.75 x 100= 75%
Changing percents to fractions
To change a percent to a fraction follow these rules.
1. Percents are ALWAYS out of 100. Your denominator will ALWAYS be
100.
2. Your percent becomes your numerator.
47% =
47
100
Changing decimals to percents
To change a decimal to a percent follow these rules.
1. Move your decimal point 2 places to the right. This is the same
as multiplying by 100.
.36 = .36 x 100= 36%
25
Changing percents to decimals
To change a percent to a decimal you must follow these rules.
1. Start where the original decimals point is in the percent.
2. Move the decimal 2 places to the left. This is like dividing by 100.
96% = 96.0= 96/100=
.96
Percent Proportions
This is how you compare numbers and percents. It is
important to remember that percents can be written as fractions and
ratios can be written as fractions. We are comparing the part to whole
ratio of percents and numbers.
_PART_
WHOLE
or
is
of
-Tip- When you see of _____, whatever is after of will be your WHOLE number
When you see is that will be your part.
If it says what number or what percent, that will be your variable
This usually works.
Example 1
Find 70% of 90.
Step 1
Set up % proportion
70
x

100 90
Step 2
Cross Multiply
100x  6300
Step 3
Solve 1 Step Equation
(Divide each side by 100)
100 x  6300
26
x  63
Example 2
What number is 75% of 600?
Step 1
Set up % proportion
75
x

100 600
Step 2
Cross Multiply
100x  45000
Step 3
Solve 1 Step Equation
(by dividing)
100x  45000
x  450
Example 3
What percent of 250 is 25?
Step 1
Set up % proportion
x
25

100 250
Step 2
Cross multiply
250x  2500
Step 3
Solve 1 Step Equation
(by dividing)
250x  2500
x  10
27
Percent of Change
Percent of increase
This is when the price goes UP.
1. SUBTRACT to find the amount of change
2. Solve a % proportion
x
amountofchange

100 originalamount
Remember you are solving for a percent.
3. Cross multiply and divide to find the percent of increase.
Example:
The junior high school’s enrollment changed from 1200 to 1350 students.
Find the percent of change in enrollment. Round to the nearest whole
percent.
1.
1350 – 1200= 150
2.
x
150

100 1200
3.
x  12.5
Cross Multiply and Divide
Rounded is 13%
The increase in student enrollment is 13%
28
Percents of decrease
This is when the price goes down.
1. Divide the new amount by the original amount.
2. Subtract 1 from the result.
3. Write the decimal as a percent.
new
 1100  percent of decrease
old
Example:
Potatoes baked in the oven require 60 minutes to cook. A pressure
cooker can do the same job in 20 minutes. Find the percent of change in
cooking time. Round to the nearest whole percent.
1.
20  60  .333
2.
.333 – 1= -.6666
3.
-.666= -66.66 %
66.66% rounded = -67%
The percent of decrease is 67%
29
When using percent of increase or decrease, you must first decide if it’s
going up or down.
UP = % of Increase
Down = % of Decrease
Sales Tax and Discounts
Sales tax is a percent of the purchase price.
We will continue using proportions to solve these problems.
Tax _ or _ Discount %

price
100
1. Set up the proportion
2. Cross Multiply
3. Divide to find the variable
Example1: Find the total price of a $17.75 soccer ball if the sales tax is 6%.
x
6

17.75 100
106.5 = 100x
1.065 = x
Round 1.065 to $1.07
Now add the tax back to the original price to find the final total.
$17.75 + 1.07 = $18.82
30
Example 2:
Find the price of a $69.95 tennis racket that is on sale for 20% off
x
20

