Fractions 101: How to Think About Fractions, by Angela Berkeley

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So you’re approaching that time when your children will start to learn about fractions in their curricula?  You know that this is important foundational material.  You know this information will be used for all future math and science classes, for statistics classes, and throughout your children’s lives.  You have already started to give your children a concrete introduction to fractions, and now it is time to start teaching them the more abstract and complicated aspects of this subject.  How will you frame this for them?

This question can be made more challenging by the approach your math curriculum takes.  It is fairly common for math curricula to teach fractions by teaching lists of rules. Of course, knowing such rules is helpful and necessary, but memorizing them without really understanding why they work the way they do makes it difficult to detect computation or calculation errors and double check one’s work. It can also be difficult to recall those rules perfectly in order to apply them to complicated math problems.

In general,  there are ways to think about fractions that are extremely helpful in remembering how to work with them; helpful both for teaching them and for using them.  You can teach your children this whether your curriculum does so or not, and it will help them to truly understand what they are doing when they work with fractions. You can use these to brush up on fractions on your own or to have some useful concepts to teach your children.

Here is the first, most foundational thing to keep in mind:  A fraction is fundamentally a division problem.  (Note:  It is not a division EQUATION.  Rather, it is a division PROBLEM, because there is no equal sign in a fraction.  It is a division problem.)

The top part of the fraction is divided by the bottom part of the fraction.

So when  you see 1/2, it is the same as seeing 1 divided by two.

2/3 is the same as 2 divided by 3.

(1702)(3B)/4+C is the same as (1702)x(3B) divided by (4+C).

Again, the top part is always divided by the bottom part.

How is this helpful?  For one thing, if you ever need to convert a numerical fraction into a decimal, you simply work the division problem, and voila, you will have a decimal.

For another thing, it emphasizes a powerful message that the top and bottom parts of a fraction cannot ever be reversed.  In the same way that 100 divided by 10 is completely different from 10 divided by 100, 100/10 and 10/100 are completely different from each other.

Remembering that a fraction is fundamentally a division problem reminds us of that over and over.

So what are the two parts of the division problem?

By the way, these are the other foundational things to keep in mind:

The bottom number (denominator) tells us how many equal parts one whole thing is divided into.  

The top number (numerator) tells us how many *of those parts* we have.

 

OK, read that again, slowly, and think about it.

Now for some examples:

3/4 means that 1 whole thing is divided into 4 equally sized parts (because the denominator is 4), and we have 3 of those parts (because the numerator is 3.)

17/20 means that 1 whole thing is divided into 20 parts, and we have 17 of them.

298/1200 means that 1 whole thing is divided into 1200 parts, and we have 298 of them.

This always works.  It is ALWAYS true, no matter how complicated-looking the numerator or denominator is.

Whenever you see a fraction, say to yourself, “This is a division problem.  The bottom number tells me how many parts one whole thing is divided into.  The top number tells me how many of those parts I have.”

Stay tuned. We will move on to implications for four function arithmetic of fractions in the next article.

Angela Berkeley–Although Angela Berkeley wanted to homeschool her daughter, she was unable to find others to partner with in this endeavor and felt that it was unfair to homeschool an only child; so she enrolled her in kindergarten. However, because the family was facing a mid-semester cross-country move during their daughter’s first grade year, she pulled her out to homeschool until they settled into their new home. This went so well, and her daughter liked it so much, that they ended up homeschooling through 8th grade.  Using an eclectic classical style, this was an extremely successful process, producing a confident, personable, and academically well-prepared entrant into a local high school.

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