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Mathematician’s Notes

Math Notes of 1 N.Paulson, 2012

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!



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.

http://www.mathsisfun.com/temperature-conversion.html



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 to come up with

. So now we need

another category for numbers.

*

*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…



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.


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: With Arithmetic: Multiplication

Division

__________________________________________________________________________________

The Identity Property of Addition: Any number plus zero is that number. Adding zero to a number does not change its value.


_____________________________________________________________________________________

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


Add first, them multiply: √ it checks!

Multiply each separately, then add:

This is how you may see the distributive property in future math problems:

or

Notes for Mathematicians


Cancel

+ + + + + + + + +

+ + + + + + + + - - -

+ + + + - - - 0+0+0+1=1

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.


Or, you can offset the negative number by putting a parenthesis around it so it’s easier to see.


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

We DON’T “CANCEL”… We MAKE ZEROES

Vertical Number

Line



𝑎𝑏

𝑎𝑏

Cancel

Cancel

Multiplicative Inverse Property:

Any number multiplied by its reciprocal equals one. This property allows us to divide numbers.

With Algebra:

→

1 → (any number divided by itself equals 1)

With Arithmetic:

→

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:

and

, because →

and →

1

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

is 1.

→

1

Or,

→

1 1 = → (any number divided by itself equals 1)

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,”

… but don’t say that bad word!→



Zero Product Property: The product of any number and zero is zero


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.


them (pass them out) between 4 friends… how many does each friend get? Well, duh, 0. But….


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!



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.

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 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.

Build the 1st integer: - - -

Build the 2nd integer: + + + + + Pair one positive and one negativeto make zero pairs.

+ + Whatever is left over is your sum.


Say: “three negatives plus 5 negatives equals 8 negatives.”

Build the 1st integer: - - -

Build the 2nd integer: - - - - -

There are no zeroes to pair.

Combine all the negatives and count them.

- - - - - - - - equals

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. We write negative numbers using a minus sign, such as: is said “negative seven” 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.”

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

Same Result



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

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. - - -

Subtraction is the same as adding the opposite:

Change subtraction operation to addition of the

opposite. Build 6 negatives, add three positives, and make zero pairs with 1 positive and 1 negative. Whatever is not paired up is your sum.

- - -

Solve subtraction problems with tile spacers by adding zero pairs first:

Add Zeroes, then take away negatives Use the Additive Inverse Property to change

the subtraction operation to addition

Say: “5 positives, take away, 3 negatives.”

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)

5 + 0 + 0 + 0 is still 5

Now, take away 3 negatives:

+ + + + + + + +

Which leaves 8 positives: —

Say: “5 positives, take away, 3 negatives.” Use the Additive Inverse property to

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:


- - -

+ -

+ -

+ -

+ -

+ -

+ -

- - -

Same Result

Same Result



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.


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.




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:

There are more negatives than positives, the answer

will be negative. Find the absolute Value:

| | and | | The difference between 6 and 5 is 1 because

So, because there are more negatives, the answer

will is

So,

Change subtraction to addition of the

opposite.

There are more negatives than positive

numbers, so the answer is negative.

| | and | | The difference between 6 and 5 is 1 because

So, because there are more negatives, the

answer will is

So,

With Tile Spacers:

5 positives + + + + + combined with 6 negatives - - - - - - Equals

-

You can also pair numerals to make zeroes = 0 → so,

With Tiles spacers would be really time-consuming.

So Try something like this:

Take 16 negatives from the 36 negatives and pair them with the 16 positives to make a zero. Then add to the 20 remaining negatives.

plus

or

+ -

+ -

+ -

+ -

+ -

+5 -5

+5 -5

Same Result

+16 -16

+16 -16

BAD WORD ALERT! Some people say cancel here, but

really we are MAKING ZEROES



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: 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,

means “remove four groups of –3.” If you removed four groups of 3 negatives, you would

have 12 positives left. So,

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

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:

If: Then: If: Then:



Cancel

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.”

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:

+

→



Cancel

makes it easier to talk about

the total fraction

to say something

like....

𝒉 𝒅 ⬚

.....

is easier to talk about than

Renaming fractions into

the same units,

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.)

Add:

→ When adding fractions with different denominators, you can rename fractions

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.

+

___________________

_________________________________________________________________________________________ 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:

· ·

_________________________________________________________________________________________ 3) Example of the giant one when solving algebraic equations:

Divide both sides by 2

Simplify →

Multiply both sides by the reciprocal (

)

(

) (

) Simplify

Simplify

Simplify 9



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 Problem

With Words Use /Meaning

of the minus sign

Or

Five negatives take away one negative is four negatives.

The negative number can have parenthesis, or not, these equations are equal.

Subtraction Operation

Three positives plus two negatives

is one positive Identify a negative number

The opposite of ….. negative two is two

First Minus sign: Opposite of…

Second Minus Sign: Identify a negative number



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.



You can write an expression to show how you might predict how much money you could earn, depending upon how many cars you wash.

Expression:

Quantities in the 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”

You earn:

Substitute Calculate multiplication Simplify

If you wash 45 cars:

You earn:

Substitute Calculate multiplication Simplify

____________________________________________________________________________

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.

Equation:

Quantities in the 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)



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. ____________________________________________________________________________

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.



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 dividing the variable.

Constant: The numeral that doesn’t have a 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



Make Ones: Divide both sides by the coefficient

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

√

Check: does the solution make

the equation true?

(

)

(

)

(

)

(

)

Check: does the solution

make the equation true?

√

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 E M D A S Grouping Symbols Exponents

Multiplication and Division in the order they occur

Addition and Subtraction in the order they occur

Parenthesis

Square Root Symbol

Division Bar

And some other symbols that you will learn later

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


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".

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.


Some mathematicians disagree, for complicated reasons, but it is accepted that

Patterns for exponents: Exponential

form Expand it out Power What Patterns can we see in the chart?

32

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 →

=

16

8

4

2

Any number to the zero power = 1

because it follows the pattern

1

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.


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.


→

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


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.

http://www.mathwords.com/

http://www.coolmath.com/

http://www.mathsisfun.com/

http://www.purplemath.com/index.htm

http://www.cpm.org/

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