69.95 100
1399 = 100x
13.99 = x
Now subtract the 20% from the original price to find the discounted price.
$69.95 -13.99 = $ 55.96
31
32
Sequencing
There are 2 basic types of sequencing.
Arithmetic Sequences
An Arithmetic Sequence is made by adding the same value each time.
1, 4, 7, 10, 13, 16, 19, 22, 25, ...
This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time.
3, 8, 13, 18, 23, 28, 33, 38, ...
This sequence has a difference of 5 between each number.
The pattern is continued by adding 5 to the last number each time.
Geometric Sequences
A Geometric Sequence is made by multiplying by the same value each
time.
2, 4, 8, 16, 32, 64, 128, 256, ...
This sequence has a factor of 2 between each number.
The pattern is continued by multiplying the last number by 2 each time.
3, 9, 27, 81, 243, 729, 2187, ...
This sequence has a factor of 3 between each number.
The pattern is continued by multiplying the last number by 3 each time.
33
Finding Patterns
When faced with a sequence, you need to look at it and see if you can get a "feel" for
what is going on.
Find the next number in the following sequence:
1, 4, 9, 16, 25,....
12 = 1, 22 = 4, 32 = 9, 42 = 16, and 52 = 25.
The next number in the sequence is
62 = 36.
It looks as though the pattern here is squaring. That is, for the first term (the 1-st term),
they squared 1; for the second term (the 2-nd term), they squared 2; for the third term
(the 3-rd term), they squared 3; and so on. For the n-th term ("the enn-eth term"), they
will want me to square n. In particular, for the sixth term, they will want me to square 6.
Find the next number in the following sequence:
2, 5, 10, 17, 26,....
To find the pattern, I will list the numbers, and find the differences. That is, I will
subtract the numbers in pairs (the first from the second, the second from the third,
and so on), like this:
The next term is 37.
You would add 11 to 26 to find this number.
really
You may have to look
hard to find the pattern. Don’t
give up! It’ll appear.
34
Input/Output Tables
Function machines such as the one below help students visualize how a
rule associates each input value with an output value.
Imagine a machine that works like this. When a number goes in (the
input) it’s dropped into the machine. The machine changes the number
according to the rule. Then a new number comes out (the output).
For example:
This machines rule is to add 5 to what is put in.
Rule:
f ( x)  X + 5
So…….. if 4 goes in, 9 comes out.
if 7 goes in, 12 comes out.
If 53 goes in, 58 comes out.
If -6 goes in, -1 comes out.
These “in” and “out” numbers can be displayed in a table like this:
Input
4
7
53
-6
Output
9
12
58
-1
35
To solve these types of problems you may need to find missing information for
both inputs and outputs OR you may even need to find the rule. Let’s try an
example problem:
Rule: f ( x)  x  7
Input
9
12
27
51
Output
Rule: f ( x )  3 x
Input
3
Output
12
5
6
21
Rule:
Input
f ( x) 
x
5
Output
1
10
3
25
Rule: ?
____________________
Input
2
3
4
5
6
7
Output
1
1.5
2
2.5
3
3.5
36
Coordinate Plane
The plane determined by a horizontal number line, called the x-axis,
and a vertical number line, called the y-axis, intersecting at a point
called the origin. Each point in the coordinate plane can be specified by
an ordered pair of numbers.
There are four quadrants in a coordinate plane. Labeled above.
Points in Quadrant 1 have positive x and positive y coordinates.
Points in Quadrant 2 have negative x but positive y coordinates.
Points in Quadrant 3 have negative x and negative y coordinates.
Points in Quadrant 4 have positive x but negative y coordinates.
The center of the coordinate plane is called the origin. It has the coordinates of
(0,0).
37
You can locate any point on the coordinate plane by an ordered pair of
numbers (x,y) called the coordinates.
Here the ordered pair is (2,3)
The x value is 2 and the y value is 3.
When an ordered pair is used to locate a point on a grid, the two
numbers are called the 'coordinates' of the point. In the diagram above,
the point (2, 3) has been marked with a dot. The coordinates of this point
are '2' and '3'.
38
Slope and Y-Intercept
Every straight line can be represented by an equation: y = mx + b. The
coordinates of every point on the line will solve the equation if you
substitute them in the equation for x and y.
Slope = change in y
change in x
The steepness of a line is called its SLOPE.
The vertical change is called the change in y. (Up and Down)
The horizontal change is called the change in x. (Right and Left)
We often refer to slope as…
rise
run
Point Slope Form
The slope of a line can also be found by using the coordinates of any two
points on the line.
Slope= difference in y-coordinates
=
y2  y1
x2  x1
Difference in x-coordinates
For example:
Find the slope of a line that contains the points A (-1, -2) and B (-4, -3).
Slope = (-2) – (-3)
(-1) – (-4)
-2 + 3
or
1
3
-1 +4
When a line has positive slope, it rises from left to right.
When a line has a negative slope, it falls from left to right.
39
The equation of any straight line, called a linear equation, can be written
as: y = mx + b, where m is the slope of the line and b is the y-intercept.
The y-intercept of this line is the value of y at the point where the line
crosses the y axis.
Remember
Let's use these two points to calculate the slope m of this line.
A = (1,1) and B = (2,3)
Subtract the y value of point A from the y-value of point B to find the change in the y
value, which is 2. Then subtract the x value of point A from the x value of point B to find
the change in x, which is 1. The slope is 2 divided by 1, or 2.
40
Graphing Equations
Remember…
We compare the x-axis to a number line. If you move right from the origin, your x
value will be a positive number. If you move left from the origin, your x value will
be a negative number. We compare the y-axis to a thermometer. As you go up,
the y value will be a positive number. As you go down the y value will be a
negative number. Use this table as a reference
x
Move right (positive number)
y
Move up (positive number)
Move left (negative number)
Move down (negative
number)
…by plotting points on a coordinate plane
The first thing you must do is to set up a function table like below. Use the
“x” values of -3, 0, 3 for all tables unless otherwise given.
x
-3
0
3
y
(x,y)
Next you apply the Rule or Equation. For example:
Y = 2x + 2
x
-3
0
3
y
-4
2
8
(x,y)
(-3,4)
(0,2)
(3,8)
The next step will be to plot these points on a coordinate graph and draw
the line that contains them.
41
…by using slope and y intercept
The point where a graph intersects an axis is called an INTERCEPT of the
graph.
The Y-INTERCEPT is the value of an equation when x= 0.
For Example:
To find the y-intercept
let x = 0
y= x + 3
y=0+3
y=3
To graph lines you must know that:
y = mx + b
Where
For example: y =
m= slope
b= y-intercept
2
x2
3
2
and Y-intercept (b)= (0,2)
3
On the graph you will:
Slope (m) =
1. Plot y-intercept
2. Locate other points by using slope
3. Connect the points with a line
42
To graph a linear equation, we can use the slope and y-intercept
1.
Locate the y-intercept on the graph and plot the point.
2.
From this point, use the slope to find a second point and plot it.
3.
Draw the line that connects the two points.
43
Solving Inequalities
Solving linear inequalities is very similar to solving linear equations, except
for one detail: you flip the inequality sign whenever you multiply or divide
the inequality by a negative. The easiest way to show this is with some
examples:
1)
The only difference here is that you have
a "less than" sign, instead of an "equals"
sign. Note that the solution to a "less
than, but not equal to" inequality is
graphed with a parentheses (or else an
open dot) at the endpoint.
Graphically, the solution is:
2)
Note that "x" does not have to be on the
left, but it is often easier to picture how
to deal with it this way. Don't be afraid to
rearrange things to suit your taste.
Graphically, the solution is:
3)
Same ol', same ol', but with a "less than
or equal to" sign, instead of a plain
"equals". Note that the solution to a "less
than or equal to" inequality is graphed
with a bracket (or else a closed dot) at
the endpoint.
Graphically, the solution is:
44
4)
Divide both sides by a positive two.
Copyright © Elizabeth Stapel 2000-2007 All Rights
Reserved
Graphically, the solution is:
5)
This is the special case noted before.
When we divided by the negative two,
we had to flip the inequality sig
Graphically, the solution is:
Look at these examples:
45
Simplifying Expressions
Algebraic expressions contain alphabetic symbols as well as
numbers. When an algebraic expression is simplified, an equivalent
expression is found that is simpler than the original. This usually means
that the simplified expression is smaller than the original.
There is no standard procedure for simplifying all algebraic expressions
because there are so many different kinds of expressions, but they can
be grouped into three types:
(a) those that can be simplified immediately without
any preparation.
(b) those that require preparation before being
simplified.
(c) those that cannot be simplified at all.
Type A
2x + 3y - 2 + 3x + 6y + 7
This expression can be simplified by identifying like terms and then
grouping and combining like terms, like this:
·
+2x and +3x are like terms, and can be combined
to give +5x,
·
+3y and +6y combine to give +9y, and
·
-2 and +7 combine to give +5.
So after simplifying, this expression becomes:
5x + 9y + 5
Type B
Distribute into the parenthesis
Combine like terms by adding coefficients and constants
46
47
Customary Unit Conversion
Quantity measured
Length
Unit
Symbol
inch
in
foot
yard
mile
ft
yd
mi
Mass
(“weight”)*
ounce
oz
pound
ton
lb
T
Area
square inch
in²
square foot
square yard
ft²
yd²
ounce
oz
cup
pint
quart
c
pt
qt
gallon
g
Volume
Relationship
1 ft = 12 in
1 yd = 3 ft or 36 in
1 mi = 1760 yd or 5280 ft
1 lb = 16 oz
1 T = 2000 lb
1 ft² = 144 in²
1 yd² = 9 ft²
1 c = 8 oz
1 pt = 2 cups
1 qt = 2 pt
1 g = 4 qt
In general, when you convert from a smaller unit to a larger one you will
divide.
2r 8
8
2
32 12 12 32  2  2 ft
Ex. 32 in = _______ ft
12
3
20  8  oz/c   2  c/pt   2  pt/qt  
Ex. 20 oz - _______ qt
20 1 1 1 20 5
   
 or 0.625qt
1 8 2 2 32 8
**Remember that when dividing you may multiply by a number’s reciprocal-1
the reciprocal of n is .
n
***Sometimes you must repeatedly divide or multiply a given amount to convert it to
another unit of measurement.
So, when you convert from a larger unit from a smaller unit you will
multiply.
Ex.
3
c = _______ oz
4
Ex. 2 qt = _______ c
3 16oz 48


