th
8 Grade Math
Workbook
Aligned to Durham
Public Schools
Curriculum
Table of Contents
Unit 1: Real Numbers
1.1
1.2
1.3
1.4
1.5
Classifying Real Numbers
Operations with real numbers
Exponents
Scientific Notation
Equations
Unit 2: The Pythagorean Theorem
2.1 Proving the Formula
2.2 Solving for the missing side of a right triangle
2.3 Real word applications
2.4 Using the Pythagorean Theorem to find distance
2.5 Finding perimeter and area using distance
Unit 3: Geometry
3.1 Transformations by Translation
3.2 Transformations by Reflections over x or y – axis
3.3 Transformation by Rotations
3.4 Dilations
3.5 Dilations with Transformations
3.6 Special Angles created by a transversal and two
parallel lines
3.7 Volume of a Cylinder
3.8 Volume of a Cone
3.9 Volume of a Sphere
Unit 4: Functions
4.1 What is a function?
4.2 Analyzing a real – life graph
4.3 Determining functions from graphs, maps, and
points
4.4 Graphing points
Unit 5: Linear Equations
5.1 Graphing using an input/output table
5.2 Graphing using the linear equation
5.3 Graphing Horizontal and Vertical lines
5.4 Graphing by solving the linear equation
5.5 Writing the equation of a line from a graph
5.6 Writing the equation of a line from the slope
and a point
5.7 Writing the equation of a line from two points
5.8 Writing the equation of a line from a table
5.9 The linear equation in real life scenarios
5.10 Scatterplots and how do they correlate
5.11 Experimenting with Scatterplots
Unit 6: Systems of Linear Equations
6.1 Solving by graphing
6.2 Solving using substitution
6.3 Solving special situations
6.4 Solving real life scenarios
1.1
Classifying Real Numbers
When you first started to learn how to count you started with 1 then 2,
3, and so forth. Then a little later as you progressed in school you
learned that the first number was 0 then 1, 2, 3, and so forth, which are
called whole numbers. Next you learned that there were numbers
6
between 1 and 2 like 1.2 or which are called rational numbers.
5
Finally, in the sixth grade came the other side of numbers called
negative numbers like -1, -2, -3 and so forth or integers.
Now is the last set of numbers that makes up the real numbers which
are irrational numbers. These are numbers like rational numbers as
they are in between whole numbers, but cannot be turned into a
fraction like √8 or π.
Below is a diagram of real numbers:
REAL NUMBERS
Rational Numbers
Irrational Numbers
.25
-1
.3̅
-2
-3 Whole Numbers -4
2 3 4 5
-7
π
4
0 1
-5
1.625
1
Integers
1
√2
3
√15
-6
-8
13
8
or 1
5
8
√8
√32
*It is important to understand that a number can belong to more than one group and where it starts you
work your way out. For example 1 starts as a whole number but as you work your way out you see integers
and rational numbers. So 1 belongs to the set of whole, integer, and rational numbers.
Directions: Define each vocabulary word based on the reading and in your own
words on a separate sheet of paper.
1) Whole Numbers 2) Integers 3) Rational Numbers 4) Irrational Numbers
5) Real Numbers
Directions: Based on your definitions create a matching multiple choice section.
Please follow the model below:
6) _____ Whole Number
a. your definition for any vocab word
7) _____ Integer
b. your definition for any vocab word
8) _____ Rational Number
c. your definition for any vocab word
9) _____ Irrational Number
d. your definition for any vocab word
10) _____ Real Number
e. your definition for any vocab word
Directions: Classify each number as either a whole, integer, rational, or irrational.
It is possible for a number to belong to more than one group.
11) 5
19)
20
5
12) -8
13) 1.3
20) - √3
14) √84
15) 2π
22) 2.4̅
23) 3
21) 0
1
2
16) -1.8
17)
24) 0.37654
2
8
25)
18) √16
8
2
** EOG Prep**
26) Look at the following diagram and circle which numbers are not in the correct
group and explain why?
Rational
2.5
-8
√12
2
7
Irrational
√49
5π
̅̅̅̅
.78
√4
7
2
-3.25641
1.2 Operations with Real Numbers (Integers)
When adding or subtracting integers the most important thing to understand is
calculators do not apply. It is important to realize that for those of us who want
to go to college you will be expected to take either the SAT or ACT which is a
timed test and spending time to type in a basic operation in the calculator is not
going to led to a promising score. So below I will give some basic rules that you
need to practice until you have them down.
