Place value and number
Place value, rounding, ordering, negative numbers, multiples, factors and primes: what each question asks, the quick way to answer it, and the traps the wrong answers set.
Chapter 1
What are number questions?
Every digit sits in a column, and its column says what it’s worth. A 9 in the ones is worth 9; in the thousands, 9000.
The number. Commas split big numbers into threes from the right, to make them easier to read.
The question. What a digit is worth, how a number rounds, which is biggest, or what kind of number it is.
In 95,270, what is the 9 worth?
The options
- Option 1: 9 thousands
- Option 2: 9 hundred thousands
- Option 3: 9 ten thousands
- Option 4: 9 tens
A Maths question from Rush, level 2.
The columns
Each column is worth ten of the one to its right. After the point, the names end in “ths”: tenths, hundredths, thousandths.
Chapter 2
Solve it in 3 steps.
Here’s the question, one step at a time.
The question
In 95,270, what is the 9 worth?
The options
- Option 1: 9 thousands
- Option 2: 9 hundred thousands
- Option 3: 9 ten thousands
- Option 4: 9 tens
A Maths question from Rush, level 2.
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Step 1
Name the columns from the right.
The last digit is always the ones. Then work left: the 7 is in the tens, the 2 in the hundreds and the 5 in the thousands.
Start at the right: ones, then tens.
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Step 2
Read the digit’s column.
One more to the left, the 9 is in the ten thousands. So it’s worth 9 ten thousands: 9 × 10,000 = 90,000.
9 × 10,000 = 90,000.
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Step 3
Check by saying it.
95,270 is ninety-five thousand, two hundred and seventy. Ninety thousand is 9 ten thousands, so it fits. And with 5 digits, there are no hundred thousands.
ninety-five thousand, two hundred and seventy
- 90,000 + 5000 + 200 + 70
- The 9 is worth 90,000.
Three mistakes to spot
| The mistake | Do this instead |
|---|---|
| Going by the comma. 9 thousands puts the 9 with the thousands, as it’s before the comma. But 95 thousand is 9 ten thousands and 5 thousands. | Count the columns one at a time, from the right. |
| Starting at the tens. 9 hundred thousands comes from calling the last digit tens, which pushes every digit one column too far. | The last digit is always the ones. |
| Hearing ninety as tens. 9 tens: ninety-five thousand starts with “ninety”, which sounds like tens. But it’s ninety thousand. | Say the whole number. The column comes with it. |
So it’s 9 ten thousands. The 9 is fifth from the right, so it’s worth 90,000.
Remember: Ones first, then count left.
Chapter 3
Reading, rounding and ordering.
The rest of what Know Your Numbers asks about a number’s size. Each card is the rule, one example and the trap.
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- sixty-six thousand → 66
- and eighty-five → 085
- 66,085
Words into figures
Write the thousands, then exactly three more digits.
Sixty-six thousand and eighty-five: 66, then 085 (no hundreds, so a 0). That’s 66,085. Trap: 66,805 or 660,085, with a 0 moved or added.
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Rounding
Look one column to the right of the one you’re rounding to. 5 or more rounds up.
4927 to the nearest 1000: the hundreds digit is 9, so it rounds up to 5000. Trap: 4900, the nearest 100.
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15.61 → 16
8.55 → 8.6
Rounding decimals
Nearest whole number: the tenths decide. One decimal place: the hundredths decide.
15.61 has 6 tenths, so it rounds up to 16. 8.55 has 5 hundredths, so to 1 decimal place it’s 8.6. Trap: 15.6, which isn’t a whole number.
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6.806.836.906.15
Largest and smallest
Give every number the same number of decimal places. Then compare from the left.
Which is smallest: 6.8, 6.83, 6.9 or 6.15? As 6.80, 6.83, 6.90 and 6.15, it’s 6.15. Trap: 6.9, because 9 looks smaller than 15. Whole numbers: the first column that differs decides.
