**re-Algebra Topics**. keyed to a popular communityccollege remedoal text (Martin-Gay.. Fourth Edition). It is used at our local Antelope Valley College for students who need a remedial course as determined by the ACT College Placement Test. Mastery of these topics will avoid having to take such an expensive ($ and time) remedial course.

See also a list of mathematical concepts and skills at ** IXL Grade 7 Math Practice .**

Page Contents - BackSpace to Return

- 1.0 Whole Numbers
- – 1.1 Study Skill Tips for Success in Mathematics
- – 1.2 Place Value, Names for Numbers, and Reading Tables
- – 1.3 Adding Whole Numbers and Perimeter
- – 1.4 Subtracting Whole Numbers
- – 1.5 Rounding and Estimating
- – 1.6 Multiplying Whole Numbers and Area
- – 1.7 Dividing Whole Numbers
- – 1.8 An Introduction to Problem Solving
- – 1.9 Exponents, Square Roots, and Order of Operations
- 2. Integers and Introduction to Variables
- – 2.1 Inroduction to Variables and Algebraic Expressions
- – 2.2 Introduction to Integers
- – 2.3 Adding Integers
- – 2.4 Subtracting Integers
- – 2.5 Multiplying and Dividing Integers
- – 2.6 Order of Operations
- 3. Fractions and Mixed Numbers
- – 3.1 Introduction to Fractions and Mixed Numbers
- – 3.2 Factors and Simplest Form
- – 3.4 Adding and Subtracting Like Fractions, Least Common Denominator and Equivalent Fractions
- The least common denominator of a list of fractions is the smallest positive number divisible by all the denominators in the list.
- – 3.5 Adding and Subtracting Unlike Fractions
- Add unlike fractions
- – 3.6 Complex Fractions, Order of Operations, and Mixed Numbers
- – 3.7 Operations on Mixed Numbers
- 4. Decimals
- – 4.1 Introduction to Decimals
- – 4.2 Adding and Subtracting Decimals
- – 4.3 Multiplying Decimals and Circumference of a Circle
- – 4.4 Dividing Decimals
- – 4.5 Fractions, Decimals, and Order of Operations
- – 4.6 Square Roots and the Pythagorean Theorem
- 5. Ratio, Proportion, and Measurement
- – 5.1 Ratios
- – 5.2 Proportions
- – 5.3 Proportions and Problem Solving
- – 5.4 Length: U.S. and Metric Syst ems of Measurement
- – 5.5 Weight and Mass: U.S. and Metric Systems of Measurement
- – 5.6 Capacity: U.S. and Metric Systems of Measurement
- – 5.7 Conversions Between the U.S. and Metric Systems
- 6. Percent
- – 6.1 Percents, Decimals, and Fractions
- – 6.2 Solving Percent Problems Using Equations
- – 6.3 Solving Percent Problems Using Proportions
- – 6.4 Applications of Percent
- – 6.5 Percent and Problem Solving: Sales Tax, Commision, and Discount
- – 6.6 Percent and Problem Solving: Interest
- 7. Statistics and Probability
- – 7.1 Reading Pictographs, Bar Graphs, Histograms, and Line Graphs
- – 7.2 Reading Circle Graphs
- – 7.3 Mean, Median, and Mode
- – 7.4 Counting and Introduction to Probability
- 8. Introduction to Algebra
- – 8.1 Introduction to Variables
- Variable term
- – 8.2 Solving Equations: The Addition Property
- – 8.3 Solving Equations: The Multiplication Property
- – 8.4 Solving Equations Using the Arithmetic Properties
- – 8.5 Equations and Problem Solving
- 9. Geometry
- – 9.1 Lines and Angles
- – 9.2 Plane Figures and Solids
- – 9.3 Perimeter
- – 9.4 Area
- – 9.5 Volume and Surface Area
- – 9.6 Congruent and Similar Triangles
- Appendices
- Appendix A: Tables
- Appendix B: Exponents and Polynomials
- Appendix C: Inductive and Deductive Reasoning

##### 1.0 Whole Numbers

##### – 1.1 Study Skill Tips for Success in Mathematics

Use our Show-Hide Text Toggling to conveniently:

-1- Read a topic title – generally for a concept definition or procedure name

-2- Actively recall and recite the definition or procedure steps

-3- Click the title to check accuracy and completeness of your recall & recitation

-4- Toggle click on title to repeatedly show-hide the text until you can correctly recall it

-5- Link to and toggle concrete worked examples for understanding and creating your own examples,

-6- Link to prior knowledge connections, and then

-7- Practice answering interleaved review-questions,

-8- Familiarize yourself with typical questions for each skill topic

-9- Practice answering skills at **IXL Grade 7 Math Skills Practice**.

=10- P**review topics for your next-class-lesson work.**

##### – 1.2 Place Value, Names for Numbers, and Reading Tables

#### Digits

The Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to write numbers using base ten – our decimal system.

Other important numbers with different bases:The digits 0 and 1 are used to write numbers using base two – our binary system used to represent on/off states in computers and in electrical machinery.

The digits 1 through 59 were used to write numbers using base 60 – the ancient Babylonian numeration system that was convenient for its easier representation of fractions in commerce, etc.

