Primary mathematics/Fractions

In the early primary grades, fractions are generally thought of as "a part" of a whole, and are used to express numbers less that one. Yet, fractions encompass much more mathematical territory than this. They can of course be larger than one.

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Methodology

Generally speaking, the progression used when teaching fractions starts with the use of manipulatives and other visual representations that make connections to real life things: pizza, money, candy, etc. As children begin to be able to articulate these understandings both verbally and with mathematical expressions, more abstract models such as area rectangles, fraction bars and grids are introduced, permitting students to expand their understandings in more abstract ways.

The Use of Manipulatives to Teach Fractions

Candidates for manipulatives used to teach fractions are single objects or groups of the same object that can be divided equally. There are a number of commercially available "toys" that are designed just for this purpose, but these are not necessary for a resourceful teacher. Regardless, students should initially be making connections to things they "know", like pizza and candy.

Origami

Oragami used to teach fractions

By simply taking a square piece of paper and folding it in half; write the fraction ${\tfrac {1}{2}}$; fold in half again; write ${\tfrac {1}{4}}$. Continue this process to create smaller fractions. Unfold the paper and trace lines where there are folds in the paper.

Notice that in the case of the example, the manipulative (the folding paper) becomes a model (a more abstract visual representation) as students draw over the folds and fill in the values.

Money

As we all know, money is a great motivator (for any age). It is something all children are familiar and comfortable with, and perhaps most importantly - they want it! This doesn't mean you have to give it to them, but you can use it to leverage their interest.

The use of money as a manipulative/model to teach fractions (and many other concepts in mathematics) has many advantages. Thankfully, most all currencies use a base ten system. Using change, the US dollar can be broken up into fourths (quarters), tenths, twentieths and hundredths, which makes it a perfect model to use for teaching decimals and even percentages as well.

From Manipulatives to Models

While manipulatives offer a tactile and visual way for students to learn about fractions, models let them take their exploration to a higher level of abstraction, while still maintaining a visual aspect to their learning.

"Area rectangle" modeling the fraction ${\tfrac {3}{4}}$

Area Diagrams

"Area diagrams" or "area rectangles" are perhaps one of the best ways for children to model fractions. They have the advantage of visually showing "how much" of the whole a fraction takes up, and will become an indespensible tool students will eventually use for modeling operations with fractions. Students should be encouraged to create their own area diagrams to reinforce their learning.

"100 Grid" modeling ${\tfrac {2}{10}}$

"100 grids" are a type of area diagram commonly used to teach fractions. Because the fractions that 100 grids model at the single unit level are hundreths, they are usually introduced in the upper primary grades. Similar to the way that area rectangles can be used to model multiplication, 100 grids are a great way to model fractions and simultaneously facilitate students in making important connections to percentages and decimals. Be sure to have students demonstrate that they understand these connections by filling in the 100 grid and then writing down the equivalent expressions.

For instance: ${\tfrac {1}{4}}=25\%=0.25$

Notice in the above example that children should be expected to always reduce fractions to simplest terms (if they know how).

Equivalent Fractions

Because fractions represent both division and ratios, it is important to introduce all of these concepts to their web, or schema, of mathematical understanding.

Equivalent Fractions and Ratios

When students create tables such as the ones below (based on the multiplication tables of the numbers 1 and 3, and the cost of a candy bar) they are exposed to the connection between fractions and ratios.

1	2	3	4	..	7
3	6	9	12	..	21

In the table below, note that after division is used to "reduce" the ratio to ascertain the cost of a single candy bar, the rest of the table is much more easily completed:

Cost of Candy Bar(s)
1	2	3	10	?	20
?	$1.30	?	?	$16.90	?

Using the Identity Property of Multiplicaton to Create Equivalent Fractions

Students need to understand the reason that the methods they use to create equivalent fractions actually work. One very effective way to do this is for them to initially learn the Identity Property of Multiplication.

Of course, we know that ${\tfrac {a}{a}}=1$, but most primary grade students are not able to make the connection between this axiom in action and its corresponding abstract mathematical expression stated with variables, so we should first let children explore the ways they can represent the number $\mathbf {1}$: I.e., ${\tfrac {23}{23}}=1$, or ${\tfrac {0.6}{0.6}}=1$, or ${\tfrac {1,000,000}{1,000,000}}=1$...

