B. Number · Number Sense: Numbers to 10 000, Fractions and Decimal Tenths

Counting to 10 by Halves, Thirds, Fourths and More

B1.6About 35 minutes

Learning goals

I can count to 10 by halves, thirds, fourths, fifths, sixths, eighths and tenths.
I can count with fraction strips, on a number line, and without tools.
I can write amounts over 1 as a fraction (like 5/4) or a mixed number (like 1 1/4).
I can count backward by fractions.

Words to know

mixed number
A whole number and a fraction together, like 2 1/3 (two and one third).

improper fraction
A fraction where the numerator is equal to or greater than the denominator, like 5/4 or 3/3. It is 1 whole or more.

count by
Add the same amount again and again: 1/4, 2/4, 3/4, ...

Remember from Grade 3

In Grade 3 you added groups of unit fractions: 1/4 + 1/4 + 1/4 = 3/4, and you learned that 2 halves make 1 whole. You also met mixed numbers like 1 1/2 when you shared things fairly. Now we'll count by fractions all the way to 10!

Let's start with a story

Zoe walks her dog every day. Each walk is 1/2 km long. How far has she walked after each walk?

1/2, 1, 1 1/2, 2, 2 1/2, 3, ...

Every 2 halves make 1 whole km. To get to 10 km, she needs 20 walks, because 10 wholes × 2 halves = 20 halves.

Counting by halves from 0 to 3. Each jump is 1/2. Two jumps make 1 whole.

Counting by fourths

Count by fourths, and say the wholes when you reach them:

1/4, 2/4, 3/4, 1, 1 1/4, 1 2/4, 1 3/4, 2, 2 1/4, ...

When you have 4 fourths, you have 1 whole. Keep going and you make another whole every 4 counts.

Counting by fourths from 0 to 2. The dot shows 5/4 (five fourths). Above the dot is its other name: 1 1/4.

Two ways to write amounts over 1

After 1 whole, we can write an amount in two ways:

as an improper fraction: 5/4 (five fourths)
as a mixed number: 1 1/4 (one and one fourth)

They are the same amount. 5 fourths = 4 fourths + 1 fourth = 1 whole and 1 fourth.

Count by	Fractions in 1 whole	Counts to reach 10
halves	2/2	20
thirds	3/3	30
fourths	4/4	40
fifths	5/5	50
sixths	6/6	60
eighths	8/8	80
tenths	10/10	100

How many counts it takes to reach 10 wholes. The denominator tells how many counts make each whole.

Worked example

Example 1: Count by thirds

Count by thirds from 0 to 3.

1/3, 2/3, then 3/3 = 1.
1 1/3, 1 2/3, then 2.
2 1/3, 2 2/3, then 3.

1/3, 2/3, 1, 1 1/3, 1 2/3, 2, 2 1/3, 2 2/3, 3. That's 9 counts, because 3 wholes × 3 thirds = 9 thirds.

Worked example

Example 2: Start in the middle

Count by fifths from 8 3/5 to 10.

8 3/5 → 8 4/5
8 4/5 → 8 5/5, which is 9
Keep going: 9 1/5, 9 2/5, 9 3/5, 9 4/5, 10

8 3/5, 8 4/5, 9, 9 1/5, 9 2/5, 9 3/5, 9 4/5, 10

Worked example

Example 3: Count backward

Count backward by tenths from 2 to 1 5/10.

Going back from 2, the first count is 1 and 9/10: 1 9/10.
Then 1 8/10, 1 7/10, 1 6/10, 1 5/10.

2, 1 9/10, 1 8/10, 1 7/10, 1 6/10, 1 5/10

Worked example

Example 4: Change between the two ways

Write 11/4 as a mixed number. Write 2 1/3 as an improper fraction.

11/4: every 4 fourths is 1 whole. 8 fourths make 2 wholes, and 3 fourths are left. So 11/4 = 2 3/4.
2 1/3: each whole is 3 thirds. 2 wholes = 6 thirds. Add 1 more third: 6 + 1 = 7/3.

11/4 = 2 3/4 and 2 1/3 = 7/3.

Count out loud

When counting by eighths or tenths, say the wholes louder: "... 6/8, 7/8, ONE, 1 1/8 ...". It helps you notice every time you make a new whole.

Just for fun

Riddle: How many half-kilometre dog walks does it take to walk 10 km?

Answer: 20 walks. By then the dog is tired, and so is the person holding the leash!

Hands-on activity

Measuring cup count

You need: a 1/2 cup and a 1/4 cup measuring cup (or one of them), a big bowl, dry rice, oats or water, and a grown-up. Do this over a sink or tray.

Fill the 1/4 cup and pour it into the bowl. Say "one fourth".
Keep going: "two fourths, three fourths, one cup, one and one fourth..." Pour and count until you reach 3 cups.
How many fourths did you pour? (12.)
Empty the bowl. Now use the 1/2 cup and count by halves to 3 cups. How many halves? (6.)

Without tools: close your eyes and count by fourths to 5 out loud. Your grown-up can check with the table in this lesson.

Math talk: "Why did you need more fourths than halves to make 3 cups?"

Your turn

Count by halves from 0 to 4.

Count by fourths from 1 to 2.

Count by sixths from 5/6 to 1 2/6.

How many thirds are in 10 wholes?

Write 7/4 as a mixed number.

Write 1 2/5 as an improper fraction.

Count backward by halves from 10 to 8.

More practice: warm-up

What comes next? 1/4, 2/4, 3/4, ...

How many halves make 1 whole?

How many tenths make 1 whole?

What comes next? 2 1/2, 3, 3 1/2, ...

Is 3/3 less than 1, equal to 1, or more than 1?

More practice: on your own

Count by eighths from 6/8 to 1 3/8.

Count by fifths from 9 2/5 to 10.

Write 9/4 as a mixed number.

How many eighths are in 3 wholes?

Count backward by thirds from 4 to 2 2/3.

Challenge

Ali counts by tenths: "... 1 8/10, 1 9/10, 1 10/10, 1 11/10." What should he say instead of 1 10/10 and 1 11/10?

Lin counts by sixths from 0 and Raj counts by thirds from 0. Who reaches 10 with fewer counts? How many counts does each need?

A snail crawls 3/4 m every hour. Count by 3/4 to find how far it crawls in 4 hours.

Check your understanding

Quiz

What comes next? 2/3, 1, 1 1/3, ...

How many fourths make 10 wholes?

Which is the same as 6/5?

Counting by tenths, what comes after 4 9/10?

Count backward by halves: 6, 5 1/2, 5, ...

Which is the same as 2 3/8?

For parents and teachers

Big idea: B1.6 asks Grade 4 students to count to 10 by halves, thirds, fourths, fifths, sixths, eighths and tenths, with and without tools. Counting by fractions shows that fractions are numbers that sit between whole numbers, and leads to mixed numbers and improper fractions.

Common difficulties:

Not noticing the whole: saying "4/4, 5/4" without realizing 4/4 = 1, or "1 10/10" instead of 2.
Counting backward across a whole (from 2 to 1 9/10). Practise on a number line.
Mixed numbers vs improper fractions: both are correct ways to write amounts over 1. Grade 4 students should see they are the same amount.

How to help: Count by fractions while measuring in the kitchen (cups and spoons), while walking ("every block is 1/4 km"), or with a clock ("every 15 minutes is 1/4 hour"). Take turns counting and ask your child to catch you if you skip a whole.