What you'll learn
Place value is the idea that a digit's worth depends on where it sits in a number: in 435 218 the 4 is worth 400 000, not 4. This topic covers reading, writing, comparing, ordering and rounding whole numbers up to one million, and it is the foundation the rest of the SEA Mathematics paper is built on. Number is the largest section of the paper, worth 19 of the 40 items, and place value questions appear both on their own and hidden inside money, measurement and problem-solving questions. A child who is secure here makes far fewer careless errors everywhere else.
Key terms and definitions
Digit — one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A number is built from digits, in the same way a word is built from letters.
Place value — the value a digit has because of its position. In 7 400, the 7 means 7 thousands and the 4 means 4 hundreds.
Value of a digit — what that digit is actually worth. In 264 583 the 6 has a value of 60 000, because it sits in the ten thousands place.
Standard form (ordinary notation) — the number written the usual way, such as 405 060.
Expanded notation — the number written as the sum of the value of each digit: 405 060 = 400 000 + 5 000 + 60.
Word form — the number written in words: four hundred and five thousand and sixty.
Ascending order — arranged from smallest to largest.
Descending order — arranged from largest to smallest.
Rounding — replacing a number with a nearby, easier number to a stated place, such as the nearest ten, hundred or thousand.
Estimate — an answer that is close enough to be useful, usually found by rounding first.
Core concepts
The place value columns up to one million
Reading from the right, the columns run: ones (units), tens, hundreds, thousands, ten thousands, hundred thousands, and then millions. Each column is worth ten times the one to its right. This is why 1 000 000 has six zeros after the 1 — you have multiplied by ten six times.
Writing the columns above the digits is the single most useful habit a child can build:
| Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|
| 4 | 3 | 5 | 2 | 1 | 8 |
So 435 218 is 4 hundred thousands, 3 ten thousands, 5 thousands, 2 hundreds, 1 ten and 8 ones.
The job of zero
Zero is a place holder. It does not add anything to the number, but it keeps the other digits in the right columns. In 405 060 the zeros tell you there are no ten thousands, no hundreds and no ones. Take a zero out and every digit to its left shifts down a column, which changes the number completely. Children who write "four hundred and five thousand and sixty" as 4005060 have heard the words correctly but lost track of the columns.
Reading and writing large numbers
Trinidad and Tobago follows the convention of grouping digits in threes from the right, with a space: 435 218, not 435218. Reading the number aloud in those groups makes it manageable — "four hundred and thirty-five thousand, two hundred and eighteen."
When writing from words back to digits, work from the largest group down and fill every empty column with a zero before moving on. Say the columns out loud as you write them.
Comparing and ordering
To compare two whole numbers:
- Count the digits first. More digits means a bigger number, so 100 002 beats 98 990 without any further work.
- If the digit counts are equal, compare from the left. The first column where they differ decides it.
So for 70 400 and 70 040: both have five digits; the ten thousands match (7 and 7); the thousands match (0 and 0); the hundreds are 4 and 0, so 70 400 is the larger.
The symbols are > (is greater than) and < (is less than). The wide end always faces the bigger number.
Rounding
To round to a given place:
- Find the digit in that place.
- Look at the digit immediately to its right — and only that digit.
- If it is 5 or more, round the place digit up by one. If it is 4 or less, leave the place digit alone.
- Replace every digit after the place with zero.
Rounding 46 782 to the nearest thousand: the thousands digit is 6; the digit to its right is 7; 7 is 5 or more, so the 6 becomes 7 and everything after becomes zero, giving 47 000.
The rule looks only one place to the right. Children often try to round 46 449 to the nearest hundred by noticing the 9 at the end and rounding upwards in stages; that gives 46 500 instead of the correct 46 400.
Estimating
Rounding is most useful as a way of checking. Before working out 612 × 39, round to 600 × 40 = 24 000. If the final answer is nowhere near 24 000, something has gone wrong. Estimating first turns a wrong answer into a caught mistake.
Worked examples
Example 1: Value of a digit
What is the value of the digit 6 in 264 583?
Write the columns out: 2 is hundred thousands, 6 is ten thousands, 4 is thousands, 5 is hundreds, 8 is tens, 3 is ones.
