not whole numbers i: fractions presented by frank h. osborne, ph. d. © 2015 emse 3123 math and...

Not Whole Numbers I:Fractions

Presented byFrank H. Osborne, Ph. D.

© 2015

EMSE 3123Math and Science in Education

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Teaching the Meaning of Fractions• So far, we have studied the teaching of whole

number concepts. Frequently we need to express those that are not whole numbers.

• These are called rational numbers.

• One way to express rational numbers is via fractions.

• Using manipulatives, we need first to have something we can consider as a whole, or ‘1’.

• This can be divided in numerous ways.

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Teaching the Meaning of Fractions• Example: Take a piece of clay and find its

mass. – Now, break off a piece amounting to ¾ of the

whole clay. – How would you demonstrate that you have in

fact broken off ¾?

• To do this, you need to break the clay into 4 equal parts and then take 3 of these. We express this in the following form.

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Teaching the Meaning of FractionsWe write number of parts (numerator) over

the number of equal parts (denominator).

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Teaching the Meaning of FractionsA way to show this pictorially is to take a square

and identify it as the “whole”. The square can then be divided into any number of equal parts.

We use shading to indicate how much our fraction has.

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Teaching the Meaning of FractionsWe use shading to indicate how much our

fraction has.

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Teaching the Meaning of FractionsIf more than one of the parts of is shaded, we

still use our definition to express the number as

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Teaching the Meaning of FractionsFormal instruction for associating a fraction

with a part of the whole should begin in third grade asking the students to express each not-whole-number as a fraction.

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Teaching the Meaning of FractionsFormal instruction for associating a fraction

with a part of the whole should begin in third grade asking the students to express each not-whole-number as a fraction.

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Teaching the Meaning of FractionsChildren should also have experiences

working with objects, where a certain number of objects represents a whole, and each object is a fractional part of the whole.

For example, we have 8 pears.

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Teaching the Meaning of Fractions

Each pear is ?/8 of the whole?

Five pears are ?/8 of the whole?

?/8 of the pears are colored?

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Each pear is ?/8 of the whole? 1/8

Five pears are ?/8 of the whole? 5/8

?/8 of the pears are colored? 4/8 = 1/2

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Teaching the Meaning of FractionsCircular or rectangular regions can also be

used as models to illustrate fractions. Divide them into equal parts.

Some examples of rectangular regions are shown in the activity on the next slides.

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Teaching the Meaning of FractionsWrite down different size relationships

among fraction from the strips below. Use terminology of greater than, less than, equals.

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Just a few: 1/9 is less than 1/6

1/6 is greater than 1/9

1/4 is greater than 1/9.

1/8 is less than 1/3.

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Teaching the Meaning of Fractions• Children need to realize through use of

manipulatives that fractions with the same numerator and denominator are all equal to 1.

We can show that 9/9=6/6=3/3=8/8=4/4=1.

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Teaching the Meaning of Fractions• With the aid of manipulatives, children

should be able to order simple fractions, first with the same numerator:

1/9, 1/8, 1/6, 1/4, 1/3,

• and then different numerators.

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Teaching the Meaning of Fractions• After mastering size relationships for

fractions less than 1, they can be introduced to fractions greater than 1.

• For example, circular regions, like these below, can be used to introduce fractions greater than 1.

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Teaching the Meaning of FractionsRectangles, like those shown below, can

demonstrate that a fraction can be expressed as a mixed number.

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Developing Concepts for Fractions With Denominators of 10 and 100• When working with fractions it is very

important to introduce fractions in which the denominator is 10 or 100.

• This work prepares students for decimals which are the next topic.

• Children must learn that there is another way to write tenths. This permits early introduction of decimals as another way of expressing fractions with 100 or 100 as the denominator.

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Developing Concepts for Fractions With Denominators of 10 and 100

c. Use grids to answer each question:

1/10 = ?/100 2/10 = ?/100 10/10 = ?/100

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1/10 = 10/100 2/10 = ?/100 10/10 = ?/100

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1/10 = 10/100 2/10 = 20/100 10/10 = ?/100

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1/10 = 10/100 2/10 = 20/100 10/10 = 100/100

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d. How would you demonstrate on the grids which is bigger, 45/100 or 5/10?

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d. How would you demonstrate on the grids which is bigger, 45/100 or 5/10?

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e. Demonstrate the answer to50/100 + 12/100 = ?

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e. Demonstrate the answer to 50/100 + 12/100 = 62/100

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Once children understand the differences between 1/10’s and 1/100’s, simple decimal notation can be introduced as a shorter way of writing fractions that have denominators of tenths or hundredths, e.g.,

1/10 = .1

1/100 = .01

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Children should then be able to express any number of tenths or hundredths in terms of decimals e.g.,

2/10 = .2 2/100 = .02

3/10 = .3 3/100 = .03

. .

. .

9/10 = .9 9/100 = .09

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Addition and Subtraction of Fractions

Addition and subtraction of fractions using manipulatives is a natural extension of the activities we have already done.

