Understanding

Whole numbers tell us how many whole, big ol’ natural things we have (even if there aren’t any – that’s what “0” is for). Fractions are for when we’ve got parts of whole things, or parts of groups of things.

With big ol’ whole numbers… Multiplying gets you a bigger answer. 2 x 8 = 16 bananas.

“OF” is the word we use to multiply Eight bunches *of* 2 bananas Means 8 x 2 bananas (or 16 )

But…sometimes we do things in smaller chunks.

My friend charges ten dollars/hr to mow lawns. My friend charges ten dollars/hr to mow lawns.

Four hours of work? Four hours of work?

But this yard… half an hour.

Half an hour… half the ten dollars.

½ an hour of work… ½ of ten dollars… split it into two pieces get the value of one piece.

Here’s where the language gets tricky. Are we multiplying or dividing? To multiply by ½ … take ½ *OF* something… We divide by two. The denominator of a fraction tells us how many even sections we divide our “whole thing” by.

Think of that fraction bar as a machete, SLICING something into groups.

Counting calories? This deliciousness has 400.

Six times three (six groups of three) 16

Six times two (Six groups of two) 17

Six times one 18

19 6 times 3 6 times 2 6 times 1 Would six times ½ be bigger or smaller? Why?

Six times ½ means the same thing as ½ *of* six. 20 Either way It’s three!

Of … means multiply… if we multiply by a *fraction,* we are cutting something into pieces… so things are getting *smaller.* The denominator tells us how small the pieces are… how many times we had to divide.

Multiplying fractions ½ of 12 ½ of 100 ½ of 48 Same thing as 1 x

Multiplying fractions ½ of 12 6 ½ of ½ of Same thing as 1 x

What about a fourth? You’ve got a dozen eggs that you’re going to share with your three friends… but you’re not giving them all away. The 4 of you are going to share them evenly. You’re dividing by 4… … or splitting into fourths… … or multiplying by 1/4.

Division Start with a group, then break it into smaller groups of equal size.

Fractions Are another way of writing division

12/4 … it’s 3, eggs or oranges or miles 12/4 … it’s 3, eggs or oranges or miles

What’s 1/3 of 12? But what if one of my friends doesn’t want any? Now we get more, right ?

1/3 of 12 xxxx xxxx xxxx 4 people would get 3 oranges each… 3 people would get 4 oranges each.

So… don’t think of a fraction as just a number. It is a relationship!!! Yes, it’s a relationship that DIVIDES – but that’s not always a bad thing… sometimes dividing is SHARING

Why is 12 the same as 12 1? What if you divide 12 into… one group? It’s got everybody in it; all /1 is the same as 12.

What’s 1/3 of a dozen eggs?

What’s 1/4 of a dozen eggs?

What’s 1/12 of a dozen eggs?

What’s 1/3 of a dozen eggs?

What’s 1/4 of a dozen eggs?

What’s 1/12 of 12 eggs?

1 x 12 eggs gives me 1 egg x 12 eggs gives me 1 egg What’s 1/12 of 12 eggs?

Here it is in all math symbols ;) What’s ½ of this dozen eggs? ½ x 12 = 6 12 ÷2 = /3 of 12 =4 12 ÷ 3 = /12 of 12 = 1 12 ÷ 12 = 1 1/12 x 12 = 1 =6 =4

The idea to remember – the bigger the denominator, the smaller the pieces. The denominator is what you divide by. 1 is much, much bigger than (and 1/100 is teeny teeny tiny compared to 100)

½ of 80 is  8  20  40  320

¼ of 80 is  8  20  40  320

1/10 of 80 is  8  20  40  320

½ of 80 is  8  20  40  320

¼ of 80 is  8  20  40  320

1/10 of 80 is  8  20  40  320

1/20 of 80 is  4  8  20  40  320

That’s nice, but those aren’t what math problems look like. Here are a few for your practice and entertainment

1/3 of 30 = ____ 1/7 of 14 = ____ ½ of 100 = ____½ of 14 = _____ ¼ of 100 = _____ 1/14 of 14 = ____ 1/10 of 100 = _____ ½ of 10 = _____ 1/8 of 80 = _____1/5 of 10 = _____

1/3 of 30 = __10 1/7 of 14 = 2 ½ of 100 = 50½ of 14 = 7 ¼ of 100 = 25 1/14 of 14 = 1 1/10 of 100 = 10 ½ of 10 = 5 1/8 of 80 = 101/5 of 10 = 2

And that’s what taking one piece of a group is about… you can have fun with this. Look over spaces and groups through the day, and imagine dividing them into fourths… which is the same as dividing BY four… or multiplying BY one fourth. Practice the language on the simple ones, and the more complicated ones will make much more sense!