Mathematician’s Notes
BE A
MATHEMATICIAN!
All you have to do is… THINK!
Imagination is more important than knowledge.
- Albert Einstein
We only think when confronted with a problem.
-John Dewey
It’s not that I’m so smart it’s just that I stay with problems longer.
-Albert Einstein
______________________________________________________________________________
Let’s talk about math…
Math is about thinking! It is what mathematicians do first when presented with a problem to
solve. Almost all of the problems that mathematicians solve begin with a real world problem to
solve. You might think of them as “word” problems. Mathematicians think of “word” problems
as “world” problems. The best tool you have for solving math problems is your brain!
You can use your brain to think flexibly about math problems and whether your solutions make
sense. There are many ways to solve problems. You need to be able to choose a way that
works for you and you need to be able to explain and justify your method to others.
Understanding how others solve math problems gives you more tools to solve them. As you
become more skilled at solving problems, you will begin to choose the most efficient way to
solve problems.
Many people have been solving problems for a long time and so algorithms have developed for
solving math problems. Algorithms are step-by-step procedures for solving problems. Once
you understand why an algorithm works, you may want choose it as the most efficient way to
solve the problems, but you may also have an efficient way that works for you better,
depending upon the problem situation. Being able to choose different methods makes you a
skilled mathematician, which prepares you to function in, and contribute to, the society in
which you live. Math is everywhere in your life. It is beautiful, puzzling, and logical all at the
same time. You can be a successful mathematician, even if you struggle with it. We learn and
grow by engaging in and overcoming difficulties!
You can be Mathematician! All you have to do is THINK!
Math Notes Page 1 of 1
N.Paulson, 2012
Mathematician’s Notes
It’s all in the numbers: The Real Numbers!
Pictures from Mathis fun.com.
Number: a count or measurement
They are really an idea in our minds. We write or talk about numbers using numerals
such as "5" or "five". We could also hold up 5 fingers, or tap the table 5 times. These are
all different ways of referring to the same number.
There are also different types of numbers, such as whole numbers (1,2,3) decimals (1.48,
50.5), fractions (1/2, 3/8), and more.
____________________________________________________________________________________________________________________
Numeral: A symbol or name that stands for a number. Examples 3, 49, and twelve are all numerals.
Digit: A symbol used to make numerals. 0,1,2,3,4,5,6,7,8, and 9 are the ten
digits used to make numerals.
____________________________________________________________________________________________________________________
Types of Numbers:
The Counting Numbers: the counting numbers have been around for thousands of years.
}
We can use numbers to count: {
Zero: the idea of zero was not natural to early humans, but it represents a quantity, so it needed to be
added to our number system. Not only does it represent a quantity, but it is a placeholder. “5 2”
means 502 (5 hundreds, no tens, and 2 units). If you didn’t have the zero to hold the place of the tens,
then this number might look like 52.
Whole numbers: include the Counting Numbers and Zero:
Whole numbers:
{
}
Negative Numbers: Around the 16th century, mathematicians decided they needed negative numbers in
order to make their algebraic solutions work. So, counting backwards from zero gives us negative
numbers. A number less than zero is a negative number.
Sometimes it’s hard to understand how numbers can be negative.
A simple example is temperature.
We define zero degrees Celsius (0° C) to be when water freezes ... but if we get
colder we need negative temperatures.
So -20° C is 20° below Zero
So, negative numbers along with whole numbers become called Integers.
Math Notes Page 2 of 2
N.Paulson, 2012
Mathematician’s Notes
Vertical
Number
Line
Integers
Integers are the whole numbers and their opposites. The Integers include zero,
the counting numbers, and the negative of the counting numbers, to make a list
of numbers that stretch in either direction indefinitely. Zero is neither positive,
nor negative and
Numbers are opposites if they are the same distance from zero on either side of
a number line.
and
When you add a number to its opposite, the sum is zero.
A number line is helpful when operating with integers. Sometimes a horizontal
number-line makes sense, but sometimes a vertical number-line makes more
sense to use. Choose what is best for the problem you are trying to solve.
Horizontal Number Line
Rational Numbers: Numbers that can be written as a fraction. They include all the integers and all
fractions.
If you have one orange and want to share it with someone, you need to cut
it in half. You have just invented a new type of number! You took a number
and divided it by another number
another category for numbers.
to come up with
. So now we need
*
*Remember the fraction bar means division. You cannot divide a number by zero. Since we
now write division problems as:
b cannot be zero.
Irrational numbers: Irrational numbers are numbers that are not rational… what? If the definitions
above don’t fit the number, then it is irrational.
