Math 3
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MATHEMATICS UNIT 3 Factor Trees Greatest Common Factor Lowest Common Multiple Fractions and Mixed Numbers Equivalent Fractions Addition of Fractions
EXPRESSING A NUMBER AS A PRODUCT OF PRIME FACTORS FACTOR TREES
Numbers can be written as products of a series of prime numbers multiplied together. To do this, we can use a factor tree.
Let’s use
Continue to express all composite numbers as products of two factors until all the factors are prime numbers.
This can also be written as 3
2 x 3 = 24
GREATEST COMMON FACTOR Many numbers have factors that are common. For example, let’s look at
1 x 24 2 x 12 3x8 4x6
1 x 36 2 x 18 3 x 12 4x9 6x6
Factors of 24 are 1,2,3,4,6,8,12,24 Factors of 36 are 1,2,3,4,6,9,12,18,36 There are factors that are common to both 24 and 36. They are 1,2,3,4,6, and 12 Of these common factors, 12 is the largest, or greatest. It is called the
LOWEST COMMON MULTIPLE Any time you take a number and multiply it by another number, you get a multiple of the original number. Let’s look at the number 3.
3x1=3 3x2=6 3x3=9 3 x 4 = 12 3 x 5 = 15 3 x 6 = 18 3 x 7 = 21 3 x 8 = 24
3,6,9,12,15,18,21, and 24 are all multiples of 3. There are many more. All you have to do is multiply 3 by any number to get a multiple of 3. Also, the number 3 will divide evenly into all of its multiples.
Many numbers have common multiples. For example: Multiples of 2: 2,4,6,8,10,12,14,16,18,20... Multiples of 3: 3,6,9,12,15,18,21,24,27,30... Common Multiples: 6,12,18...
...
The symbol means “and so on.” It refers to an infinite set of numbers
The number 6 is the smallest multiple common to both 2 and 3. It is called the Lowest Common Multiple.
FRACTIONS A fraction is a number that refers to a part of a whole or a part of a group.
NUMERATOR: This
number refers to the number of parts of the whole you are dealing with.
DENOMINATOR: This number refers to the total number of pieces in the whole.
Seven people in total applied for the job. Four of those people were granted interviews.
Four of the seven received interviews.
FRACTIONS
P Fractions where the numerator is a smaller number than the denominator are called proper fractions. P Fractions where the numerator is larger than the denominator are called improper fractions. These types of fractions indicate an amount greater than one whole
IMPROPER FRACTIONS AND MIXED NUMBERS
14 3
This type of fraction means that there is more than 1 whole.
Improper fractions can also be expressed as mixed numbers. A mixed number is a number that has a whole number and a fraction together.
14 3
} This line means “divided by.”
To change an improper fraction to a mixed number, you must divide the numerator by the denominator.
14 divided by 3 is Number of Wholes
Remainder
The original denominator. We are talking about thirds. Fourteen thirds is the same as four and two thirds
To change a mixed number to an improper fraction, you must follow the following procedure:
Step 2: Add the answer from step 1 to the numerator
12 + 2 = 14 Step 1: Multiply the denominator by the whole number
3 x 4 = 12
Step 3: Place the answer from step 2 over the original denominator
EQUIVALENT FRACTIONS Any time the numerator and denominator of a fraction are both multiplied by the same number, the result is called an equivalent fraction. Although the numbers in an equivalent fraction are different, it still represents the same amount of the original whole. These fractions are
X
equivalent.
They both represent the same amount of the whole.
ADDING FRACTIONS +
+
=
=
When the denominators of the fractions are the same, simply add the two numerators together and place your answer over the original denominator.
When the denominators are not the same, you will have to find a common denominator before you can begin adding.
To find a common denominator for these two fractions, you need to find the lowest common multiple for 5 and 3
The lowest common multiple of 5 and 3 is 15
+
+
=
=
Look at the relationship between the original denominators of the fractions and the new common denominator. What number would you have to multiply the old denominator by to get the new one? Whatever that number is, multiply the old numerator by it as well and place your answer over the new common denominator. Do this for both fractions. Add the two numerators together and put your answer over the new denominator.