#### Whole Numbers

The whole numbers for our decimal system are 0, 1, 2, 3, 4, 5 6, 7, 8, 9, 10, 11, … etc.The smallest whole number is 0.

Each whole number is one larger than the previous one.

There is no largest whole number.

#### Place value in whole numbers

The place value of a digit is the value of a digit’s position in a number determined by its position in the number.

The value of a digit is given by the place value.

Place values are determined from, right to left as: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred million; billions, ten billions, hundred billions, trillions — and so forth.

#### Standard form for whole numbers

A whole number written in standard form is a string of digits or numerals, separated into groups of threes by commas.

Each group of three digits is called a period.

Beginning from right to left we have the following periods: ones; thousands, millions; billions; trillions; and so forth.

#### Write** a** whole number i**n** words

To write a whole number in words, write the number in each period (excepting the ones period).

#### Read a whole number in words

To read a whole number in words, read the number in each period with the name of the period – from left to right.

Do not name the ones period.

#### Write a whole number in standard form

To write a number in standard form, write the number in each period followed by a comma.

#### Expanded form of whole numbers

The expanded form of a whole number shows each digit of the number with its place value.

#### Write a whole number in expanded form

Write each digit with its place value named.

#### Whole numbers graphed on the number line

Whole numbers are pictured as equally spaced points on a number line in counting order from 0 to the right.

#### One whole number greater than another whole number

The whole number graphed to the right of another whole number on the number line is the greater of the two.

#### One whole number smaller than another whole number

The whole number graphed to the left of another whole number on the number line is the smaller of the two.

#### Compare whole numbers

We use the symbol “ > ” to state that one number, a, is greater than another, b:

a > b.

We use the symbol “ < “ to state that one number, a is smaller than another, b:

a < b.

##### – 1.3 Adding Whole Numbers and Perimeter

#### Sum of whole numbers

The sum is the result of adding the addends when adding whole numbers.

#### Addends

Addends are the numbers added when you add whole numbers.

#### Add two or more whole numbers with no carrying

Write the addends so that the digits with the same place value position are vertically aligned.

To add two or more whole numbers with no carrying, we add the digits of each addend in the ones place, then the tens place, then the hundreds place, and so on, from right to left.

No “carrying” is necessary if the sum of each of the digits in its place adds to 9 or less.

#### Add two whole numbers with carrying

Write the addends so that the digits with the same place value position are vertically aligned.

To add two whole numbers with carrying, we add the digits in each place and carry 1 if the sum of the digits is greater than 9. This means we add 1 when summing the digits in one place to the left of the one we are now adding.

We do this when summing the digits in each place value position, left to right.

#### Addition property of 0

The sum of 0 and any number is that number.

#### Commutative property of addition

We can add any two numbers in any order without changing the sum.

#### Associative property of addition

When adding whole numbers we can change the grouping of the addends without changing the sum.

#### Polygon

A polygon is a flat figure formed by line segments connected at their ends.

#### Perimeter of a polygon

The perimeter of a polygon is the distance around the polygon.

The perimeter of a polygon is the sum of the lengths of its sides.

#### Key words in addition word problems

Key words in addition word problems may include: added to; plus; increased by; more than; total, sum.

##### – 1.4 Subtracting Whole Numbers

#### Difference of whole numbers

The difference of two numbers results from subtracting one whole number, the subtrahend, from another, the minuend.

#### Minuend

The minuend is the number from which one subtracts.

#### Subtrahend

The subtrahend is the number that one subtracts from the minuend to get the difference.

#### Subtraction properties of 0

The difference of any number and that same number is 0.

The difference of any number and 0 is that same number.

#### Subtract whole numbers without borrowing

Write the minuend and subtrahend so that the digits with the same place value position are vertically aligned.

Find the difference between the digits with the same place value position.

Borrowing will not be necessary if the minuend digit is greater than the corresponding subtrahend digit.

#### Subtract whole numbers with borrowing

Write the minuend and subtrahend so that the digits with the same place value position are vertically aligned.

Find the difference between the digits with the same place value position.

Borrowing is necessary when a digit in the second number (the subtrahend) is larger than the corresponding digit of the first number (the minuend).

We borrow 10 for this place value digit subtraction by subtracting (“borrowing”) 1 from the minuend digit one place value position higher – when next subtracting in that position.

#### Key words in subtraction word problems

The following key words or phrases may occur in subtraction word problems: subtract; difference; less; take away; decreased by; subtracted from.

##### – 1.5 Rounding and Estimating

#### Rounding whole numbers

Whole numbers are rounded or approximated to better use, understand, simplify computation or remember the numbers.

#### Round whole numbers to a given place value

If this digit is 5 or less, replace it and each digit to its right by 0.

If this digit is 6 or more, add 1 to the digit on left and replace all digits to its right to 0.

#### Round whole numbers to estimate a sum

Round addends and add to estimate sum and check credibility of exact sums.

#### Round whole numbers to estimate a difference

Round minuends and subtrahends to estimate differences and check for incorrect determination of them using exact calculation.

##### – 1.6 Multiplying Whole Numbers and Area

#### Multiplication of whole numbers

Multiplication is repeated addition but with different notation.

#### Factors

Factors are the numbers being multiplied to produce the product of a multiplication.

