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

One Third and Two Sixths Are Equal

B1.7 B1.6About 30 minutes

Learning goals

I can show that one third and two sixths of the same whole are the same amount.
I can explain why, by cutting each third into two equal parts.
I can use this idea in a fair-sharing problem.

Words to know

equal
The same amount, even if it looks different or has a different name.

one third
1 of 3 equal parts of a whole.

two sixths
2 of 6 equal parts of a whole.

Two pizza parties

Party 1: 3 friends share 1 pizza. They cut it into 3 equal parts. Each friend gets one third.

Party 2: 6 friends share 2 pizzas that are the same size as the first one. They cut each pizza into 6 equal parts. Each friend takes one piece from each pizza, so each friend gets two sixths of a pizza.

Who gets more pizza, a friend at Party 1 or Party 2? Let's find out!

Left: one third of a pizza. Right: two sixths of the same size pizza. The coloured parts cover the same amount.

Look closely. The coloured part of each pizza is the same size!

One third and two sixths of the same whole are equal. Everyone at both parties gets the same amount.

Why are they equal?

Start with a whole cut into thirds. Now cut each third into 2 equal parts.

The whole now has 6 equal parts. Those are sixths.
Each third became 2 sixths.

So one third is the same as two sixths. We didn't add or take away any pizza. We just cut it into smaller pieces!

One third (top) and two sixths (bottom) of the same strip line up exactly. The dashed line shows they end at the same place.

Remember from Grade 1

You already know another pair like this: one half and two fourths of the same whole are equal. Cut each half into 2 equal parts and you get fourths!

Worked example

Example 1: Fold a paper strip

Show that one third and two sixths are equal using two paper strips of the same length.

Fold strip A into thirds. Colour one third.
Fold strip B into thirds, then fold it in half. Open it. It has 6 equal parts: sixths. Colour two sixths.
Put strip B under strip A. Line up the left ends.
The coloured parts end at the same place.

One third and two sixths cover the same amount, so they are equal.

Worked example

Example 2: Granola bars

6 kids share 2 granola bars fairly. How much does each kid get? Name it in two ways.

Way 1: Cut each bar into thirds. That makes 6 thirds (3 + 3), one for each kid. Each kid gets one third of a bar.
Way 2: Cut each bar into sixths. That makes 12 sixths. Deal them out to 6 kids: each kid gets 2 sixths.
Both ways give each kid the same amount of granola bar.

Each kid gets one third of a granola bar, which is the same as two sixths.

Worked example

Example 3: Is it fair?

Grandma cuts two loaves of banana bread that are the same size. Noor gets one third of the first loaf. Her cousin gets two sixths of the second loaf. Noor says, "That's not fair, he has more pieces!" Is she right?

Her cousin has 2 pieces, but they are sixths, so each piece is smaller.
Two sixths is the same amount as one third.

It is fair. They both got the same amount of banana bread. More pieces does not always mean more food!

Just for fun

A man ordered a pizza. The cook asked, "Should I cut it into 3 slices or 6 slices?"

The man said, "Better make it 3. I could never eat 6!" (Was he right? Nope. Same pizza!)

Hands-on activity

Fold, colour and compare

You need: 2 paper strips the same length (or 2 slices of bread, or 2 tortillas that are the same size), crayons, and a table knife with a grown-up.

Fold or cut the first one into 3 equal parts. Colour or take one third.
Fold or cut the second one into 3 equal parts, then cut each part in half. Count the parts: there should be 6. Take two sixths.
Put the one third on top of the two sixths. Do they match?
Bonus: Is two thirds the same as four sixths? Try it!

Math talk: "Why does cutting each third in half make sixths?"

Your turn

True or false: one third of a pizza is the same as two sixths of the same pizza.

A whole is cut into thirds. Then each third is cut into 2 equal parts. How many parts are there now? What are they called?

How many sixths are the same as one third?

Mia has one third of a sub. Jack has two sixths of the same size sub. Who has more?

6 friends share 2 pizzas fairly. Name each friend's share in two ways.

Which is the same as one half: two fourths or two sixths?

More practice: warm-up

How many equal parts are in sixths?

How many equal parts are in thirds?

Is one third the same amount as two sixths of the same whole?

Which has bigger pieces: thirds or sixths of the same cake?

If you cut one third into 2 equal pieces, what is each piece?

More practice: on your own

Ryan says two sixths is more than one third because 2 is more than 1. What would you tell him?

3 kids share 1 cake. 6 kids share 2 cakes of the same size. Compare the shares.

Draw a rectangle cut into 6 equal parts. Colour two sixths. How much of the rectangle is that in thirds?

Lucy ate two sixths of a pie. Sam ate one third of the same size pie. Who ate more?

Which is the same as one third of a sandwich?

Challenge

Is two thirds the same as four sixths of the same whole? Explain.

How many sixths make one half of the same whole? Use a strip to check.

6 friends share 4 sandwiches fairly. One friend says each gets two thirds. Another says each gets four sixths. Who is right?

Check your understanding

Quiz

One third is the same as how many sixths of the same whole?

Which is true?

You cut each third of a granola bar in half. What do you get?

6 kids share 2 pizzas fairly. Each kid gets...

Mo has two sixths of a pie. Zoe has one third of the same size pie. Who has more?

Which has bigger pieces: a pie cut into thirds or the same pie cut into sixths?

For parents and teachers

Big idea: One third and two sixths of the same whole are equal. When each third is cut into two equal parts, the whole has six parts, and each third becomes two sixths. The amount does not change, only the number of pieces.

Common difficulties:

Thinking more pieces means more food. Put one third on top of two sixths (paper or bread) so your child can see they match.
Comparing different-sized wholes. Use two strips or two slices that are exactly the same size.
Folding thirds. This is tricky. A grown-up can mark the folds with a ruler, or use a pre-cut paper strip.

Note: The Grade 2 expectation is specifically about one third and two sixths in fair-sharing situations. One half and two fourths (from Grade 1) is a helpful reminder. Two thirds and four sixths is an optional extension.

How to help: When cutting food, ask, "If I cut each of these thirds in half, how many pieces will there be? Will you get more food?"