EXPRESSING A NUMBER AS A PRODUCT OF PRIME FACTORS FACTOR TREES
Numbers can be written as products of a series of prime numbers multiplied together. To do this, we can use a factor tree.
Let’s use
Continue to express all composite numbers as products of two factors until all the factors are prime numbers.
This can also be written as 3
2 x 3 = 24
GREATEST COMMON FACTOR Many numbers have factors that are common. For example, let’s look at
1 x 24 2 x 12 3x8 4x6
1 x 36 2 x 18 3 x 12 4x9 6x6
Factors of 24 are 1,2,3,4,6,8,12,24 Factors of 36 are 1,2,3,4,6,9,12,18,36 There are factors that are common to both 24 and 36. They are 1,2,3,4,6, and 12 Of these common factors, 12 is the largest, or greatest. It is called the
LOWEST COMMON MULTIPLE Any time you take a number and multiply it by another number, you get a multiple of the original number. Let’s look at the number 3.
3x1=3 3x2=6 3x3=9 3 x 4 = 12 3 x 5 = 15 3 x 6 = 18 3 x 7 = 21 3 x 8 = 24
3,6,9,12,15,18,21, and 24 are all multiples of 3. There are many more. All you have to do is multiply 3 by any number to get a multiple of 3. Also, the number 3 will divide evenly into all of its multiples.
Many numbers have common multiples. For example: Multiples of 2: 2,4,6,8,10,12,14,16,18,20... Multiples of 3: 3,6,9,12,15,18,21,24,27,30... Common Multiples: 6,12,18...
...
The symbol means “and so on.” It refers to an infinite set of numbers
The number 6 is the smallest multiple common to both 2 and 3. It is called the Lowest Common Multiple.
FRACTIONS A fraction is a number that refers to a part of a whole or a part of a group.
NUMERATOR: This
number refers to the number of parts of the whole you are dealing with.
DENOMINATOR: This number refers to the total number of pieces in the whole.
Seven people in total applied for the job. Four of those people were granted interviews.
Four of the seven received interviews.
FRACTIONS
P Fractions where the numerator is a smaller number than the denominator are called proper fractions. P Fractions where the numerator is larger than the denominator are called improper fractions. These types of fractions indicate an amount greater than one whole
IMPROPER FRACTIONS AND MIXED NUMBERS
14 3
This type of fraction means that there is more than 1 whole.
Improper fractions can also be expressed as mixed numbers. A mixed number is a number that has a whole number and a fraction together.
14 3
} This line means “divided by.”
To change an improper fraction to a mixed number, you must divide the numerator by the denominator.
14 divided by 3 is Number of Wholes
Remainder
The original denominator. We are talking about thirds. Fourteen thirds is the same as four and two thirds
To change a mixed number to an improper fraction, you must follow the following procedure:
Step 2: Add the answer from step 1 to the numerator
12 + 2 = 14 Step 1: Multiply the denominator by the whole number
3 x 4 = 12
Step 3: Place the answer from step 2 over the original denominator
EQUIVALENT FRACTIONS Any time the numerator and denominator of a fraction are both multiplied by the same number, the result is called an equivalent fraction. Although the numbers in an equivalent fraction are different, it still represents the same amount of the original whole. These fractions are
X
equivalent.
They both represent the same amount of the whole.
ADDING FRACTIONS +
+
=
=
When the denominators of the fractions are the same, simply add the two numerators together and place your answer over the original denominator.
When the denominators are not the same, you will have to find a common denominator before you can begin adding.
To find a common denominator for these two fractions, you need to find the lowest common multiple for 5 and 3
The lowest common multiple of 5 and 3 is 15
+
+
=
=
Look at the relationship between the original denominators of the fractions and the new common denominator. What number would you have to multiply the old denominator by to get the new one? Whatever that number is, multiply the old numerator by it as well and place your answer over the new common denominator. Do this for both fractions. Add the two numerators together and put your answer over the new denominator.
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