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TRANSCRIPT
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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
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Fractions Presentation Part 1
www.njctl.org
2011-11-29
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Fractions
Presentation 1
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· Meaning of Fractions
· Equivalent Fractions
· Lowest Term Fractions
· Improper Fractions and Mixed Numbers
· Using Fractions in Measurement
· Adding Fractions with Common Denominators
· Adding Mixed Numbers with Common Denominators
· Subtracting Fractions with Common Denominators
· Subtracting Mixed Numbers with Common Denominators
· Finding Common Denominators
· Comparing Fractional Numbers
Table of Contents
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· Adding Fractions with Unlike Denominators· Subtracting Fractions with Unlike Denominators· Adding Mixed Numbers with Unlike Denominators· Subtracting Mixed Numbers with Unlike Denominators· Multiplying Fractions
· Multiplying Fractions and Whole Numbers
· Multiplying with Mixed Numbers
· Dividing Fractions· Dividing with Whole Numbers and Mixed Numbers
Table of Contentsfor Presentation 2
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Meaning of Fractions
Return toTable of Contents
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Fraction - An expression that indicates the quotient ( ) of two quantities.
Numerator - The number above the fraction bar. The numerator answers the question"How many parts?"
Denominator - The number below the fraction bar. The denominator answers the question "How many total?"
Key Terms
Numerator 3 Denominator 7
Fraction = =
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Proper Fraction - A fraction where the numerator (top number) is less than the denominator (bottom number)
5 7 11 9 13 17
Improper Fraction - A fraction where the numerator (top number) is greater than or equal to the denominator (bottom number)
8 12 9 3 7 9
Key Terms
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Equivalent Fractions - Fractions that represent the same number or are equal to each other.
2 4 6 8 10 12 14 3 6 9 12 15 18 21
Mixed Number - A fraction with a whole number and a proper fraction.
5 1 2 2
= = = = = =
2
ImproperFraction
MixedNumber
Key Terms
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What fractions are represented below?
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Halves
Pull
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Thirds
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Fourths
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Fifths
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Sixths
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Sevenths
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Eighths
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1 What fraction of the whole is shaded?
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2 What fraction of the whole is white?
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Internet links for more practice
Fractions Naming Model link
Fractions Parts of a Whole Model link
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Equivalent Fractions
Return toTable of Contents
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19
19
19
What do you notice about the denominators in each set of equivalent fractions?
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Click below to use this interactive number line.
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25
= ??
Use the multiplication table to make equivalent fractions.
Pull
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To create equivalent fractions, multiply(or divide) the numerator and denominator by the same number.
2 x 3 6 x 2 12 x 3 36 7 x 3 21 x 2 42 x 3 126
36 / 6 6 / 3 2 126 / 6 21 / 3 7
= = =
= =
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3 Which set of fractions is equivalent?
A
B
C
D
1 2 2 2
=
4 1 8 4
1 3 3 6
3 9 7 21
=
=
=
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4 Write a fraction equivalent to 4 7
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5 Which fraction is equivalent to ?
A
B
C
D
3 8
16 6
12 24
12 32
9 16
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6 Which set of fractions is equivalent?
A
B
C
D
3 9 4 16
=
9 2 12 3
6 1 18 3
3 9 7 14
=
=
=
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7 Write a fraction equivalent to 5 9
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Lowest Term Fractions
Return toTable of Contents
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Reducing Fractions to Lowest Terms
Once you complete operations with fractions, you must write your answer in lowest terms.
The easiest way to determine if your answer is in simplest form is to check:1. Do the numerator and denominator have any common factors? 2. If they do not, the answer is in simplest form. 3. If they do, divide both the numerator and denominator by common factors until there are none left.
40 10 4 2 2 220 10 22 2 11
= =
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Remember: Whatever you do to the numerator, you must do to the denominator.
18 24
In this problem, 18 and 24 are divisible by 2, 3 and 6. If you divide by 6 (the GCF), you will reach the simplest form in one step.
18 6 3 24 6 4
=
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If you divide by 2 and 3 (common factors), then you will reach the simplest form in two steps. Either way, the simplest form is the same.
18 3 6 2 3 24 3 8 2 4
= =
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8 Simplify to lowest terms.
21 35
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9 Simplify to lowest terms.
3 9
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10 Simplify to lowest terms.
21 29
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11 Simplify to lowest terms.