 12 oz
4 1c
4
2  2 pt/qt   2  c/pt  = 8 c
48
Metric Unit Conversion
Quantity measured
Length, width,
distance, thickness,
girth, etc.
Mass
(“weight”)*
Area
Volume
Unit
Symbol
Relationship
millimeter
mm
10 mm = 1 cm
centimeter
meter
kilometer
cm
m
km
100 cm = 1 m
milligram
mg
gram
kilogram
metric ton
g
kg
t
1 kg = 1000 g
1 t = 1000 kg
square meter
m²
1 m² = 10,000 cm²
square kilometer
km²
1 km² = 1,000,000 m²
milliliter
mL
cubic centimeter
liter
cm³
L
1 km = 1000 m
1000 mg = 1 g
1000 mL = 1 L
1 cm³ = 1 mL
1000 L = 1 m³
The same general rules apply when converting between units in the
metric system: you will be dividing when changing from a smaller unit to a
larger one and multiplying when changing from a larger unit to a smaller
one.
**Remember here that you may use the shortcut of moving the decimal point to the
right when multiplying or to the left when dividing.
Ex. 135 cm = _______ m
Ex. 0.75 kg = _______ mg
135  100 cm/m   1.35 m
0.75  1000  g/kg   1000  mg/g   750000 mg
**You may also express your answer in scientific notation: 750000  7.5105
Ex. 1.5 L = _______ mL
1.5  1000 mL/L   1500mL
49
US Customary
Length
Weight
1 ft = 12 inch
1 yd = 3 ft
1 mi = 1760 yd
1 mi = 5280 ft
Capacity
1 lb = 16 oz
1 T = 2000 lb
1 gal = 4 qt
1 gal = 8 pt
1 gal = 16 c
1 qt = 2 pt
1 qt = 4 c
1 pt = 2c
1 c = 8 fl oz
When converting from a big unit to a small unit you multiply.
When converting from a small unit to a big unit you divide.
Metric Conversions
Kangaroos
Hop
Down
km
hm
dam
kL
hL
daL
kg
hg
dag
Mountains
Base Unit
Meter
(m)
Liter
(L)
Gram
(g)
Drinking
Chocolate
Milk
dm
cm
mm
dL
cL
mL
dg
cg
mg
When going from a big unit to a small unit you move to the right, therefore
you move your decimal the same amount of spaces to the right.
When going from a small unit to a big unit you move to the left, therefore
you move your decimal the same amount of spaces to the left.
50
Geometric Shapes
A polygon is a closed figure made by joining line segments, where each
line segment intersects exactly two others.
Geometric Shape
(polygon)
Triangle
(a) Equilateral
(b) Isosceles
(c) Scalene
Quadrilateral
(a) Trapezoid
(b) Parallelogram
(c) Rectangle
(d) Square
(e) Rhombus
Description
Diagram
A closed three-sided figure
All sides are congruent
Two congruent sides
No congruent sides
(b)
(a)
A closed four-sided figure
Two parallel sides
Two pairs of parallel sides
with the opposite sides
having the same length
A parallelogram with four
right angles
All sides are congruent with
four right angles
A parallelogram with all
sides congruent
(a)
(c)
(b)
(e)
(c)
(d)
Perimeter
The perimeter of a polygon is the distance around the outside of the
figure.
To find the perimeter of triangles, quadrilaterals, or other polygons all you
need to do is find the sum of the lengths of all the sides.
Ex. Find the perimeter of the triangle.
10 in
6 in
6  8  10  24 in
8 in
7 cm
Ex. Find the perimeter of the trapezoid
10 cm
7  10  10  16  43cm
10 cm
16 cm
51
Area
The area of a polygon is the measure of the region inside the figure. It is
the number of square units needed to cover its surface.
To find the area of triangles, quadrilaterals, or other polygons is given by
the general formula: A = b  h , where A is the area; b = the length of the
base; and h = the height of the figure.
Ex. Find the area of the rectangle.
2m
A=b  h
A = 6.2  2
A = 12.4m 2
6.2 m
**Note that the formula for the area of a square is A = s 2
s
Ex. Find the area of the triangle.
**Note that if you draw a diagonal line from
corner to corner, you will divide the
rectangle into two congruent triangles.
2 cm
1
2
The formula for finding the area of a
trapezoid is: A = 12  b1  b2  h
A=
A=
A=
 b1  b 2  h
1
2  7 15  5 
1
2  22  5 
1
2 110 
1
2
A=
1
2
b h
 2  2 
1
2  4
A = 2cm 2
b×h
Ex. Find the area of the trapezoid.
A=
1
2
A=
2 cm
Therefore, the formula for
the area of a triangle is
A=
A=
7 in
5 in
15 in
A = 110
52
***Circles
radius
diameter
The diameter is a straight line passing
through the center of a circle to touch both
sides of the circumference.
The radius is the distance from the center
of a circle to its circumference. It is half
the diameter
The distance around the outside of a circle is called the circumference
rather than the perimeter.
The formula for finding the circumference of a circle is:
C = 2 r or C =  d
The area of a circle is the measure of the region
inside the circle. It is the number of square units
needed to cover its surface.
The formula for finding the area of a circle is:
A =  r2
In calculations, we will
typically use the
fraction
22
or the
7
decimal 3.14 for pi ().
Ex. What is the circumference of a circle with a radius of 3 cm?
C = 2 r
C  2 3.14  3 
C  2 9.42 
**Note that you use the  symbol because you
C  18.84 cm
are using an approximate value for pi
.
Ex. Find the area of a circle with a diameter of 6 ft.
A =  r2
6
A =  3.14  
 2
A 
A 
2
 3.14  3 
 3.14  9 
2
**Remember that you need
to divide the diameter by 2
to get the radius.
6 ft
A  28.26ft 2
53
Surface Area
Three-dimensional shapes can be solid or hollow. They have width, height
and length.
The surface area of a three-dimensional figure is the total area of the
surface of a solid, including the area of the base(s) of the figure. That
means that you will be finding the areas of all the surfaces of a threedimensional figure and adding them up.
When using the formulas below you must remember that the width of a
rectangle may be the same as the base of a triangle – so it may be
helpful to redraw the three-dimensional shape in “pieces”.
Three-dimensional Shape
Rectangular
Prisms
h
Surface Area Formula
SA = 2lw +2lh +2wh
If you have a cube the formula for surface
area is: SA = 6s2
w
l
Triangular Prisms
SA = area of 2 triangular bases
+ the areas of the rectangular sides
Pyramids
SA = the area of the base (whatever shape)
+ the area of the faces or sides
Cylinders
radius
SA = 2 r 2   2 r  h
54
SA =  r 2 +  rs
where s = the slant height of the curved surface
Cones
s
r
Ex. Find the surface area of a rectangular prism with a
length or 5 in, width of 2 in and height of 4 in.
4 in
SA = 2lw +2lh +2wh
2 in
5 in
SA = 2  5  2   2  5 4   2  2  4 
SA = 2  10   2  20   2  8 
SA = 20 +40+16
SA = 76 in 2
Ex. Find the surface area of the triangular prism.
SA = area of 2 triangular bases
+ the areas of the rectangular sides