Adding or Subtracting
 Do not worry about if it tells you to add or subtract simply look at the two
numbers and if the signs are the same you add and if they are different
you subtract.
Same
Example 1: 4 + 5 = 9
Example 2: -4 – 3 = -7
Both the 4 and 5 are positive
both 4 and 3 are negative
Different Example 3: 5 – 8 = -3
Example 4: -6 + 8 = 2
The 5 is positive but the 8 is negative The 6 is negative and the 8 is
positive
Subtract 8 and 5 which is 3 but it is a
Subtract 8 and 6 which is 2 but it
negative 3 because 8 is bigger than 5 is a positive 2 because 8 is bigger
** The only problem you have to be careful on is when you are subtracting
a negative number for example: 5 – (-2) = 7 because when you have
two negative signs side by side they actually turn into a positive.
5 – (-2)  5 + 2 = 7
Multiplying or Dividing
 When multiplying or dividing if the numbers have the same sign the
answer will be positive. If the numbers have different signs then the
answer will be negative. *Only true looking at 2 numbers at a time.*
Example 5: (-5) • (-8) = 40
Both the 5 and 8 are negative so they
have the same signs so the product
was a positive 40
Example 6:
−32
4
= -8
The 32 is negative and the 4 is positive so
the signs are different so the quotient was
a negative 8
Directions: State whether you will add or subtract the following problems. You
do not need to solve the problems. Put all your answers on a separate sheet of
paper.
1) -6 + 10
2) 5 + 9
3) -2 – 8
4) -4 + (-3)
5) -3 – (-2)
6) 9 – 12
Directions: Create 2 addition and 2 subtraction problems that equal -4.
7)
8)
9)
10)
Directions: Solve the following operations.
11) -5 – 8 = ___ 12) -6 + (-2) = ___ 13) 7 – 9 = ___ 14) 8 + (-7) = ___
15) 1 – (-3) = __ 16) 12 – 5 = ___ 17) -9 + 3 = ___ 18) -5 – (-9) = ___
19) 2 – 5 = ___ 20) -7 + (6) = ___ 21) 3 – 5 + 9 + 1 = ___ 22) 6 – 5 – 4 – 2 = ___
Directions: State whether you will have a positive or negative product or
quotient. You do not need to solve the problems.
23) (-4) • (-7)
24) 5 • (-3)
25) -5 • 2
26) 48 ÷ (-9)
27)
40
10
28)
−18
−3
Directions: Create 2 multiplication and two division problems that will equal – 6.
29)
30)
31)
32)
Directions: Solve the following operations.
33) -9 • 8 = ___ 34) (-3) (-4) = ___ 35) 7 • (-5) = ___ 36) 12 • 6 • (-2) = ___
37)
−42
7
= ___ 38)
−72
−6
= ___ 39)
108
9
= ___ 40) 36 ÷ -3 ÷ -2 = ___
1.2 Operations with Real Numbers (Fractions)
Fractions for whatever reason make many students nervous or even force
students to simply give up (IDK). However the fact remains that if you know your
multiplication table at least up to 12 dealing with fractions may take a little more
time, but they are not hard. The keys to dealing with fractions are below:
 If adding or subtracting you need to find a common denominator first
 If multiplying simply multiply the numerators then the denominators
2
 Always make a mixed number like 1 into an improper fraction by
5
multiplying the whole number to the bottom number and adding the top
7
number so you get 1 • 5 = 5 + 2 = 7 
5
Example 1
Example 2
Converting a decimal to a fraction.
Adding or Subtracting Fractions
̅̅̅̅
.45
.25
3
25
Repeating is
100
Reduce by dividing
4
5
+ =?
9
Step 1: Find a common denominator by mult. the
not over 100
bottom numbers or finding the first number that 4 and 9
÷25 biggest number it is 1 less
45
that goes into both
both go into.
÷9
99
4 • 9 = 36
Step 2: Rename (new numerators) the two fractions
with 36 as their denominator
1
5
4
11
3
4
Example 3

27
36
because 4 • 9 = 36 so I had to multiply 3 • 9 to get 27
Multiplying
3
5
Dividing (KCF)
5
3
6
5
• =?
Multiply Numerators then
denominators.
3
So you get
15
30
reduce ÷ 15 
5
1
2

20
5
9
6
Because 9 • 4 = 36 so I had to multiply 5 • 4 to get 20
÷ =?
Keep the first fraction change
division to mult. and flip 2nd fraction
3 • 5 = 15 and 5 • 6 = 30
5
Step 3:
6
18
27
5
25
36
• =
36
+
20
36
Add the renamed fractions
=
47
36
and that is it because it cannot be reduced
Directions: Convert each decimal to a fraction. Make sure to show your work and
always reduce when necessary.