Irrational numbers:
 Numbers that are decimal fractions (decimals) that don’t repeat in a pattern, or
don’t ever end.
 Non-perfect square roots: i.e. the √
 π Is the symbol for Pi, it never ends or repeats… We’ll learn more about Pi in
middle school, but we generally think of
Real Numbers: All of these types of numbers make up the big category of real numbers.
Imaginary Numbers: What?.... Yes, there is a set of numbers called imaginary numbers, but we’re
going to wait until high school to learn about those…
Math Notes Page 3 of 3
N.Paulson, 2012
Mathematician’s Notes
Properties of Numbers:
AF 1.3 (6/7)
Properties are truths, or proven conjectures, that work in every case. People have been doing math for
so long they have found truths about how numbers work. They are proven by many people, over and
over again. If you think of math as a game, these are the legal moves. Just like Soccer (only goalie can
touch the ball with her hands) and Baseball (you run around the bases in order from 1st, to 2nd, to 3rd,
then to home plate) have rules, Math has rules. If you understand these rules, you will win in the game
of math. The letters a, b, and c just stand for “any number”.
Commutative Property: of Addition and Multiplication
You can change the order of an addition problem or a multiplication problem and get the same result.
With Algebra:
With Arithmetic:
Addition:
Multiplication:
______________________________________________________________________________
Associative Property: of Addition and Multiplication
You can change the grouping of an addition problem or a multiplication problem and get the same
result.
With Algebra:
With Arithmetic:
Addition:
Multiplication:
_____________________________________________________________________________________________
The Identity Property of Multiplication: Any number multiplied or divided by one is that number.
Multiplying a number by one, does not change its value.
With Algebra:
Middle School Way:
Multiplication
With Arithmetic:
Division
__________________________________________________________________________________
The Identity Property of Addition:
Any number plus zero is that number. Adding zero to a number does not change its value.
With Algebra:
With Arithmetic:
_____________________________________________________________________________________
Distributive Property:
Says that you get the same product when you:
 Multiply a number by a group of numbers added together …or
 Do each multiply separately, then add them
With Algebra:
With Arithmetic:
Add first, them multiply:
Multiply each separately, then add:
√ it checks!
This is how you may see the distributive property in future math problems:
or
Math Notes Page 4 of 4
N.Paulson, 2012
Notes for Mathematicians
Vertical
Number
Line
Additive Inverse Property:
A number plus its inverse, has a sum of zero. In addition a number’s inverse is its
opposite. The operation of Subtraction is the inverse of the operation of Addition. That
means it undoes addition. This property allows us to subtract numbers.
With Algebra:
With Arithmetic:
Or, you can offset the negative number by putting a parenthesis around it so it’s easier to see.
With Algebra:
With Arithmetic:
Opposites are numbers that are the same distance from zero on either side of a number
line. Sometimes a horizontal number line makes sense to use, however sometimes a
vertical number line makes more sense to use. You’ve been moving back and forth on a
number line since Kindergarten, it’s the same here. Choose what is best for your situation.
Horizontal Number Line
The opposite of
and the opposite of
The Additive Inverse Property
allows us to subtract
“REAL MATHEMATICIANS DON’T SUBTRACT, THEY ADD THE OPPOSITE.”
The additive inverse property shows that subtraction gets the same result as adding the
opposite. Remember:
and
Example:
4 positives
+ + + +
take away 3 positives
+ + + +
equals 1 positive
+
And
Example:
4 positives
+ + + +
plus 3 negatives
+ + + +
- - -
equals 1 positive
+ + + +
- - 0+0+0+1=1
Cancel
We DON’T “CANCEL”… We MAKE ZEROES
Math Notes Page 5 of 5
N.Paulson, 2012
Notes for Mathematicians
Multiplicative Inverse Property:
Any number multiplied by its reciprocal equals one. This property allows us to divide numbers.
→
With Algebra:
1 →
→
With Arithmetic:
𝑎𝑏
𝑎𝑏
(any number divided by itself equals 1)
1
1=→
(any number divided by itself equals 1)
Reciprocal: A number multiplied by its reciprocal equals one. The reciprocal is the multiplicative
inverse of the number.
Examples of Reciprocals:
1=→
and →
and , because →
1
*(any number divided by itself equals 1)
*Some people call this “canceling” a number… But it’s not cancelling! It’s making a one!
And,
because 4 groups of
Or,
→
→
is 1.