SA = 2  bh   2 lw two congruent sides   lw third side 
1
2
5 mm
4 mm
15 mm
6 mm
SA = 6  5  2  15  4    15  6 
SA = 30 + 2  60   80
SA = 30 + 120 + 80
SA = 230 mm 2
Ex. Find the surface area of the pyramid.
6 cm
SA = the area of the base (whatever shape)
+ the area of the faces or sides
SA = s 2  4  12 bh 
SA = 5  4  12  5  6  
5 cm
2
SA = 25 + 4 
 30  
SA = 25 + 4  15 
5 cm
1
2
SA = 25 + 60
SA = 85 cm 2
Ex. Find the surface area of a cylinder with a diameter of 3 in and a
height of 6 in.
SA = 2 r 2   2 r  h
SA = 2  1.5    2  1.5   6
SA   2  3.14  2.25    2  3.14  6 
3 in
2
SA  14.13   37.68 
SA  51.81in 2
6 in
55
**Remember that you need
to divide the diameter by 2
to get the radius.
Ex. Find the surface area of a cone with a radius of 4 in
and slant height of 7 in.
SA =  r 2 +  rs
SA =   4     4×7 
4 in
7 in
2
SA  3.14  16   3.14  28 
SA  50.24 + 87.92
SA  138.16 in 2
56
Volume
The general formula for finding the volume of a three-dimensional figure is:
V = Bh, where B is the area of the base
Three-dimensional Shape
Rectangular
Prisms
Volume Formula
V = lwh
h
w
l
Triangular Prisms
V = 12 lwh
Pyramids
V = 13 Bh
B = the area of the base
which may be a rectangle, triangle, or other shape
Cylinders
V =  r 2h
radius
h
Cones
V = 13  r 2h
h
r
Ex. Find the volume of a rectangular prism with a
length or 5 in, width of 2 in and height of 4 in.
V  lwh
V  5 24
4 in
5 in
2 in
V  40 in 3
57
Ex. Find the volume of the triangular prism.
V  12 lwh
6  4  15 
V  12 360 
V
5 mm
1
2
4 mm
15 mm
6 mm
V  180 mm 3
Ex. Find the volume of the pyramid.
V  13 Bh
5  5  7 
V  13 175
V
1
3
7 cm
5 cm
V  58.333 mm 3
V  58.3 mm
5 cm
3
Ex. Find the volume of a cylinder with a diameter of 3 in and a
height of 6 in.
V  13 Bh
**Remember that you need
to divide the diameter by 2
to get the radius.
V  r 2h
3 in
6 in
V  3.141.5  h
2
V  3.142.25 6
V  42.39 mm 3
Ex. Find the surface area of a cone with a radius of 4 in
and height of 6 in.
V  13 r 2h
V
1
3
3.141.5 6
2
4 in
6 in
3.142.25 6 2
3
V  14.13 mm 3
V
58
Scale Factor/Proportions
Scale drawings are the same shape but not the same size as the objects
they represent. The scale factor is a ratio that shows how much the
drawing has been magnified or reduced.
Indirect measurement is a method used to calculate the measurement of
extremely large or hard to measure objects such as a tree, building or
mountain; using a proportion.
Ex.
The height of the person is 6 ft tall and his shadow has a length of 2 ft.
If a tree has a shadow that measures 6ft, we can determine the tree's height
with this proportion:
man's shadow
man's height
=
tree's shadow
tree's height
2
6
=
= 2h = 36
6
h
h = 18 ft
59
Types of Triangles
In the earlier section on polygons we discussed how triangles are classified
according to the lengths of their sides. Triangles may also be classified
according to what kind of angles they contain.
Description
All angles are less than 90o
Triangle Name
Acute Triangle
All angles are 60c
Example
Equiangular
60o
60o
One angle is greater than 90c
One angle is exactly 90c
60o
Obtuse
Right
Ex.
60o
60
Ex.
o
This triangle may be classified as obtuse
triangle and also as an isosceles triangle.
This triangle may be classified as a right
triangle and also as a scalene triangle.
60
Pythagorean Theorem
In any right triangle, the area of the square whose side is the hypotenuse
(the side of a right triangle opposite the right angle) is equal to the sum of
areas of the squares whose sides are the two legs (i.e. the two sides other
than the hypotenuse).
If we let c be the length of the hypotenuse and a and b be the
lengths of the other two sides, the theorem can be expressed as the
equation: a 2  b2  c 2
The side of the triangle opposite the right angle (c) is the hypotenuse. The
other sides are called legs (a) and (b).
hypotenuse
leg
a
c
b
leg
A = c2
A = a2 a
b
c
A = b2
61
Ex. The Pythagorean Theorem may be used to
find the height of a ladder leaning against a wall. If the
bottom of an 11 m ladder is placed 4 m from a wall and
leaned in, how far up the wall does the ladder reach?
a 2  b2  c 2
42  x 2  112
16  x 2  121
16   16  x 2  121  16
x 2  105
x 2  105
x  10.25 m
Ex. What is the diagonal length of a TV screen
whose dimensions are 80 x 60 cm?
a 2  b2  c 2
602  802  x 2
3600  6400  x 2
10, 000  x 2
10, 000  x 2
100 cm  x
62
Translation, Rotation, Reflection, Dilation
There are four different kind
of transformation.
Remember that a
transformation is a change in
position or size. The four
transformation are: rotation,
translation, reflection, and
dilation.
Rotation is turning of a figure
in certain direction and angle
around the fixed point.
Rotation
Translation can be thought of as a slide of a figure in a plane.
Translation
63
Reflection is like a mirror image.
Reflection
Dilation is when you enlarge or shrink a figure.
Dilation
64
65
Measures of Central Tendency
Mean- the average of a set of data
Median- the middle number of a set of data
Mode- The number that appears the most in the set of data
Range- The difference between the highest and lowest numbers in
a set of data
Upper Quartile- the median of the upper half of numbers in the set
of data
Lower Quartile- the median on the lower half of the set of data
REMEMBER!! WHEN CALCULATING MEDIAN (AND QUARTILES) IF YOU HAVE TWO
NUMBERS
ADD TOGETHER AND DIVIDE BY 2
66
Scatterplots
Statisticians and quality control technicians gather data to determine correlations
(relationships) between events. Scatter plots will often show at a glance whether a
relationship exists between two sets of data.
Do you think the amount of time Regent would spend studying will affect his grade?
Let's decide if studying longer will effect Regent’s grades based upon a
specific set of data. Given the data below, a scatter plot has been
prepared to represent the data. Remember when making a scatter plot,
do NOT connect the dots.
Study
Hours
3
5
2
6
7
1
2
7
1
7
Regents
Score
80
90
75
80
90
50
65
85
40
100
The data displayed on the graph resembles a line rising from left to right. Since the slope
of the line is positive, there is a positive correlation between the two sets of data. This
means that according to this set of data, the longer he studies, the better grade Regent
will get on his examination.
If the slope of the line had been negative (falling from left to right), a negative
correlation would exist. Under a negative correlation, the longer Regent studies, the
worse grade he would get on his examination. YEEK!!
If the plot on the graph is scattered in such a way that it does not approximate a line (it
does not appear to rise or fall), there is no correlation between the sets of data. No
correlation means that the data just doesn't show if studying longer has any affect on
Regent’s examination scores.
67
Histogram
A histogram is a graphical method for displaying the shape of a
distribution. It is particularly useful when there are a large number of
observations.
The first step is to create a frequency table.
The left side is the intervals and the right side is the frequency or how often
the event occurs in that interval.
Here is a set of data:
8, 12, 13, 19, 21, 22, 25, 27, 27, 29, 33, 36, 38, 39, 44, 48
Data Range Frequency
0-10
1
10-20
3
20-30
6
30-40
4
40-50
2
**Sometimes tally marks are used to count frequency**
Then create a histogram by having the frequency on the y axis and the
intervals of data on the x axis.
68
Box and Whisker Plots
Vocabulary
Box and Whisker Plot – A way to display and summarize data.
Upper Quartile – The median of the upper half of numbers from the set of data.
Lower Quartile - The median of the lower half of numbers from the set of data.
Upper Extreme – The highest number in the set of data that is not an outlier.
Lower Extreme – The lowest number in the set of data that is not an outlier.
Interquartile range – The value achieved by subtracting the upper quartile and the lower
quartile.
Outlier – A number that is more than 1.5 times the interquatile range from the upper or
lower quartiles.
Median – the middle number from the set of data.
The first step in constructing a box-and-whisker plot is to first find the median, the lower
quartile and the upper quartile of a given set of data.
Example: The following set of numbers are the amount of marbles fifteen different boys
own (they are arranged from least to greatest).
18 27 34 52 54 59 61 68 78 82 85 87 91 93 100
68 is the median; 52 is the lower quartile; 87 is the upper quartile
Now you would construct your box and whisker plot.
69
Step by Step instructions on making a box and whiskers plot
We know that the median of a set of data separates the data into two equal parts.
Data can be further separated into quartiles.
The first quartile is the median of the lower part of the data.
The second quartile is another name for the median of the entire set of data.
The third quartile is the median of the upper part of the data.
Quartiles separate the original set of data into four equal parts.
Each of these parts
contains one-fourth of the data.
The data: Math test scores 80, 75, 90, 95, 65, 65, 80, 85, 70, 100
Write the data in numerical
order and find the first quartile,
the median, the third quartile,
the smallest value and the
largest value.
median = 80
first quartile = 70
third quartile = 90
smallest value = 65
largest value = 100
Place a circle beneath each of
these values on a number line.
Draw a box with ends through
the points for the first and third
quartiles. Then draw a vertical
line through the box at the
median point. Now, draw the
whiskers (or lines) from each
end of the box to the smallest
and largest values.
70
Stem and Leaf Plot
A stem-and-leaf plot is a display that organizes data to show its shape and
distribution.
In a stem-and-leaf plot each data value is split into a "stem" and a "leaf".
The "leaf" is usually the last digit of the number and the other digits to the
left of the "leaf" form the "stem". The number 123 would be split as:
stem
leaf
12
3
Constructing a stem-and-leaf plot:
The data: Math test scores out of 50 points: 35, 36, 38, 40, 42, 42, 44, 45, 45,
47, 48, 49, 50, 50, 50.
Writing the data in numerical
order may help to organize the
35, 36, 38, 40, 42, 42, 44, 45, 45, 47, 48, 49, 50,
data, but is NOT a required
50, 50
step. Ordering can be done
later.
Separate each number into a
The number 38 would be represented as
stem and a leaf. Since these are
two digit numbers, the tens digit
Stem
Leaf
is the stem and the units digit is
3
8
the leaf.
Group the numbers with the
Math Test Scores
same stems. List the stems in
(out of 50 pts)
Stem
Leaf
numerical order. (If your leaf
3
568
values are not in increasing
4
022455 789
order, order them now.) Title
5
000
the graph.
Prepare an appropriate legend
Legend: 3 | 6 means 36
(key) for the graph.
A stem-and-leaf plot shows the shape and distribution of data. It can be
clearly seen in the diagram above that the data clusters around the row
with a stem of 4.
71
Notes:



The leaf is the digit in the place farthest to the right in the number,
and the stem is the digit, or digits, in the number that remain when
the leaf is dropped.
To show a one-digit number (such as 9) using a stem-and-leaf plot,
use a stem of 0 and a leaf of 9.
To find the median in a stem-and-leaf plot, count off half the total
number of leaves.
Special Case:
If you are comparing two sets of data, you can use a back-to-back stemand-leaf plot.
Data Set A
Leaf
320
Stem
4
Data Set B
Leaf
1567
The numbers 40, 42, and 43 are from Data Set A.
The numbers 41, 45, 46, and 47 are from Data Set B.
72
Bar Graph
Bar graphs are the simplest way to display data.
Remember:
Bar graphs display data.
When making a bar graph, be sure to include
all the correct parts.
First, draw the x and y axis.
Next, number the scale.
axis
axis
3
2
scale
1
Then, label the x and y axis.
After that, give the bar graph a
title.
axis
axis
label
label
3
2
3
1
2
1
TITLE
scale
scale
label
label
73
Finally, draw the bars using your
data.
axis
label
Your finished bar graph should
look
like this.
TITLE
3
2
bars
1
scale
label
74
Circle Graphs
Circle graphs, also called pie charts, are a type of graph used to
represent a part to whole relationship.




