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Introduction to Fractions -Section 1

What is a fraction? Why is this type of number needed? The answers to these questions depend on who answers them. From a mathematician’s point of view, the defining of fractions (or more formally rational numbers) provides a type of number that by their very nature have properties that whole numbers and integers do not possess. From a fifth grader’s perspective, fractions allow us to address problems related to measurement and fair share division. This section will focus on the various definitions and conceptualizations of fractions, various ways to concretely model fractions, different representational forms of fractions, ordering fractions, and problem solving using fractions in a concrete, contextual sense.

Definition and conceptualizations of fractions

There is no standard definition of a fraction. Some sources will use the term fraction and rational number interchangeably. In some cases, the term fraction is used in reference to rational expressions, which can be composed of variables and polynomial expressions, or to make

comparisons of real numbers, such as . For the purposes of this textbook, fraction and

rational number will be defined as related, yet distinct, constructs. The definition used here relates to how fractions and rational numbers are usually taught in school mathematics. Before formal definitions of the terms are given, a review some of the different ways to conceptualize these types of numbers is presented.

The following are ways the representation can be interpreted from these different

perspectives.

Parts of a whole: From this perspective means 3 equal sized parts out of 4 equal parts of a

unit whole. This is the conceptualization with which most elementary students are introduced to fractions.

Quotient: From this perspective we view the fraction bar as another symbol representing

division, hence means 3 divided by 4 and the quotient of this division is what is being

represented. This is the perspective we use when we teach students to convert fractions to their decimal representation.

Ratio and Rates: Ratios are a broader concept of which fractions are a special case. Ratios and rates will be discussed in more detail in Chapter XX but for the purposes of the discussion here,

means 3 units compared to 4 units of the same or different measure.

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The ‘Parts of a Whole’ conceptualization is the one most easily supported by children’s understanding of division and fair-sharing, so this is the conceptualization that will be utilized for defining the terms ‘fraction’ and ‘rational number’.

Definition. A fraction is a number represented by two whole numbers, a and b, b ≠ 0 using the

notation or a/b indicating that some unit whole has been divided or partitioned into b equal

sized parts and a of them are under consideration. The a part of the fraction is called the

numerator and the b part is called the denominator. The meaning of is relative to what unit

whole it is referencing. A rational number is a number represented by two integers, a and b,

b ≠ 0 using the notation or a/b. We will expand this definition more in a later section.

Modeling fractions

As children are taught the definition of a fraction, one important consideration is what type of concrete and pictorial models will be used to represent this concept. Three types of common models used to represent fractions are set models, area models, and number line models.

A set model is appropriate when examples involving discrete objects are being used. If examples involve things that cannot be subdivided, but can be grouped in various ways, then this type of model is appropriate. For instance, when discussing a fraction in the context of a group of people or other living things, it is not appropriate to split a person or a dog into smaller pieces. For instance, having a problem with 2 ½ ponies just doesn’t make sense. An example of a set model for ¾ might look like this:

Three-fourths of the happy faces are light.

An area model is another way that fractions are commonly represented for students. An area model is appropriate when giving examples involving objects that can be subdivided into any number of pieces. The model consists of some type of two-dimensional figure, such as a circle, square, rectangle, etc. While any type of figure can be used as an area models, some shapes are better than others. Circles are often used in elementary school textbooks, but have some limitations when used to represent operations with fractions. Squares or rectangles are

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easier to subdivide into odd numbers of equal sized pieces and are excellent models when used to teach fraction operations.

_____________________________________________________________________________

An example of an area model might look like this:

Three-fifths of the large rectangle is shaded.

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A number line model is appropriate when using examples involving linear measurement. An advantage of number lines is that most children are already familiar with this instructional aid. A disadvantage is that number lines, for some children, are very abstract and if not presented carefully can actually lead to confusion. Consider this example of a common question that might be found in a homework set or standardized test over fractions.

Name the fraction represented by the diagram.

0 1 2 3 4

Some students will interpret this diagram as 11/25 because they will count all of the represented subdivisions, making that the denominator and then counting those represented by the arrow and making that the numerator. What they will fail to understand is that with a number line model, the distance from 0 to 1 is always the unit whole and any representation using a number line must be interpreted with that understanding.

