Place value, rounding, negative numbers and Roman numerals

Place value to ten million

Year 6 pupils read, write, order and compare numbers up to 10,000,000 and know the value of every digit. In 4,503,217, the 5 is worth 500,000 and the 3 is worth 3,000. The commas group the digits in threes (millions, thousands, ones), and reading a large number one group at a time is the key skill: four million, five hundred and three thousand, two hundred and seventeen.

• 4,503,217The 5 is in the hundred-thousands place: 500,000.
• four million and five thousand = 4,005,000Zeros hold the empty places.
• four million and five thousand = 4,5000Lost zeros: the most common place value error.

When ordering, compare the number of digits first. If they match, work from the left until the digits differ.

Rounding

Rounding replaces a number with a nearby simpler one. To round to a given place, look at the digit one place to the right: 5 or more rounds up, 4 or less rounds down. Everything after the rounding place becomes zero.

• 3,467 to the nearest 100 → 3,500Look at the tens digit: 6, so round up.
• 72,490 to the nearest 1,000 → 72,000Look at the hundreds digit: 4, so round down.
• 6.35 to 1 decimal place → 6.4The hundredths digit is 5, so round up.

Number lines make rounding visible: mark the two possible answers, find the midpoint, and see which side the number falls. The same skill is needed to read scales with unlabelled divisions, where children work out what each mark is worth before placing a number.

Negative numbers

Negative numbers appear in temperatures, bank balances and depths. In Year 6, children calculate intervals across zero, such as the difference between −4°C and 7°C. The reliable method is to count in two jumps: from −4 to 0 is 4, from 0 to 7 is 7, so the difference is 11.

• −4 to 0 is 4, 0 to 7 is 7: difference = 11Two jumps across zero.
• 7 − 4 = 3Ignoring the sign: the most common error.

A vertical number line, like a thermometer, works better than a horizontal one for most children, because “colder” and “warmer” make the direction obvious. Another frequent misconception is that −8 is greater than −3; the thermometer fixes this too.

Roman numerals

Year 6 pupils read Roman numerals to 1,000 (M) and recognise years written in them. The symbols are I (1), V (5), X (10), L (50), C (100), D (500) and M (1,000). Symbols are written largest first and added, except that a smaller symbol before a larger one is subtracted. Only six subtraction pairs are allowed: IV, IX, XL, XC, CD and CM.

• MCMXCV = 1000 + 900 + 90 + 5 = 1995Break it into thousands, hundreds, tens and ones.
• IC for 99Not a valid pair. 99 is XCIX: 90 + 9.

In the SATs

Place value questions appear on both reasoning papers: writing numbers in words or digits, ordering, giving the value of a digit, rounding to a given place, and finding differences across zero. They also underpin the arithmetic paper, where place value errors are behind many lost marks in column methods and decimal calculations. Roman numerals typically appear as a short reading or writing question.

Common mistakes

• Losing or adding zeros when writing large numbers from words.
• Rounding by looking at the wrong digit, or changing digits to the left of the rounding place.
• Calculating intervals across zero by subtracting the numbers as written.
• Thinking a negative number with bigger digits is larger.
• Writing four of the same Roman numeral in a row (IIII) or inventing subtraction pairs (IC, VX).

Teaching ideas

Place value grids

Use a grid with columns from millions to ones. Children place digit cards, read the number aloud, then make the largest and smallest numbers possible from the same digits.

Human number line

Give pupils cards with numbers to stand in order across the room, including negatives. Then ask two children to find the distance between them by counting steps through zero.

Rounding with context

Use real data: town populations, football attendances, distances to planets. Round each to different places and discuss when each level of accuracy makes sense.

Roman years

Show film credits, clock faces and building inscriptions with Roman numerals. Pupils decode the years and write their own birth year.

Try it: five quick questions

SATs-style questions to use as a starter or a quick check. Tap to see each answer.

1. What is the value of the 7 in 2,374,051?

   Show answer

   70,000

2. Round 48,652 to the nearest 1,000.

   Show answer

   49,000

3. The temperature is −6°C and rises by 9 degrees. What is the new temperature?

   Show answer

   3°C

4. Write 1,999 in Roman numerals.

   Show answer

   MCMXCIX

5. Order from smallest: −3, −11, 2, −1

   Show answer

   −11, −3, −1, 2