#### Multiplication property of 0

The product of 0 and any number is 0.

Multiplication property of 1

The product of 1 and any number is that same number.

#### Commutative property of multiplication

When multiplying two numbers the order of these numbers can be changed without changing the product.

#### Associative property of multiplication

When multiplying two numbers you can change the grouping of factors without changing the product.

#### Distributive property of multiplication

Multiplication distributes over addition.

#### Partial product

The intermediate product of a single digit in the first factor by the digits of the second factor keeping account of the 0’s for the place value of the digit in the first factor.

Partial products are aligned vertically corresponding to place values and summed to form the complete product of a whole number multiplication.

#### Product of two whole numbers

The product of two whole numbers is the result of multiplying two whole numbers.

#### Multiply two or more whole numbers

Write each factor to vertically align place value positions.

Multiply each digit of the first factor by the digits of the second factor to form partial products, carrying where necessary.

Align partial products vertically so that place value positions correspond.

Add partial products to determine the product of the factors.

Use Associative property of multiplication to group factors and replace the multiplication of any two by its product and the next remaining factor to be multiplied.

#### Multiply any whole number by 0

Any whole number multiplied by 0 is 0 according to the Multiplication property of 0

#### Apply commutative property of multiplication for whole numbers

Change the order of the factors to be multiplied without changing the product.

#### Apply distributive law of multiplication for whole numbers

Distribute the multiplication over the addition without changing the product.

#### Apply associative law of multiplication

Change the grouping of factors without changing the product.

#### Rectangle

A rectangle is a four sided figure with two equal and opposite sides, called the length and width.

The sides at the corners are perpendicular or at right angles to each other.

#### Area

Area is a measure of the amount of surface of a region. Thus, it is the count of unit square measures covering that region.

It is a special application of multiplication.

#### Find area of rectangle

The area of a rectangle is the product of its length and its width.

#### Key words for multiplication word problems

Several words and phrases that indicate the operation of multiplication include:

multiply, product, times.

##### – 1.7 Dividing Whole Numbers

#### Division of a whole number by another whole number

Division is the process of separating the quantity into b equal parts and considering a of them.

Division of a whole number a by another whole number b is written as:

a division-sign b or or

a / b (using parentheses as a most convenient way to write division using a single line of type).

#### Fraction bar or slash

The bar separating a and b in the division a by b ( or a / b)

is called a fraction bar or the fraction slash – it symbolizes division.

#### Quotient of two whole numbers

The quotient of two whole numbers is the result of the division process, consisting of another whole number which counts the whole number of multiples of b in a.

#### Divisor

The divisor is the number that is used to divide into a dividend. It is b in the division a/b.

#### Dividend

The dividend is a, the whole number to be divided in the division a/b.

#### Remainder

The remainder of a division q = a/b is the whole number quantity left after determining an even whole number quotient.

#### Find quotient of any whole number by itself

a / a = 1 if a is any whole number

#### Find quotient of any whole number by 1

a / 1 = a where a is any whole number

#### Find quotient of 0 by any whole number other than 0

0 / b = 0 if b ≠ 0 an db = a whole number

#### Find quotient of any whole number by 0

a / 0 = undefined

#### Perform long division of any whole number by another whole number

Write the divisor to the left of the division radical, include the dividend in the radical, and write digits of the long division solution digit by digit over the corresponding place value.

Test each digit as a trial dividend until a whole number quotient can be found. Write the quotient digit above and in alignment with the place values of the rightmost digits.

Multiply the quotient digit by the divisor and write under corresponding place value.

Determine the difference and bring down the next number from the dividend.

Determine a trial quotient using the difference; if no trial quotient is possible, bring down the next digit of the dividend to augment the number of digits in the trial dividend.

Continue until the last digit of dividend is considered.

Write any difference a remainder.

#### Key words in division word problems

Key words and phrases that may indicate the operation of division include:

divide, quotient, divided by, divided or shared equally among.

#### Average

The average of a list of numbers is the sum of the numbers divided by the number of listed numbers.

Find average of a list of numbers

The average of a list of numbers is found by adding the numbers in the list and dividing the total by the number of list items.

##### – 1.8 An Introduction to Problem Solving

##### – 1.9 Exponents, Square Roots, and Order of Operations

#### Exponential notation

Exponential notation is shorthand for the repeated multiplication of a number.

#### Exponent

An exponent is a shorthand notation for the number of repeated multiplications of a factor, also called the base of the exponent.

#### Base

The base in exponential notation is the number or factor that is to be repeatedly multiplied.

#### Factor

The factor in exponentiation is the number to be repeatedly multiplied when multiplying.

Write using exponential notation

To write repeated multiplication of a factor using exponential notation – use the factor as the base and use the number of multiplications for the exponent.

Evaluate an exponential expression

To evaluate an exponential expression: write out the repeated multiplication by using the base as the common factor, write out as many multiplications as the exponen, and

then multiply – left to right.

#### Simplify expressions with whole numbers using order of operations

The order of operations for whole numbers is as follows:

• Do all operations within grouping symbols such as parentheses or brackets.

• Evaluate any expressions with exponents

• Multiply and divide in order from left to right.

• Add or subtract in order from left to right.

#### Find area of square

A square is a four sided polygon with equal sides perpendicular to each other.

