Tag: denominator

  • Why It’s Okay to Add Denominators when Adding Fractions

    Why It’s Okay to Add Denominators when Adding Fractions

    I did some videos for avoiding finding a common denominator and why this trick works. Ever since then I’ve pondered what it would look like if you added without a common denominator.

    A mom on the Living Math forum was asking for advice on how to teach adding numerators and not adding denominators in fractions. So this topic is again in the front of my mind.

    First note, that you can add both numerators and denominators – this is called piecewise addition – but then you are getting into a whole new world of stuff. Actually, you’ll be experimenting with some goodies from Abstract Algebra (my field of study). We’ll talk about how and why this is good in a few paragraphs.

    What is adding fractions, anyway?

    For us in the “real world,” adding fractions is adding values that can’t be counted with fingers. It’s adding bits of things together – maybe whole things with bits, even.

    We have agreed to use things that look like

    some number on topsome number on bottom that isn’t zero\frac{\text{some number on top}}{\text{some number on bottom that isn’t zero}}

    to represent fractional pieces of a whole.

    When we add 12+14=34\frac{1}{2} + \frac{1}{4} = \frac{3}{4} we have said that our total is

    3 pieces of a whole pizzawhen the pizza is cut into 4 equal pieces\frac{\text{3 pieces of a whole pizza}}{\text{when the pizza is cut into 4 equal pieces}}
    Total: 3/4 of a “pizza” is blue.

    Quantity over value is important to young children.

    I remember taking money from Little Brother when we were kids. You might think I was stealing, but it was more of conning the innocent. He would have 3 dollar-bills. I would have 20 pennies.

    He would quite willingly trade his 3 monies for my 20 monies. For him, this was a matter of quantity. I knew the value, though. (Bad girl – sorry, LB I owe you. xo)

    This measuring of quantity, rather than value was important in LB’s development. Instead of someone teaching him how to measure value, he learned it through experience.

    This is why we should allow kids to add the denominators.

    Let’s use the example above and add the numerators and the denominators to see what that means.

    Adding 14+12=26\frac{1}{4} + \frac{1}{2} = \frac{2}{6} means that our total is

    2 pieces that we have6 pieces that were possible to have\frac{\text{2 pieces that we have}}{\text{6 pieces that were possible to have}}

    Notice here you can’t “reduce” the fraction – because this isn’t the value  as we know it.

    If you teach adding fractions using manipulatives or toys and allow adding denominators, soon your children will see that your   (on the top in the picture below) is very different than having their  , on the bottom.

    We have six total pieces.

    This will inspire them to work toward equality – and invent fractional addition (correctly) on their own!

    Tell us how it goes in the comments!

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  • What’s the Domain, Why You Need It and How You Get It

    What’s the Domain, Why You Need It and How You Get It

    As you teach domain and range, do you get the question, “Why are we doing this?”

    No doubt the question, “When am I ever going to use this?” comes up too, right?

    I’ve asked myself that question my whole teaching and tutoring life. And now’s the time for an answer.

    A function is really a question.

    As I wrote in a previous post, a function is a question with only one answer to a valid question.

    When I write:

    y = 3x + 2 where x = 4

    I mean: “What is three times a number (that number is four), plus two?”

    The domain is all the possible questions:

    • What is three times a number (that number is five,) plus two?
    • What is three times a number (that number is six,) plus two?
    • What is three times a number (that number is seven,) plus two?
    • What is three times a number (that number is eight,) plus two?
    • <how long will I have to do this – Egad!>

    Not only do the questions go on forever, but they also have all the fractions and decimals in between. And all the negative versions of those numbers (and zero).

    So, what’s the point, you might ask. Looks like the questions go on forever and you can just pick any number.

    The domain might not include all the numbers.

    The two sticky points for the definition of “function” are bolded:

    A function is a question with only one answer to a valid question.

    The “valid question” part is where the domain comes in.

    The numbers that make  “valid” – meaning we actually can get some answer – are the numbers that aren’t negative.

    Many functions have “all real numbers” as a domain. There are no limits on the things you can put in, other than numbers that aren’t imaginary or alligators.

    For the most part, there are only two places where you have to be careful of limited domains. Those are

    • Numbers that cause a zero to turn up in the denominator
    • Numbers that cause negatives to turn up in square roots.

    Here are two videos tackling each:

    What do you think? Is this easier to teach when you consider “analyzing” the function rather than “solving” it? Share your thoughts and tips in the comments!