1
1=→
1
(any number divided by itself equals 1)
Cancel
We DON’T “CANCEL”… We “MAKE ONES”
____________________________________________________________________________
The multiplicative Inverse Property
allows us to divide
“REAL MATHEMATICIANS DON’T DIVIDE, THEY MULTIPLY BY THE RECIPROCAL.”
(In Middle School, this is how we write division)
→
______________________________________________________________________________
“Cancel” is a bad word to say in math class!
You say that you are “making a one” or “making a zero,”
Cancel
… but don’t say that bad word!→
Math Notes Page 6 of 6
N.Paulson, 2012
Notes for Mathematicians
Zero Product Property:
The product of any number and zero is zero
With Algebra:
With Arithmetic:
And, this might seem like “duh”, but you are going to want to know about this for future algebra classes:
If a times b is zero, then either a equals zero, or b equals zero
So, if ab=0, then either a=0 or b=0, or both =0
So, if
then
because
Can we talk… about dividing by zero?
It is impossible to divide a number by zero. If you try to use a calculator to divide by zero, it will get all upset
and give you an “E” for error.
Think!
About it!
Let say the problem is:
, as we say in middle school. This means you have 8 pencils and you are going to divide
them (pass them out) between 4 friends… how many does each friend get? Well, duh, 2.
, as we say in middle school. This means you have 0 pencils and you are going to divide
them (pass them out) between 4 friends… how many does each friend get? Well, duh, 0.
But….
, as we say in middle school. This means you have 4 pencils and you are going to divide
them (pass them out) between 0 friends… how many does each friend get? Well, uhhhhh…..?
You can’ pass out 4 pencils to zero people.
Got it?
Good!
Math Notes Page 7 of 7
N.Paulson, 2012
Notes for Mathematicians
Operating With Integers:
The additive inverse property allows subtraction. It says that subtraction gets the same result as
adding the opposite.
Remember the parenthesis sets a number off as a negative number,
but it can also be written without parenthesis.
Same Result
Adding and Subtracting integers with tile spacers:
To add two integers using tile spacers, a positive number is represented by the appropriate number of (+)
tiles and a negative number is represented by the appropriate number of (–) tiles.
To add two integers start with a tile representation of the first integer in a diagram and then place into the
diagram a tile representative of the second integer. Any equal number of (+) tiles and (–) tiles makes “zero”
and can be removed from the diagram. The tiles that remain represent the sum.
If you are at home and don’t have any tile spacers, use pennies for negatives and dimes for positives, or beans
for positives and rice for negatives, or green Apple Jacks for positives and Red Apple Jacks for negatives… you
get the idea!
Example #1: Adding Integers



nd
Build the 2 integer:
Pair one positive and one negativeto make zero
pairs.
+

+++++
- - - - - -
Build the 1st integer:
+
-
+
-
+
-
+
-
_
Whatever is left over is your sum.
Example #3: Adding Integers
Say: “three negatives plus 5 negatives equals
8 negatives.”
 Build the 1st integer:
- - - - - --
Example #2: Adding Integers
 Build the 1st integer: - -  Build the 2nd integer: + + + + +

Pair one positive and one negativeto make zero pairs.
+
-
+
-
+
-
+ +
Whatever is left over is your sum.
Example #4: Adding Integers
Say: “4 plus 3 equals 7”
You don’t need to build this with integers because you
have been adding these numbers since 1st grade.
 Build the 2nd integer:
We write negative numbers using a minus sign, such as:
 There are no zeroes to pair.
is said “negative seven”
 Combine all the negatives and count them.

- - -
- - - - equals
But, with positive numbers, we don’t have to include
the positive sign. It is assumed.
So, is said “eight”, though it means positive
“REAL MATHEMATICIANS DON’T SUBTRACT, THEY ADD THE OPPOSITE.”
Math Notes Page 8 of 8
N.Paulson, 2012
Notes for Mathematicians
Subtracting Integers Tile Spacers: Sometimes you may have to Make Zeroes
Here are a couple of things you need to remember for subtracting integers.
 Additive Inverse Property: A number plus its opposite equals aero
With Algebra:
With Arithmetic: and
 Identity Property of Addition: If you add zero to a number, it doesn’t change the value.
With Algebra:
With Arithmetic:
Using Tile Spacers
Using Additive Inverse Property and tile spacers
Subtraction is the same as adding the opposite:

Say: “Negative six, take away, negative three.”
------

Build the the first integer.

Take away 3 negatives.
(the arrow represents take away)

Three negatives are left.
Build 6 negatives, add three positives, and make zero pairs
with 1 positive and 1 negative. Whatever is not paired up is
your sum.