When choosing any model, it is important that students have been given some guidance in how to interpret it. One of the key concepts in understanding fractions is relating the

representation to its unit whole.

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Consider this example:

What fraction is being represented by the shaded part of this diagram? The answer to that question depends upon how the unit whole for the diagram is interpreted. If the large rectangle is considered to be the unit whole, then the diagram represents 3/12. If the bold line in the middle of the diagram is seen as making two rectangles – then the unit whole consists of 6 small rectangles and the diagram is representing 2/6 in the left rectangle and 1/6 in the right rectangle or 3/6 altogether. This example not only shows the importance of identifying the unit whole for the situation, it is an example of how pictorial representations used to teach fractions need to be well defined or students may draw conclusions that the teacher hasn’t intended.

What if the above diagram is redrawn to look like this?

If the large rectangle only has 4 equal parts instead of 12 and the shaded sections were shifted to line up, what fractional part of the large rectangle is shaded? What has changed? Is the relationship ¼ the same as 3/12 for this rectangle? These are all questions related to understanding the definition of a fraction and lead to the next concept.

Equivalent or equal fractions

Fractions that represent the same relative amount are called equivalent or equal fractions. Every fraction (actually every real number) has an infinite number of representations.

Let’s consider the fraction . This fraction can be represented with the following area model:

One third of the large rectangle is shaded. By dividing each of the one third sections into two

equal sections, the model can be changed to represent the fraction .

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This process of dividing each of the sections into equal sized subsections can continue to represent an infinite number of different fractions that all relate to the same relative amount.

This model could represent . But all of the diagrams have the same sized rectangle (the unit

whole), with the same amount of the area of that rectangle shaded blue and the same amount of the area of the rectangle not shaded. The only difference between the three diagrams is how many divisions of the area are shown. This is the concept behind equivalent or equal fractions.

Symbolically this concept can be defined as follows: Given any fraction , for which

b ≠ 0, we can represent the same relative amount in an infinite number of ways by multiplying both the numerator and denominator by the same natural number n, therefore

. When is such that a and b have no common factors greater than 1, then we say that

the fraction is in simplest form, reduced form, or in lowest terms.

There are several methods for determining if two fractions are equivalent. Method 1 – Use cross products

For example, determine if and are

equal? Using cross products it can be shown that , so the two fractions are representing the same relative amount.

Method 2 – Simplify both fractions

Another way to determine if and are equal fractions is to simplify both

fractions and show they are equal to the same fraction.

and . Since both and equal , they

must equal each other.

Method 3 – Convert both fractions to their decimal representations (Note: this method is not appropriate for students who haven’t studied decimals)

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The conceptualization that = , leads to the use of the decimal

representation of this quotient to determine if the two fractions are equal. Since both

and are equal to , then by the transitive property the two fractions must equal the

same relative amount. (Decimal representations of fractions will be discussed more in the next chapter.)

Ordering fractions

Research on how children learn about fractions has revealed that given opportunities to interact with various types of fraction manipulatives and interesting and challenging problem situations to work through, children will develop their own techniques for ordering fractions. These ‘intuitive ordering techniques’ are grounded in a developing fraction number sense. Traditional textbooks use more computationally oriented techniques of ordering fractions. Both are presented here.

Intuitive ordering techniques

• Same denominator method. If the two fractions have the same denominator, that means the ‘parts’ are all the same size, so comparing the fractions consists of determining which fraction is describing more parts, so compare the numerators. The fraction with the larger numerator is larger.

Rule – given

Example: Given and , since 2 < 5, we determine that .

• 1/n type fractions. If the two fractions both have a numerator of 1, then compare the denominators. The larger the denominator, the smaller the size of 1 part. The fraction with the smaller denominator is the larger fraction.

Rule – given

Example: Given and , since 3 < 8 then .

• Same numerator method. This is a generalization of the 1/n method.

Rule - given

Example: Given and , since 5 < 13 then

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• (n-1)/n type fractions. This method examines how close to being a unit whole each fraction is. The fraction with the smaller missing piece is the larger fraction.

