Understanding Fractions

For fractions with the same numerator, as the denominators get smaller, the fractions get

Larger

For fractions with the same numerator, the smallest fraction is the fraction with the

Largest denominator

For fractions with the same denominator, the largest fraction is the fraction with the

Largest numerator

The smallest common multiple of two numbers is called

Least common multiple (LCM)

less than

Less than is used to compare two numbers when the first number is smaller than the second number. The symbol for less than is < Example 4 < 12

How do you find the least common denominator or LCM of two fractions?

List multiples of the larger denominator. Stop when you find one that's also a multiple of the smaller denominator. Sometimes the LCD will be one of the given denominators. Sometimes the LCD will be the product of the given denominators.

What are the steps for dividing fractions?

Reciprocals are used to divide by a fraction. Dividing by a fraction gives the same answer as multiplying as the fraction's reciprocal. Rewrite division by a fraction as multiplication by its reciprocal. 1. Copy the first fraction 2. Change division to multiplication 3. Change the second fraction to its reciprocal 4. Multiply (cross cancel before or after multiplying) Ex. 1/2 % 3/4 1/2 X 4/3 1/1 X 2/3 = 2/3

How do you subtract mixed numbers with like denominators?

Rename the fraction by borrowing from the whole number if necessary. Subtract the fractions. Subtract the whole numbers. Simplify if necessary and write your answer in simplest form. Ex. 2 1/4 - 1 3/4 Rename 1/4 as 5/4 by borrowing a whole number from 2 1 5/4 - 1 3/4 = 2/4 = 1/2

For fractions with the same denominator, as the numerators get smaller, the fractions get

Smaller

For fractions with the same numerator, as the denominators get larger, the fractions get

Smaller

For fractions with the same numerator, the largest fraction is the fraction with the

Smallest denominator

For fractions with the same denominator, the smallest fraction is the fraction with the

Smallest numerator

simplify fractions

Sometimes when you add, subtract, multiply or divide fractions, you will be able to simplify your answer. To simplify a fraction, first find the greatest common factor (GCF) of the numerator and denominator and then divide both by the GCF.

To find a fraction's reciprocal

Switch a fraction's numerator and denominator

How do you find a reciprocal of a fraction?

Switch a fraction's numerator and denominator to find its reciprocal

GCF (Greatest Common Factor)

The largest number that divides evenly into both the numerator and denominator of a fraction. The largest number that is a factor of two or more numbers. Ex. If the factors of 24 are 1,24,2,12,3,8,4,6 and the factors of 12 are 1,12,2,6,3,4 then the greatest common factor of 24 and 12 is the number 12

fraction line

The line that separates the numerator and denominator. It is also a division symbol.

numerator

The number written above the line in a fraction. It tells how many parts are being counted.

denominator

The number written below the line in a fraction. It tells how many equal parts are in the whole.

What are quarters?

The parts that you get when a unit is divided into four equal parts. Also called fourths

halves

The parts you get when a unit is divided into two equal parts

Least Common Multiple (LCM)

The smallest multiple (other than zero) that two or more numbers have in common. Ex. The multiples of 6 are 2,3,6,1 The multiples of 4 are 1,4,2 The least common multiple of 6 and 4 is 2

How do you compare fractions with different denominators?

To compare fractions without pictures, the denominators must be the same. When you have unlike denominators, find the least common multiple (LCM) and rename the fractions. Change each fraction so it has the same denominator by finding equivalent fractions, multiply both the numerator and denominator by the same number. Look at the numerator to determine the larger fraction. Ex. Which is larger 1/7 or 2/3? First find the LCM and rename the fractions. The LCM of 3 and 7 is 21. 1/7 x 3/3 = 3/21 2/3 x 7/7 = 14/21 Compare the numerators to determine the larger fraction. 14 is larger than 3, so 14/21 is larger than 3/21

How do you add or subtract fractions with a common (like) denominator?

You add or subtract the numerators and write the difference or sum over the common denominator. Always keep the common denominator in your answer. After you add or subtract fractions, always check to see if you can simplify your answer. Always express your answer in its simplest form. Ex. 2/5 + 3/5 = (2+3)/5 = 5/5 = 1

common denominator

A denominator that is the same in two or more fractions.

improper fraction

A fraction whose numerator is greater than its denominator. 12/3 is an improper fraction

Factors are the numbers multiplied together to make what?

A product

What is a prime number?

A whole number that has exactly two factors, 1 and itself.

How do you add mixed numerals with like denominators?

Add the fractions. Add the whole numbers. Reduce to simplest form.

If you multiply the numerator and denominator of a fraction by the same number (larger than 0), what will you make?

An equal fraction

When solving word problems, what are the first two steps?

Read the problem. Write the equation.

What is a Common Denominator?

When two or more fractions have the same denominator.