They are circular shaped graphs with the entire circle
representing the whole.
The circle is then split into parts, or sectors.
Each sector represents a part of the whole.
Each sector is proportional in size to the amount each
sector represents, therefore it is easy to make
generalizations and comparisons.
Circle graphs represent 100%
75
Probability
Definition
Example
An experiment is a situation involving chance
or probability that leads to results called
outcomes.
In the problem below, the
experiment is spinning the
spinner.
An outcome is the result of a single trial of an
experiment.
The possible outcomes are
landing on yellow, blue, green or
red.
An event is one or more outcomes of an
experiment.
One event of this experiment is
landing on blue.
Probability is the measure of how likely an
The probability of landing on blue
is one fourth.
event is.
Experiment 1:
A spinner has 4 equal sectors colored yellow,
blue, green and red. After spinning the spinner,
what is the probability of landing on each color?
Outcomes:
The possible outcomes of this experiment are
yellow, blue, green, and red.
Probabilities:
P(yellow) =
number of ways to land on yellow 1
=
total number of colors
4
=
number of ways to land on blue
1
=
total number of colors
4
P(green) =
number of ways to land on green
1
=
total number of colors
4
P(blue)
P(red)
=
number of ways to land on red
1
=
total number of colors
4
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Theoretical Probability
Theoretical probability is the ratio of the number of ways the event can
occur to the total number of possibilities in the sample space.
The theoretical probability of an event is based on the assumption that
each of a number of possible outcomes is equally likely. The theoretical
probability of an event can be defined as the ratio of the number of
favorable outcomes to the total number of outcomes in the sample space.
For example, if a die is tossed, the probability of getting a 4 is because
4 is one of six possible outcomes. Similarly, the probability of getting a
number less than 5 is , because either 1, 2, 3, or 4 is favorable.
Experimental Probability
The experimental probability for equally likely events is the fraction:
# favourable outcomes
total outcomes
If you toss a coin 50 times and you end up with 20 heads and 30 tails, the
experimental probability is:
P(H) = 20/50 = 0.4
P(T) = 30/50 = 0.6
Theoretical experiment changes from experiment to experiment. If the
experiment is fair, the theoretical and experimental probability should be
very similar if a large number of trials are made. Try flipping a coin
yourself. Flip it 50 times and record the numbers of heads and tails. Divide
your totals by 50 to get the experimental probability.
77
Difference between experimental and theoretical
probability
Think about tossing a coin. You should get a head or a tail 50% or ½ of the
time. This is the theory or theoretical probability. If you actually toss a coin
20 times will your probability be ½?
The theoretical probability of the The experimental probability for equally
event is the fraction:
likely events is the fraction:
# ways the event can occur
total possible outcomes
# favourable outcomes
total outcomes
In the Experiment, the theoretical The experimental probability for equally
probability of the event that heads likely events is the fraction:
comes up is:
# favourable outcomes
P(H)
1/2 = 0.5 = 50%
total outcomes
=
P(T)
1/2 = 0.5 = 50%
In the Experiment, if heads comes up 13 of
=
20 times, the experimental probability of the
event that heads comes up is:
P(H) = 13/100 = 0.65 = 65%
Think about this:
The more times you complete an experiment, the closer it’s probability
should come to being the same as the theoretical probability.
78
Tree Diagram
When attempting to determine a sample space (the possible outcomes from an
experiment), it is often helpful to draw a diagram which illustrates how to arrive at the
answer.
One such diagram is a tree diagram. In addition to helping determine the number of
outcomes in a sample space, the tree diagram can be used to determine the probability
of individual outcomes within the sample space. The probability of any outcome in the
sample space is the product (multiply) of all possibilities along the path that represents
that outcome on the tree diagram
Show the sample space for tossing one penny and rolling one die.
(H = heads, T = tails)
By following the different paths in
the tree diagram, we can arrive at the
sample space.
Sample space:
{ H1, H2, H3, H4, H5, H6,
T1, T2, T3, T4, T5, T6 }
The probability of each of these
outcomes is 1/2 • 1/6 = 1/12
A family has three children. How many outcomes are in the sample space that indicate
the sex of the children? Assume that the probability of male (M) and the probability of
female (F) are each 1/2.
Sample
space:
{ MMM
MMF
MFM
MFF
FMM
FMF
FFM
FFF }
There are 8
outcomes in the
sample space.
The probability
of each outcome
is
1/2 • 1/2 • 1/2 =
1/8.
79
Simple Probability
Probabilities are numbers expressed as ratios, fractions, decimals, or
percents. They are determined by considering the results of an
experiment. Simple probability is when we conduct a one stage or one
object experiment. We choose an event, then the probability of that
event is found by counting the number of times the event is true
(favorable) and dividing by the total number of possible and equally likely
outcomes.
P(event) =
number of true outcomes
total number of equally likely outcomes
COIN
A common example is tossing a fair coin. There are two possible outcomes
in the sample space S = {heads, tails}. The probability of event A = "tossing
a head" is found by considering the number of true outcomes (head)
divided by the number of possible outcomes (head/tail), or 1/2. Notice
that probability of tossing a tail is also 1/2. We denote this as P(H) = 1/2 or
P(T) = 1/2.
We can describe several probabilities using a spinner numbered with eight
equal sections (so there are equally likely outcomes). Here the sample
space is S = {1,2,3,4,5,6,7,8}.
P(5) = 1/8
The probability of spinning a five is one out of eight
P(even) = 4/8
The probability of spinning an even number is four out of eight
P(~ 3) = 7/8
The probability of spinning anything but three is seven out of eight
P(> 2) = 6/8
The probability of spinning a number greater than two is six out of eight
P(10)
=0
SPINNER
Spinning a ten is an impossible event
P(<10) = 8/8
Spinning a number less than ten is eight out of eight is certain
DICE
The probability of rolling a specific number on one fair die is 1/6. However,
we also consider the probability of rolling a two or a six. We use the
addition rule to add the probability of rolling a two (1/6) and the
probability of rolling a six (1/6) to get the total probability P(2 or 6) = P(2) +
P(6) = 2/6.
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Compound Probability
The theory that the probability that two independent events will occur is
equal to the probability that one independent event will occur times the
probability that a second Independent event will occur. For example, on
a single toss of two dimes (each dime having a head and a tail), the
probability that both will land on their tails is equal to 1/4 (1/2 x 1/2).
A compound event involves the use of two or more items such as:




two cards being drawn
three coins being tossed
two dice being rolled
four people being chosen
Amy has 5 tank tops, 3 pairs of jeans, and 2 pairs of sneakers. How many
different outfits consisting of one tank top, one pair of jeans, and one pair of
sneakers are possible?
Counting Principle : 5 x 3 x 2 = 30 outfits
How many different 4 letter words can be formed from the letters in the word
MATH?
Using Permutations: 4P4 = 4 • 3 • 2 • 1 = 24
81
82
Absolute value - The absolute value of an
integer is its distance from 0 on a number
line. The absolute value of n is written as |n|.
Adjacent - Adjacent means 'next to'.
Adjacent angles are immediately next to
each other and share a common side.
Adjacent sides are also immediately next to
each other and share a common vertex.
Box & Whisker Plot - A box-and-whiskerplot divides a data set into four parts: the
lower quartile, which is the median of the
lower half of the data the upper quartile,
which is the median of the upper half of the
data the lower extreme is the least value the
upper extreme is the greatest value
Algebraic expression - Algebraic
expressions are combinations of variables,
numbers, and at least one operation.
Altitude - The altitude is the perpendicular
distance from the vertex of a
triangle to the opposite side.
A
Chord – A straight line joining two points
on the circumference of a circle.
Chord
Arc
Arc – An arc is part of a
circle or curve between two
points.
B
Circle graph – Also known as a pie chart or
graph displays data as parts of circle.
Area -- The measure of the
region inside a figure. It is the number of
square units needed to cover its surface.
Arithmetic - An area of mathematics that
includes: addition/subtraction/multiplication
and division of whole number, decimals and
fractions.
Associative Property – Also known as the
Grouping Property, changing the grouping
does not change the result of the operation.
Addition
(a + b) +c = a + (b + c)
Multiplication
(ab)c = a(bc)
Circumference – The distance around a
circle is called the circumference.
Bar Graph – A graph in which information
is shown using columns or bars. Bar graphs
make it easy to compare data..
Sales
Coefficient – In a mathematical expression,
it is the number that multiplies the
variable(s).
Ex. In 3x 2  5 y , 3 is the coefficient of
x 2 term and 5 is the coefficient of the y
term.
100
80
60
East
40
West
20
North
Combination – An arrangement, or listing,
of objects in which order is not important is
called a combination. You can find the
number of combinations of objects by
0
1st Qtr 2nd 3rd Qtr4th Qtr
Qtr
A
83
B
dividing the number of permutations of the
entire set by the number of ways each
smaller set can be arranged.
Commutative Property – Also known as
the Order Property, changing the order of
the numbers does not change the result of
the operation.
Addition
a+b=b+a
Multiplication
ab = ba
Complementary angle – One of two angles
whose sum is 90o.
Compound events – A compound event
involves the use of two or more items such
as:




two cards being drawn
three coins being tossed
two dice being rolled
four people being chosen
a c
 then ad  bc.
b d
5 20
if
 , then 5  48  12  20. 240 = 240.
12 48
if
Cross simplifying – The process of dividing
out common factors diagonally across two
ratios.
Ex.
If
17
h
17
h
1
h

