B. Number · Number Sense: Numbers to 200 and Fractions

Shares Smaller Than One: Halves, Thirds, Fourths and Sixths

B1.6About 35 minutes

Learning goals

I can cut a whole into equal parts: halves, thirds, fourths or sixths.
I can tell if parts are equal or not equal.
I can count fractional parts, like one third, two thirds, three thirds.
I can solve a fair-share problem where each person gets less than one whole.

Words to know

fraction
A number that names equal parts of a whole, like one third or three fourths.

whole
One complete thing, like a whole pizza or a whole sandwich.

thirds
3 equal parts of a whole. Each part is one third.

sixths
6 equal parts of a whole. Each part is one sixth.

fourths
4 equal parts of a whole. Each part is one fourth (also called one quarter).

Not enough for everyone!

Three friends want to share 2 sandwiches. There isn't even 1 whole sandwich for each friend. How can they share fairly?

In this lesson you will learn about shares that are smaller than one whole.

Equal parts only!

A fraction names equal parts of a whole. All the parts must be the same size.

If you cut a sandwich into 3 pieces, but one piece is huge and the others are tiny, those are not thirds. That would not be fair!

Left: 3 equal parts, so each part is one third. Right: 3 parts that are not equal, so they are not thirds.

Names for equal parts

The name of the part comes from how many equal parts make the whole.

Equal parts in the whole	Each part is called	Count the parts to make one whole
2	one half	one half, two halves
3	one third	one third, two thirds, three thirds
4	one fourth	one fourth, two fourths, three fourths, four fourths
6	one sixth	one sixth, two sixths, ... six sixths

Halves, thirds, fourths and sixths

The same whole cut into halves, thirds, fourths and sixths. One part of each is coloured.

Counting fractional parts

We can count equal parts, just like we count apples!

one third, two thirds, three thirds

When you have three thirds, you have the whole thing. Three thirds make one whole.

Counting thirds. Three thirds make one whole.

Counting sixths. Six sixths make one whole.

Counting past one whole

You can keep counting past one whole!

one half, two halves, three halves, four halves ...

Two halves make 1 whole. So three halves is 1 and a half, and four halves is 2 wholes.

Sharing when there's less than one each

When there are fewer things than sharers, cut every thing into as many equal parts as there are sharers. Then give each sharer one part from each thing.

Worked example

Example 1: One pizza, four friends

4 friends share 1 pizza fairly. How much does each friend get?

There are 4 sharers, so cut the pizza into 4 equal parts.
Each part is one fourth.
Each friend gets one part.

Each friend gets one fourth of the pizza.

Worked example

Example 2: Two sandwiches, three friends

3 friends share 2 sandwiches fairly. How much does each friend get?

There are 3 sharers, so cut each sandwich into thirds.
Now there are 6 thirds in all (3 + 3).
Give each friend one third from the first sandwich and one third from the second sandwich.
Count each friend's pieces: one third, two thirds.

Each friend gets two thirds of a sandwich.

2 sandwiches, each cut into thirds. Each friend gets 2 pieces: two thirds.

Worked example

Example 3: Pancakes for four

4 children share 3 pancakes fairly. How much does each child get?

4 sharers, so cut each pancake into fourths.
Each child gets one fourth from each of the 3 pancakes.
Count: one fourth, two fourths, three fourths.

Each child gets three fourths of a pancake.

3 pancakes cut into fourths. Each child gets one fourth from every pancake: three fourths. Each colour shows one child. The 4th child gets the uncoloured pieces.

Just for fun

What do you call a sandwich cut into 2 equal parts by a pirate?

A haaarrrlf! (Okay, okay. A half.)

Hands-on activity

Paper sandwich sharing

You need: some paper rectangles (these are your "sandwiches"), scissors (with a grown-up), crayons, and 3 to 6 stuffed animals.

Fold a paper rectangle in half, then in half again. Open it. You made fourths! Colour and count them: one fourth, two fourths, three fourths, four fourths.
Fold another paper into thirds (like folding a letter). Count the thirds.
Now share 2 paper sandwiches among 3 toys. Fold each one into thirds and cut. How much does each toy get?
Share 3 paper sandwiches among 4 toys. How much does each toy get?
Share 1 paper sandwich among 6 toys. (Hint: fold into thirds, then fold in half.)

Math talk: "How do you know the parts are equal?"

Your turn

A whole is cut into 6 equal parts. What is each part called?

How many thirds make one whole?

Count by fourths: one fourth, two fourths, __, __

3 friends share 1 granola bar fairly. How much does each get?

2 people share 1 apple fairly. How much does each get?

4 children share 3 sandwiches fairly. How much does each get?

A pie is cut into 3 parts. One part is much bigger. Are the parts thirds?

More practice: warm-up

A whole cut into 2 equal parts makes ___.

A whole cut into 4 equal parts makes ___.

How many sixths make one whole?

6 kids share 1 pizza fairly. How much does each kid get?

Count: one half, two halves. How many wholes is that?

More practice: on your own

3 children share 2 pancakes fairly. How much does each child get?

Count by sixths from one sixth to six sixths.

6 friends share 5 granola bars fairly. How much does each friend get?

How many wholes are four halves?

Sara cut a tortilla into 4 pieces and says each piece is one fourth. What must be true for her to be right?

Challenge

How much is five halves? Use wholes and halves.

4 friends share 2 pizzas fairly. Each pizza is cut into fourths. How much does each friend get? Can you say it another way?

Amir has some sixths of a cake. He counts: one sixth, two sixths, ... and stops at four sixths. How many more sixths does he need to make one whole cake?

Check your understanding

Quiz

How many equal parts are in thirds?

6 friends share 1 cake fairly. Each friend gets...

How many fourths make one whole?

3 friends share 2 sandwiches fairly. Each friend gets...

Which shows halves?

What comes next? one sixth, two sixths, three sixths, __

For parents and teachers

Big idea: A fraction names equal parts of a whole. The number of equal parts gives the name (halves, thirds, fourths, sixths). We can count fractional parts: one third, two thirds, three thirds = one whole. When there are fewer items than sharers, cut every item into as many parts as there are sharers.

Common difficulties:

Calling unequal parts "thirds" or "fourths". Always ask, "Are the parts the same size?"
Cutting only one item when sharing several (sharing 2 sandwiches among 3 by cutting just one). Ask, "Did everyone get a piece of each?"
Thinking a sixth is bigger than a third because 6 is bigger than 3. The next lesson focuses on comparing.
Folding thirds is hard. Fold paper like a letter, or a grown-up can mark the folds.

Note: In Grade 2, children say and write fractions in words. They do not need to write fractions with numbers yet. "One fourth" and "one quarter" mean the same thing.

How to help: Cut sandwiches, tortillas and pizza at home into halves, thirds, fourths and sixths. Count the pieces out loud as they're eaten.