---
---
Change subtraction operation to addition of the
opposite.
---
+
-
+
-
+
-
- - -
Same Result
Solve subtraction problems with tile spacers by adding zero pairs first:
Use the Additive Inverse Property to change
the subtraction operation to addition
Add Zeroes, then take away negatives
Say: “5 positives, take away, 3 negatives.”
Say: “5 positives, take away, 3 negatives.”
 Use the Additive Inverse property to
+++++

Build the the first integer.

You need to take away 3 negatives, but your first number
only has positives. So you need to make a few zeroes
until you get enough negatives to take away.
+ + + + + plus
+
-
+
-
+
-
(3 zeroes)
change the subtraction operation to
addition of the opposite.
 So, instead of:
 Change to expression to:
Which is the same as:
From first grade, you know that:
5 + 0 + 0 + 0 is still 5
 Now, take away 3 negatives:
+++++
+++
---
Which leaves 8 positives:
—
Same Result
“REAL MATHEMATICIANS DON’T SUBTRACT, THEY ADD THE OPPOSITE.”
Math Notes Page 9 of 9
N.Paulson, 2012
Notes for Mathematicians
Using a number Line for Adding and Subtracting Integers
To add two integers using a number line, start at the first number and then move the appropriate number of
spaces to the right or left depending on whether the second number is positive or negative. Your final
location is the sum of the two integers.
If you are adding positive numbers, you add to the positive side. If you are adding a negative numbers, you
travel to the negative side.
Example #1: Adding Integers
Example #2: Adding Integers
Subtraction is the opposite (inverse) of addition so it makes sense to do the opposite (inverse)
actions of addition. When using the number line, adding a positive integer moves to the right so subtracting
a positive integer moves to the left. Adding a negative integer move to the left so subtracting a negative
integer moves to the right.
Example #3: Subtracting Integers
Example #4: Subtracting Integers
—
Summary of Integer addition:
When you add integers using the tile model, zero pairs are only formed if the two numbers have different signs. After
you circle the zero pairs, you count the un-circled tiles to find the sum. If the signs are the same, no zero pairs are
formed, and you find the sum of the tiles. Integers can be added without building models by using the rules below.
 If the signs are the same, add the numbers and keep the same sign.
 If the signs are different,
o Before calculating, Think!... Are there more negatives or more positives? This will decide the sign of
your answer.
o Then, ignore the signs (that is, use the absolute value of each number.)
o Subtract the number closest to zero from the number farthest from zero. The sign of the answer is the
same as the number that is farthest from zero, that is, the number with the greater absolute value.
(see more about absolute value on the next page)
Summary of Subtraction with integers:
To find the difference of two integers, change the subtraction operation sign to an addition sign. Then change the sign of
the integer to the opposite of what the original problem asked you to subtract, then apply the rules for addition of
integers.
“REAL MATHEMATICIANS DON’T SUBTRACT, THEY ADD THE OPPOSITE.”
Math Notes Page 10 of 10
N.Paulson, 2012
Notes for Mathematicians
Absolute Value:
The absolute value of a number is the measurement of its distance from zero. It tells how far a particular
number is from zero.
For Example: the number
is 6 away from zero, but
is also 6 away from zero, just on the
other side. So the absolute value of
and
is 6. Since you don’t measure distance in negative
amounts, you just say that it is 6 away from zero, not
away from zero.
These are absolute value symbols. They are two vertical bars on either side of a number.
|
|
so, |
|
| |
and
So, if you are using absolute value to add and subtract integers, you find the difference between the absolute
values (ignore the signs) and then give the answer the sign of the number with greatest absolute value.
Example:
 Change subtraction to addition of the
opposite.
 There are more negatives than positives, the answer
 There are more negatives than positive
will be negative.
 Find the absolute Value:
|
|
numbers, so the answer is negative.
| |
and
|
|
and
|
|
 The difference between 6 and 5 is 1 because
 The difference between 6 and 5 is 1 because
 So, because there are more negatives, the answer
 So, because there are more negatives, the
will is
answer will is
So,
So,
With Tile Spacers:
5 positives
+
-
+++++
+
-
+
-
combined with
+
-
+
-
6 negatives
Equals
= 0 → so,
+5
-5
-
You can also pair numerals to make zeroes +5
-5
With
- - - - - -
Tiles spacers would be really time-consuming.
So Try something like this:
Same Result
Take 16 negatives from the 36 negatives and pair them with the 16 positives
add to the 20 remaining negatives.