##### 2. Integers and Introduction to Variables

##### – 2.1 Inroduction to Variables and Algebraic Expressions

#### Variable

A variable is a letter used to represent a number.

#### Algebraic expression

An algebraic expression (or an expression) is a combination of operations on variables and numbers.

#### Expression

See algebraic expression. It is a combination of operations on variables and numbers.

#### Evaluate (algebraic) expression for the variable

To evaluate an expression for a variable is to replace a variable by a number, translate the verbal phrases into variable expressions and then finding the value of that expression..

##### – 2.2 Introduction to Integers

#### Positive numbers

A positive number is a whole number greater than 0.

#### Negative numbers

A negative number is a whole number less than 0.

#### Signed numbers

Signed numbers consist of positive numbers, 0, and the negative numbers.

#### Integers

Integers are signed whole numbers, to include 0.

#### Graph integers

Use the left to right increase-directed number line to graph the negative numbers to the left of 0 and graph the positive whole numbers to the right of 0.

#### Compare integers

An integer a is greater than another integer b if the graph position of a on the number line is to the right of b.

We use the notation a > b to say that “a is greater than b”.

We use the notation b < a to say that b is less than a

If you think of “<“ and “ > “ as the points of an arrowhead, notice that the “greater than” or “less than” symbol always points to the smaller integer.

#### Absolute value of an integer

The absolute number of a number is its distance from 0 on the number line.

The absolute value is always positive.

#### Opposite integers

Opposite integers are two integers that are the same distance from 0 on the number line and on opposite sides of 0.

The minus sign ” – ” is used to state “opposite of”.

#### Find opposite of a given integer

The opposite of “a” is “-a”.

If a > 0 then a is to the right of 0 on the number line, “a” units. The opposite of “a” is a units to the left of 0, “a” units.

If a < 0, then a is to the left of 0, ”a” units – and the opposite of “a” is then “a” units to the right of 0.

##### – 2.3 Adding Integers

#### Add two integers with the same sign

To add integers on a number line start from 0 on the line and draw an arrow representing the first number.

From the tip of this first arrow draw another arrow representing the second number.

The tip of the second arrow represents their sum.

For larger numbers – add their absolute values and use their common sign as the sign of the sum.

#### Add two integers with different signs

To add two integers a and b with different signs – first find the larger absolute value minus the smaller absolute value.

Use the sign of the number with the larger absolute value as the sign of the sum.

#### Apply associative property of integer addition

If a, b and c are integers then

a + (b +c) = (a + b) + c

#### Apply commutative property of integer addition

If a and b are integers then

a + b = b + a.

#### Evaluate addition of expressions using integer replacement values

Rewrite expressions showing the positive and negative signs of the numbers.

Add expressions keeping separately the “+” sign for the addition operation.

##### – 2.4 Subtracting Integers

#### Subtraction of two integers

We rewrite subtraction problems as integer addition problems.

If a and b are numbers, then a – b = a + ( -b ).

#### Add and subtract more than two integers

Rewrite the differences as additions of signed numbers and add left to right.

Evaluate adding and subtracting expressions using integer replacement

Substitute integers explicitly for the variables in the expression – being explicitly mindful of each sign.

Rewrite subtractions as additions and add left to right.

##### – 2.5 Multiplying and Dividing Integers

#### Multiplication of integers

Multiplication of integers is similar to that for whole numbers – excepting that one must be mindful of the signs of each factor.

The product of two numbers having the same sign is a positive number.

The product of two numbers of different sign is a negative number.

#### Multiply two like-signed integers

(+a) • (+b) = +(a • b) = a • b

or

(-a) (-a) = + (a • b) = a • b

#### Multiply two unlike-signed integers

If a and b are whole numbers, then

(-a) • (+b) = – (a • b)

or

(+a) • (-b) = – (a • b)

#### Division of integers

The quotient of two numbers having the same sign is a positive number.

The quotient of two numbers having different signs is a negative number.

#### Divide two like-signed integers

If a and b are whole numbers then – if b is not 0 –

-a /-b = +(a / b)

or

+a / +b = +(a / b)

#### Divide two unlike signed integers

If a and b are whole numbers then – if b is not 0 –

-a / +b = -(a / b)

or

+a / -b = -(a / b)

##### – 2.6 Order of Operations

#### Order of Operations Review (GEMDAS)

• Look for and do all operations within a **G** rouping symbol (such as absolute values, fraction bars, parentheses and bracket)

• Evaluate any expressions with **E** xponents

• **M **ultiply and **D **ivide in order from left to right

• **A **dd or **S **ubtract in order from left to right

##### 3. Fractions and Mixed Numbers

##### – 3.1 Introduction to Fractions and Mixed Numbers

#### Fractions

A fraction is a part of a whole.

It is written in terms of a denominator and a numerator separated by a fraction bar or slash.

The value of the fraction is the quotient of the numerator by the denominator with division signified by the fraction bar or slash.

0 cannot be a denominator because division by 0 is undefined.

The fraction bar is traditionally the horizontal bar – or in single-line math type – the backslash symbol “/” – and it signifies division of the numerator by the denominator.

#### Numerator

The numerator of a fraction defines how many parts of the whole are being considered. It is the number or expression above the fraction bar.

