B. Number · Number Sense: Numbers to 1000 and Fractions

Same Amount, Different Names: Thirds and Sixths, Fifths and Tenths

B1.7 B1.6About 35 minutes

Learning goals

I can show that one third = two sixths and two thirds = four sixths.
I can show that one fifth = two tenths, two fifths = four tenths, and more.
I can solve fair-share problems and name each share two ways.
I can explain equivalent fractions with a drawing.

Words to know

equivalent fractions
Fractions that name the same amount of the same whole.

thirds and sixths
A whole cut into 3 or 6 equal parts. Each third is 2 sixths.

fifths and tenths
A whole cut into 5 or 10 equal parts. Each fifth is 2 tenths.

Remember

In the last lesson, you saw that cutting each piece in half makes twice as many pieces, but the same amount. That works for every fraction family!

Cut each third in half and you get sixths.
Cut each fifth in half and you get tenths.

The class party

10 kids at a class party share 2 long sub sandwiches.

If each sub is cut into fifths, there are 10 fifths. Each kid gets one fifth.
If each sub is cut into tenths, there are 20 tenths. Each kid gets two tenths.

Same subs, same kids, same amount for each kid. So one fifth = two tenths.

Two thirds and four sixths of the same strip end at the same place. They are equivalent.

Thirds	Sixths
one third	two sixths
two thirds	four sixths
three thirds (1 whole)	six sixths (1 whole)

The thirds and sixths family.

Two fifths and four tenths of the same strip end at the same place. They are equivalent.

Fifths	Tenths
one fifth	two tenths
two fifths	four tenths
three fifths	six tenths
four fifths	eight tenths
five fifths (1 whole)	ten tenths (1 whole)

The fifths and tenths family. The number of tenths is always double the number of fifths.

Worked example

Example 1: Sharing pizzas

6 kids share 4 pizzas. How much does each kid get? Name it two ways.

Way 1: Cut each pizza into thirds: 4 × 3 = 12 thirds. Share 12 thirds among 6 kids: 2 thirds each.
Way 2: Cut each pizza into sixths: 4 × 6 = 24 sixths. Share 24 sixths among 6 kids: 4 sixths each.

Each kid gets two thirds, which is the same as four sixths.

Worked example

Example 2: Fruit bars

5 friends share 4 fruit bars. How much does each friend get? What if the bars are cut into tenths?

Cut each bar into fifths. Each friend takes one fifth from each bar: four fifths.
If the bars are cut into tenths: 4 × 10 = 40 tenths. Share 40 tenths among 5 friends: 8 tenths each.

Each friend gets four fifths, which is the same as eight tenths.

Worked example

Example 3: Using tenths to find fifths

5 people share 3 pies. Kim cuts each pie into tenths. How many tenths does each person get? What is that in fifths?

3 pies × 10 = 30 tenths.
Share 30 tenths among 5 people: 6 tenths each.
Every 2 tenths make 1 fifth. 6 tenths is 3 groups of 2 tenths, so it is 3 fifths.

Each person gets six tenths, which is the same as three fifths.

Worked example

Example 4: Are they equal?

Is five sixths the same as two thirds?

Change two thirds into sixths: each third is 2 sixths, so two thirds = four sixths.
Five sixths is not four sixths. It is one sixth more.

No. Five sixths is more than two thirds.

A fun discovery

Did you notice? One half has names in these families too: one half = three sixths = five tenths. Try it with your fraction strips!

Just for fun

Two thirds and four sixths walk into a bakery.

The baker says, "Sorry, I can't tell you two apart!"

Hands-on activity

Measure, cut and compare

You need: 4 paper strips, a ruler, a pencil, crayons and scissors (ask a grown-up for help).

Make two strips 18 cm long. On one, mark every 6 cm to make thirds. On the other, mark every 3 cm to make sixths.
Make two strips 20 cm long. On one, mark every 4 cm to make fifths. On the other, mark every 2 cm to make tenths.
Colour two thirds and four sixths. Put them one above the other. Do they match?
Colour three fifths and six tenths. Do they match?
Find one more pair that matches. Write it down.

Math talk: "How does the ruler help me make equal parts?"

Your turn

One third = ___ sixths

Two thirds = ___ sixths

One fifth = ___ tenths

Three fifths = ___ tenths

Ten tenths = ___ fifths

3 kids share 2 pizzas. Each gets two thirds. How many sixths is that?

Which is equal to four fifths: four tenths, eight tenths or five tenths?

More practice: warm-up

How many sixths make one whole?

How many tenths make one whole?

If you cut each third in half, what do you get?

If you cut each fifth in half, what do you get?

Two tenths = one ___

More practice: on your own

Jo says two thirds and four sixths are not equal because 4 is more than 2. What would you tell Jo?

5 friends share 2 pizzas cut into tenths. How many tenths does each get? What is that in fifths?

6 friends share 2 trays of lasagna. Name each share two ways.

Six tenths = ___ fifths

Use a 20 cm strip to explain why one fifth = two tenths.

Challenge

10 kids share 8 pizzas. How much does each kid get? Name it two ways.

Which is more: three fifths or five tenths? Explain.

6 friends share 5 granola bars. Each gets five sixths. Lucas says that is the same as two thirds. Is he right?

Check your understanding

Quiz

One third is the same as:

Two fifths is the same as:

10 kids share 2 sub sandwiches. How much does each kid get?

Two thirds is the same as:

5 friends share 3 pizzas cut into tenths. How many tenths does each friend get?

Which pair is NOT equal?

For parents and teachers

Big idea: Equivalent fractions name the same amount. In each family (thirds and sixths, fifths and tenths) the smaller pieces are made by cutting each bigger piece in half, so the number of pieces doubles. The curriculum (B1.7) lists the families halves-fourths-eighths, thirds-sixths and fifths-tenths for Grade 3.

Common difficulties:

Thinking more pieces means more (four sixths > two thirds).
Mixing families: trying to change thirds into tenths. Stick to the families in this lesson.
Measuring strips: help your child measure carefully with a ruler so parts are equal.

How to help: Use a 6-cup egg carton (or one row of a 12-cup carton) for sixths and thirds: 2 cups are one third, or the measured strips. Ask "How else can you name that share?" whenever food is shared.