Rule – given

Example: Given and , since the missing fifth sized piece is larger than the missing

eighth sized piece, then .

• Comparison to ½ method. This is a benchmarking method where the two fractions are compared to the value ½. Most children can easily divide by two, even if the dividend is an odd number. One way to use this method is to divide each denominator by 2, then compare the numerator to this result. If the numerator is less than the denominator divided by 2, then the fraction is less than ½. If the numerator is more than the denominator divided by 2, then the fraction is more than ½. Another way to use this method follows.

Rule -

Example: Given and , we determine that < because 4 < 9÷2 and >

because 7> 12÷2. So using the transitive property, .

Traditional textbook methods• Find a common denominator. This reflects the first of the intuitive techniques listed above. A representation that gives both fractions a common denominator is found and the fraction with the larger numerator is the larger fraction. For example, which is smaller 11/20 or 4/7?

Therefore, 11/20 < 4/7.

Convert to decimal. As discussed before, this method is not appropriate for students who haven’t studied decimals, but this is a common method of comparing fractions once students understand the concept of fractions as representing the division of the numerator by the

denominator.

AN IMPORTANT PROPERTY

Fractions have an interesting property that natural numbers, whole numbers, and integers do not possess. Consider this question. Name a whole number between 0 and 1? The answer, of course, is there are no whole numbers between 0 and 1. But if the questions is changed to “Can you name a fraction between 0 and 1?”, the answer is “yes”. What is amazing is that there are an

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infinite number of fractions between 0 and 1 and between any two fractions in that interval, there are an infinite number of fractions. This concept is called the density property.

Given any two numbers a and b in a set, if there exists another number c such that a < c < b, then the set of numbers exhibits the density property.

Improper fractions and mixed numeralsOne misconception that students often develop is that all fractions are less than 1.

Obviously, this is not the case. Fractions that are between 0 and 1 are called proper fractions.

Another way to define this is to say that given the fraction if a < b, then the fraction is

considered a proper fraction. Fractions for which the numerator and denominator are equal can be simplified and are also whole numbers or non-negative integers. Fractions in which the numerator is greater than, or equal to, the denominator are called improper fractions. More

formally, given the fraction , then the fraction is an improper fraction.

It is important that students have concrete experience with both proper and improper fractions. Consider this diagram:

First it is important that the unit whole in this situation is well defined. If the diagram is considered to represent a set (unit whole) of 12 small squares, then the interpretation for this diagram would be 11/12. But if the unit whole is defined as one large square consisting of 4 smaller squares, then this diagram represents the value 11/4 or 2 whole squares and ¾ of another. When we represent 11/4 as 2 ¾ we are using what is called the mixed numeral form of representation. 11/4 and 2 ¾ represent the same relative amount, but there are advantages and disadvantages of both forms. When doing computations with fractions, the improper form of the fraction is desirable. When the answer is related to a real world situation, often the mixed numeral form is a better way to represent the amount. For instance, most people would say it is 2 ¾ miles to the store, instead of saying it is 11/4 miles to the store. Both statements are correct, but in everyday usage, the mixed numeral form is more accepted.

Students need to be able to easily move between improper representations of fractions and mixed numeral representation of fractions.

, so for example . When working from improper form

to mixed numeral form, the numerator is divided by the denominator. The quotient of this division becomes the whole number part of the representation and the remainder of the division

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becomes the numerator of the fraction part, with the divisor becoming the denominator. For

example

Another thing that needs to be understood about the mixed numeral form is the implied

addition between the whole number part of the answer and the fraction part. . If

students don’t understand this implied addition relationship, they will often make errors when performing calculations with mixed numerals. Finally many students confuse simplifying fractions with putting improper fractions into mixed numeral form. The two processes are not the same. Simplifying fractions means the fraction’s numerator and denominator share on common factors other than 1. It has nothing to do with whether the numerator is larger than the denominator.

A summary of concepts about fractions

• There are various ways to conceptualize and define fractions– The most commonly used conceptualization in the K-8 curriculum is the part-whole concept

• When using the part-whole definition of fractions, two things must be stressed.