17 51
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12 Simplify to lowest terms.
24 96
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Internet links for more practice
Simplifying Fractions link(interactive at bottom of page)
Rename in Lowest Terms interactive
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Return toTable of Contents
Improper Fractions and
Mixed Numbers
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In a proper fraction, the numerator is always less than the denominator. The value of a proper fraction is always less than 1.
3 5
In an improper fraction, the numerator is equal to or greater than the denominator. The value of an improper fraction is equal to or greater than 1.
5 7 5 5
Proper or Improper?
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An improper fraction can be expressed as a whole number and a fraction. This is called a mixed number.
"Seven Halves"
3 1 2
7 2 =
Seven halves can fill up 3 whole strips and of another whole strip. That's and we say "three and one half", meaning "three plus one half".
3 1 2
1 2
Mixed Numbers
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Converting Improper Fractionsto Mixed Numbers
Remember, the fraction bar symbolizes division! Therefore:
· Divide the numerator by the denominator to see how many "wholes" there are.· Write the remainder over the denominator
Let's look at "seven halves" again.
7 2 3 1
2 7 2 =
32 7 -6 1
2
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Let's look at another example.
Now try this one. 17 8
13 5
25 13 -10 3
5
2 3 5
13 5
=
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Converting Mixed Numbersinto Improper Fractions
To do this, complete a multiplication problem.
· Multiply the whole number by the denominator to create an improper fraction (of the original whole number).· Add the new fraction to the original fraction (from the mixed number).
Let's look at "seven halves" again.
3 1 2
7 2
== 6 2
+ 1 2
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Let's look at another example.
Now try this one.
4 2 3
14 3
== 12 3
+ 2 3
6 3 7
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13 is a proper fraction.
True
False
8 5
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14 Change to a mixed number.
1 7 8
A
15 8
2 7 8
B
1 3 8
C
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15 Change this mixed number to an improper fraction.
1 3 4
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16 Change this mixed number to an improper fraction.
3 9 10
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17 Change this improper fraction to a mixed number.
28 6
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Return toTable of Contents
Using Fractionsin Measurement
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Each one inch segment on this ruler is like one strip folded into four parts. For every inch, there are , and inch markings. 1
4 1 2
3 4
1 4
1 4
1 4
1 4
Fraction Ruler
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Move the pink arrows to two locations on the ruler. Then hit the large blue arrow for the distance between the arrows to be measured.
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18 What is the length between the two arrows?
1 9 16
in.A 1 5 8
in.C
3 4
in.D1 3 4
in.B
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19 What is the length between the two arrows?
1316
in.A 3 4
in.C
7 8
in.D1 in.B
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Return toTable of Contents
Adding Fractionswith Common Denominators
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Adding Fractions with Common Denominators
To add fractions with common denominators, add the numerators and leave the denominator the same. Make sure your answer is in simplest form.
The denominator indicates the number of parts of the whole. If the fractions have a common denominator, they are the same "size" so we can add the numerators (or number of parts).
2 6 3 6 5 6
+
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Try these!Move the boxes to see work and answers.Be sure to simplify all answers.
2 4 1 4
3 4
+
3 7 1 7
4 7
+
5 12 4 12
9 12
+
3 4
11301330
2430
+
4 5
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20 3 10 2 10
+
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21 5 8 1 8
+
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22 7 14 3 14
+
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23 5 12
2 12
+
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24 8 20
6 20
+
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Return toTable of Contents
AddingMixed Numberswith Common Denominators
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Adding Mixed Numbers withCommon Denominators
To add mixed numbers with common denominators, add the fractions then add the whole numbers. Make sure your answer is in simplest form.
2 1 6
+ 1 4 6
3 5 6
5 1 9
+ 2 2 9
7 3 9
= 7 1 3
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25 Is the equation below true or false?
True False
1 1 4
+ 3 2 4
4 3 4
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26 Is the equation below true or false?
True False
4 1 4
+ 4 1 4
8 2 4
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27 Find the sum.
2 5 12
+ 3 2 12
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28 Find the sum.
5 3 10
+ 7 5 10
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Adding Mixed Numbers withCommon Denominators
Sometimes after you add the mixed numbers, the fraction is improper. When this occurs, you must rename the improper fraction as a mixed number and add it to the whole number.