The denominator defines the total number of equal parts of the whole being considered. It is the number or expression below the fraction bar.

#### Denominator

The denominator of a fraction defines the total number of equal parts in the whole.

It can never be 0.

#### Proper fractions

A proper fraction is a fraction whose numerator is less than its denominator.

#### Improper fractions

An improper fraction is a fraction whose numerator is greater or equal to its denominator.

#### Write fractions to represent shaded areas of figures

Let a figure be divided into “a” equal parts. Let “b” be the number of parts in the figure that are shaded.

Then the fraction that represents the shaded areas of the figure is the fraction b/a.

This is one way to help one visualize the concept “fraction”.

#### Graph fractions on the number line

Fractions are graphed on the number line between 0 and 1 or 0 and -1, depending on its sign.

If 1 (one) unit represents the whole, then the fraction “a/b” is plotted “a” units from 0, where a unit is one of “b” equal parts connecting from 0 to 1,if “a” is positive; —– if “a” is negative, it will be “a” units to the left of 0 towards -1..

#### Equivalent fractions

Fractions that represent the same proportion of a whole or the same point on the number line are said to be equivalent.

#### Multiply numerator and denominator by the same non-zero number to get equivalent fraction

If a, b, and c are numbers, then a / b= (a • c) / (b • c).

This is true because any number multiplied by 1 is equal to that number.

And since c/c = 1,

(c/c) • (a/b) = 1 •(a / b) = a/b = (a • c) / (b • c)

using the distributive rules for multiplication.

Divide numerator and denominator to get the equivalent fraction.

#### Simplify to equivalent fraction: a/a = 1

If a is any non-zero number then a / a = 1

i.e., a non-zero number, divided by itself is 1,

or,

a non-zero number multiplied by its reciprocal is 1.

#### Simplify to equivalent fraction: a/1

If a is any number then a/1 = a.

#### Simplify to equivalent fraction: 0/a

Any non-zero number a divided into 0 is 0;

so 0/a = 0

#### Simplify a/0?

Not possible. Division by 0 is undefined.

##### – 3.2 Factors and Simplest Form

#### Prime number

A prime number is a whole number greater than 1 whose only divisors are 1 and itself.

The first nine prime numbers are: 2, 3, 5, 11, 13, 17, 19, 23, 29, ….

#### Composite number

A composite number is a number that is greater than 1 and not prime.

#### Factor

A factor is any number that divides a number evenly – that is, with a remainder of 0.

### Factorization

Factorization is the writing of a number as a product of factors.

#### Prime factorization

The prime factorization of a number is a factorization in which all the factors, other than 1 and itself, are prime numbers.

### Factor tree

A factor tree is a tree with a top node that is the number to be factored and next lower level nodes represent factors in its factorization

Each such lower level node is again subdivided into lower level nodes each again representing a factor in its factorization

The lowest level nodes are factors that are prime numbers, thus halting the factor tree division process. This is so because prime numbers, by definition, cannot be factored further.

#### Divisibility tests

A whole number is divisible by

- 2 — if the ones digit is even (or 0, 2, 4, 6, 8)
- 3 –- if the sum of the digits is divisible by 3
- 5 – if the ones digit is divisible by 0 or 5

Find the prime factorization of a number

Determine the prime factors of a small number (say less than a hundred) by creating a factor tree and using the found prime factors as the factors in the factorization.

For larger numbers – or for showing all work – begin with the lowest prime number to determine if it can be a divisor –

if so, keep dividing by it to determine the largest number of times it divides into the number and using that frequency as its exponent – then, determine how often the remaining factor can be divided by the next highest prime number and using that frequency as its exponent – and so on.

At each step – divisibility tests are useful in determining whether the next higher prime is even a factor.

The prime factorization will be the product of each of the prime numbers raised to exponent numbers that state their frequency as a repetitive divisor.

#### Fractions in simplest form or lowest terms

A fraction is in simplest form, or lowest terms, when the numerator and the denominator have no common factors other than 1.

#### Simplify a fraction – write a fraction in simplest form

To write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors.

In simplifying a product it may be possible to identify a common factor without writing out its prime factorization.

If the denominator of a fraction contains a variable, it is assumed that the variable is such that the denominator is always non-zero – since division by zero is always undefined.

– 3.3 Multiplying and Dividing Fractions

#### Multiply two fractions

To multiply two fractions you multiply the numerators to form the numerator of the resulting fraction and you multiply the denominators to form the denominator of the resulting fraction.

If a, b, c and are numbers and b and d are not 0, then

(a / b) • (c / d) = (a • c) / (b • d)

Whenever you can, simplify the fractions before – and after – the multiplication.

#### Exponential expressions with fractional base

An exponential expression can have a fractional base, so that:

(a / b)^^{m }= (a/b) • (a/b) • … • (a/b) =

(a^^{m}) / (b^^{m}) = (a • a • a • … • a) / (b • b • b • … • b)

— multiplied m times in both numerator and denominator – and as fractions – where m is a whole number.

In the above notation the carrot signifies exponentiation, so that … a^^{2 } = a • a .

#### Reciprocal of a fraction

Two numbers are reciprocals of each other if their product is 1.