, then



35 102
35 102 35 6
Cubic/cubed – to be raised to the third
power. n3
Degrees – 1) A unit for measuring the size
of angles.
2) A unit for measuring temperature.
Dependent event – If the result of one event
affects the result of a second event, the
events are called dependent events.
Congruent – Two objects are congruent if
they have the same shape and the same size.
Diagonal – A diagonal is a line joining two
non-adjacent vertices (corners) of a polygon.
Conversion/convert – To change into
another form, such as converting a mixed
number to an improper fraction or
converting a fraction to a decimal.
Diameter – The diameter is a straight line
passing through the center of a circle to
touch both sides of the circumference.
1
2
Ex. 1 
3
2
3
 0.75
4
Coordinate grid – Also known as the
coordinate plane, see below.
Coordinate system/Coordinate plane – A
coordinate system has a horizontal number
line (called the x-axis) and a vertical number
line (called the y-axis). These lines cross at
right angles at a point called the origin (0,
0).
Cross multiplying – [Cross Product
Property]. The process of multiplying
diagonally across two ratios.
Distributive Property – The property of
distributing one operation over another and
the resulting answer is the same.
a(b+c) = ab + ac
5(2 + 7) = (5  2) + (5  7)
Divisible – One number is divisible by
another number if the second number
divides "evenly" into the first. That is, when
the first number is divided by the second
number, there is a remainder of zero.
Domain – The set of xcoordinates of the set of
84
Factoring (polynomial) – Writing a
polynomial as the product of monomials
and/or polynomials.
points on a graph; the set of x-coordinates of
a given set of ordered pairs. The value that is
the input in a function or relation.
Ex. 2 x 2  4 x  6  2  ( x 2  2 x  3)
Ex.
The set of ordered pairs for the relation
is {(2, 3), (3, 2), (4, 2), (0, 0)}. The domain is the
first number of the ordered pairs {2, 3, 4, 0}.
Factors – When two or more integers are
multiplied, each integer is a factor of the
product. "To factor" means to write the
number or term as a product of its factors.
Elimination –To remove a term (an
unknown quantity) by combining like
terms.
Ex.
18  1  18; 18  2  9; 18  3  6.
1, 2, 3, 6, 9 and18 are factors.
Equivalent – 1. equal in value, measure,
force, effect, significance, etc. 2.
Mathematics (of two sets) able to be placed
in one-to-one correspondence. 3. Geometry
having the same extent, as a triangle and a
square of equal area.
Formula – A formula is a mathematical rule
written using symbols, usually as an
equation describing a certain relationship
between quantities.
Event -- an experiment, an event is the
Frequency – The number of times a
particular item appears in a set of data is the
frequency.
result that we are interested in.
Function – A function is a relation in which
each element in the domain is paired with
exactly one element in the range.
Experimental probability – The
experimental probability of an event is the
estimated probability based on the number
of positive outcomes in an experiment.
Fundamental Counting Principle – also
called the Basic counting principle is used to
determine the number of outcomes for an
experiment. The number of ways each
choice can be made are multiplied together.
Ex.
Exponent – An exponent is a small number
placed to the upper-right of a number
showing the number of times that number is
multiplied by itself.
Exterior angles – 1) the angle formed
outside a polygon when one side is
extended. Angles 1 and 2 are exterior
angles
Home
2
1
Movie
Theater
House
2) an angle outside two lines when they are
crossed by a third line (a transversal).
Angles 1, 2, 7 and 8 are exterior angles.
2 paths to my friend’s house times 3 paths to the
movie theater equals 6 different ways to walk to the
movie theater. 2  3  6
t
1
3
5
7
My Friend’s
6
8
2
l
4
Height (altitude of
a triangle) – the
perpendicular
distance from
altitude or
height
m
85
the vertex of a triangle to the opposite side.
Histogram – A histogram is a bar graph
representing frequency distribution.
Hypotenuse – The longest side of a right
triangle is called the hypotenuse.
Identity Property – (1) For addition, the
identity number is 0 because 0 added to any
number is that number.
(2) For multiplication, the identity number is
1 because any number times 1 is that
number.
Interquartile range – The interquartile
range is calculated by subtracting the lower
quartile from the upper quartile.
Interior angles – 1) an angle within a
polygon.
2) An angle within two lines when they are
crossed by a third line (a transversal).
Intersecting lines – Lines that have one and
only one point in common.
Ex.
a  0  a or 0  a  a
a 1  a or 1 a  a
Independent event – If the result of one
event is not affected by the result of another
event, the events are called independent
events.
Inequality – (1) not equal in size, amount or
value.
(2) A statement showing that one quantity is
not equal to another.
a < b means a is less tha b a
a > b means a is greater that b
a  b means a is not equal to b
Integer – An integer is any number from the
set
{… , -3, -2, -1, 0, +1, +2, +3, …}.
Intercepts – An intercept is the point where
the graph of an equation crosses an axis
AB intersects PQ at point C
Inverse operations – Inverse operations are
“opposite” operations. They are used to
solve equations and inequalities.
addition and subtraction are inverse operations
multiplication and division are inverse operations
Irrational number – A real number that
can be written as a non-repeating or nonterminating decimal, but not as a fraction is
an irrational number.
Isolation – the complete separation of the
variable term(s) from the other terms in an
equation.
Levels of symmetry [lines of symmetry] –
The line of symmetry is the line that divides
the figure into two mirror images.
Ex.
In this example, 2 is the
x-intercept and  2 is
the y-intercept.
86
Linear – relating to a line
Line graph – uses lines to join points which
represent data
Mode – In a set of data, the mode is the
piece of data that occurs the most.
Monomials – A monomial is a polynomial
containing one term which may be a
number, a variable or a product of numbers
and variables, with no negative or fractional
exponents.
Negative numbers – Integers less than 0 are
negative numbers. They are the opposites of
the whole numbers.
Non-linear – Not in a straight line.
Lowest terms – Also called Simplest Form
of a fraction. In a fraction, the numerator
and denominator are relatively prime,
meaning that they save no
common factor other than 1.
Ex.
1
is in lowest terms.
2
6
6 2 3
is not in lowest terms, and must be reduced.  
8
8 2 4
Mass – Mass is the quantity of matter in an
object.
Mean – The mean of a data set is the
arithmetic average. To find the mean, add all
of the numbers in the data set and then
divide by the number of items in the set.
Measures of center (measures of central
tendency) – The three most common
measures of the center r middle of a
distribution are the mean, median and mode.
Median – The median is the middle number
in a data set when the data are arranged in
numerical order. When there is an even
number of items in a data set, the median is
the mean of the two middle numbers.
Metric unit – The basic units of
measurement in the metric system are:
meter (m) for length; gram (g) for mass and
liter (L) for capacity.
Order of operations [PEMDAS] – 1. Do
all operations within grouping symbols first.
2. Multiply and divide in order from left to
right.
3. Add and subtract in order from left to
right.
Ordered pair – The first number in an
ordered pair is the x-coordinate. It tells how
far the point is to the right or left of the
origin. The second number in an ordered
pair is the y-coordinate. It tells how far the
point is up or down from the origin.
Origin – The point where the x-axis and yaxis meet.
Outcome – An outcome is one of the
possible results of a probability experiment.
Outliers – An outlier is a value far away
from most of the rest in a set of data.. Data
that are more than 1.5 times the interquartile
range from the quartiles are outliers. The
limits on outliers are the points beyond
which data are considered to be outliers.
Parallel – Parallel lines are lines that are
always the same distance apart. They never
intersect (like railroad tracks).
Percent – A permutation is an arrangement,
or listing, of objects in which order is
important.
87
Perimeter – The distance around the outside
of a figure.
Permutations – An arrangement of objects
in particular order
Quadrants – Any
of the quarters of a
coordinate plane
are called
quadrants.
Perpendicular – Intersecting at or forming
right angles.
Pi – Pi () is the ratio of the circumference
of a circle to its diameter.
Polygon -- A closed figure made by joining
line segments, where each line segment
intersects exactly two others.
Polynomials – The prefix poly means
many--a polynomial is an expression that
contains many terms. It is an expression
that can be a monomial of the sum of
monomials.
Radius – The radius is the
distance from the center of a
circle to its circumference. It is
half the diameter
diameter
radius
Range – 1) The range of the data is the
difference between the greatest and least
numbers in the data.
2) The range of a
function is the set of
second numbers.
Prime number – A counting number that
has exactly two factors, itself and 1.
Ex. The set of
ordered pairs for the
relation is {(2, 3), (3, 2),
(4, 2), (0, 0)}. The
range is {3, 2, 0}.
Prime factor – A factor which is a prime
number is called a prime factor.
Prime factorization – Expressing a number
as a product of prime factors.
Rate – A rate is a ratio of two measurements
with different units.
Ex.
180  2  2  3  3  5 or 180=22  32  5
Rate of change – A change in one quantity
with respect to another quantity is called a
rate of change.
Probability – Probability is the chance that
a particular outcome will occur.
Ratio – A ratio is a comparison of two
numbers by division.
Proportion – 1) A proportion is a part to
whole comparison.
2) A proportion also states the equality of
two ratios written as an equation.
Rational number – A rational number is a
real number that can be written as: a ratio of
two integers, excluding zero as a
denominator; a repeating or terminating
decimal; or an integer.
Pythagorean Theorem – In a right triangle,
the sum of the squares of the lengths of the
legs (a and b) is equal to the square of the
length of the hypotenuse (c).
a2 + b2 = c2, where a and b are the legs and c
is the hypotenuse.
c
Real numbers – All numbers, rational and
irrational make up the set of real numbers.
Reduce – 1) make smaller in size or number
2) to simplify a fraction by dividing the
numerator and denominator by its greatest
common factor (GCF).
a
88
b
Reflection – A reflection is a mirror image
of a figure across a line of symmetry.
congruent and their corresponding sides are
in proportion.
Relation – A relation is a set of ordered
pairs. The set of the first coordinates is the
domain of the relation. The set of second
coordinates is the range of the relation. You
can model a relation with a table or graph.
Similar figures – Objects or figures that
have the same shape. They do not
necessarily have the same size.
Ex.
Right angle – A right angle is one whose
measure is exactly 90o.
Rotation – 1) to turn an object
2) one of three basic rigid motions og
geometry
Sample – A sample is a section of a whole
group. To get data from part of a group and
use it to give information about the whole
group.
Scale – 1 an instrument with graduated
spaces, as for measuring.
(2) a graduated line, as on a map,
representing proportionate size.
(3) the proportion that a representation of an
object bears to the object itself.
Scatter plots – Graphs that show the
relationship between two quantities.
Ex.
ABC
DEF
Simple event – or single event involves the
use of ONE item such as:




