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Number and place value

Hello! I am Bea. Numbers are built from places, and every digit has a job. Let us find out how.

Cheat sheet: Number and place value

Place value

Columns, right to left: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions.

After the point: tenths, hundredths, thousandths.

Digit 5 in 8,153,260 is worth 50,000.

×10 and ÷10

  • ×10: each digit moves 1 place left.
  • ÷10: each digit moves 1 place right.
  • 36 × 10 = 360. 3.4 × 10 = 34 (not 3.40).

Rounding

  • Look at the digit to the right of the place you round to.
  • 5 or more: round up. 4 or less: stay.
  • 7,462 → 7,460 (10) → 7,500 (100) → 7,000 (1,000)

Comparing

  • More digits means bigger (whole numbers).
  • Same length? Compare from the left.
  • Symbols: < smaller, > bigger.

Negative numbers

  • Further down or left = smaller.
  • −8 < −3 < 0 < 4
  • 4 − 7 = −3 (cross zero).

Roman numerals

I 1, V 5, X 10, L 50, C 100, D 500, M 1,000.

Small before big means take away, but only these six pairs are used: IV 4, IX 9, XL 40, XC 90, CD 400, CM 900. So 99 is XCIX, not IC.

Scales and steps

  • Step size = (difference of labels) ÷ (number of gaps).
  • Count the gaps, not the marks.
  • Counting in 25s: 150, 175, 200, 225, 250.

Words to know

digit, value, round, estimate, negative, interval, multiple, ascending (smallest first), descending (biggest first).

Mini lessons

Place value for whole numbers and decimals

Skills: NPV-01, NPV-02, NPV-03

The place of a digit tells you its value. In 8,153,260 the 5 is in the ten thousands place, so it is worth 50,000. After the decimal point the places are tenths, hundredths and thousandths. Multiplying by 10 moves every digit one place to the left.

ThHTO36036 × 10 = 360
The ones column would be empty, so a 0 holds the place. 36 × 10 = 360. The digits moved; no zero was just stuck on.
Worked example

In 8,153,260, what is the value of the 5? In 8.153, what is the value of the 3?

  1. Read the places from the right: ones 0, tens 6, hundreds 2, thousands 3, ten thousands 5, hundred thousands 1, millions 8.
  2. The 5 is in the ten thousands place: 5 × 10,000 = 50,000.
  3. In 8.153 the places after the point are tenths 1, hundredths 5, thousandths 3.
  4. The 3 is in the thousandths place: 3 ÷ 1,000 = 0.003.

Answer: 50,000 and 0.003

Comparing, ordering and rounding

Skills: NPV-04, NPV-05

To compare numbers, count the digits first, then compare from the left. To round, find the place you are rounding to, then look at the digit just to its right. If it is 5 or more, round up. If not, stay the same.

3,4003,4503,5003,5003,468 rounds to 3,500
So 3,468 rounded to the nearest 100 is 3,500.
Worked example

Put in order, smallest first: 4,305; 4,035; 4,350; 3,999. Then round 7,462 to the nearest 100.

  1. 3,999 starts with 3 thousands, so it is the smallest.
  2. The rest all have 4 thousands. Compare hundreds: 0, 3, 3. So 4,035 is next.
  3. 4,305 and 4,350 both have 3 hundreds. Compare tens: 0 and 5. So 4,305 is before 4,350.
  4. For 7,462 to the nearest 100, look at the tens digit: 6. It is 5 or more, so round up: 7,500.

Answer: 3,999; 4,035; 4,305; 4,350, and 7,500

Negative numbers and intervals across zero

Skills: NPV-06, NPV-07

Negative numbers are below zero. Picture a thermometer: the lower you go, the smaller the number. When you count or subtract and pass zero, keep going into the negatives. To find the gap between a negative and a positive number, go up to zero first, then on to the positive number, and add the two parts.

630−3−6−9−3°C(warmer)−8°C(colder)0°C

Going down the line, numbers get smaller. −8 is lower than −3.

Worked example

(a) Count on in steps of 6 from −15. Write the first five numbers. (b) How far is it from −6 to 9? (c) Find the number half way along the line from −10 to 6. (d) How many whole numbers lie on the line from −4 to 3, including both ends?

  1. (a) −15, then −15 + 6 = −9, then −9 + 6 = −3, then −3 + 6 = 3 (we crossed zero), then 3 + 6 = 9.
  2. (b) From −6 up to 0 is 6. From 0 on to 9 is 9. The gap is 6 + 9 = 15.
  3. (c) The gap from −10 to 6 is 10 + 6 = 16. Half of 16 is 8. −10 + 8 = −2.
  4. (d) List them: −4, −3, −2, −1, 0, 1, 2, 3. That is 4 negatives, zero and 3 positives: 4 + 1 + 3 = 8.

Answer: (a) −15, −9, −3, 3, 9 (b) 15 (c) −2 (d) 8

Roman numerals

Skills: NPV-08

Roman numerals use letters. Add the values from left to right. If a smaller letter comes before a bigger one, take it away instead (IV is 5 − 1 = 4). You never use the same letter more than three times in a row. Only these six pairs are used: IV, IX, XL, XC, CD, CM. For example 99 is XC + IX = XCIX (not IC).

IVXLCDM
1510501005001,000
Worked example

Write 449 in Roman numerals.

  1. Split it up: 400 + 40 + 9.
  2. 400 is CD (500 − 100).
  3. 40 is XL (50 − 10).
  4. 9 is IX (10 − 1).
  5. Join them: CD + XL + IX.

