Identifying Integers and their Opposites                                                                                       Module 1

What is an Integer?

A number that is positive or negative

What is a positive integer?

A number to the right of zero

What is a negative integer?

A number to the left of zero

What are opposite integers?

Numbers that have the same absolute value EX: +5 and -5, -8 and +8

What is absolute value?

The distance of a number from zero

*Distance is always positive, so absolute value is always positive.

Absolute Value symbol

The symbol is two straight lines on the sides of a number. Ex: |-8|, |6|

Keywords for (+) numbers

Increase, up, rise, above, deposit

Keywords for (-) numbers

Decrease, down, below, spend

Which number is sea level?

The number zero

   

 

Comparing and Ordering Integers                                                                                           Module 1

What are Rational Numbers?

They are anything that can be written in the form of a fraction and where the

Denominator cannot be zero.

Rational Numbers include…

Natural Numbers, Whole Numbers, Integers, Decimals, and Fractions

What are Natural Numbers?

Sometimes called counting numbers. Examples: 23, 810, 12, and 100

What are Whole Numbers?

They include all the natural numbers but also include zero.

What are Integers?

They include all whole numbers but also include negative numbers

Integers DO NOT include decimals or fractions. Examples: -45, 87, -901

What are inequalities?

Inequalities show a difference in size using the symbols <, >, or =

(<) less than       (>) greater than      (=) equal to

What are number lines?

Number lines show the location of a number in relation to zero

Vertical Number Lines

Show elevation. They go up and down.

Horizontal Number Lines

Show distance. They go side to side

What are negative numbers?

Numbers to the left of zero. Example: -5, -62, and -89

What are positive numbers?

Numbers to the right of zero. Example: 6, 85, and 13

How to compare numbers

The number more to the left on a number line has less value.

The number more to the right on a number line has  greater value

Greatest Common Factor                                                                                                          Module 2

What are factors?

Numbers that multiply together to make a bigger number

Example: 2x4; 2 and 4 are factors of 8

What are prime numbers?

Numbers that only have two factors: 1 and itself

Example: 3 is prime, it only has two factors 1 and 3

what are composite numbers?

Numbers that have 3 or more factors

Example: 12; 1, 2, 3, 4, 6, 12 are all factors of twelve

What are common factors?

Factors that go into two or more numbers.

Example:

the factors of 8 are: 1,2,4,8

the factors of 24 are: 1,2,3,4,6,8,12,24

The common factors of 8 and 24 are 1,2,4,8

What is GCF?

GCF means greatest common factor

The Greatest Common Factor (GCF) of 8 and 24 is 8

What is distributive property?

The distributive property lets you multiply a sum by multiplying each

addend separately and then add the products.

example: 5 (6+2) is the same as 5x6 + 5x2  They both equal 40

Find the sum of the numbers

Example: 56 + 64

as the product of their GCF

step 1) find the GCF, for 56 & 64 the GCF is 8

and another sum

step 2) write each number as the product of the GCF and another number

   

(8 x 7) + (8 x 8)

Step 3) rewrite the expression using distributive property

8 (7 + 8)

Prime factorization

the determination of the set of prime numbers which multiply together to

give the original integer

GCF using factor trees

Example: Find the GCF of 14 & 56

Step 1) Prime factorize

Step 2) List the prime factors from least to greatest for each number

14: 2 x 7

56: 2 x 2 x 2 x 7

Step 3) Find the pairs for the factors

Step 4) Write one number from each pair and multiply to get the GCF

2 x 7 = 14           GCF= 14  

   

Least Common Multiple                                                                                                        Module 2

What is a multiple?

A multiple is the result of multiplying a number by an integer (not a fraction).

Example: 12

3 × 4 = 12, so 3 and 4 are factors of 12

What is a Least Common

The smallest positive number that is a multiple of two or more numbers.

Multiple?

Example: the Least Common Multiple of 3 and 5

 is 15, because 15 is a multiple of 3 and also a multiple of 5. Other common multiples include 30 and 45, etc, but they are not the smallest (least).  Also called Lowest Common Multiple

   

Title:         Rational Numbers                                                                                                       Module 3

What are Rational Numbers?

a real number that can be written as a simple fraction

In mathematics, a rational number is any number that can be expressed as a fraction, or the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Basically, numbers that CAN be written as a fraction, or terminating decimals, or predictable fractions like 1/3

Rational numbers can be categorized into other subgroups within the Real Number System

Natural Numbers: These are the positive whole numbers on the number line starting with 1.

Whole Numbers: These are the positive whole numbers on the number line, including zero.

Integers: These are the positive and negative whole numbers on the number line.

What are  Irrational Numbers?

Numbers that CANNOT be Written as a fraction.

Fraction Operations                                                                                                               Module 4

How to Multiply Fractions? WITH SAME OR DIFFERENT DENOMINATORS

To multiply fractions, we multiply the numerators and then multiply the denominators.