The reciprocal of the fraction a/b is: b/a, since

(a / b) • (b / a) = (a • b) / (b • a) = 1.

Every number has a reciprocal – except 0 – there is no number so that

0 • a = 1.

#### Find the reciprocal of a fraction

The reciprocal of a fraction “ a/b ” is obtained by exchanging the value of numerator with the denominator – so that it becomes equal to: “ b/a ”.

#### Divide two fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

If b, c and d are not 0, then:

(a / b) / (c / d) = (a / b) • (d / c) = (a • d) / (b • c)

When dividing by a fraction rewrite the division as a multiplication – THEN look to simplify by canceling common factors.

#### Solving applications by multiplying and dividing fractions

- Understand the problem by reading and re-reading problem.
- Determine what is to be found or proved
- Determine what is given
- Determine how to connect what is given to what is wanted
- Formulate an equation by translating key words and word phrases to symbol operations.
- Solve for the unknown in the formulated equation.
- Interpret the results by checking the work and state conclusions that answers the posed questions.

##### – 3.4 Adding and Subtracting Like Fractions, Least Common Denominator and Equivalent Fractions

#### Like fractions

Fractions that have the same denominator are called like fractions.

#### Unlike fractions

Fractions that have different denominators are called unlike fractions.

#### Add like fractions

To add two or more like fractions (each having the same denominator as the others) add the sum of the numerators and place the difference over the common denominator.

#### Subtract like fractions

To subtract two or more like fractions (each having the same denominator as the others) – subtract the numerators, left to right, and place the difference over the common denominator

#### Least common denominator (LCD)

##### The least common denominator of a list of fractions is the smallest positive number divisible by all the denominators in the list.

#### Find the LCD (Divisibility of Multiples of Larger Denominator)

See whether the largest denominator is divisible by smallest.

If so, choose larger denominator as LCD.

If not, check consecutive multiples of larger denominator for divisibility by smallest.

If so, choose larger- denominator- multiple as LCD.

When large multiples are needed, use the algorithm “Find LCD (Division By Primes of All Denominators)”.

#### Find the LCD (Cancel Common Factors after Prime Factorization of Numerators and Denominators)

Write the numerator as a product of primes.

Write the denominator as a product of primes.

Cancel common factors in numerator and denominator.

#### Find the LCD (Product of Primes Divisible into all Denominators)

Write all denominators on a line – perform

Begin with 2, smallest prime, to check divisibility into each denominator.

Divide into each number and record multiple factors of primes as above.

Continue the same with 3, 5, 7 etc.

#### Least common multiple

See Least Common Denominator (LCD).

##### – 3.5 Adding and Subtracting Unlike Fractions

##### Add unlike fractions

Find the LCD of the denominators of the fractions.

Write each fraction as an equivalent fraction whose denominator is the LCD.

Add the like fractions.

Write the sum in simplest form.

#### Subtract unlike fractions

Find the LCD of the denominators of the fractions.

Write each fraction as an equivalent fraction whose denominator is the LCD.

Subtract the like fractions.

Write the difference in simplest form.

#### Add and subtract with given fractional replacement values

Substitute fractions as replacement values and add, subtract, multiply and divide as specified.

##### – 3.6 Complex Fractions, Order of Operations, and Mixed Numbers

#### Complex fractions

A complex fraction is a fraction whose numerator and denominator is a fraction.

#### Simplify complex fractions (Multiply reciprocal of denominator)

Simplify numerator and denominator of fraction each to be a single fraction.

Multiply numerator fraction by reciprocal of denominator fraction.

Simplify.

#### Simplify complex fractions

Simplify by;

finding the common LCD for all of the denominators in both numerator and denominator

then multiply both numerator and denominator to remove all denominators in them.

#### Order of operations

Start by looking for grouping symbols such as absolute values and fraction bars

Do all operations within grouping symbols such as parentheses and brackets Evaluate any expressions with exponents and find any square roots.

Multiply in order from left to right.

Add or subtract in order from left to right

##### – 3.7 Operations on Mixed Numbers

#### Mixed number

A mixed number is a number consisting of a whole number and a proper fraction.

#### Write a mixed number as an improper fraction

To write a mixed number as an improper fraction:

Multiply the whole number by the denominator of the fraction.

Add the numerator of the fraction to the product above.

Write the sum of the previous step as the numerator of the improper fraction over the denominator of the old fraction which then becomes the denominator of the improper fraction.

#### Write an improper fraction as a mixed number

To write an improper fraction as a mixed number or a whole number:

Divide the denominator into the numerator

The whole number of the mixed number is the whole number quotient.

The proper fraction part of a mixed number is formed as:

(Remainder) / (Denominator of improper fraction)

When the remainder is 0 the improper fraction is a whole number.

#### Multiply two mixed numbers

Convert both mixed numbers into improper fractions and multiply the fractions as usual.

#### Divide two mixed numbers

Convert both mixed numbers into improper fractions and divide the fractions as usual.

You can always write a whole number as a fraction with denominator 1.

#### Add two mixed numbers

Add the whole number parts to form the whole number portion of an initial mixed number sum.

Add the fractional parts separately.

If the added fractional parts sums to a proper fraction, retain it as the proper fractional part of the mixed number sum. Also retain the whole number portion of the initial mixed number sum for the whole number portion of the final mixed number sum.