– Students must understand what the ‘whole’ is– Students must understand that the fraction must have equal sized parts of the ‘whole’

• There are an infinite number of representations for any single fraction amount – we call these equivalent or equal fractions.• A fraction is considered simplified if the numerator and denominators have no factors in common, other than 1.• There are intuitive and traditional methods for ordering fractions, both have value for students.• Students need to be proficient with moving from improper fraction form to a mixed numeral form.

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Addition and Subtraction of Fractions (Section 2)

In this section, the operations of addition and subtraction of fractions will be explored. The

following will be discussed: developing a conceptual understanding of the algorithms for

addition and subtraction of fractions, common denominators and how they relate to the Least

Common Multiple concept, and properties of addition and subtraction of fractions.

Adding Fractions

Most students that have had some previous exposure to adding and subtracting fractions

will recall to do these operations a common denominator is required. But when these students are

asked why a common denominator is needed, they generally have no answer. This reveals a lack

of conceptual understanding of fractions and the operations of addition and subtraction.

Consider the following problem and an area model that represents it.

The model shows two equal sized squares. The first square is cut into 2 even sized pieces and 1

piece (or ½ of the whole square) is shaded. The second square is cut into 3 even sized pieces and

1 is shaded. Using the most common context for addition – the join or union context, we could

draw a diagram of adding these two shaded regions that would look like this:

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The answer to the problem is all of the shaded parts of this square, but how much is that?

In order to identify a single fraction to describe this amount of the whole square, it is necessary

to have a common denominator to identify this amount. If the fractions in the above problem are

considered as sixths, then it is possible to answer the question ‘how much is this amount?’

This diagram shows the result is .

Another reason a common denominator is necessary has to do with a general concept about

addition and subtraction. In some sense the mathematical expressions that are added or

subtracted must be alike. (Other examples of this concept are adding like radicals and adding like

terms.) With whole number situations, this was never a problem, but now that the focus is on

numbers that can represent different sized pieces, it is important that when adding or subtracting

these pieces be the same size – hence the need for a common denominator.

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Algorithms for fraction addition

If the problem to be solved involves fractions that have the same denominator, then

addition of those fractions consists of counting how many pieces of a certain size you have

altogether. Since the numerators of the fractions indicate how many pieces of the size indicated

by the denominator are involved, this process becomes a matter of adding the numerators of the

two fractions as follows: Let a/b and c/b be fractions then: .

If the fractions involved do not have a common denominator, then the process becomes

more involved. One all purpose algorithm that will work to add fractions without a common

denominator is this: Let a/b and c/d be fractions, then:

This algorithm doesn’t focus on finding the lowest common denominator, just the handiest one.

An area model makes this method clearer. If we do the problem again we could think

about it this way with the model. The columns in the figure represent the halves and the rows

represent the thirds. By dividing the whole in this way, the smallest rectangles in the figure

represent sixth-sized pieces of the whole. So the equivalent of one column would be three of the

small rectangles ( ) and the equivalent of one row would be two small rectangles ( ). So

adding is like adding .

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The process of dividing the whole vertically for the denominator of one of the fractions and

horizontally for the second fraction, is essentially the same as multiplying the two denominators

together and goes back to the area context for multiplication.

On aspect of this method is that the process does not always give the least common

denominator. Consider the example . Using the algorithm given above the process for

adding these two fractions would look like this:

Since the least common denominator wasn’t used in the addition process an additional step at the

end of the process to simplify the answer was necessary. So the pedagogical question - is it

really necessary for today’s students to find the least common denominator before adding

fractions?

If it is determined that using the least common denominator is a desired process, then the

algorithm for adding two fractions would be as follows.

Given the following steps would be in the process.

1. If b = d are equal, then .

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2. If b ≠ d, then find LCM(b,d). Then determine x so that x(b) = LCM(b,d) and y so that y(d) =

LCM(b,d). Once this is done the work looks like this:

Here is an example using this method.