+16
-16
plus
+16
to make a zero. Then
-16
BAD WORD ALERT!
Some people say cancel here, but
really we are MAKING ZEROES
or
Math Notes Page 11 of 11
N.Paulson, 2012
Notes for Mathematicians
Multiplying and Dividing Integers:
Remember 3rd grade? That’s when you learned how to multiply. You learned that 4 groups of 3 pencils gave
you a total of 12 pencils. Or:
. And, if you had 12 pencils and divided them between you and 3
friends you would each get 3 pencils. Or:
.
You may remember that multiplication and division are inverse operations, they undo each other.
Well, multiplication and division of Integers follows the same structure as multiplying and dividing whole
numbers.
Multiplication by a positive integer can be
represented by combining groups of the same
number:
In both examples, the 4 indicates the number of
groups of 3 (first example) and –3 (second example) to
combine.
and
Multiplication by a negative integer can be
represented by removing groups of the same
number:
Division is the Inverse of Multiplication, so we
can follow the patterns and divide. See if you
notice any patterns when using numerals instead
of tile spacers.
 If:
Then:
 If:
Then:
Modeling with tile spacers: If you begin with an equal
number of positive and negatives, that gives you a
neutral field. So let’s say you have 12 zeroes.
+ + + + + + + + + + + +
- - - - - - - - - - - means “remove four groups of 3.”
If you removed four groups of 3 positives, you would
have 12 negatives left.
So,
 If:
Then:
means “remove four groups of –3.”
If you removed four groups of 3 negatives, you would
have 12 positives left.
So,
 If:
Then:
In all cases, if there are an even number of negative
factors to be multiplied, the product is positive; if there
are an odd number of negative factors to be
multiplied, the product is negative.
This pattern also applies when there are more than
two factors. Multiply the first pair of factors, then
multiply that result by the next factor, and so on,until
all factors have been multiplied.
=12
Math Notes Page 12 of 12
N.Paulson, 2012
Notes for Mathematicians
Making Ones,… Giant Ones!
“Making ones” is a huge part of being flexible with numbers and being successful in higher level
Mathematics. The Identity Property of one and the equality properties help you make ones! Some
people say the word “cancel” when they really mean that they are “Making Ones.”
Cancel
The Many Uses of the Giant One:
1) Renaming Fractions for Simplifying when Adding: (Equivalent fractions and Simplified Fractions)
You can multiply a fraction, or any number, by one and the result is that same fraction, or number
a) Example of the giant one when renaming fifths to something in tenths:
→Divide both denominator and numerator by the same divisor
b) Example of the giant one when simplifying fractions:
→Multiply both denominator and numerator by the same factor
c) Example of the giant one when changing mixed numbers to fractions:
→
+
+
+
=
→
d) Example of the giant one when changing fractions to mixed numbers:
+
→
Math Notes Page 13 of 13
N.Paulson, 2012
Notes for Mathematicians
e) Example of the giant one when adding fractions: (This is for subtracting fractions too, but that is
really adding the opposite of a number… so, it’s still adding.)
→ When adding fractions with different denominators, you can rename fractions
Add:
with the same denominator. The Least Common Multiple (LCM) between 4 and 3 is 12, so you need
to rename the fractions into twelfths by making equivalent fractions.
Renaming
fractions into
the same units,
+
___________________
.....
is easier to talk
about than
makes it easier
to talk about
the total
fraction
to say
something
like....
𝒉 𝒅
⬚
_________________________________________________________________________________________
2) Example of the giant one when dividing exponents: some people call this “canceling” (which you already
know is a bad word), but really we are “making ones.”
Simplify:
Cancel
·
·
_________________________________________________________________________________________
3) Example of the giant one when solving algebraic equations:
Divide both sides by 2
Multiply both sides by the reciprocal ( )
Simplify →
(
)
( )
Simplify
Simplify
Simplify
9
Math Notes Page 14 of 14
N.Paulson, 2012
Notes for Mathematicians
Three Uses and Meanings of the Minus Sign:
Many students get confused with problems which involve a minus sign. The first step is to figure out
how the minus sign is being used and what it means in an expression or an equation.
1.
Subtraction Operation:
The problem is asking you to find the find the difference
between two numbers. In other words, the problem is asking you to subtract one number from
another. You can still think of subtraction as “take away”.
(It’s what you learned in first Grade)
→
6 take away 1 is 5
_____________________________________________________________________________________
2.
Negative number: Shows the location of a negative number on the number line.
(It’s what you learned in fifth Grade)
Smaller numbers on the left
Larger numbers on the right
The dot identifies
_____________________________________________________________________________________
3.