1) .43
2) .7
̅̅̅̅
4) .23
3) .6
̅̅̅̅
5) .57
6) .6̅
7) .28
8) 2.3
Directions: Convert each fraction to a decimal. Make sure to simply divide the
numerator by the denominator until there is no remainder left.
2
Example:
5
5 2
5 cannot go into 2 so you put a decimal behind it and
add a 0  5 2.0 5 goes into 20 exactly 4 times so the answer is 0.4 or .4
9)
4
10)
5
5
8
11)
7
12)
5
3
*13)
10
5
6
Directions: Simplify each fraction by either adding or subtracting. Remember you
must have a common denominator to add or subtract.
1
4
2
5
14) + = ?
7
2
8
3
18) –
=?
2
1
3
4
15) + = ?
6
1
7
4
19) – = ?
5
3
8
4
16) + = ?
20)
9
10
3
– =?
4
3
1
5
6
17) 1 + 3 = ?
2
5
5
8
21) 2 – = ?
Directions: Simplify each fraction by either multiplying or dividing.
2
5
3
6
6
2
7
3
22) • = ?
26) ÷ = ?
8
1
5
2
8
1
9
3
23) • = ?
27) ÷ = ?
4
6
9
11
5
3
2
8
24) •
=?
28) ÷ = ?
2
1
7
4
25) • 3 = ?
29)
**EOG Prep**
̅̅̅̅ as a fraction in lowest terms.
30) Write the decimal .36
12
5
1
÷3 =?
2
1.3 Exponents (Simplifying, expanded form, 0, and negative exponents)
When you began learning math you learned that 2 + 2 + 2 = 6 and then you
learned instead of adding all these you could have simply multiplied. Since there
are three two’s you could have done 2 • 3 which still equals six and means the
same thing. Now you are going to learn a shortcut for multiplication which is
exponents. For example instead of saying 2 • 2 • 2 = 8 you could have said 23
which is still 8 since there are three two’s. Below is a diagram that represents
exponents.
2
5
Base
Exponent
25 is in exponent form to put it in expanded form you
simply need to write out how many times the base is
getting multiplied.
25 expanded form = 2 • 2 • 2 • 2 • 2 and simplified form
would be doing the math.
Exponent form: 25 = Expanded form: 2 • 2 • 2 • 2 • 2 = Simplified: 32
** Exponent Rule 1: Anything to the 0 power = 1**
Example
20 = 1
-20 = -1
Directions: Put each expanded form problem into exponent
form.
1) 3 • 3 • 3 • 3 2) –(2 • 2 • 2 • 2 • 2 • 2) 3) 5 • 5
4) –(4 • 4)
Directions: Write each exponential expression in expanded
form.
4) 15
5) 43
6) -37
7) 51
8) 28
9) -34
10) 72
Directions: Simplify each exponential expression.
11) 24
12) 33
18) -70
19) 22
13) -43
20) 53
14) -26
21) .250
15) 92
16) 27
17) 40
22) -1.2520
**Exponent Rule 2: Negative exponents mean the base is in the wrong
place. In order to make the exponent positive you must flip the base in other
words if it is in the numerator take the base and exponent to the denominator
and the exponent will be positive. If it is in the denominator take the base and
exponent to the numerator and the exponent will now be positive.**
Examples 1
Example 2
2-3 because the exponent is negative
you must flip the base. The 2 is in the
numerator so take it to the denominator
then the -3 will become positive. This
happens because a negative exponent
means the base is in the wrong place.
1
1
1
2-3  23  2•2•2  8
Exponent
Expanded Simplified
1
2−4
Negative
exponent so flip base
24  2•2•2•2  16
Directions: Simplify the following problems.
Show the exponent form, expanded form,
and simplified form.
23) 3-2 24) 4-3 25) 7-2 26)
1
2−5
27)
1
4 −2
1.3 Exponents Operation Rules
The purpose for rules in math is to allow short – cuts to simplify or solve problems
faster. However the key thing to any rule is the fact that it requires
memorization. In life you are going to memorize many things such as social
security numbers, passwords for school or email, and the list goes on. Now you
could simply write all these on a piece of paper and carry it with you at all times
to prevent memorizing it, but in the end it is just easier to memorize the needed
information. Math is no different memorization is something in your control that
just makes math easier.
Rule 3: Multiplying with like bases
Example: 23 • 22 = ?