If the fractional part sum is an improper fraction, write that as a mixed number, add the whole number part of it to finalize the whole number portion of the mixed number sum, and retain the fractional part as the proper fractional part of the mixed number sum.

In more advanced algebra courses you will be asked to leave the result as a improper fraction properly reduced.

##### 4. Decimals

##### – 4.1 Introduction to Decimals

##### – 4.2 Adding and Subtracting Decimals

##### – 4.3 Multiplying Decimals and Circumference of a Circle

##### – 4.4 Dividing Decimals

##### – 4.5 Fractions, Decimals, and Order of Operations

##### – 4.6 Square Roots and the Pythagorean Theorem

##### 5. Ratio, Proportion, and Measurement

##### – 5.1 Ratios

##### – 5.2 Proportions

##### – 5.3 Proportions and Problem Solving

##### – 5.4 Length: U.S. and Metric Syst ems of Measurement

##### – 5.5 Weight and Mass: U.S. and Metric Systems of Measurement

##### – 5.6 Capacity: U.S. and Metric Systems of Measurement

##### – 5.7 Conversions Between the U.S. and Metric Systems

##### 6. Percent

##### – 6.1 Percents, Decimals, and Fractions

##### – 6.2 Solving Percent Problems Using Equations

##### – 6.3 Solving Percent Problems Using Proportions

##### – 6.4 Applications of Percent

##### – 6.5 Percent and Problem Solving: Sales Tax, Commision, and Discount

##### – 6.6 Percent and Problem Solving: Interest

##### 7. Statistics and Probability

##### – 7.1 Reading Pictographs, Bar Graphs, Histograms, and Line Graphs

##### – 7.2 Reading Circle Graphs

##### – 7.3 Mean, Median, and Mode

##### – 7.4 Counting and Introduction to Probability

##### 8. Introduction to Algebra

##### – 8.1 Introduction to Variables

#### Variable

A variable is the assignment to a letter symbol of a number selected as a placeholder.

The assignment represents a selection of any one of a collection of numbers – but, once chosen, it represents the same specific number in under consideration.

#### Terms

The addends of an algebraic expression consisting of a combination of operations with numbers and with letters that represent variables

#### Constant (term)

A constant term is a term in an expression that is only a number.

##### Variable term

A variable term is a term that includes a variable.

#### Numerical coefficient

A numerical coefficient of a term is the number factor of a term.

#### Like terms

Like terms are terms that have the same variable in common.

#### Unlike terms

Unlike terms are terms that do not have any variables in common.

#### Combine like terms

Combine the coefficients of like terms using the rules of integer addition and subtraction and use the resultant sum or difference as the coefficient for the common variable.

#### Apply distributive property to addition of two variables

Distribute multiplication over addition, so that, if a, b and c are any numbers (but each a specific number for the sake of this discussion), then:

a (b + c) = a • b + a • c

#### Apply distributive property to subtraction of two variables

Distribute multiplication over addition and subtraction, so that, if a, b and c are any numbers (but each a specific number for the sake of this discussion), then:

a • (b – c) = a • b – a • c

#### Apply commutative property to addition of two variables

If a and b be any numbers (once chosen, each must be some specific number – it can be any must be some number), then:

a + b = b + a

#### Apply commutative property to multiplication of two variables

If a and b are any numbers (once chosen, each must be some specific number – usually the same for the discussion at hand – again – it can be any number but must be some specific number), then:

a • b = b • a

#### Apply associative property to addition of variables

If a, b and c are any numbers (once chosen, each can be any number but it must be some specific number), then their order or grouping in an expression can be changed without altering their sum — so that:

(a + b) + c = a + (b + c)

#### Apply associative property to multiplication of two variables

If a, b and c are any numbers (once chosen, eachcan be any one but it must be some specific number for the discussion), then their order or grouping in an expression can be changed without altering their product — so that:

(a • b) • c = a • (b • c)

#### Add algebraic expressions

Add two or more algebraic expressions by adding the terms of each expression.

#### Multiply algebraic expressions

Multiply two or more algebraic expressions by applying the associative and distributive properties to the operation

Write the terms of each of the algebraic expressions, just as if they were variables or numbers.

Don’t forget to use the rules of “order of operations”.

#### Simplify expressions

Use the multiplication distributive property for removing parentheses – then multiply expressions and combine like terms.

#### Order of Operations Review

- Start by looking for
**G**rouping symbols such as absolute values and fraction bars - Do all operations within a grouping symbols or nested symbols including parentheses and bracket
- Evaluate any expressions with exponents
- Multiply and divide in order from left to right
- Add or subtract in order from left to right

#### Value of an Algebraic Expression

The value of an algebraic expression is equal to the value of its simplified form.

#### Addition of algebraic identities

Any algebraic expression is evaluated to be equal to the sum of itself and 0.

##### – 8.2 Solving Equations: The Addition Property

#### Equation

An equation is a statement of the equality in values of two expressions.

It consists of a left side expression, a right side expression and an equal sign “=” that expresses the equality of values of the expressions.

#### Equal sign

The equal sign “=” in an equation is used to assert that the values for the two expressions being compared are the same.