The Opposite of….:
This means the opposite of a number or quantity. It is sometimes the
most difficult to figure out. Opposite numbers are the same distance from zero on a number
line.
(You are going to learn more about this in middle school)
With Words:
With Symbols:
o
The opposite of two is negative two
o
The opposite of ten is negative ten
or
o The opposite of is negative
or
______________________________________________________________________________
Math Problems With the Minus Sign You Will See in Middle School:
Problem
With symbols
Or
Problem
With Words
Five negatives take away one negative is
four negatives.
The negative number can have parenthesis,
or not, these equations are equal.
Three positives plus two negatives
is one positive
The opposite of ….. negative two is two
Use /Meaning
of the minus sign
Subtraction Operation
Identify a negative number
 First Minus sign: Opposite
of…
 Second Minus Sign: Identify a
negative number
Math Notes Page 15 of 15
N.Paulson, 2012
Notes for Mathematicians
Three Uses for a Variable… that you’ll see in Middle school:
Variables are not just letters. They are symbols, or letters, that are used to represent quantities, either
unknown or varying (changing). They have many different meanings depending on how they are used
and what is the purpose. A variable can be a specific number, it can be used to explain (generalize) a
pattern (a relationship between varying quantities), or a variable can be used as a “placeholder” for a
formula.
Here’s some vocabulary you will need to know to understand how variables work:
Variable:
 A quantity that can change, or that may
take on different values.
 A letter or symbol representing such a
quantity
Quantity: Something that is measured. It can
be:
 An exact number
 A number that varies (changes)
 Two quantities that vary (or change)
together
Expression:
 a mathematical calculation using
numbers and/or variables, or other
mathematical operations
 don’t have equal signs or an inequality
symbol
 You simplify or evaluate expressions.
simplify: combine like terms as much
as you can (adding, multiplying… etc.)
evaluate: substitute a number in for a
variable and simplify the equations
Equation:
 a mathematical sentence using an equal
sign to separate two expressions
 You solve equations
solve: find the value of the variable
Common Variable Uses in Middle School:
Unknown: Solving for one specific quantity
Generalize: To explain a pattern that works for
any stage in the pattern, and can
predict future stages.
Placeholder: Variables used in proven
mathematical formulas.
____________________________________________________________________________
How are variables used?
1. As an unknown quantity: A variable is a letter, or other symbol, that is used to represent a
specific number. An algebraic equation can be written and solved for one specific answer.
Example:
or
⬚
This equation says that “some quantity” plus 2 is 10. The only answer in this case can be 8.
______________________________________________________________________________
2. To generalize (explain) a pattern or situation: A variable can be used to show the
relationships in a pattern that varies (changes).
Example: Let’s say that you are going to raise money for your soccer club by having a
car wash. You are going to earn $5 for each car that you wash. You are going to have to
spend (a minus quantity) $40 for supplies such as sponges, soap, towels, window
cleaner, etc.
Math Notes Page 16 of 16
N.Paulson, 2012
Notes for Mathematicians
You can write an expression to show how you might predict how much money you could earn,
depending upon how many cars you wash.
Quantities in the expression:
Expression:
This expression can be used to say “$5 times the number of cars washed, minus the $40 for
supplies” will tell you the money earned for your soccer club.
You can evaluate the expression for any number of cars you wash:

If you wash 30 cars, you would substitute “30” for “c” in the expression and simplify
the expression “do the math”
 Substitute
 Calculate multiplication
You earn:
 Simplify

If you wash 45 cars:
You earn:
 Substitute
 Calculate multiplication
 Simplify
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You can use an equation to figure out how many cars you need to wash to earn a certain
amount of money. You would use two variables.
Quantities in the equation:
Equation:
Evaluate an equation:
 Let’s say, you want to earn $300 dollars, how many cars do you have to wash?
You substitute 300 for the
which represents the amount of money you want to
earn, then solve the equation to find out how many cars to wash:
THINK!
What amount of money minus $40 is $300? So, we got
that amount of money by washing cars at $5 each… how many cars did we wash?
Or:
You can add the $40 to both sides of the equation because you are going to have to wash some cars to
earn money to pay your coach back for the supplies. Next, divide the remaining amount by $5 to get
the number of cars you need to wash to earn $300 for your club, and $40 to pay for the supplies.
Or:
You can solve this problem using an Algorithm (step-by-step procedure) you can use. It is explained in
more detail in the next pages. (see page 11)
Math Notes Page 17 of 17
N.Paulson, 2012
Notes for Mathematicians
Solve the equation to find out how many cars you need to wash in order to earn $300.
