Part 4ii: Introduction to Fractions
Filling in the Gaps
After re-watching the clips I realized that there were glaring omissions between clip 4ii (fraction basics) and 4iii (adding/subtracting fractions with unlike denominators). Conspicuously absent from clip 4ii (on this page) is a discussion of equivalent fractions, and an explanation of adding/subtracting fractions with like denominators.
To a student who is learning about fractions for the first time, the fact that the fraction is somehow equal to the fraction could be quite confusing; after all, the numerators of both fractions are different as are the denominators. As teachers, we understand that ‘renaming’ a given fraction as an equivalent fraction is a crucial step in the addition/subtraction process, thus before we introduce fraction arithmetic we must not only convince students that two ‘different looking’ fractions can in fact be equivalent, but we also need to justify the correct procedure for renaming fractions. A model that can be used to achieve both of these goals is illustrated below:
To illustrate , one might begin with four bars that look like the top bar above. At this stage, since all four bars are exactly the same, there is no question that the same fraction of each bar is shaded, and therefore all bars represent the same fraction (at this stage, ). To illustrate equivalent fractions, we then simply subdivide the second, third, and fourth bars into 4, 6, and 10 total pieces respectively. Note that since we didn’t change any of the shading, all four bars still represent the same fraction of the whole, but the act of subdividing has ‘renamed’ the second, third, and fourth bars , and respectively. Thus it must be the case that .
So fraction strips can be used to demonstrate that ‘different’ looking fractions can actually be equivalent, but this model also illustrates the correct procedure for obtaining equivalent fractions. For example, if we begin with the top bar, the act of subdividing each piece into two pieces multiplies the total number of pieces by 2, but also multiplies the number of shaded pieces by 2. But recall in a fraction, the number of fractional units that make the whole (in this case, the total number of pieces in the bar) is just the denominator of the fraction, and the number of fractional units we are interested in (in this case, the total number of shaded pieces of the bar) is just the numerator of the fraction. Therefore, in this case we created an equivalent fraction by multiplying the numerator and denominator by 2. Of course there is nothing special about the number 2; if we multiply both the numerator and denominator of a fraction by the same (nonzero) number, we will obtain an equivalent fraction.
Adding/Subtracting Fractions with Like Denominators
The most common mistake students make when adding and subtracting fractions is to simply add numerators and add denominators; such a student would think . So the first thing we must teach students when we introduce fraction arithmetic is that it makes no sense to add or subtract fractions if they have unlike denominators. Once they are convinced, we must teach them how to add/subtract when the denominators are the same. This can be done by using very precise language and by appealing to mathematics that is already known to students, namely addition and subtraction of whole numbers. (The final step of demonstrating how to add/subtract fractions with unlike denominators is shown in the clip on this page)
A nice way of approaching fraction addition/subtraction is to appeal to students’ knowledge of language. We can introduce this idea by considering the following problem: Suppose we order a pizza and cut it into seven pieces. From our basic understanding of a fraction, our fractional unit is a ‘seventh’. So we can describe one ‘slice’ of pizza as one ‘seventh’ of the pizza; note that both ‘slice’ and ‘seventh’ are nouns in this case. Now suppose that you eat three pieces. In this case, three is describing the number of pieces, or sevenths, of the pizza you ate, so it can be thought of as an adjective.
Now consider the following early grade addition problems involving whole numbers:
- John has 3 trucks, James has 2 trucks. Together they have trucks.
- John has 3 marbles, James has 2 marbles. Together they have marbles.
- John has 3 dollars, James has 2 dollars. Together they have dollars.
- John has 3 apples, James has 2 oranges. Together they have pieces of fruit.
In the first three problems, the ‘units’ of the addends are the same; as such, when we add them together the unit of the sum is also the same. But in the fourth example, the units of the addends differ; thus in order to perform addition we must first find a common unit. So from a very early age students understand that it makes no sense to add (or subtract) unless the units (nouns) of the original numbers are the same, and this idea can be extended to fraction addition/subtraction.
Once students understand that it makes no sense to add/subtract fractions when the denominators differ, we must convince them of the proper way to add/subtract fractions when the denominators are the same. Let’s use the pizza example from above, and suppose in addition to the three pieces you ate, I ate 2 pieces. The arithmetic problem describes the fraction of the pizza that we ate altogether. The only conceivable mistake that could be made when performing this addition is the mistake described earlier, namely . Let’s use language to illustrate the correct process.
- You ate 3 pieces, I ate 2 pieces. Together we ate pieces.
- You ate 3 ‘sevenths’, I ate 2 ‘sevenths’. Together we ate ‘sevenths’.
- You ate , I ate . Together we ate .
Therefore , and this idea can be extended to illustrate the general process for adding/subtracting fractions with like denominators.