Just specify a piece to represent the whole, then represent each fraction in the addition or subtraction problem with fraction pieces. Bring the pieces together for addition, or take away for subtraction, and express the answer in terms of the whole.

We will start with ½ + ¾ as a demonstration.

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Demonstration of ½ + ¾ .

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When studying addition and subtraction it is helpful to have all of the fraction pieces for a particular set of problems lined up.

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We can do many fraction manipulations with the set of fraction pieces. For example, to add ½ and 1/3, we bring the corresponding pieces together.

How much is this?

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To find the answer we compare it to the whole by lining it up with the complete set.

We see that the answer is less than the whole but more than one-half of the whole.

The set also gives us the exact answer.

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For the exact answer we use the other fraction pieces.

We see that our answer is 10 of the twelfths (10/12) or 5 of the sixths (5/6).

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We could also express each of the fractions in the problem in terms a smaller piece. One-half is the same lengths as 3 of the sixths while 1/3 is the same as 2 of the sixths.

The longest rod that will fit into both pieces is called the common denominator. This is a concrete demonstration of how it is found.

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Subtraction proceeds in a similar way. For example, in order to subtract 1/6 from ½

(1/2 – 1/6)

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To see how much the answer is we line it up with the rest of the pieces.

We find that it represents 2 of the sixths, or 1 of the thirds.

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Or we could have also used our common denominator method seeing that ½ is the same as 3 of the sixths. Take 1 sixth away and you have two sixths left which is 1/3.

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Teaching Fractions with Cuisenaire RodsSize Relationships

• We can assign any rod to represent a whole or the number one.

• The other rods will have a corresponding value in terms of the whole.

• Assume that the dark green (6) rod equals one. What are the number values of all the other rods?

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Line the other rods up against the dark green one to see the answer.White = Dark Green =

Red = Black =

Green = Brown =

Purple = Blue =

Yellow = Orange =

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Line the other rods up against the dark green one to see the answer.White = 1/6 Dark Green = 1

Red = 1/3 Black = 7/6

Green = 1/2 Brown = 4/3

Purple = 2/3 Blue = 3/2

Yellow = 5/6 Orange = 5/3

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Using the rods to answer, which is larger?a.2/3 or ½?

b.2/3 or 5/6?

c.3/2 or 4/3?

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Using the rods to answer, which is larger?a.2/3 or ½?

b.2/3 or 5/6?

c.3/2 or 4/3?

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Now we assume that brown (8) represents the number one. What are the values of all the rods?White = Dark Green =

Red = Black =

Green = Brown =

Purple = Blue =

Yellow = Orange =

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Now we assume that brown (8) represents the number one. What are the values of all the rods?White = 1/8 Dark Green = 3/4

Red = 1/4 Black = 7/8

Green = 3/8 Brown = 1



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Using the rods to answer, which is larger?a.¾ or 5/8?

b.½ or 3/8?

c.¾ or 7/8?

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Using the rods to answer, which is larger?a.¾ or 5/8?

b.½ or 3/8?

c.¾ or 7/8?

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How many eighths make up 1 1/4?

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How many eighths make up 1 1/4?

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Teaching Fractions with Cuisenaire RodsAddition and Subtraction of Fractions

Let the Orange (10) rod represent the number one. What will represent

a. ½

b. 1/5

c. 1/10

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Let the Orange (10) rod represent the number one. What will represent

a. ½

b. 1/5

c. 1/10

Note that ½ = 5/10 and 1/5 = 2/10. We need to keep the common denominator in mind when we add or subtract fractions.

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Use your rods to verify that ½ + 1/5 = 7/10.

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Use your rods to verify that ½ + 1/5 = 7/10.

In terms of the common denominator, we have 5/10 + 2/10 = 7/10.

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Use your rods to answer the following

a. ½ + 2/5

b. 3/5 + 1/10

c. 1/5 + 7/10

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Use your rods to answer the following

a. ½ + 2/5

b. 3/5 + 1/10

c. 1/5 + 7/10

Answers are: a. 5/10 + 2/10 = 9/10

b. 6/10 + 1/10 = 7/10

c. 2/10 + 7/10 = 9/1057


Finding a common denominator is most important. You cannot add fractions unless the denominators are the same size.

Example: We wish to add 1/3 + 1/2.

We see that the denominators are different. So we need to find a common denominator.

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We take threes and twos and lay them end to end until they line up evenly.

It takes 2 threes and 3 twos to make two even rows. Therefore, the common denominator is 6. Now we can deal with the numerators.

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We used 2 threes, so we need to have 2 ones. We used 3 twos, so we need 3 ones.

When we line them up, we get a total numerator value of 5.

Answer: 1/3 + 1/2 = 2/6 + 3/6 = 5/6

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As the children practice this procedure they can move up to more complicated examples. However, this is done the exact same way.

Example: We wish to add 2/7 + 3/5.


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We take sevens and fives and lay them end to end until they line up evenly.

It takes 5 sevens and 7 fives to make two even rows. Therefore, the common denominator is 35. Now we can deal with the numerators.

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We used 5 sevens, so we need to have 5 twos. We used 7 fives, so we need 7 threes.