A "Giant One" expressing the value ${\tfrac {7}{7}}$

We can explain to students that if ${\frac {7}{7}}=1$, and anything multiplied by one doesn't change its value, then ${\frac {3}{5}}\times {\frac {7}{7}}=$ shouldn't change it's value, but this is very difficult for many students to understand because of the level of abstraction. Models and visual representations become necessary. One very effective visual representation we can use is the Giant One.

Students also need to be reminded of the fact that anything times 1 doesn't change its value (The Identity Property of Multiplication).

The Giant One can be used to set up the equation, which helps students to visualize a way to find equivalent fractions. Students should draw these themselves to reinforce their understanding of the concept. Even students with poor motor skills can draw Giant Ones; it doesn't matter how neatly they are drawn:

Using Area Rectangles to Create Equivalent Fractions

As we have seen earlier, area rectangles have the advantage of visually showing students "how much" a fraction represents. And because it is possible to show a fraction using either horizontal or vertical lines, by using both horizontal and vertical lines it is possible to simultaneously show two superimposed fractions.

Note how this exercise is executed almost in the same fashion as multiplying fractions, except that the second fraction, because it is equivalent to 1 is not filled in. The super-imposition of thirds and forths creates a total of 12 equal rectangles, or 12^ths, and 9 filled in rectangles. Students should become adept at creating these. Students eventually should be able to look at the image on the right and know what equivalent fractions are being modeled.

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The same thing should be modeled with a Giant One problem to reinforce the connection:

Basic Operations

The more standard algorithms that most of us use to add, subtract, and multiply fractions, are to a child's perspective, a collection of rules, which at best "work", but at worst become all mixed up in students' minds. This is usually because these 'rules' have no underpinnings in the students' web of mathematical knowledge. In other words, if students don't understand why the "rules" for using these methods exist in the first place, it becomes very difficult to apply them at the right time, not to mention execute them consistently. For this reason, this "traditional" approach (just teaching them how to do things, but not why they are doing them the way they are) to teaching operations with fractions requires a "drill and kill" methodology.

Thankfully, the contemporary approach to teaching these methods has students explore these concepts in many different ways at a fundamental level.

Adding Fractions

In the lowest grades, it may be best to only add fractions with a common denominator. This can be explained as "you can't add apples and oranges together", or, since those could be added together to get the total pieces of fruit, you could use an example of adding items which are even more different, like weeks and sheets of paper. For example, "if you have 2 weeks and 10 pieces of paper to make an art project, you don't add them together to get 12". In this analogy, the denominator of the fractions describes the type of fractions they are, which determines if they can be added.

In later grades, students can convert one or both fractions to have a common denominator, before adding. This should wait until after multiplication of fractions has been taught, as this is a necessary step in finding a common denominator.

Subtracting Fractions

The same logic applies to teaching subtraction of fractions as addition, with one additional element; it is now possible to get a negative result. This can be explained with real world objects and "owing". For example, "if you said you would give your brother half a cookie, but only had a quarter cookie to give him, you still owe him another quarter cookie. This is the same as saying you now have a negative quarter cookie".

Multiplying Fractions

Most students when first confronted with the problem ${\tfrac {2}{3}}\times {\tfrac {1}{4}}$ have difficulty understanding that the product of these numbers will be smaller than either of them. This is a natural conjecture, born of their years (not too many) of experience with multiplication.

The use of the word "of" to describe multiplication needs to be used to realign their understanding. Children know that $5\times 6=30$ because five groups of six is thirty. Asking them "what ${\tfrac {1}{4}}$ of 100 is" helps them to better understand why multiplying with fractions changes their earlier understanding. As they begin to understand the implications of this they can then be introduced to the concept using two fractions. I.e., what is 1/2 of 1/2?

At this point models should be used to reinforce their understanding of this. Area models can be very effective towards this end, where the area of the intersecting fractions represents the product.