one card being drawn
one coin being tossed
one die being rolled
one person being chosen
Simplify – to make less complex or
complicated; make plainer or easier.
Simplest form – A fraction is in simplest
form when the GCF of the numerator and
denominator is 1.
Skew lines – Lines that do not lie in the
same plane in space, so that they cannot be
parallel or intersect.
Slope – The steepness of a line. Given by
the ratio of the y-coordinates over the xcoordinates. This ratios is sometimes called
rise-over-run.
y y
m 2 1
x2  x1
Square number – The square of a number
is product of a number times itself.
Similar polygons – Two polygons are
similar if their corresponding angles are
n2  n  n
89
Square root – A number when multiplied
by itself gives the original number.
Standard unit – (1) In the English or
customary system: mass = pound, length =
feet, and volume = gallon.
(2) In the metric system: mass = gram,
length = meter, and volume = liter.
Statistics – 1) Statistics is the collection,
organization, presentation, interpretation and
analysis of data.
2) A statistic is a single number, computed
from a sample that summarizes some
characteristic of a population.
Stem & leaf plot – In a stem-and-leaf plot,
the last digit in each data item is the leaf.
The digits in front of the leaf become the
stem.
Ex.
Systems of inequalities – or Simultaneous
inequalities are two or more inequalities in
two or more variables that are considered
together or simultaneously. The system may
or may not have common solutions.
Terms – 1) one of the numbers in a
sequence, e.g. 1, 3, 5, 7, …
2) one of the numbers in a series, e.g. 2 + 4
+ 6 + 8…
3) one part of an algebraic expression which
may be a number, a variable or a product of
both.
Tessellation – A tessellation is a repetitive
pattern of regular polygons
that fit together with no
overlaps or gaps. In a
tessellation, the sum of the
measures of the angles
where the vertices of the
polygons meet is 360°.
Theoretical probability – Theoretical
probability is the ratio of the number of
ways an event can occur to the total number
of possible outcomes.
Transformation – A change in position or
size, including: reflection, translation,
rotation, or dilation/enlargement.
Substitution – an event in which one thing
is substituted for another; such as replacing
a variable in an expression with a number.
Translation – A translation is sliding part of
a drawing to another place without turning
it.
Supplementary angle – One of two angles
whose sum is 180o.
Transversal – A line (t) that intersects tow
or more lines ( m and n) at different points.
Surface area – The total area of the surface
of a solid, including the area of the base(s)
of the figure.
Systems of equations – or Simultaneous
equations are two or more equations in two
or more variables considered together or
simultaneously. A system of equations may
have no common solution, one common
solution, or an infinite number of common
solutions.
t
m
n
Tree diagram – A diagram shaped like a
tree used to display sample space by using
one branch for each possible outcome.
90
Unit rate – A unit rate is a rate in which the
denominator is 1 unit.
Volume - The volume of a solid figure is the
measure of the space it occupies
Variable – Variables, usually letters, are
used to represent numbers in some
expressions.
Vertical angles – A pair of angles directly
opposite each other, formed by the
intersection of straight lines. Vertical angles
are congruent. Angles 1 and 4; 2 and 3; 5
and 8; and 6 and 7 are vertical angles.
t
1
3
5
7
6
2
l
4
m
8
91
92
Websites that are useful for looking up
information
The Math Forum
http://mathforum.org/dr.math/
WebMath
http://www.webmath.com/
Math.com
http://www.math.com/
Free Math Help
http://www.freemathhelp.com/
Math League
http://www.mathleague.com/help/help.htm
Algebra Help
http://www.algebra.com/
S.O.S Mathematics
http://www.sosmath.com/
A+ Math
http://www.aplusmath.com/
About.com
http://math.about.com/
Purple Math
http://www.purplemath.com/
Online Math Help
http://www.onlinemathlearning.com/
Math Nerds
http://www.mathnerds.com/mathnerds/
Math Goodies
http://www.mathgoodies.com/
Your Teacher
http://www.yourteacher.com/
AAA Math
http://www.aaamath.com/
Algebra Help
http://www.algebrahelp.com/
Math Power
http://www.mathpower.com/
Discovery Education
http://school.discoveryeducation.com/homeworkhelp/homework_help_h
ome.html
Yahoo Education
http://education.yahoo.com/homework_help/math_help/
93
Printable Worksheets
Ed Helper
http://edhelper.com/math.htm
Purple Math
http://www.purplemath.com/
Super Kids
http://www.superkids.com/aweb/tools/math/
Drill Sheets
http://donnayoung.org/math/drills.htm
Math Slice
http://www.mathslice.com/
Math Worksheets
http://www.mathworksheets.org/
Math Drills
http://www.math-drills.com/
Math Café
http://www.mathfactcafe.com/
Free Worksheets
http://www.freemathworksheets.net/
Aplus Math
http://www.aplusmath.com/Worksheets/index.html
The Math Worksheets Site
http://themathworksheetsite.com/
Online Math Manipulatives
National Library of Virtual Manipulatives
http://nlvm.usu.edu/
Shodor
http://www.shodor.org/interactivate/
NCTM
http://illuminations.nctm.org/ActivitySearch.aspx
94
Online Math Games
Ed Helper
http://edhelper.com/math.htm
Cool Math
http://www.coolmath.com/
Fun Brain
http://www.funbrain.com/
A+ Games
http://www.aplusmath.com/games/
Cool Math 4 Kids
http://www.coolmath4kids.com/
Math Playground
http://www.mathplayground.com/games.html
Gamequarium
http://www.gamequarium.com/math.htm
Primary Games
http://www.primarygames.com/math.htm
Math Games
http://www.multiplication.com/interactive_games.htm
Play Kids Games http://www.playkidsgames.com/mathGames.htm
Video Console Math Games
Brain Age
Are you Smarter than a 5th Grader?
Math Software
Math Type
http://www.dessci.com/en/products/mathtype/
Exam View
http://www.fscreations.com/
Geometry Expressions
http://geometryexpressions.com/index.php
95
Geometers Sketchpad
http://mathbits.com/MathBits/GSP/GSP.htm
Handy Graph
http://www.handygraph.com/download.php
Algebra Ace
http://www.mathsupport.com/algebra.htm
Software Available for Purchase
Academy of Math (K-12)
Accelerated Math (diagnostic for
grade equivalency)
Acceleration Station (1-12)
Acuity (formative assessment, 38, and algebra, 6-12)
Adaptive Curriculum (5-8)
Aha!Math (K-5)
ALEKS (K-12)
Algebra Problem Solvers
AMATH (K-4, 5-12+)
A+nyWhere Learning System (112)
Apex Learning Math (6-12)
Carnegie Learning Cognitive
Tutor (6-12+)
Carnegie Learning Math Prep
(for high school high stakes
testing, 9-12)
Classworks