Answer: CDXLIX

Counting in steps and reading scales

Skills: NPV-09, NPV-10

To read a scale, do not guess that each mark is 1. Find two labelled numbers, work out the gap between them, and divide by the number of small gaps. Then you know what each mark is worth.

020406080100?

Between 60 and 80 there are 4 equal gaps, so each gap is (80 − 60) ÷ 4 = 5. The arrow is one small gap after 60, at 65.

Worked example

What number does the arrow show on the scale above? Then count in 25s from 150 for five numbers.

  1. The labels are 60 and 80. The difference is 80 − 60 = 20.
  2. There are 4 small gaps between them, so each gap is 20 ÷ 4 = 5.
  3. The arrow is one gap after 60: 60 + 5 = 65.
  4. Now count in 25s: 150, 150 + 25 = 175, 200, 225, 250.

Answer: 65, and 150, 175, 200, 225, 250

Worked examples across the levels

Level 1 is the easiest. Level 5 is a real challenge. Try each one yourself before you read the steps.

Level 1 (Year 3): one step

What is the value of the 6 in 563?

  1. The digits are 5 hundreds, 6 tens, 3 ones.
  2. 6 tens = 60.

Answer: 60

Level 2 (Year 4): one or two steps

Write 2,000 + 300 + 5 as one number. What is 10 more?

  1. 2,000 + 300 + 5 = 2,305 (the tens column is empty, so it is a 0).
  2. 10 more than 2,305 is 2,315.

Answer: 2,305, then 2,315

Level 3 (Year 5): two steps

Round 2,449,712 to the nearest 100,000.

  1. The hundred thousands digit is 4.
  2. Look at the next digit, the ten thousands: it is 4. That is less than 5, so stay.
  3. Replace the rest with zeros: 2,400,000 (two million four hundred thousand).

Answer: 2,400,000

Level 4 (exam standard): several steps or working backwards

At midnight it is −6°C. By 6 am it rises 4 degrees. By noon it rises 9 more. By 6 pm it falls 11. What is the temperature at 6 pm?

  1. −6 + 4 = −2.
  2. −2 + 9 = 7.
  3. 7 − 11 = −4.

Answer: −4°C

Level 5 (challenge): three to five steps

I am a 4-digit number. Rounded to the nearest 100 I am 6,500. Rounded to the nearest 10 I am 6,450. My digits add up to 17. What number am I?

  1. Nearest 100 is 6,500, so I am from 6,450 to 6,549.
  2. Nearest 10 is 6,450, so I am from 6,445 to 6,454. Both clues together: 6,450 to 6,454.
  3. Digit sums: 6,450 → 15; 6,451 → 16; 6,452 → 17; 6,453 → 18; 6,454 → 19.
  4. Only 6,452 has a digit sum of 17.

Answer: 6,452

Common mistakes

Thinking −8 is bigger than −3

Mistake code: NEG_MAGNITUDE

The mistake: Saying −8 > −3 because 8 is bigger than 3.

Why it is wrong: For negative numbers, the bigger the digit, the further below zero you are, so the smaller the number.

How to avoid it: Picture a thermometer. Which is colder? −8°C is colder, so −8 is the smaller number.

−8 is below −3, so −8 < −3.

“Just add a zero” when multiplying by 10

Mistake code: PV_ADD_A_ZERO

The mistake: Writing 3.4 × 10 = 3.40.

Why it is wrong: Adding a zero does not change 3.4 at all. Multiplying by 10 makes every digit worth ten times as much.

How to avoid it: Move each digit one place to the left. Then fill any gap with a 0 place holder.

3.4 × 10: the 3 moves to the tens and the 4 to the ones, so the answer is 34.

Reading every mark as 1

Mistake code: MEAS_SCALE_STEPS_AS_ONE

The mistake: Reading the arrow in the scale as 61 because it is one mark after 60.

Why it is wrong: The marks may be worth 2, 5, 10 or more. Counting them as 1 gives the wrong number.

How to avoid it: Work out the step: (difference of two labels) ÷ (number of gaps).

Labels 60 and 80 with 4 gaps: each gap is 5. One gap after 60 is 65, not 61.

Quick check

Five questions, easy to hard. Use paper and have a go before you open the answer.

  1. What is the value of the 8 in 58,214?

    Show answer

    8,000

    The digits are 5 ten thousands, 8 thousands, 2 hundreds, 1 ten, 4 ones. The 8 is in the thousands place.

  2. Round 6,572 to the nearest 100.

    Show answer

    6,600

    The tens digit is 7, which is 5 or more, so the hundreds digit goes up from 5 to 6.

  3. Which is the coldest: 3°C, −2°C, −7°C or 0°C?

    Show answer

    −7°C

    Negative numbers are below zero, and −7 is the furthest below zero.

  4. Write 94 in Roman numerals.

    Show answer

    XCIV

    90 is XC (100 − 10) and 4 is IV (5 − 1). Join them: XCIV.

  5. A sequence starts 19, 12, 5 and keeps going down by 7. What is the 6th number?

    Show answer

    −16

    Continue: 5 − 7 = −2 (4th), −2 − 7 = −9 (5th), then −9 − 7 = −16 (6th).

For grown-ups

  • Ask your child to say numbers aloud with their place names, such as “two hundred and five thousand”. Reading big numbers out loud builds confidence.
  • Use a real thermometer or a lift display (with a basement) to talk about negative numbers. Ask “which is colder?” before you ask “which is bigger?”.
  • When reading a scale, ask “what is each small mark worth?” before reading the arrow. It is the single most useful habit for scale questions.