#### Left side of equation

The left side of an equation is an algebraic expression being compared to the right side of the equation.

#### Right side of equation

The right side of an equation is an algebraic expression being compared to the left side of that equation.

#### Solution of an equation

A solution to an equation is a value for the variable (or variables) that makes an equation a true statement.

#### Solving an equation

To solve an equation is to determine the solution of the equation.

#### Equivalent equations – Simplest equivalent equation

An equation equating expressions of the form:

“x = a number”

or

” ’a number’ = x “.

Both forms correctly express the simplest equivalent equation.

At the very end we conventionally rewrite the equation so that the variable “x“ is on the left side of the equation.

#### Addition property of equality

The same number may be added to both sides of an equation without changing the solution to an equation.

If a, b and c represent numbers, and, if a = b then a + c = b + c.

#### Simplify an equation by adding the same number to both sides

To simplify an equation seek to get the variable x alone to one side of the equation.

You can add the same number to both sides of the equation without changing the solution value using the Addition property of equality.

Look to add a number on the side of an equation where there is a subtracted number. By adding the same number to both sides you add the two terms to get 0 on this side – which can be then neglected in rewriting the equivalent equation.

Don’t forget to add the number on the other side of the equation.

##### – 8.3 Solving Equations: The Multiplication Property

#### Subtraction property of equality

The same number may be subtracted from both sides of an equation without changing the solution to an equation.

If a, b and c represent numbers, and, if a = b — then a – c = b – c.

#### Simplify an equation by subtracting the same number to both sides

To simplify an equation seek to get the variable x alone to one side of the equation.

You can subtract the same number to both sides of the equation without changing the solution value using the Subtraction property of equality.

Look to subtract a number on the side of an equation where there is an added number. By subtracting the same number to both sides you subtract the two terms to get 0 on this side – which can be then neglected in rewriting the equivalent equation.

Don’t forget to add the number on the other side of the equation.

##### – 8.4 Solving Equations Using the Arithmetic Properties

#### Multiplication property of equality

Both sides of an equation may be multiplied by the same number without changing the solution of the equation.

If a, b, and c are numbers, and if, a = b then a • c = b • c.

#### Simplify an equation by multiplying both sides by the same number

To simplify an equation seek to get the variable x alone to one side of the equation.

You can multiply the same number on both sides of the equation without changing the solution value using the Multiplication property of equality.

Look to multiply the same number on the side of an equation where there is a division by number. By multiplying the same number on both sides you get 1 as the coefficient of the variable or 1 as a term.

The resultant rewritten equation can then be further simplified by using the Addition or Subtraction properties of equality.

Don’t forget to multiply the same number on the other side of the equation.

#### Division property of equality

Both sides of an equation may be divided by the same number without changing the solution of the equation.

If a, b, and c are numbers, and if, a = b then a / c = b / c.

#### Simplify an equation by dividing the same number into both sides

To simplify an equation seek to get the variable x alone to one side of the equation.

You can divide both sides of the equation by the same number without changing the solution value using the Division property of equality.

Look to divide by the same number on the side of an equation where there is a multiplication by a number. By dividing the same number on both sides you get 1 as the coefficient of the variable or 1 as a term.

The resulting rewritten equation can then be further simplified by using the **Addition or Subtraction** properties of equality.

Don’t forget to add to, subtract from, multiply or divide into the same expression(s) on the other side of the equation.

#### Check to verify the solution of equation

To check that a number is the solution to an equation – substitute the number in the variable of the original equation and verify that it is a true statement.

Verify the truth of the statement by checking that the values of the original equation’s left and right side expressions are the same.

#### Write some phrases as expressions

Twice a number, increased by -9

= 2x + (-9)

Three times the difference of a number and 11

= 3(x – 11)

The quotient of 5 times a number and 17

= (5x) / 17

Remember that 5x = 5 • x and that the backslash fraction bar / signifies division.

##### – 8.5 Equations and Problem Solving

#### Linear equations in one variable

A linear equation in one variable is an equation containing only one variable that is not below the fraction bar of any expression and whose exponent is 1 – whenever it appears.

#### Steps for solving a linear equation in one variable

- If parentheses are present use the distributive property.
- Combine like terms on each side of the equation.
- Use the addition and subtraction property of equality to rewrite the equation to an equivalent one – in which the variable is on one side of the equation and the constant terms are on the other side.
- Combine all terms in x.
- Divide both sides of the equation by the coefficient of x.
- You have left x = (some simplified number or expression).

#### Write sentences in word problems as equations

- Translate “a number” to x.
- Translate to equal sign words such as: equal, gives, is/was, yields, amounts to, is equal to.
- Translate into addition words such as: sum, plus, added to, more than, increased by total.
- Translation into subtraction words such as: difference, minus, subtracted from, less than, decreased by, less.
- Translation to multiplication words such as: product, times, multiply, twice, of, double.
- Translation to division words such as: quotient, divided by, into, per.

#### Steps to solve word problems

- Understand the problem by becoming comfortable with it through reading and re-reading it.
- Choose a variable to represent the unknown
- Construct a drawing if appropriate
- Propose a solution and check it
- Translate the problem into an equation.
- Solve the equation.
- Interpret the results.
- Check the proposed solution in the stated problem and state your conclusion.