Adding Fractions Wrong

Everyone knows… (that’s a bad way to start a post)

To add fractions:

• if they have common (same) denominators (bottoms), we add the numerators (tops) and keep the denominators: and in general
• If they do not have common denominators (bottoms are different), we change the fractions to equivalent fractions (same value, different look, like and ) with common denominators, keep those denominators:

Or, instead of making two separate cases, multiply each numerator by the other denominator, and take the sum (that’s the numerator) and multiply the denominators: and in general . And yes, that might not give the least common denominator. In the numeric example, that fraction can be reduced. But that’s ok. The answer is correct, and , so we are fine.

Adding wrong

I see kids add fractions wrong. Hell, in the US we can see adults add fractions wrong. Numerator plus numerator, denominator plus denominator. Tops with tops, bottoms with bottoms. If you haven’t seen it, you’re not looking.

If we want to add 1/2 a pizza and 1/8 of a pizza (a slice) we don’t get 2/10 of a pizza (less than half). The idea that this is adding is ridiculous.

But tops + tops over bottoms + bottoms DOES produce a result. Let’s name this process – I’ll call it MISADDING – and see what MISADDING actually does.

MISADDING

Being formal, define MISADDING as

A few days ago, Dave Marain published this:

NSFS
(Not Suitable For Students)
1. Have Ss write 3 more like this and have another member of group check it
2. Discuss why it works and when it doesn't!@MrValencia24 #iteachmath @ExploreMTBoS pic.twitter.com/3GKqVyAIa7

— David Marain (@dmarain) November 30, 2023

It reminded me of MISADDING. On Saturday I played around with MISADDING with some kids, and talked about it a little bit with a mathematician. Here is some of what we decided:

• MISADDING is commutative, since the underlying operation is regular addition.
• MISADDING is associative, as long as we don’t reduce the results along the way (we will come back to this reducing issue, which is huge).
• Take a fraction, and MISADD something to it. Can we get the original fraction back? In other words, is there a MISADDITIVE identity? Sort of? Well, turns out that this only happens when we MISADD equivalent fractions. or and in general .
• MISADDING is not well-defined. If we add fractions, with real addition, it does not matter what form those fractions are in – our answer will be equivalent. For example,
  
  and .
  Those answers are equivalent.
  But MISADDING? Uhn-uh.
  
  but ,
  hardly the same thing.
• MISADDING always produces a result between the two values being MISADDED (if those values are different. If they are equal, MISADDING returns the same value, as we saw above). This means that if then . There’s a pretty straightforward proof, if you’d like to try it.
• MISADDING gives the average of the two numbers under a very specific condition. I’ll leave you to figure that out. But observe: . Nice, right?
• MISADDING with whole numbers is fun. Just write 1 as the denominator. But remember, you will get different answers if you MISADD with than if you MISADD with .
• If we drop the notation, and write the fractions in parentheses with a comma, MISADDING becomes vector addition: (a,b) + (c,d) = (a+c,b+d) or (1,2) + (4,7) = (5,9) (except the restriction on zero denominators is lost).
• MISADDITION is closed on an interval. For example, MISADD two fractions between 0 and 1, and the result will be a fraction between 0 and 1. That “in-betweenness” of the results is interesting – since we are always in-between, we can never break out.
• Combining the vector idea with the in-betweenness idea, we could see the pairs (or the fractions) as an expression of slopes… Rewrite this as (b,a) + (d,c) = (b+d,a+c), and now think of (b,a) as the line from (0,0) to (b,a) and its slope will be . Vector addition will give us a point with an associated slopes between the two given slopes, UNLESS the points are on the same line, and then the result will be on the same line (equivalent to MISADDING two equivalent fractions, and getting an equivalent fraction as the result).

Too long, didn’t read version:

When you misadd fractions by adding the tops and adding the bottoms – the result will not be greater than the two originals (which is what you expect for adding positive numbers) but IN BETWEEN them.

And if you came here for a post about Mulgrew… (I’m stopping here)