Many people seem a bit phobic about “fractions”. This anxiety likely has two sources: not really understanding what a fraction represents, and having memorized a bunch of rules way back in elementary school without understanding why they work.
Revisiting fractions using variables as well as constants, with complicated as well as simple problems, helps conquer anxieties about this topic.
What Is A Fraction?
can be interpreted as
- The number three divided by the number five
- Three of the quantity one fifth
- A fifth of the quantity three
- Three parts of a five part “whole”
- A rational number
A rational number is a number that can be expressed as the ratio (or quotient) of two integers. Three fifths satisfies this definition, and is therefore a rational number.
“The number three divided by the number five” describes a division problem, with three in the numerator and five in the denominator. This is perhaps the most “rote” and least “intuitive” interpretation of a fraction, but it is also the simplest. Every fraction is a division problem, and every division problem involving two quantities can be expressed as a fraction.
“Three parts of a five part whole” is illustrated above, and is the interpretation that is probably used most often in elementary school. The entire object above represents the quantity one. The denominator instructs us to divide the object into five equal parts, and the numerator instructs us to use three of them.
The above illustration also fits the description “three of the quantity one fifth”. Once again, the entire bar represents the quantity one, and it has been divided into five equal squares, so each square represents the quantity “one fifth”. Using three of the squares therefore represents three “fifths”.
Note that when read the way fractions are usually read (as “three fifths”), the phrase can be translated into algebraic notation as either (using the first interpretation) or (using the second interpretation). It is useful to be confident rewriting a fraction with the reciprocal of the denominator in either the first or second position:
An illustration of the final fraction above is:
Each row above represents a “whole” quantity divided into five parts. The three rows therefore add up to a total of three “whole” quantities. The left column is one fifth of the entire grid, or one fifth of three. Note that the size of the darker three squares on the left above is exactly the same as the size (after rotating it by 90 degrees) of the darker three squares on the left of the previous illustration, which demonstrates that both interpretations produce the same result.
Each of the four arithmetic operations (addition, subtraction, multiplication, and division) work with fractions too. Some operations are easy to compute with fractions, others more complex. I believe that multiplying two fractions is the easiest of the four operations.
To multiply two fractions, multiply their numerators and multiply their denominators. Why does this rule work? What is really going on when we follow this rule? To explore these questions a bit, let’s consider an example:
One way of simplifying the above is to rewrite division as multiplication by the reciprocal (scaling by a number between zero and one), then use the commutative property of multiplication to rearrange things a bit:
While this now makes it easy to calculate the two times four part, it still requires us to evaluate the product of two fractions. How is the product of these two reciprocals calculated?
Intuitively, if you are asked to divide a whole pizza into three equal parts, you will get something like the illustration below. Each third of the pizza has a different color:
Now, divide each color (third) into five equal parts. Looking at the result of slicing the pizza into thirds, then each slice into fifths, the pizza has been cut into fifteen slices: three large slices (each with a different color above) were the result of the first division, then each large slice became five smaller slices as the second division was carried out, creating a total of fifteen small slices (as numbered above). So, “a fifth of a third” will be one of the above slices, which represents one fifteenth of the original pizza:
Another way to envision the process of taking “a fifth of a third” relies on interpreting multiplication as scaling. The vertical line on the left of the illustration below has a length of one, and the green segment at the bottom of that line has a length of one third.
The horizontal line at the bottom of the above illustration also has a length of one, and the red segment on the left of it has a length of one fifth. Multiplying one quantity by another is done by stretching the first quantity (the green line segment) into a new dimension (to the right) by the length of the second quantity (the red line segment). The result is the yellow rectangle with “1” in its center.
As you can see in the above illustration, the yellow rectangle is identical in size to the other 14 rectangles shown. Together, all fifteen rectangles create a 1 by 1 square, whose area is 1. The yellow rectangle, which is the product of “one third” and “one fifth”, has an area equal to one fifteenth of the square’s area, so the product of these two numbers equals one fifteenth.
A non-geometric approach to the problem can also be taken. Convert both reciprocals into their decimal equivalents, then multiply them:
one third is
and one fifth is
and the product of the two is
which is the decimal equivalent of one fifteenth.
And lastly, a physical approach could be based on a room that contains 15 chairs (similar to the square in the illustration above). If you divide them into three equal groups, there will be 5 chairs in each group. If you then divide each group into five equal sub-groups, there will be 1 chair in each sub-group. The result of dividing 15 by three and then five is 1. What proportion (or “fraction”) of the original number of chairs does this represent? One fifteenth… the same answer arrived at via the other approaches.
Some useful insights hopefully came to mind while thinking about the above:
- Dividing by three then dividing by five will produce the same result as dividing by five then dividing by three. When rewritten as multiplication by the reciprocal, the commutative property of multiplication allows us to rearrange the factors without changing the result.
- Dividing by one number (three), then dividing by another (five), produces the same results as dividing by their product (fifteen).
Think about how we read the original problem: “two thirds times four fifths”. This could be rephrased as “two of the quantity a third times four of the quantity a fifth”, which not only is equivalent to the first statement grammatically but also mirrors the way division can be rewritten as multiplication by the reciprocal (remember that “of” indicates multiplication).
By rewriting the entire problem as a multiplication problem, and using the commutative property of multiplication to change the order of the factors, the original problem becomes “two, times four, times a third, times a fifth”. Hopefully you are now comfortable with the reasoning that two times four is eight, and a third times a fifth is a fifteenth, so we have now simplified the problem down to the product of eight and a fifteenth. As our last step, we can now reverse the process of turning division into multiplication, and rewrite a number times a reciprocal as the quotient of the two numbers:
So, the product of two fractions will always end up being the product of their numerators divided by the product of their denominators.
One note of caution… if any numerator or denominator has more than one term, put the entire numerator or denominator in parentheses before multiplying. The vinculum (the horizontal line indicating division) is also a grouping symbol that indicates that both the numerator and the denominator should be treated as a single factor – as if they were in parentheses:
And lastly, I find multiplication by a fraction to be the easiest time to re-introduce and discuss the notion of multiplication as scaling (instead of repeated addition). Since fractions seldom represent whole numbers, the interpretation of multiplication as scaling by a factor is somehow easier to comprehend in examples involving fractions.
Earlier posts in this series:
Algebra Intro 1: Numbers and Variables
Algebra Intro 2: Addition
Algebra Intro 3: Subtraction
Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Algebra Intro 5: Addition, Subtraction, and Terms
Algebra Intro 6: Multiplication
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Posts that continue this series:
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions
What? No pictures?
I do appreciate, however, the level of complexity you are willing to allow this topic. Too often, the topic is reduced to “of means multiply and here’s the algorithm for multiplication” (and by the way, kids, don’t try to think too hard about what this might have to do with the multiplication of whole numbers).
Might a diagram or two add something to your text?
GeoGebra (of which I understand you are quite fond) is a lovely tool for good, quick mathematical diagrams.
Thank you for the suggestion. I have reworked the posting to include some graphical illustrations, along with some additional verbiage to describe them. See what you think!