When we line them up, we get a total numerator value of 31.

Answer: 2/7 + 3/5 = 10/35 + 21/35 = 31/35

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The same concept applies to subtraction.

You can only subtract fractions with equal size denominators.

Example: We wish to subtract 2/3 – 1/5.


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We take threes and fives and lay them end to end until they line up evenly.

It takes 5 threes and 3 fives to make two even rows. Therefore, the common denominator is 15. Now we can subtract the numerators.

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We used 5 threes so we need 5 twos. We used 3 fives so we need 3 ones which will be subtracted from the twos.

Answer: 2/3 – 1/5 = 10/15 – 3/15 = 7/15

Remember that addition and subtraction of fractions require a common denominator.

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Multiplication of Fractions

• Just as we did for whole number multiplication, we want to teach the meaning of multiplying by a whole number or another fraction before teaching algorithms.

• This is done using concrete objects.

• We can use the repetitive addition model, or the area model.

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The addition model for multiplying fractions.

Using repetitive addition, multiplying a whole number by a fraction such as 3 x ½ is the same as

3 x ½ = ½ + ½ + ½ = 3/2 = 1 ½

For another example,

2 x 3 ½ = 3 ½ + 3 ½ = 3 + 3 + ½ + ½ = 7

These can easily be demonstrated using fraction pieces or Cuisenaire rods.

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How would we teach the meaning of multiplying a fraction by a whole number such as ½ x 3?

½ x 3 = 3 x ½ = 1 ½

A fraction multiplied by a whole number is the same as taking the fractional part of the whole number

½ x 3 = ½ of 3 = 1 ½

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So, 3 x ½ is the same as taking half of the number 3, which is 1 ½.

This means that if we take three wholes and divide into two equal parts, each part is 1 ½.

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As another example, multiplying the fraction 1/3 by 3

1/3 x 3 = 1/3 of 3 = 1

This means that if we have three wholes, and divide them in to three equal parts, each is 1.

Multiplications such as 2/3 x 3 follow logically.

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In words, the problem 2/3 x 3 means that we find the fraction piece which is 1/3 of 3 and take two of these pieces. As each third is 1, then 2/3 is equal to 2.

1/3 of 3 = 1 2/3 of 3 = 2 (two 1/3 pieces) 3/3 of 3 = 3 (three 1/3 pieces)

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• Essentially, in multiplying a fraction by a whole number, a certain number of objects represents your whole, and you are taking a fractional part of these objects as we demonstrated above.

• After students understand the meaning of multiplying a whole number by a fraction they can be introduced to the algorithm which is cross-cancellation and multiplication.

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Cross-cancellation and multiplication.

Example: ¾ x 12

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Multiplication of two fractions can follow the same approach.

½ x 1/3 = ½ of 1/3

Which means to take a 1/3 fraction piece and divide it into two equal parts (or, take ½ of this 1/3 piece).

Each part represents what part of the whole?

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½ x 1/3 = ½ of 1/3

Each part represents what part of the whole?

We see that if we take ½ of the 1/3 piece we get a piece that corresponds to 1/6 of the whole.

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In Lab, we will work with exercises based on this pattern.

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The area model for multiplying fractions.

We used the area approach in the multiplication of whole numbers. We can use the same method for multiplying fractions.

½ x 1/3 =

As shown above, the answer is 1/6.

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As another example, we multiply

2/3 x 3/5 =

As shown, the answer is 6/15.

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• Division of fractions has the same interpretation as division of whole numbers.

X ÷ Y means, “How many Ys fit into X?”

In the case of a whole number divided by a fraction, such as

2 ÷ ½ means, “How many 1/2’s fit into 2?”

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Division of Fractions


One way to demonstrate the answer is by repetitive subtraction:

2 – ½ - ½ - ½ - ½ = 0

We see that four ½’s fit into 2.

This can be illustrated by fraction pieces or Cuisenaire rods. We will use the purple (4) to represent the whole.

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Or, “How many ½’s fit into 2 wholes?”

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Dividing two fractions together can be done in a similar manner. ½ ÷ ¼ means how many ¼’s fit into ½. Using successive subtraction

½ - ¼ - ¼ = 0

Using Cuisenaire rods it is

done this way. We can see

that two ¼’s fit into ½.

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For a set of exercises we will use orange 10 and red 2 together to make a whole. What is the value of each color?

White = Dark Green =

Red = Black =

Green = Brown =

Purple = Blue =

Yellow = Orange =

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For a set of exercises we will use orange 10 and red 2 together to make a whole. What is the value of each color?

White = 1/12 Dark Green = 1/2

Red = 1/6 Black = 7/12

Green = 1/4 Brown = 2/3



We can use the rods to divide

1÷1/6. How many reds (1/6) make up one whole?

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Simple activities can begin as early as 2nd or 3rd grade. Have the students learn the meaning of mathematical operations using manipulatives before teaching the algorithms.

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The End

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not whole numbers i: fractions presented by frank h. osborne, ph. d. © 2015 emse 3123 math and...

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