The 6 sits in the ten thousands column, so its value is 6 × 10 000 = 60 000.
The common wrong answer is simply "6", which names the digit rather than its value. The question asks what the digit is worth, not what it looks like.
Example 2: Ordering five-digit and six-digit numbers
Arrange in ascending order: 98 990, 110 002, 109 870, 100 002.
First count digits. 98 990 has five digits and the rest have six, so it is the smallest.
Now compare the three six-digit numbers from the left. All start with 1. In the ten thousands column they read 1, 0 and 0, so 110 002 is the largest of the three. Between 109 870 and 100 002, the thousands column gives 9 against 0, so 100 002 is smaller.
Ascending order: 98 990, 100 002, 109 870, 110 002.
Example 3: Rounding in a real context
A stadium recorded 46 782 spectators over a weekend. Round this to the nearest thousand.
The thousands digit is 6. The digit to its right is 7. Since 7 is 5 or more, the 6 rounds up to 7 and the remaining digits become zeros.
The answer is 47 000 spectators.
Notice that the rounded figure is larger than the true one. Rounding is not the same as "about" in a vague sense — it follows a fixed rule, and the rule sometimes rounds up even when the number is only just past halfway.
Common mistakes and how to avoid them
Naming the digit instead of its value. Asked for the value of the 4 in 435 218, a child writes 4. Fix: always finish the sentence "the 4 is worth …" out loud.
Losing zeros when writing from words. "Four hundred and five thousand and sixty" becomes 4005060. Fix: write the six empty columns first, then fill them in from the left.
Comparing by the first digit alone. A child decides 98 990 is larger than 100 002 because 9 is bigger than 1. Fix: count the digits before looking at any of them.
Rounding in stages. Rounding 46 449 to the nearest hundred by first rounding 49 up to 50. Fix: the rule looks at exactly one digit — the one immediately to the right of the place being rounded.
Forgetting to replace with zeros. Rounding 46 782 to the nearest thousand and writing 47 782. Fix: everything after the rounded place becomes zero, without exception.
Reading spaces as decimal points. Treating 46 782 as a decimal number. Fix: in whole numbers the space is only there to group digits in threes and makes no difference to the value.
How parents can help at home
You do not need to remember any of this yourself to be useful. Three things help more than anything else.
Ask for the value, not the digit. Point at a digit in a number on a receipt or a bill and ask, "what is that one worth?" The answer should be a number like 60 000, not 6. This single question catches the most common error in the topic.
Use real numbers around the house. Populations, distances on road signs, prices of large items and cricket attendances are all six-digit numbers. Ask your child to read one aloud, then write it down, then round it to the nearest thousand.
When they are stuck, ask them to write the columns. Almost every error in this topic comes from losing track of which column a digit is in. Saying "draw the columns" is more helpful than giving the answer, and it is the same prompt a teacher would use.
Do not worry if your child is slow at first. Speed comes from confidence with the columns, not from being rushed.
Exam technique for place value, ordering and rounding
Read the question twice and underline the place being asked for. "Round to the nearest thousand" and "round to the nearest hundred" look similar under time pressure and produce completely different answers.
Where a question asks you to order numbers, check whether it says ascending or descending before writing anything down. Marks are regularly lost by giving a correctly ordered list in the wrong direction.
In multiple-choice items, the wrong options are usually built from the common errors — one will be the digit rather than its value, one will be rounded the wrong way, one will have a zero missing. If your answer matches one of those patterns exactly, check it again before moving on.
Estimate before you calculate on any question involving large numbers. It costs a few seconds and it catches answers that are ten times too big or too small.
Quick revision summary
- A digit's value depends on its column: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions.
- Each column is worth ten times the column to its right.
- Zero holds a place open; removing it changes the whole number.
- Group digits in threes from the right when writing: 435 218.
- To compare, count the digits first, then work from the left.
- The symbol > means greater than and < means less than; the wide end faces the larger number.
- To round, look at one digit only — the one immediately right of the place named.
- 5 or more rounds up; 4 or less stays the same; everything after becomes zero.
- Estimate by rounding before calculating, then check your answer against the estimate.