2•2•2 2•2 if you count the
number of 2’s you get 5 so the answer is 25
The short – cut to this problem is when you
have the same base and are multiplying all
you need to do is to add the exponents and
keep the same base (do not change the
base)
Rule 4: Dividing with like bases
Example:
27
23
 expanded form below
2•2•2•2•2•2•2  24
2•2•2
The short – cut to this problem is when
you have the same base and are dividing
all you need to do is to subtract the
exponents and keep the same base.
Rule 5: Raising an exponent to an exponent
Example: (24)3  The three as the exponent outside the parenthesis is simply saying you
multiplying three sets of 24  (24)•(24)•(24) (2•2•2•2)(2•2•2•2)(2•2•2•2) if you count
up the 2’s you should see 12 so the exponential answer is 212
The short – cut to this problem is to simply multiply the exponents and keep the same
base.
Directions: Simplify each question making sure to give the exponential answer
and simplify it to a number in standard form.
1) 33 • 32
7)
29
23
14) (23)2
2) 2-3 • 2-4
8)
37
32
9)
15) (34)2
3) 3 • 32
85
87
10)
16) (58)0
4) 53 • 5-5
23
2−3
11)
5) 42 • 4-6
3−1
12)
33
17) (4-1)3
6) 210 • 2-7
25
13)
29
18) (9-2)-1
60
6−2
19) (24)-3
Review Sections 1.1 – 1.2
20) Which group or groups does – 5 belong? 21) Which group or groups does √20 belong?
22) 8 – 11 = ? 23) -2 – 4 = ? 24) -7 + (-2) = ? 25) -9 • 8 = ? 26)
−84
−12
27) (-4)(-2)(-3)
**EOG Prep**
28) In the list belong group the numbers as whether they are rational or irrational.
̅̅̅̅, 3, 9, - 1 , √64
2−3 , √40 , -3π, 1.37
5
2
̅̅̅̅ as a fraction in lowest terms.
29) Write .21
30) Write .3̅ as a fraction in lowest terms.
31) Write .35 as a fraction in lowest terms.
32) Simplify 23 • 25 • (22)-3
33)
35 4−2
33 42
34)
2−3 25 38
26 32 34
35)
24 5−2
2−1 5−5
1.4 Scientific Notation (Writing, standard form, operations)
Scientific notation is basically an easier way to write a very large or small number.
For example when measuring a large distance like from one planet to another
planet scientific notation does not require you to write out all the 0’s instead you
can used a shortened form called scientific notation. For example the distance
from earth to Neptune is 2,900,000,000 miles. In scientific notation I can write
the same number like this: 2.9 x 109. Meanwhile for a small number like the size
of a nucleus which is 0.0000000000000025 meters can be written in scientific
notation as 2.5 x 10-15. Below is a more descriptive analysis of scientific notation.
The Exponent tells you how many times you
have to move the decimal in order to make it a
single digit number. Also a positive exponent
means a big number while a negative
exponent means a small number.
8.2 x 107
The first number is always between 1
and 9.9 in other words it must be a
single digit number.
The x 10 relates to moving the
decimal because we use a
base 10 number system
Multiplying with Scientific Notation
Dividing with Scientific Notation
(2.3 x 105) (5.2 x 106)
Remembering your exponent rules this is a
multiplication problem with like bases as they
are both 10 so you can add the exponents.
Then you need to multiply the single digit
numbers.
5.2
X 2.3
156
After multiplying we have
two numbers after the decimal
so your answer should have
1040
two numbers after the decimal
1196
1011
11.96 x
The only problem is 11.96 is a two digit number
so we have to move the decimal one more time
which will also change the exponent by 1
1.196 x 1012
1.2 𝑥 104
1.5 𝑥 107
Remembering your exponent rules this is a division
problem so exponents must be subtracted.
Then you need to divide the single digit numbers.
15 12.0
First 1.5 turns to 15 by moving the
120
decimal 1 to the right so we do the
0
same thing for 1.2. Now 15 does not
go into 12 so we put a decimal carry
it up and add a 0. 15 goes into 120
exactly 8 times so we get 0.8
0.8 x 10-3  Must move decimal 1 time and change
the exponent by making it 1 smaller 8.0 x 10-4
Directions: Write each number that is in standard form in scientific notation.
1) 5,100,000 2) 423,000,000,000 3) 185 4) 50,200 5) 1,750 6) 807,540
7) .00034 8) 0.024 9) 0.00000000000423 10) .0000072 11) 0.49 12) .0007
Directions: Write each number that is in scientific notation in standard form.