Students should be both asked to create these models from word problems as well as create word problems that fit with the model. For instance,

2/3 of the students in the class are wearing yellow shirts.  1/4 of the class goes to the library.
What fraction of the class is wearing yellow shirts in the library?

or

2/3 of my garden was planted with yellow flowers, and 1/4 of all of the flowers were sold.
What fraction of all of the flowers sold from the garden were probably yellow?

Note that in this case the wording of the problem has created a lesson with a connection to probability.

The image below models the problem ${\frac {4}{10}}\times {\frac {5}{10}}$ using a 100 grid.

Note that the use of a 100 grid permits the answers to problems to be expressed as fractions, decimals and even percentages (E.g., $.4\times .5$ and 40% of 1/2.)

Dividing Fractions

Teaching students how to divide with fractions by simply showing them how to multiply the dividend and the reciprocal of the divisor offers practically no insight to their mathematical understanding.. because no connections have been made to what they already know about division. Before students learn to divide whole numbers, they need to understand how to multiply them. The same concept applies with fractions. Students should first explore division of whole numbers by fractions at a basic level. Models should be used whenever possible. For instance, the problem "How many 1/2s are there in 4? can be modeled with any graphic representation, such as.

The next level of difficulty involves two numbers with fractions, such as can be found in the problem, "How many ${\tfrac {2}{3}}$ are there in $2{\tfrac {2}{3}}$?

Similar models can be used to explore division of fractions with different denominators. I.e., "How many ${\tfrac {1}{6}}$ are there in $2{\tfrac {2}{3}}$?

At this time students should also be exploring and learning how to find reciprocals. It will be necessary for them to understand what reciprocals do before they start to use them to accomplish division problems with fractions.

Rulers

The above method of teaching division with fractions can be accomplished with rulers as well. Most rulers that are incremented in inches provide fractions down to sixteenths. Rulers are a "real world" tool and familiarity with the specific fractions encountered when using a ruler have an exponential component. I.e, ${\frac {1}{2}}$ ${\frac {1}{2^{2}}}$ ${\frac {1}{2^{3}}}$ ${\frac {1}{2^{4}}}$

Division with Giant One

Though to a teacher, it may at first seem complex, students who are comfortable with the Giant One find this method is useful if not preferable when first solving division problems that include fractions. The best way to teach this method is to start with a whole number dividend and express the problem as a fraction(which students understand is the same as division). Thus, the equation $4\div {\tfrac {2}{3}}$, turns into the fraction ${\frac {4}{\tfrac {2}{3}}}$. Multiplying this fraction by a Giant One such that the number of the divisor of the bottom fraction is inserted into the Giant One creates a model like the one at right, which becomes $12\div 2$.

Using the Giant One to solve ${\tfrac {4}{5}}\div {\tfrac {2}{3}}$

Division with two fractions requires the use of the reciprical of the divisor of the problem to be inserted into the Giant One:

Eventually, students who use this method recognize that the top fracton of the problem is being multiplied by the reciprical of the bottom fraction. The process of discovering this "trick" for themselves gives them ownership of the larger conceptual understanding. In other words, students who learn to divide fractions in this way understand at a fundamental level "why" the trick works

Adding fractions

Fractions that have the same or "Common" denominator are called "Like" fractions.

1/3, 2/3

(one-third, two-thirds)

To add Like fractions together such as these:

1/3 + 2/3 = ?

(one-third plus two-thirds equals what?)

1. Add the numerators (the top numbers):

1 + 2 = 3

(one plus two equals three)

2. Use the common denominator (the bottom numbers):

/3

All together it looks like this:

1/3 + 2/3 = 3/3

(one-third plus two-thirds equals three-thirds)

3. Then simplify the answer as much as you can by dividing the denominator into the numerator.

3/3 = 1

(three divided by three equals one)

If the numerator is now larger than the denominator it might look like this:

4/5 + 3/5 = 7/5

(four-fifths plus three-fifths equals seven-fifths)

simplified:

5 goes into 7 one time, with 2 left over. Put the remainder (2) over the denominator (5).

7/5 = 1 2/5

(seven-fifths equals one and two-fifths)

To Add or Subtract fractions with different denominators, you must first convert them to equivalent fractions with common denominators.