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Advanced place value and decimal columns
Tenths, hundredths and thousandths, and empty placeholder columns.
In Lesson 1.1, every column was for whole numbers. But some numbers are smaller than 1, like half a pizza, or the 0.5 in 3.5 metres.
To write them, we need the decimal point. In this lesson you will learn:
- what the decimal point does
- what tenths, hundredths and thousandths are
- when a zero matters, and when it doesn’t
- how to compare decimals and put them in order
- what happens to decimals when you multiply or divide by 10, 100 and 1000
1. The decimal point
Each column is still 10 times smaller than the one on its left. So after the ones come three new columns:
| Column | Short name | A 1 here is worth | As a fraction |
|---|---|---|---|
| Tenths | t | 0.1 | 110 (1 of 10 equal parts) |
| Hundredths | h | 0.01 | 1100 (1 of 100 equal parts) |
| Thousandths | th | 0.001 | 11000 (1 of 1000 equal parts) |
Here is 3.762 in a place value chart. The 6 is in the hundredths column, so it is worth 6 hundredths, which is 0.06.
Build a decimal: press ▲ and ▼ to change the digits
3.762
Three point seven six two
- Tap any digit to see what it is worth.
- Make 0.5. Then make 0.05. Which one is bigger?
- Make the smallest number the chart can show that is more than zero.
2. Tenths and hundredths
- 10 tenths make 1 whole.
- 10 hundredths make 1 tenth.
- 100 hundredths make 1 whole.
- 10 thousandths make 1 hundredth.
The hundred square: slide or tap to shade it
The whole square is 1. Each column is 1 tenth. Each little square is 1 hundredth.
0.37
0.37 is 3 tenths and 7 hundredths. That is the same as 37 hundredths.
- Tap a little square to shade up to it.
- Shade exactly one column. What decimal is that?
- Shade the whole square. What happens to the number?
3. Reading and writing decimals
Example: read 3.25
3.25
You can also say what the digits are worth: 3.25 is 3 ones, 2 tenths and 5 hundredths. Some questions say it as “3 and 25 hundredths”.
How to write a decimal from words
- Write the whole number, then the point
- Find the column the words name“seven hundredths” means the 7 goes in the hundredths column.
- Fill any empty column before it with a 0
Example: write four and seven hundredths
The tenths column is empty, so it gets a 0. Without it, you would have 4.7, which is four and seven tenths.
The number display: drag the digits and the decimal point
- Drag the red decimal point to a different gap. Or drag a digit key into the number.
- Drag anything off the display to take it out.
- No dragging? Tap the point or a digit, then tap a gap. Del takes out the one you picked.
- Drag the point one place to the left. Is the number bigger or smaller now?
- Put a 0 between the point and the 7. How do the words change?
- Can you make the display say four point zero five?
4. Zeros: when do they matter?
| Numbers | Does the zero matter? |
|---|---|
| 0.5 and 0.05 | Yes. In 0.05 the zero holds the tenths column, so the 5 is only worth 5 hundredths. |
| 3.4 and 3.40 | No. They are the same number. The zero on the end adds 0 hundredths. |
| 7.2 and 07.2 | No. A zero at the very front of a number adds nothing. |
| 6 and 6.0 | No. 6.0 is 6 ones and 0 tenths: still 6. |
One more rule: if there are no ones, we still write a 0 before the point. We write 0.6, not .6, so nobody misses the point.
Tap a zero to take it away. Does the number change?
5. Comparing and ordering decimals
How to compare two decimals
- Line up the decimal points
- Fill any empty columns on the end with 0Now both numbers have the same number of digits after the point.
- Compare from the left. The first column that is different decides it.
Example: which is bigger, 0.5 or 0.45?
- Line them up and fill the gap: 0.50 and 0.45.
- Ones: 0 and 0. The same.
- Tenths: 5 and 4. Different! 5 is bigger.
Which is bigger? Pick < or >
Put them in order: drag the cards, smallest at the top
- Drag a card up or down. Or use its ▲ and ▼ buttons.
6. Multiplying and dividing decimals by 10, 100 and 1000
Count the zeros to know how far: × or ÷ 10 is 1 place, 100 is 2 places, 1000 is 3 places.
Example 1: 3.45 × 10
- Every digit moves 1 place leftThe 3 moves from ones to tens. The 4 moves from tenths to ones. The 5 moves from hundredths to tenths.
Example 2: 6.2 ÷ 100
- Every digit moves 2 places rightThe 6 moves from ones to hundredths. The 2 moves from tenths to thousandths.
- The empty ones and tenths columns get a 0
The place value machine: watch the digits slide past the point
3.4
- Press ÷ 10 again and again. How small can you go?
- Which button turns 3.4 into 340?
- Press × 100, then ÷ 100. Where do you end up?
7. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Reading the digits after the point as one number. 3.25 is not “three point twenty-five”. say each digit on its own: three point two five.
- Thinking a longer decimal is bigger. 0.45 is not bigger than 0.5. line up the points, fill the gaps with 0, then compare from the left.
- Leaving out a placeholder zero. Four and seven hundredths is not 4.7. put a 0 in the empty tenths column: 4.07.
- Mixing up tens and tenths, or hundreds and hundredths. remember the “th” columns are after the point.
- “Adding a zero” to multiply a decimal by 10. 2.5 × 10 is not 2.50. move every digit one place left: 25.
- Moving the digits the wrong way, or the wrong number of places. multiply moves left, divide moves right, and count the zeros.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 1.2.
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