13) 5.6 x 10-2
14) 7.5 x 108
15) 2.35 x 105
18) 2.8 x 103
19) 6.408 x 107
16) 8.257 x 10-4
20) 9.9 x 10-6
21) 5.0002 x 102
17) 9.4 x 106
22) 3 x 104
Directions: Simplify the following scientific notation multiplication problems.
23) (3 x 104) (2 x 103)
24) (2.4 x 105) (1.2 x 106)
26) (2.3 x 10-2) (4 x 107)
27) (2.6 x 106) (5 x 107)
25) (3.5 x 10-4) (2 x 10-3)
28) (6 x 10-2) (5 x 108)
Directions: Simplify the following scientific notation division problems.
29)
33)
9 𝑥 104
4 𝑥 108
8.4 𝑥 107
2.5 𝑥 10−4
30)
34)
8 𝑥 105
−3
2 𝑥 10
9.2 𝑥 109
4.5 𝑥 102
31)
35)
6.2 𝑥 10−3
2.1 𝑥 107
7.22 𝑥 10−8
5 𝑥 106
32)
36)
7 𝑥 10−5
3.5 𝑥 10−3
3 𝑥 109
5 𝑥 10−3
**EOG Prep**
37) In the list belong group the numbers as whether they are rational or irrational.
9
- 18, √4 , π, 3.14, - , 9, √18 , 2√81
5
̅̅̅̅ as a fraction in lowest terms.
38) Write 0.24
39) Simplify
40) Simplify
23 •2−5 •52
25• 54
(2-2)3
41) The moon has 3.2 x 109 cubic feet of moon rock on its surface if the surface
area is 1.4 x 1015 feet2 how many moon rocks are on the moon?
1.5 Equations (one and two – step equations)
Equations are the foundations of Algebra as an equation is simply solving for
something that is not known. In math and life you have many unknowns, but for
math there is a definite solution for the variable while in life many people simply
just go for it. The only problem with that is if you make the wrong choice you
simply cannot go back and correct the problem like in math. The idea behind
equations at this point in time is to get a student to think logically about all the
possible ways to solve a problem so that can be taken forth in life to make the
right choices by thinking about all the possible consequences behind each choice.
3 Rules to Solve Equations
1. Isolate the variable (Draw a line down the equal sign and determine what
side the letter is on either the left or the right of the line)
2. Use Inverse operations (Once you know what side the letter is on begin
getting rid of everything on that side by using opposite operations)
3. Do the Same thing to both side (If you do something to one side of
the equation you must do the same thing on the other side)
One – step
Two – step
1) x + 4 = -2
2) 5 = x – 8
3) 3n = 12
4)
x + 4 = -2
5=x–8
3n = 12
(2)
- 4 -4
x
=-6
+8
13 = x
+8
3
3
n=4
𝑛
2
𝑛
2
= -7
= -7 (2)
n = - 14
𝑥
5) 2x + 3 = -5 6) 4 = – 2
7
2x + 3 = -5
-3
-3
2x = -8
2
2
x = -4
𝑥
4= –2
7
+2
6=
+2
𝑥
7
𝑥
(7)6 = (7)
7
42 = x
Directions: Solve the following one – step equations.
1) n + 3 = -7
6) 3p = 6
12)
𝑚
3
= -5
2) 12 = n – 3
7) -2p = -14
13) -
𝑚
2
3) 4 + n = 1
8) 12 = -p
=8
14) -6 =
4) -2 = 6 + n
9) -20 = 5p
𝑚
𝑚
15) -3 = -
4
6
5) 7 = n + 4
10) 32 = -2p
16)
𝑚
5
= 12
11) -40 = -8p
17) – 9 =
𝑚
8
Directions: Solve the following two – step equations.
18) 2w – 7 = 9
𝑛
22) + 8 = -2
7
1
27) 4 – p = 7
2
19) -14 = 4m + 6
𝑧
23) 12 = + 7
3
20) 8 – 2x = 10
𝑘
24) 6 - = - 2
2
28) 13 = x + 5
3
4
21) -17 = 5 – 2y
25) 9 = 5 +
29) -1 = 8 + 4m
𝑡
8
26) 5 – n = -1
3
30) w – 9 = 3
4
Directions: Solve for y. Use the same rules to get y all by itself.
31) ay = t
32) y + t = r
33) 4 = y – x
34) h =
𝑦
𝑛
35) x – ay = t
36)
𝑦
𝑤
+r=7
Directions: Write the equation then solve.