 Change a subtraction operation to the
addition of the opposite.
 Make Zeroes: by adding 40 to each side.
Calculate the addition.
 Simplify
 Make Ones: Divide each side by 5. Calculate
the division.
 Simplify
So, you would have to wash 68 cars in order to earn $300 for your club.
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3. As a Placeholder, as in a formula: Formulas use letters (variables) that stand given
quantities in a formula. When you know a value for one of the variables, then you can
substitute (plug in) that value in for the letter in the formula that is holding the place for
that value.
Example: The formula for finding the volume of a box is:
V = bwh
V stands for volume, b for base, w for width and h for height.
When b=10, w=5, and h=4, then V = 10× 5 × 4 = 200
The letters a, b, and c in mathematical properties stand for “any number,” so don’t let them
scare you. Check out this one that you will learn in Algebra: The Quadratic Formula….
√
is what you are trying to find out
stand for values (numbers) that
you will know, and can just plug (substitute)
the values into the formula.
Math Notes Page 18 of 18
N.Paulson, 2012
Notes for Mathematicians
Solving One-Step Algebraic Equations:
Here’s a standard algorithm (step-by-step procedure) solving one-step algebraic equations.
Some steps may not be necessary, but mathematical properties say that this is the order you
should use.
Vocabulary:
Coefficient: The number multiplying or
Constant: The numeral that doesn’t have a
dividing the variable.
variable.
Step 1: THINK! Cover up the variable and think about what value you could put in there to
make true equation.
Step 2: Additive inverse, change any subtraction operation to addition of the opposite,
because then you can use the commutative property to add in any order.
Step 3: Balance Scale: Keep the equation balanced using mathematical properties. Move
constants to one side and variables to the other
a. Make Zeros: (if equation is adding or subtracting) add the opposite of the constant
to each side
b. Make Ones: (if equation is multiplying or dividing)
If the equation shows Multiplication: divide by the coefficient
If the equation shows Division: Multiply each side by the reciprocal of the
coefficient
Step 4: Check your answer in the original equation by substituting the value for the variable
back into the equation and see if it makes a true sentence.
______________________________________________________________________________
Example:
Step 1: THINK! Cover up the variable and think about what value you could put in there to
make a true equation.
THINK! What number plus 4 equals 15?
⬚
“I’m thinking it might be 11….”
Step 2: Additive Inverse is not necessary in this example
Step 3: Balance Scale: Keep the equation balanced!
 Make Zeros: (add the opposite of the constant to each side) Move constants to one
side and variables to the other.
 or Make Ones: (Multiply each side by the reciprocal… or divide by the coefficient)
Make Zeroes: Add the opposite of the
constant to both side of the equation.
Step 4: Check your answer in the original
equation to see if it makes a true
equation.
If
, then…
Is this true?
√ check
Math Notes Page 19 of 19
N.Paulson, 2012
Notes for Mathematicians
Check: does the solution make
the equation true?
Make Ones: Divide both sides by the
coefficient
√
Check: does the solution
make the equation true?
Make Ones:
Multiply both sides by the
reciprocal of the coefficient
( )
( )
( )
( )
√
_____________________________________________________________________________________________________________
Solving Two-Step Algebraic Equations:
This is the same as the standard algorithm for solving one-step algebraic equations, just adding
one more step. Some steps may not be necessary, but mathematical properties say that this is
the order you should use.
Step 1: THINK! Cover up the variable and think about what value you could put in there to make true
equation.
Step 2: Additive inverse, change any subtraction operation to addition of the opposite, because then
you can use the commutative property to add in any order.
Step 3: Combine Like terms on each side of the equal sign.
Step 4: Balance Scale: Keep the equation balanced using mathematical properties. Move constants to
one side and variables to the other
Step 5: Make Zeros: Undo addition, add the opposite of the constant to each side.
Step 6: Make Ones: Undo Multiplication or division:
a. If the equation shows Multiplication: divide by the coefficient
b. If the equation shows Division: Multiply each side by the reciprocal of the
coefficient
Step 7: Check your answer in the original equation by substituting your solution for the variable back
into the equation and see if it makes a true sentence.
Example:
 THINK! ⬚
What number times 2
plus 4 equals 20?
 Make Zeroes: Add the opposite of the constant
to each side. Calculate addition.
 Simplify
 Make Ones: Divide each side by the coefficient.
Calculate the division.