Example: 5/6 × 5/7

5/6 × 5/7 = 5×5/6×7 = 25/42

Adding & Subtracting Fractions with Like Denominators

EXAMPLE:

Like Denominators

2/4  + 1/4 = 3/4

•Keep Denominator

•Add Numerators           ADD 2 + 1=3

Adding & Subtracting Fractions with Unlike Denominators

EXAMPLE:

Unlike Denominators

1/2 + 1/6 =

•Find the LCD (LEAST COMMON DENOMINATOR)

•6 is the LCD

•Multiply 1/2 × 3/3=3/6

•Add 3/6 + 1/6=4/6

•SIMPLIFY TO 2/3

Dividing Fractions

Multiplying by the reciprocal

Example with a fraction being divided by a fraction

To get the reciprocal of a fraction, just turn it upside down.

In other words swap over the Numerator and Denominator

problem:   \frac{5}{8}\div \frac{2}{4}

Example with a whole number being divided by a fraction

Problem:   4\div\frac{3}{4}

*Before you begin, understand that every whole number has an “invisible” denominator of “1”

For example, 8=\frac{8}{1}, 6=\frac{6}{1}, etc.

Multiply using the reciprocal

4\div\frac{3}{4}=\frac{4}{1}\times\frac{4}{3}=\frac{16}{3}

Doing so, we get an answer of  \frac{16}{3}

The fraction \frac{16}{3}is an improper fraction (the numerator is greater than the denominator).

While there is nothing incorrect about this, an improper fraction is typically

simplified further into a mixed number.

The whole number part of the mixed number is found by dividing the 16 by the 3.

In this case we get 5

The fractional part of the mixed number is found by using the remainder of the division,

which in this case is 1 (16 divided by 3 is 5 remainder 1).

The final answer is: 5\frac{1}{3}

Example with a mixed number being divided by a fraction

problem:   2\frac{5}{9}\div\frac{3}{5}

To start, we first need to convert 2\frac{5}{9} to an improper fraction.

To convert 2\frac{5}{9} to an improper fraction:

multiply the 9 (the denominator), and the 2 (the whole number). To this product, add the 5 (the numerator)

giving 23, to form the new numerator, and use the 9 as the new denominator.

So, 2\frac{5}{9} as an improper fraction is \frac{23}{9}

The problem here is to divide  and

This problem can be solved by multiplying \frac{23}{9}  by the reciprocal of \frac{3}{5}

To find the reciprocal of \frac{3}{5}

Simply exchange the numerator and denominator, or just “flip” the fraction

upside-down.

The reciprocal of \frac{3}{5} is \frac{5}{3}

The problem here is to multiply \frac{23}{9} and \frac{5}{3}

This problem can be solved by multiplying together the two numerators (the 23 and 5),

giving 115, which will be the numerator in our answer.

Also, we’ll multiply together the two denominators (the 9 and 3),

giving 27, which will be the denominator in our answer.

Doing so, we get an answer of \frac{115}{27}

The fraction \frac{115}{27} is an improper fraction (the numerator is greater than the denominator).

While there is nothing incorrect about this, an improper fraction is typically

simplified further into a mixed number.

The whole number part of the mixed number is found by dividing the 115 by the 27.

In this case we get

The fractional part of the mixed number is found by using the remainder of the division,

which in this case is 7 (115 divided by 27 is 4 remainder 7).

The final answer is: 4\frac{7}{27}

   

Operations with Decimals                                                                                              Module 5

Adding & Subtracting Decimals

RULE:  line up the decimals & if there is an empty place value, you should add a zero

Example: 5.9 + 123

               5.90           (zero place holders)

           123. 00   (the decimal point on a whole number is behind)

        5 . 90

 + 123 . 00

     128 . 90

Multiplying Decimals

When multiplying decimals, how do you know where to place the decimal point in the product?

Example: 4.2 X 6.7

Write the problem out with the number with the most digits on top.

  • Count how many numbers are to the right of the decimal in the entire problem.
  • Multiply like normal.
  • Make sure the product has the same amount of numbers behind the decimal as the original problem did.

      4 . 2     ← one decimal place   1+ 1 = 2 numbers behind the decimal

   × 6 . 7     ←one decimal place

   2  9  4

+2 5 2 0

  2 8.1 4        ← two decimal places

Dividing Decimals

  1. We cannot divide by a decimal. If the divisor is not a whole number, move decimal point to right to make it a whole number and move decimal point in dividend the same number of places.
  2. Divide as usual. Keep dividing until the answer terminates or repeats.
  3. Put decimal point directly above decimal point in the dividend.
  4. Check your answer. Multiply quotient by divisor. Does it equal the dividend?

Example:

Multiply the divisor and dividend by powers of ten until divisor is a whole number

      Divide              Check your work                                                                                                                             

   

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