Given it is noted that the two fractions have different denominators, so we find

LCM(6,15) = 30. We note that 6(5) = 30 and 15(2) = 30, so

Note: the use of this method does not guarantee that the answer won’t need to be simplified.

Properties of fraction addition

Fraction addition has the same properties as whole number addition.

Closure : The sum of any two fractions is also a fraction

Commutative : a/b + c/d = c/d + a/b

Associative: (a/b + c/d) + e/f = a/b + (c/d + e/f)

Additive Identity: a/b + 0 = 0 + a/b = a/b

Understanding these properties can facilitate computing answers to problems such as this:

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Order of operations would indicate the addition of the 1/5 and 4/7 first. This would entail finding

a common denominator. But if original problem is rewritten using the properties of addtion, the

computation can be done mentally without the need to find a common denominator

= (commutative property)

= (associative property)

=

Subtraction of fractions

Subtraction of fractions has similar algorithms to addition.

Fractions with common denominators:

Let a/b and c/b be fractions then:

Fractions without common denominators:

Let and be fractions where , then:

Or if a least common denominator is desired:If b ≠ d, then find LCM(b,d). Then determine x so that x(b) = LCM(b,d) and y so that

y(d) = LCM(b,d). Once this is done the work looks like this:

Here is an example using this method. Given it is noted that the two fractions have

different denominators, so we find LCM(6,15) = 30. We note that 6(5) = 30 and 15(2) = 30, so

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Properties of fraction subtraction. Based on the definition of fractions being non-negative quantities, it is clear that fraction subtraction has none of the properties of fraction addition. It

can be noted that . This is not a true identity property, since .

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Multiplying and Dividing Fractions (Section 3)

Can the conceptualizations for whole number multiplication translate to fractional

quantities in ways that students can develop a solid understanding of fraction multiplication?

How can teachers conceptualize multiplication of quantities that are less than one in such a way

that students will develop a well grounded fraction number sense? What does 2/3 x ½ mean?

What about fraction division and the difficulties associated with developing a conceptualization

of measurement and partitive interpretations of this operation? Can an explanation of ‘invert and

multiply’ be developed that will enable students to understand and remember that algorithm?

These questions (and others) associated with teaching fraction multiplication and division will

guide the development of this next section.

Conceptual understanding of fraction multiplication

The multiplication fact a x b = c can be interpreted in multiple ways. One of those ways

is to interpret a x b as meaning a groups of size b (or as repeated addition of the quantity b, a

number of times). Using this interpretation of 3 x ¾ it can be considered as:

3 x ¾ = ¾ + ¾ + ¾ = 9/4 or 2 ¼.

This interpretation of 3 x ¾ provides an easy conceptualization on which to construct a story

problem. For instance: Beth has 3 jars of red paint. Each jar is 3/4 full. If she combines all of her

red paint, how many full jars of paint does Beth have? The answer is 2 full jars and ¼ of another.

Beginning the discussion of fraction multiplication with the ‘whole number x fraction’ situation

is a good way to help students begin to develop conceptual understanding of fraction

multiplication.

One thing to note about the previous situation is that the product is smaller than one of

the factors in the problem. When multiplying whole numbers with factors larger than 1, the

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product is always larger than the factors. When multiplying by 1 the product is equal to the other

factor, and in the special case of multiplication by zero, the product is always zero. With

multiplication of fractions, students are seeing products that are smaller than one or both of the

factors and this is counterintuitive to their previous experience with multiplication.

In the case of a ‘fraction x whole number’, if students have a firm understanding of the

commutative property, they can conceptualize 3/5 x 4 as if it were 4 x 3/5, so the idea of

multiplication as repeated addition still holds. Where the repeated addition conceptualization of

multiplication gets awkward, is in the case of a ‘fraction x fraction’ situation, such as 3/5 x ¾. It

can be interpreted as 3/5 of a group of size ¾. This involves subdividing a group of size ¾ into 5

even sections and selecting 3 of those. This is less transparent than the variations of ‘whole

number x fraction’. Perhaps another conceptualization of multiplication would be less difficult

for introduction of this concept.