 Simplify and Check:
√Check
Math Notes Page 20 of 20
N.Paulson, 2012
Math Notes
Order of Operations – AF 1.3(6), AF 1.4(6), AF1.2(7) & AF 2.1(7)
You may have learned to remember the steps for order of operations as PEMDAS, but now that you are
in middle school it really should be GEMDAS.
G
Grouping Symbols
E
Exponents
 Parenthesis
 Square Root Symbol
 Division Bar
 And some other symbols that you
will learn later
M
D
A
S
Multiplication and Division
in the order they occur
Addition and Subtraction
in the order they occur
From left to right
From left to right
The order of operations is the conventional order in which to calculate number problems. It just
gradually developed as the modern symbols for algebra and arithmetic developed. This order is
determined according to the rules of math; i.e number properties and number relationships.
One Way to remember the order of operations is to use an acronym:
GEMDAS: Graciously Excuse My Dear Aunt Sally….
Grouping Symbols; Exponents; Multiplication and Division (in the order they occur); Addition and
Subtraction (in the order they occur)
Representation 1:
Step-By-Step (using an algorithm)
Parenthesis (grouping)
Exponents
Multiplication & Division
Addition & Subtraction
Simplified
Representation 2:
You can simplify the operations between the
plus and minus signs first
+
+
59
Math Notes Page 21 of 21
N.Paulson, 2012
Math Notes
Patterns and Properties for Exponents – AF2.1 (7)
A number in exponential form is written with a base and an exponent. When the exponential
form is simplified, the result is a power of the base.
Vocabulary:
Base: is the number to be used as a factor
Exponent: is the number of times to use the
base as a factor
Exponential form:
4 is the
3 is the
Expanded form:
Power: The process of using exponents is called
"raising to a power", where the exponent is the
"power".
Patterns for exponents:
Exponential
form
Expand it out
Power
32
16
8
4
2
Any number to the
zero power = 1
because it follows
the pattern
1
What Patterns can we see in the chart?
 The base is the same
 The exponents increase and decrease in sequential order
 Going up the pattern multiplies the previous power by 2
(doubles it)
 Going down the pattern divides the previous power by 2,
(halves it)
 The base to the power of 1 is equal to the base
 The base to the power of 0 equals 1
 Negative exponents can be found by continuing down
the pattern. They make fractions, not negative numbers.
 Negative exponents are related to positive exponents:
 Multiplying and dividing powers are related to each
other:
→
, so…..
=8
→
=
Properties for exponents (that come from the patterns of exponents):
Raised to the Power of One: AF 2.2(7)
Any number raised to the power of one equals that number. If there is no exponent written, we know
that it is a power of one.
With Algebra:
With Arithmetic:
__________________________________________________________________________________
Raised to the Power of Zero: AF 2.2(7)
Any number raised to the power of zero equals 1.
With Algebra:
With Arithmetic:
Some mathematicians disagree, for complicated reasons, but it is accepted that
Math Notes Page 22 of 22
N.Paulson, 2012
Math Notes
Multiplying Exponents with the Same Base: AF 2.2(7) & ns 2.3(7)
“When in doubt, expand it
out!” When multiplying exponents with the same base, you add the exponents.
With Algebra:
With Arithmetic:
Expand it out→
√
____________________________________________________________________________________________________________________
Dividing Exponents with the Same Base:
AF 2.2(7) & ns 2.3(7) “When in doubt, expand it
out!”
When dividing exponents with the same base you can subtract the exponents. Or, you can make giant
ones.
With Algebra:
With Arithmetic:
→
Expand it out→
or
Make ones:
____________________________________________________________________________________________________________________
Finding a Power of a Power:
AF 2.2(7) & ns 2.3(7)
“When in doubt, expand it out!”
With Algebra:
With Arithmetic:
Expand it out→
√ check
Math Notes Page 23 of 23
N.Paulson, 2012
Math Notes
RESOURCES:
 Mathwords A to Z: www.mathwords.com
o Conceptually accurate definitions
 Cool Math: www.CoolMath.com
o Good explanations about concepts and procedures in language students easily
understand
o Challenging Math Games
 Math is Fun: www.mathsisfun.com
o This site has a great dictionary, with pictures and animations.
o Fun games and puzzles
 Purple Math: http://www.purplemath.com/index.htm
o Great lessons for “How do you really do this stuff?” Good for higher level concepts.
 College Preparatory Math: www.CPM.org
o This is a great site for conceptual explanations of the mathematics.
o The Parent guides are very helpful for parents, and students.

Math Notes Page 24 of 24
N.Paulson, 2012