3 3 5
+ 2 4 5
5 7 5
= 5 + 1 2 5
= 6 2 5
6 5 9
+ 1 7 9
7 12 9
= 7 + 1 3 9
= 8 1 3
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29 Is the equation below true or false?
True False
1 8 12
+ 1 5 12
3 1 12
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30 Find the sum.
2 4 9
+ 5 2 9
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31 Find the sum.
3 3 14
+ 2 4 14
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32 Find the sum.
4 3 8
+ 2 3 8
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Return toTable of Contents
Subtracting Fractionswith Common Denominators
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Subtracting Fractions withCommon Denominators
To subtract fractions with common denominators, subtract the numerators and leave the denominator the same. Make sure your answer is in simplest form.
The denominator indicates the number of parts of the whole. If the fractions have a common denominator, they are the same "size" so we can subtract the numerators (or number of parts).
5 6 4 6 1 6
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Try these!Move the boxes to see work and answers.Be sure to simplify all answers.
2 4 1 4
1 4
3 7 1 7
2 7
11 12 3 12
8 12
2 3
19301330
6 30
1 5
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33 7 8 4 8
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34 7 10 3 10
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35 5 6 3 6
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36 9 14
5 14
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37 7 9
5 9
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Return toTable of Contents
SubtractingMixed Numberswith Common Denominators
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Subtracting Mixed Numbers withCommon Denominators
To subtract mixed numbers with common denominators, subtract the fractions then subtract the whole numbers. Make sure your answer is in simplest form.
2 4 6
1 3 6
1 1 6
5 7 9
2 4 9
3 3 9
= 3 1 3
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38 Is the equation below true or false?
True False
4 5 9
3 9
3 2 9
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39 Is the equation below true or false?
True False
2 7 9
1 9
1 2 3
1
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40 Find the difference.
4 7 8
2 3 8
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41 Find the difference.
6 7 12
1 4 12
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42 Find the difference.
13 5 8
5 2 8
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Return toTable of Contents
FindingCommon
Denominators
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How many halves make a whole circle?
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How many fourths make half of this circle?
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How many sixths make 1/3 of this circle?
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How many eighths can fit in 1/4 of this circle?
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How many different combinations can you make to fill the circle?Keep track of what pieces you use. (You may need to rotate your pieces.)
1/8
1/4
1/21/3
1/61/7
1/5
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Fix the Sticks
You can use the set of Skip Counting Sticks to find common denominators for two fractions with unlike denominators. If you don't have a set of sticks, you can create them by listing the multiples of the denominator.
For the fractions and , line up the sticks this way for the denominator of each fraction:
Find the smallest number in the "denominator" sticks that is common in both fractions.
It's 12. The least common denominator of and is 12.
3 4
4 8 12 16 20 2428 32 36
6 1218 24 30 36 42 48 54
...
...
1 6
3 4 1 6
3 4
1 6
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A quick way to find LCDs...
List multiples of the larger denominator and stop when you find a common multiple for the smaller denominator.
Ex: and
Multiples of 5: 5, 10, 15
Ex: and
Multiples of 9: 9, 18, 27, 36
2 5
1 3
3 4
2 9
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43 Find the LCD of this pair of fractions.
2 4
1 6
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44 Find the LCD of this pair of fractions.
5 6
3 8
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Return toTable of Contents
Comparing Fractional Numbers
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Comparing Fractional Numbers
Common DenominatorsWhen you have two fraction with common denominators, all you have to do is compare the numerators.
Unlike DenominatorsTo compare fractions with unlike denominators, you have to rewrite both fractions with a common denominator. Then compare the numerators.
> 8 9
7 9
2 3 7 10
2 3
2030
=
7 10
2130
=< 2
3 7 10
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Compare the fractions
1. 4 2 7 5
4 7
2035
=
2 5
1435
=
4 2 7 5
>
2. 11 1317 17
11 1317 17
<
3. 4 3 5 4
4 5
1620
=
3 4
1520
=
4 3 5 4
>
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45 True or false?
2 3
3 4
>
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46 True or false?
5 6
5 8
>
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47 Compare the two fractions.
A >
8 11
3 4
B <
C =
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48 Compare the two fractions.
A >
3 12
1 4
B <
C =
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49 Compare the two fractions.
A >
4 9
5 8
B <
C =
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Internet links for more practice
Finding fractions on a number line link
Comparing Fractions Model
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