37) Four less than a number is six
38) Five more than twice a number is eleven
39) The quotient of a number and eight is negative five
40) Three more than the product of a number and six is twenty-one
**EOG Prep**
41) What group or groups of numbers does -21 belong to?
42) Write 0.2̅ as a fraction in lowest terms.
43) Simplify to a fraction in lowest terms.
33 61
3−1 63
44) Lake Erie has a surface area of 3.2 x 105 kilometers if there are 2.8 x 102
bottles of water for each kilometer how many bottles of water are in Lake Erie?
1.5 Equations (distribute, combine like terms, variables on both sides)
Now that you know how to solve the most basic equation it is time to build to a
more realistic equation. However it is important to remember the 3 rules as they
do not change and every equation will come down to those basic equations as
you solve it.
Example 1: Distribute
Example 2: Combine like terms
The word distribute means you need to
multiply typically the number that is
outside a parenthesis to everything in the
parenthesis.
Like terms simply means you have two
like terms on one side of the equation
that can be combined whether it is
numbers or letters.
2(x – 4) = 6
4x – 6 + 3x = 8
Step 1: Distribute
2(x – 4) = 6
4x – 6 + 3x = 8
2x – 8 = 6
7x – 6 = 8
Step 2: Draw your line and begin getting
rid of everything on the same side as the
letter
2x – 8 = 6
+8 +8
2x
= 14
2
2
x = 7
Step 1: combine like terms
Step 2: get rid of all
+6 + 6 terms on same side as x
7x = 14
7
7
x =2
*Notice when I combined like terms I
circled the sign in front of the letter
because that is part of that term.
Example 3: Variables on both sides
4x – 5 = 7x + 10 Step 1: Draw the line and begin moving any term you want. I
-4x
-4x
will move the 4x first so it will be a two – step equation.
- 5 = 3x + 10
-10
-10 
-15 = 3x now divide by 3 and you find that x = -5
Directions: Solve the following equations using the distributive property.
1) 3(x + 5) = 6
5) -8 = 4(x + 1)
2) 2(3x – 4) = 4
6) 10 = -5(x + 4)
3) 4(6 – x) = 12
4) -2(x – 5) = -6
7) 13 = 2(6 – 4x)
8) 9 = -3(x – 5)
Directions: Solve the following equation by combining like terms.
9) 3x – 8 + 2x = 7
10) 12 = 6x + 7 – x
13) 8 – 2x + 13 = -1
14) 9 = 7 + 5x – 8
11) 4x – 7 – 6x = 13
15) 18 = 8 – 4x – 2
12) -6 = 8 – 3x – 4x
16) 3 + 7x – 8 = 16
Directions: Solve the following equations with variables on both sides.
17) 3n – 8 = 5n + 2
18) 7 – n = 4n + 12 19) 6n + 8 = 3n – 13
20) -7 + 2n = 8 – n
21) 12 + 6x = 4x – 14 22) n – 9 = 5n + 11 23) 2 + 7n = 6n – 3 24) 5n – 17 = 3n + 1
**EOG Prep**
25) Solve.
1
2
(4x – 12) + 3x – 10 = 6 – 4(2x – 9) + 5x + 3
26) Classify the following number into its correct group or groups. 9
27) Simplify.
25 3−2
3−4 22
̅̅̅̅
28) Write as a fraction in lowest terms. .72
29) It was found that the average home has 1.2 x 104 dust particles for every
square foot. If the average house is 1.5 x 103 square feet how many dust particles
are in the house?
30) Solve.
Five multiplied by the sum of six and a number is equal to twice the number less
than nine.
Unit 1 Basic Test
Directions: Match the word to its correct definition.
1) ___ Irrational
a. A number that can be made into a fraction
2) ___ Whole Number
b. Any number that can be a fraction or not
3) ___ Rational Number
c. A counting number that can be positive or negative
4) ___ Integer
d. A number that cannot be made into a fraction.
5) ___ Real Number
e. The counting numbers including 0.
Directions: Classify the number into the correct group or groups of real numbers.
7) √18
6) -8
8) 50
9)
5
6
Directions: Identify which numbers are rational numbers and which numbers are
irrational numbers in the list below.
4
1
7
5
16.2, - , √6 , 33 , 2-3 , 4π, 3.2̅ , 2 , -2√10
10)
Directions: Simplify the following rational numbers.