The multiplication of a x b = c can also be viewed in the context of area. In this

interpretation a and b are the width and length of a rectangle and the product c is the area of this

rectangle. With this interpretation of multiplication, 3/5 x ¾ would be the area of a rectangle

with a width of 3/5 of a unit and length ¾ of a unit. The product then is a fraction of a whole

square unit. This interpretation allows the use of a rectangular area model to arrive at the

product.

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In this figure one dimension is cut into 4ths and the other dimension is cut into 5ths. The

heaviest shaded region represents the area of a rectangle that is 3/5 unit wide by ¾ unit long.

Also illustrated is that the reference whole in this situation, which is the 1 x 1 unit square, is

subdivided into 20 equal portions and 9 of those are double-shaded. Thus the area of a 3/5 unit x

¾ unit rectangle is 9/20 of a square unit. (Note that 20 is a common denominator for 3/5 and ¾.

Using this method, even though finding common denominators isn’t necessary to multiply the

fractions, the denominator of the product is a common denominator of the factors.)

A word problem for this conceptualization could be: Bill is making a pen for his goats.

He has room to form a rectangular pen 3/5 of a mile wide and ¾ of a mile long. What is the area

of his goat pen in square miles?

The area model method used to multiply 3/5 x ¾ can also be used to develop a conceptual

understanding of mixed numeral multiplication. For example, consider the problem: 2 1/3 x 1 ½.

0

1/4

2/4

3/4

1

1/5 2/5 3/5 4/5 1

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If the two factors in this problem are considered to be the length and width of a rectangle, then

the product would be the area of that rectangle. The following model demonstrates this.

The outlined square in the lower left corner of the rectangle represents the 1 unit x 1 unit

whole in this situation. Note that 6 of the small subdivisions are contained in this square. This

means the unit whole is divided into 6ths, and 6 would be a common denominator for the

fractions in this situation. So the entire rectangle is composed of 21 of these 1/6th sized smaller

rectangles. In other words the area of the large rectangle with dimensions 2 1/3 units x 1 ½ units

is 21/6 = 3 3/6 = 3 ½ square units.

Obviously, drawing a rectangle every time the product of two fractions is computed may

not be the most practical method, so an algorithm is desired. This algorithm can be seen

concretely in the two area models just developed.

Let and be fractions. Then

0

1/2

1

3/2

1/3 2/3 1 4/3 5/3 2 7/3

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Note: Whenever multiplying mixed numerals, use the improper form of the fractions and use this

same algorithm.

It is helpful to teach students to simplify during this process, as it makes the simplification

process easier. For example:

.

Properties of fraction multiplication

The properties of multiplication on the set of fractions are similar to the properties for whole

numbers, but with the inclusion of an important new property – a multiplicative inverse.

1. Closure: the product of two fractions is a fraction.

2. Commutative:

3. Associative:

4. Multiplicative Identity:

5. Distributive property of multiplication over addition and subtraction

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*6. Multiplicative Inverse (reciprocal): for every non-zero fraction , there

exits a unique fraction such that .

Note that the number zero has no reciprocal. By the multiplication by zero

property, any quantity multiplied by zero gives a product of zero. Consider the equation

. There is no fraction which when multiplied by zero gives a product of one.

So it is important to remember that when discussing reciprocals, the number zero has no

reciprocal. To find the reciprocal of mixed numerals, write the number in improper form,

then determine the appropriate reciprocal.

Example:

The multiplicative inverse is a very important property in developing an algorithm

for fraction division as well as for solving certain algebraic equations such as .

Division of fractions

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Fraction division is a difficult concept to model in a meaningful way. The ‘invert and

multiply’ algorithm is a rote procedure that does not provide students with any meaningful

concept of fraction division. There are several, more meaningful ways to teach fraction division

rather than just asking students to memorize a rule.

Recall from the chapter on whole number division, that there are two ways a division

problem can be interpreted: as repeated subtraction from the dividend of equal sized groups

indicated by the divisor, where the quotient indicates the number of groups and as a fair-share

concept in which the dividend is equally divided into the number of groups indicated by the

divisor and the quotient indicates the number of elements in one group.