Cut the Knot Interactive
Mathematics Miscellany and
Puzzles
Data Explorer
Desktop Tutor
Destination Math (K-12)
Dimenxian (Game-based
Algebra) and Evolver PreAlgebra (6-12+)
Edutest online assessment (K-8)
Essential Math (9-12)
Fraction Attraction (3-8)
Fraction-Oids (2-8)
Go Figure? (5-7)
Google SketchUp (free)
Green Globs and Graphing
Equations (6-12)
HeartBeeps (1-11)
Hot Dog Stand (5-12)
Math Pathways (6-12)
MathMedia Educational Software
(K-20)
Math Munchers Deluxe (3-6)
Math Problem Solver (1-8)
MathRealm
Math Works
Math Workshop Deluxe (1-7)
Measures of Academic Progress
(K-10 assessment)
Mighty Math Astro Algebra (7-9)
Mighty Math Cosmic Geometry (710)
Moogie Math
NorthStar Math (3-12)
Odyssey Math (K-12)
Optimum (3-5, 9-12)
PassKey
Path to Mastery (formerly OPT for
Success)
Paws and Pyramids (4-8)
Pearson PASeries Mathematics
(3-8) and Algebra I (6-12)
PLATO Learning System and
Straight Curve Math Series
PracticePlanet (standardized test
prep, K-12)
Pre-Algebra Pathways
Pre-Algebra World (3-8)
Prentice Hall Middle Grade Math
(6-8)
PrimeTime Math-Series by Tom
Snyder (4-9) Productions
Prime Time Math by Sheppard
Software (7-12)
Pro-Ohio (standardized test prep,
2-12)
Quarter Mile Math (3-9)
SkillsBank (6-12+)
96
IntelliTools Math (preK-8)
Mathematica
Math Advantage
Math Blaster Series (preK-9)
ST MATH (K-5, 6-8)
STAR Math (3-12)
Study Island (standardized test
prep)
97
98
Middle school students often run into trouble with their mathematics homework.
Many times, this is because they have yet to form a firm understanding of math
procedures. Other times, they do not understand the homework directions. While
frustration is a natural response this, it's not the best way to work through
difficulty.
Effective Note Taking
Writing notes for math class can be extremely helpful in middle school. This is
because middle school math teachers cover a lot of material in each class
period.
Before the Lecture
Students should have plenty of notebook paper ready. It is recommended that
they draw a vertical line about 2 1/2' from the left margin all down the page, then
label the left column 'recall' and the right one 'notes.'
During the Lecture
Students do not need to follow an outline form. It can be easier to write in simple
paragraph form. Students shouldn't try to write down everything their teacher
says; instead, they should try to capture all of the main ideas. With math class,
students also need to write out examples of math problems. Examples used
during class are often the same type of problems in the daily homework.
After the Lecture
Try to review notes immediately after school. Draw boxes around key words or
phrases. Write all of the boxed words and phrases in the left hand recall section.
This helps students remember the key subjects covered in each class period.
Students can also put key formulas and equations in the left hand column. This
will make finding pertinent information easier when trying to complete math
homework..
Help from Friends and Family
Sometimes middle school students fail to understand the sample problems
provided in their class notes and textbook. When this happens, they should
consult the advice of their friends or family. Students often forget that their older
siblings or parents once studied similar topics and faced similar difficulties in their
middle school math classes.
99
While a middle schooler should never expect others to complete their homework
for them, they can ask questions on how to complete a problem. Another solution
is to contact a classmate for help. The classmate might be able to further explain
the instructions given in class. Additionally, middle school students can compare
homework answers to see if they need to rework through a problem to get the
correct answer.
Additional Help
If a student consistently has problems with math, they might also want to ask
their middle school math teacher for additional help outside of school. The
teacher should be able to provide help or recommend a tutor to help strengthen
the student's math skills.
100
101
National Standards Link:
http://standards.nctm.org/
Ohio State Standards Links:
www.ode.state.oh.us
Follow directions to Academic content standards ( Mathematics)
http://www.ode.state.oh.us/GD/Templates/Pages/ODE/ODEDetail.
aspx?page=3&TopicRelationID=1704&ContentID=801&Content=50719
Practice test and Released Materials are also available from this site.
102
Real Number Properties Review
Properties of
Addition
Commutative
Associative
Algebraic Expression
a+b=b+a
(a + b) +c = a + (b + c)
Identity
Inverse
Sign Rules
+ + + = +
-- + -- = --
a+0=a
a + -a = 0
Example
5 +3 = 8
(-6) + (-3) = -9
5 + (-5) = 0
+ + -- or -- + +
9 + (-12) = -3
(-3) + 6 = 3
Subtraction Rule
a – b = a + (-b)
7 – 3 = 7 + (-3)
Properties of
Multiplication
Commutative
Associative
Algebraic Expression
ab = ba
(ab)c = a(bc)
Identity
Multiplicative
Inverse Property
Zero
ax1=a
1
a 1
a
ax0=0
Sign Rules
+ + = +
--  -- = +
+  -- or --  +
Example
3  4 = 12
(-3)  (-5) = 15
(-3 )  2 = -6
3  (-4) -12
Meaning
The order numbers are added in does not matter.
You may regroup the numbers to make arithmetic
easier to do.
Adding zero to a number does not change it.
A number plus its opposite always equals zero.
Meaning
The sum of two positive numbers is positive.
The sum of two negative numbers is negative.
Subtract the absolute values. Then determine the
sign:
1) The sum is zero if the numbers are opposites.
2) The sum is positive if the positive number has
the greatest absolute value.
3) The sum is negative if the negative number
has the greatest absolute value.
For subtraction, keep the first number the same;
transform the minus sign to a plus sign; and change the
second number to its opposite. Use the addition rules
to complete.
Meaning
The order numbers are multiplied in does not matter.
You may regroup the numbers to make arithmetic
easier to do.
Multiplying a number by 1 does not change it.
Multiplying a number by its reciprocal always equals 1.
Multiplying a number by zero always gives a product
of zero.
Meaning
If the signs are the same the product is positive.
If the signs are the same the product is positive.
If the signs are different the product is negative.
***The sign rules for division are the same as those for multiplication.
Properties of
Addition and
Multiplication
Distributive
Algebraic Expression
a(b+c) = ab + ac
Meaning
The numbers inside the parentheses must both be
multiplied by the number outside the parentheses.
5(2 + 7) = (5  2) + (5  7)
103
Multiplication Chart
104