Fractions

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 bananasMeans 8 x 2 bananas

(or 16 )

But…sometimes we do things in smaller chunks.

My friend charges ten dollars/hr to mow lawns.

Four hours of work? 4 x 10 = 40 dollars

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.

Eat 3 of them? 3 x 400 = 1200 calories (now go ride your bike )

Eat 1/2 of one? 1/2x 400 = 200

400 ÷ 2 = 200

Six times three (six groups of three)

16

Six times two(Six groups of two)

17

Six times one18

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 12 2 1

Multiplying fractions

½ of 12 6

½ of 100 50

½ of 48 24

Same thing as 1 x 12 2 1

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

12 ÷ 4 = 3

Start with a group, then break it intosmaller groups of equal size.

Fractions

12 ÷ 4 = 3

Are another way of writing division12 = 3

4

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

1 x 12 4 1 x 12

4 1

What’s 1/3 of 12?

1 x 12 3

1 x 12 3 1

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 12. 12/1 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.12 1

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 = 612 2 1/3 of 12 =4 12 ÷ 3 = 412 31/12 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 1 2 100

(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 = 50 ½ 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 = 10 1/5 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!