11) 5 – 9 = __ 12) -3 – 7 = __ 13) -5 – (-7) = __ 14) -8 + 7 = __ 15) 2 + (-8) = __
16) (-4) • (-6) = ___ 17) (7) (-9) = ___ 18) (-2) (-3) (-5) = ___ 19)
20)
−36
= ___ 21)
−4
−2
8
18
−3
= ___
3
5
2
5
4
6
3
6
= ___ 22) 5 – 3 • 7 + 2 = ___ 23) + = ___ 24) 2 – = __
7
2
6
3
8
3
7
8
25) • = ___ 26) ÷ = ___
Directions: Write the following exponential numbers in expanded form.
27) 45 =
28) 7-2 =
29) -24 =
30) 3-6 =
Directions: Write the following numbers in standard form or exponential form if
the number is greater than 100.
31) 43 • 44 =
32) 22 • 2-5 =
33) (53)-2 =
34) (26)0 =
35)
23
=
−6
36)
2
3−1
32
=
Directions: Write the following numbers in scientific notation.
37) 1,200,000
38) .00000000973
39) 401
40) 0.0008
Directions: Write the following numbers in standard form.
41) 2.3 x 107
42) 1.52 x 10-3
43) 7.524 x 104
44) 3 x 10-8
Directions: Simplify and make sure your final answer is in scientific notation.
45) (4.2 x 105) (2 x 103)
46) (3.12 x 10-4) (2.3 x 10-8)
47)
8.2 𝑥 103
2.1 𝑥 109
48)
9.24 𝑥 10−5
1.5 𝑥 102
Directions: Solve the following equations.
49) x – 3 = -7
54) 13 = 2m – 1
58) 3(a – 6) = 9
50) 8 = -2x
51) n + 7 = -2
55) -6 = 10 – 2w
52) 9 =
56) 9 –
𝑝
5
𝑛
4
=3
59) 8 = -2(4v – 3) 60) 3x – 7 = x + 9
62) 6 – 5g = 13 + 2g
63) 5n – 8 = n
1
64) n – 7 = 5
2
𝑟
53) + 8 = -2
4
57) 4n – 7 + n = 8
61) 19 = 6 – 4h + 1
3
65) w + = -2
4
Unit 1 EOG Test
1) Solve.
1
3
(3x – 12) + 3x – 10 = 6 – 4(3x – 5) + 5x + 4
2) Write the equation and solve.
The difference of a number and four multiplied by 6 is equal to eight less than
four times the number.
3) Solve. 4 + 2(3n – 7) = n – 3(4n – 9)
4) Classify the following list of real numbers as either rational or irrational.
5
12
9
4
̅̅̅̅ , -13,
3√5 , -7π , ,0. 78
, 3.8 , √25
5) Classify the following number into the correct group or groups.
6) Simplify.
7) Simplify.
-7
28 3−5
3−3 22
23 27 32
3−4 39 25
̅̅̅̅
8) Write as a fraction in lowest terms. 0.24
9) Write as a fraction in lowest terms.
0.5̅
10) At the end of the day the average person has 2.5 x 104 bacteria in their mouth
for every square foot. If the surface area of the average human mouth is 1.2 x 105
millimeters how many bacteria are in the human mouth at the end of the day?
11) Find the value of x.
2𝑥−8
3
= 12
2.1
Proving the Pythagorean Theorem
The illustration above simply proves how the squares of the sums of each leg of a
right triangle will always equal the square of the hypotenuse.
Vocabulary (3 things to know)
Hypotenuse – is always the ___________________ side in a right triangle
Hypotenuse – is always _____________________ the right angle
Hypotenuse – is always the ___________________ side of a right triangle.
Example 1: Determine if the sides make right
triangle using the Pythagorean Theorem.
4, 8, 5 *8 must be the hypotenuse
a2 + b2 = c2  (4)2 + (5)2 = (8)2
16 + 25 = 64
41 = 64
This is not true as 41 does not equal 64 so this
is not a right triangle.
Example 2: Determine if the sides make a right
triangle using the Pythagorean Theorem.
5, 12, 13
a2 + b2 = c2  (5)2 + (12)2 = (13)2
25 + 144 = 169
169 = 169
This is a right triangle as both sides equal the same
number.
Directions: Determine if the following sides of a triangle will make a right triangle
by using the Pythagorean Theorem.
1) 5, 3, 4 2) 9, 10, 12 3) 4, 10, 11 4) 6, 10, 8 5) 7, 9, 15 6) 25, 24, 7
7) 9, 30, 31 8) 16, 12, 9 9) 2.5, 3.2, 7 10)
3
2 1
, ,
10 5 4
11) 24, 26, 10 12) 1, 2, 6
Review: Unit 1 Scientific Notation and exponents
Directions: Simplify.