What is a fraction?

a number that expresses part of a whole (equal parts). Fractions can be shown by dividing a unit into equal parts. Equal parts must all be the same size. 3/4 is a fraction, 1/2 is a fraction, 2/3 is a fraction

Sum

the answer to an addition problem

When is a fraction in simplest form?

the form of a fraction when the numerator and denominator have no common factor other than 1. The fraction that uses the smallest numbers to express a part of a whole. 3/4 is written in simplest form because there is not a number that divides evenly into both 3 or 4 except the number 1

What is the Least Common Denominator?

the least common multiple of two or more denominators

decompose

to separate into parts

What is an equivalent fraction?

two or more fractions that have the same value 2/4 and 1/2 are equivalent fractions because they both name the same amount

like denominators

when two or more fractions have the same denominator (1/5 and 2/5)

If fractions have the same denominator, they have a

Common denominator

Fractions can be shown by dividing a unit into

Equal parts

For fractions with the same denominator, as the numerators get larger, the fractions get

Larger

To divide by a fraction,

Rewrite the problem as multiplication by the reciprocal of that fraction and then multiply.

decomposing fractions

To show a fraction as a sum of its parts. Ex. 3/4 can be decomposed in two ways . . . 1/4 + 1/4 + 1/4 Or 2/4 + 1/4

simplifying fractions (reducing)

To express a fraction in its simplest form. To rename a fraction in its simplest form, you divide both the numerator and denominator by their greatest common factor or GCF.

How do you find an equivalent fraction?

To find an equivalent fraction, multiply both the numerator and denominator but the same number. If you multiply the numerator and denominator of a fraction by the same number (not zero), you will make an equal fraction. Ex. 3/4 = 3x3 /4 x3 = 9/12 Multiply the numerator by 3, multiply the denominator by 3 3/4 and 9/12 are equivalent fractions

reciprocal

Two fractions that when multiplied together equal one. Two numbers whose product is one. Ex. 3/4 and 4/3 are reciprocal fractions

Parentheses in Formulas

Used in mathematics as grouping symbols for operations. When simplifying an expression, the operations within the parentheses are performed first. Parentheses mean "do me first" If a problem has parentheses, Do what's in the parentheses first. Write the answer above the parentheses. Complete the problem. Ex. 2/3 x (2/5 - 1/5) = 2/5 - 1/5 = 1/5 2/3 x 1/5 = 2/15

What does it mean to cross cancel when multiplying fractions?

You can save some work in multiplication problems if you simplify before you multiply. Simplify and then multiply. Find one numerator and one denominator with a common factor. Divide these numbers by their common factor. Try to divide by their greatest common factor or you may need to divide more than once. Multiply. Ex. 1/6 x 3/5 3 into 6 is 2 3 into 3 is 1 So, 1/2 x 1/5 = 1/10

equal to

having the same value. The symbol for equal is = Example 3=3

Equations

mathematical statements that show two quantities that are equal Ex. 2/4 + 1/4 = 3/4

What are equal parts?

parts of a whole that are of equal size. Fractions must be divided into equal parts

thirds

The parts you get when a unit is divided into three equal parts

How can you represent a whole number as a fraction?

A whole number can be written as a fraction by writing the number over a denominator of 1. Ex. 6 = 6/1

The fraction in simplest form is the fraction that uses

The smallest numbers

numeral

a symbol used to represent a number

List the prime numbers that are less than 20

2, 3, 5, 7, 11, 13, 17, 19 Remember, 1 is not a prime number!

List the prime numbers that are less than 60

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 Remember, 1 is not a prime number!

factor

A number that is multiplied by another number to get a product. Also,the numbers that divide evenly into another number. Ex., 3x4 = 12, 3 and 4 are both factors of 12 Ex., 12/3 = 4, 3 is a factor of 12 Ex., 12/4 = 3, 4 is a factor of 12

What is a multiple?

A multiple of a number is the product of that number and a whole number. Ex. A multiple of 4 is 8, 2 X 4 = 8

What is a mixed number? (Also called a mixed numeral)

A number made up of a whole number AND a fraction 2 1/3 is a mixed number

common factor

A number that divides evenly into both numbers in a pair. A number that is a factor of two or more given numbers. Ex. If 3 X 3 = 9, and 3 X 4 = 12, then 3 is a common factor of 9 and 12

How do you multiply fractions?

Cancel or reduce if possible, also called cross-canceling, then multiply straight across. Multiply their numerators together and then multiply their denominators together. Some people like to multiply straight across and then reduce. Either way, if you can reduce, you MUST REDUCE. Ex. 3/4 X 8/9 = cross cancel, reduce 3 and 9, cross cancel, reduce 8 and 4 - multiply numerators together 1 x 2, then multiply denominators together 1 x 3 - the answer is 2/3

How do you add or subtract fractions with different denominators?

Find the least common denominator of the two fractions. Rename both fractions using the LCD. Find the missing numerators to make equal fractions. Add or subtract the numerators and write the answer (sum or difference) over the common denominator. Ex. Add 3/4 + 2/3 Rename both fractions using the LCD of 12 9/12 + 8/12 Add the numerators and write the sum over the common denominator 17/12 Always simplify your answer = 1 5/12

greater than

Greater than is used to compare two numbers when the first number is larger than the second number. The symbol for greater than is > Example 12 > 4

product

the answer to a multiplication problem Ex. 3x4 = 12, 12 is the product of this multiplication problem

difference

the answer to a subtraction problem

whole numbers

whole numbers count units (whole things) 1, 2, 3 & 4 are all whole numbers