Consider = Q. If this problem is viewed as a repeated subtraction problem, then

the quotient Q represents how many groups of size 2/3 can be subtracted from 8/3. A model of

this might look like the following:

In this model each rectangle is divided into three equal pieces. Two whole rectangles are shaded

and 2/3 of another, so 8/3 = 2 2/3.

If the goal is to see how many groups of size two-thirds can be removed from this model, then

the solution might look like this:

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The different shadings of the model indicate groups of size 2/3 and it can be seen that there are

exactly 4 of them, therefore . It is important to note that in this problem the two

fractions had a common denominator. When this is the case, the quotient can always be

determined by dividing the numerators. This provides an alternative division algorithm, which

can be stated as follows:

Here is an example of using this algorithm for division of fraction on two fractions which do not

initially have common denominators:

Returning to = Q, what if this problem is viewed as a partitive division problem?

Then with this interpretation, 8/3 is 2/3 of a whole group and Q is the size of one whole group.

Consider this interpretation with this diagram:

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This represents 2/3 of a whole group in this situation. So 1/3 of the whole must equal 4 of the

smaller rectangles in the diagram, therefore the whole must contain 12 of the small rectangles.

One whole group would be:

The dark gray rectangles represent the parts that would need to be added to the 2/3 already

represented to make a whole group of 12/3 or 4. This interpretation of the division is related to

the missing factor interpretation of division. In other words the question to answer is “2/3 of

what amount is equal to 8/3?”

Neither of the interpretations of division just given lend themselves to an easy

understanding of the ‘invert and multiply’ algorithm, so how can its use be justified by more than

a “because it gives you the right answer” justification?

There is a property of division that is helpful in developing a justification for the ‘invert

and multiply’ algorithm which may be called Maintaining the Quotient. It can be stated as:

Given

This property can be explored by students by giving them simple whole number division facts

and having the students select various numbers, multiplying those numbers by the dividend and

divisor of the fact and observing that the result of the division is unchanged from the original

quotient. For example: if is chosen as the division fact, then observe the following:

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This property provides the key to a justification for the “invert and multiply” algorithm.

Given , this reasoning would follow:

1) If the divisor in this problem was equal to 1, then the division process would be easy, so

what could be done to mathematically to turn it into a 1?

2) Multiplying numbers by their reciprocals produces a product equal to one, so multiplying

by would result in a product of 1.

3) By the property of maintaining the quotient, as long as the dividend and the divisor in the

problem are multiplied by the same amount, the answer to the problem will be unchanged, so the

work would look like this:

.

Note that when the dividend is multiplied by the reciprocal of the divisor, the result is essentially

the ‘invert and multiply’ algorithm taught as the traditional algorithm for fraction division. Thus

‘invert and multiply’ is a shortcut step of this longer process. The process explains and justifies

why a division problem can be solved using multiplication and the reciprocal of the divisor. Here

is an example of using this process to divide two fractions:

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Therefore the ‘invert and multiply’ algorithm can be justified and we define division of fractions as follows:

Closure for division

Consider the following?

For any fractions

Is there closure for division on the set of fractions?

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Putting it all together Fraction Manipulatives Activity

You will work in groups of 3-4 students. Each group will be given a type of manipulative used to teach fraction concepts to students. Work through the problems using the manipulatives to represent your thinking and answer the questions below for your particular type of manipulative. Be prepared to present to the entire class how you would use this type of manipulative to teach various fraction concepts. Each group should turn in one copy of the work on all parts with everyone’s name included.1. On a piece of paper, trace the piece (or pieces) you decide will represent the ‘whole’ or ‘unit’, then trace the piece (or pieces) that represent the following fractions: ½, 1/4, 1/3, 1/6, ¾, 2/3, 5/6.Example: This would be the work for pattern blocks and 1/6

2. Use your pieces to represent ½ + 1/3. Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.3. Use your pieces to represent 5/6 – 2/3. Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.4. Use your pieces to represent 3 x ¾. Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.5. Use your pieces to represent 2/3 x 3/4 Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.6. Use your pieces to represent ¾ ÷ 1/2. Draw that presentation on a piece of paper. Develop a word problem that goes with this math fact.=====================================================================