-4
-5
13) (3.5 x 10 ) (3 x 10 )
17)
42
14)
8.45 x 108
18) (4.1 x 103) (2.3 x 102)
4 −5
15) 2-5 • 22
2.5 x 103
19)
7 𝑥 104
16) (32)-3
20) 33 • 3
4 𝑥 105
Review: Unit 1 Real Numbers
21) -5 – 8 = ?
3
5
4
6
8
2
9
3
25) – = ?
29) ÷ = ?
22) (-3) (-5) (-6) = ?
1
3
3
5
26) 2 + = ?
23) -4 + (-2) = ?
7
4
8
9
27) • = ?
24) 6 – (-1) = ?
1
28) .28 • = ?
2
30) What group or groups of real numbers does 12 belong?
Review: Unit 1 Equations
31) 4 – 7x = -10
34) 6(n – 4) = -6
32) -8 = 3t + 10 – t
𝑛
35) + 6 = -2
5
33) 4n – 9 = 7n + 12
36) 2 – 3(x – 5) = -7
2.2
Finding the missing side
n
15
12
The diagram above shows a right triangle with an unknown side. Since it is a right
triangle immediately we should go to the Pythagorean Theorem. The KEY to
using this formula is to make sure you know what to plug in for the hypotenuse as
mixing up the legs will not change your solution.
a2 + b2 = c2
n2 + (12)2 = (15)2
n2 + 144 = 225
at this point it is like solving an equation so let’s get the letter by itself
- 144 - 144
n2
= 81
in order to get the letter by itself we must get rid of the exponent (2)
n2 = 81
n =9
by doing the opposite which is called the square root (√
)
the square root gets rid of the 2 as an exponent and breaks 81 down to 9
because 9 • 9 or 92 is 81 so the 2 is now gone and what is left is 9
Directions: Fill in the table that shows square roots and perfect squares
1)
#
#•#
1
2
1 •1
2•2
Perfect
Square
Square
Root
1
4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Directions: Define each vocabulary term in your own words based on your
knowledge of the example and table.
2) What is a Perfect Square?
3) What is a Square Root? Symbol?
4) What is a Perfect Cube?
5) What is a Cube Root? Symbol?
6) What 3 things do you know about the hypotenuse?
Directions: Find the square root of the following numbers.
7) 25
8) 64
9) 121
10) 81
11) 196
12) 1
13) Which 2 numbers in the table are perfect cubes? (when you do the cube root
you will get a whole number)
Directions: Estimate what two whole numbers between the square root of the
following irrational numbers.
Example
√32
Look at the table and find the 2
perfect squares that 32 is between.
25 and 36
The √25 = 5 and the √36 = 6
So the √32 is between 5 and 6
14) √8
17) √96
15) √45
16) √72
18) √118
19) √24
Directions: Find the missing sides.
20)
12
21)
13
24
25
2.3 Real World Applications of the Pythagorean Theorem
Typically most people do not even give the Pythagorean Theorem any thought in
life instead we simply guess and test most of the time like when putting a ladder
against a wall or dropping an anchor. With these examples we simply guess and
test it out then again and again until we have these in the right spot. This seems
simply enough, but time might have been saved if you would have used the
Pythagorean Theorem to determine exactly where you wanted the ladder or
anchor. If you look up you will see something you cannot guess and test. The
roof above many of your heads is based on the Pythagorean Theorem based on
the load (weight) it can hold using this formula along with the pitch (slant) needed
to repel weather. The wrong pitch of the roof can actually lead water to defy
gravity and curl up the roof until it destroys the roof.
Example 1
Example 2
A firefighter has to reach a window that is
12 feet off the ground. If the ladder is 13
feet how far from the house will he need
to put the ladder?
An anchor is dropped from a boat and the
length of the anchor is 15 feet. When the
anchor was dropped the boat floated
another 9 feet before stopping, what is
the depth of the water?
**Key** A ladder is laid slanted against a
house and the hypotenuse is always
slanted.
house
12
**Key** The length of the anchor will be
slanted as the boat floats till it stops and
slanted is the hypotenuse.
13
n
Ladder
9
n
2
2
2
a + b = c plug in what you know
(12) + n = (13)  144 + b = 169
2
2
2
2
- 144
15
a2 + b2 = c2 plug in  n2 + (9)2 = (15)2
n2 + 81 = 225
- 81
- 144
b2 = 25  sq. root n = 5
anchor
n2
- 81
= 144 sq. root  n = 12
Directions: Solve the following real life scenarios using your knowledge of the
Pythagorean Theorem.