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Topic: How would YOU simplify arithmetic?
Replies: 41   Last Post: Jul 11, 2012 12:56 AM

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 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
Re: How would YOU simplify arithmetic?
Posted: Jun 25, 2012 6:15 PM

A few months ago

"Re: Understanding Division of Fractions"
http://mathforum.org/kb/message.jspa?messageID=7744265

I shared a way of making parts of fraction addition using the least common denominator (LCD) easier, by making it so that after we get the least common denominator we can go straight from the different fractions to be added to a single fraction in just one written step. (By "one written step" I mean that we can write out the algorithm as a single equation as a sum of fractions on the left side of an equation and write out a single fraction with the LCD on right side of the equation.)

We multiply or divide fractions by using an algorithm that can be stated as a single equation, but we don't teach fraction addition using the least common denominator that way. That is, when we multiply or divide fractions we can go from different fractions to one fraction in one step via this one equation. I think that the main reason why fraction addition using the least common denominator is harder than either fraction multiplication or fraction division is because it's not normally taught such that we can go from the different fractions to one fraction in one written step.

This method of teaching fraction addition using the LCD like teaching fraction multiplication or division as a single equation is like the method of fraction multiplication in that unlike the usual way of fraction addition using the LCD it can be extended very easily to any number of fractions more than two.

I find that those who most find this method of fraction addition using the LCD preferable to the usual way are algebra students who need to add or subtract rational functions or other situations in algebra where the LCD needs to be used in reducing a sum of fractions to a single fraction.

Note: In my post above, after giving some proofs of this method, I go into detail as to how to think through the algorithm while implementing it. That is, like in fraction multiplication, we do not need to actually write out the single-equation-based formula with variables to actually implement it, even though the algorithm can be written out as a single-equation-based formula with variables.

On Mon, Jun 25, 2012 at 2:53 AM, Jonathan Crabtree <sendtojonathan@yahoo.com.au> wrote:
> In Australia more students than ever leave school without basic numeracy skills.
>
> So if YOU were starting from scratch, how would you simplify arithmetic?
>
> For the sake of the question, we'll assume:
> ten numerals 0 1 2 3 4 5 6 7 8 9
> four operations + - x ÷
> equality sign =
>
> I am not seeking pedagogical replies or contributions. Instead how can arithmetic ITSELF be made simpler?