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- 2.25 × 100 = 225
- 186 ÷ 100 = 1.86
Times and divide by 10 or 100
× 10 moves every digit one column left. ÷ 10 moves them right.
× 100 moves them two columns: 2.25 × 100 = 225 and 186 ÷ 100 = 1.86. Trap: 2250 or 22.5, a column too far or too short.
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242402,40024,000
Best estimate
Round to an easy number, do the sum, then count the zeros.
304 × 8 is about 300 × 8 = 2,400. Every option starts with 24, so count the zeros. Trap: 24, which loses the zeros of 300.
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Negative numbers
Rising counts up, to 0 and past it. Falling counts down, away from 0.
−7°C rises by 13°C: 7 up to 0, then 6 more, so 6°C. Trap: −6°C, the right size on the wrong side of 0.
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C100XL40II2= 142
Roman numerals
Split it into hundreds, tens and ones. A smaller letter before a bigger one is taken away.
CXLII is C (100), XL (50 − 10 = 40) and II (2): 142. Trap: reading XL as 60. The letters: I 1, V 5, X 10, L 50, C 100, D 500, M 1000.
Chapter 4
Multiples, factors and primes.
The words, the quick tests, and a few numbers to know by heart.
Quick tests: is it a multiple?
A multiple of 6 is in the 6 times table: 6, 12, 18 and on. These tests spot one without dividing. They solve “which doesn’t belong?” too: three numbers pass one test, and one fails.
- 2, 5, 10Last digit: even for 2, 0 or 5 for 5, 0 for 10.
- 3 and 9Add the digits: 51 makes 6, so it’s in the 3s.
- 4Its last two digits are in the 4s, like 316.
- 6It’s even, and it passes the 3 test.
- 7No quick test. Know the 7 times table.
- 8Halve it three times. Still whole? It’s in the 8s.
- 11Under 100: both digits the same, like 77.
- 12It passes the 3 test and the 4 test.
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1 × 722 × 363 × 244 × 186 × 128 × 9
Factors
A factor divides in exactly. Find them in pairs, from 1 up.
Which is NOT a factor of 72? 5, because 72 ÷ 5 leaves 2 over. Trap: muddling the words. 8 is a factor of 72; 144 is a multiple of 72.
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81624
Count in 8s until 6 goes in.
The smallest common multiple
Count in the bigger number until the smaller one goes in.
6 and 8: 8, 16, 24, and 24 is in the 6s. Trap: 48 (6 × 8), a common multiple but not the smallest. In the exam it’s a story: buses every 6 and 8 minutes leave together again after 24.
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- 5 + 1 = 6, in the 3s
- 51 = 3 × 17
Prime numbers
A prime has exactly two factors: 1 and itself. Test 2, 3, 5 and 7 (and 11 above 120).
51 looks prime, but 5 + 1 = 6, so it’s in the 3s. Trap: odd isn’t enough. 49 (7 × 7) and 91 (7 × 13) need the 7s. 1 isn’t prime; 2 is.
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- 4² = 4 × 4 = 16
- 3³ = 3 × 3 × 3 = 27
- 4² + 3³ = 43
Squares, cubes and roots
The small 2 means two of it multiplied. The small 3 means three.
4² + 3³ is 16 + 27 = 43. A square root undoes a square: 11 × 11 = 121, so the square root of 121 is 11. Trap: 4² = 8, from 4 × 2.
Learn these by heart
They answer “which is prime?” and “which is NOT a square?” on sight. No square ends in 2, 3, 7 or 8.
- Primes under 50
- 23571113171923293137414347
- Squares
- 149162536496481100121144
- Cubes
- 182764125
Chapter 5: In the exam
Biggest and smallest from digits.
Rush doesn’t ask this, but the exam does. It needs columns, ordering and taking away.
The questionUse each of the digits 3, 0, 7 and 5 once. What is the largest number you can make, take away the smallest?
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Step 1
Make the largest.
Biggest digit in the biggest column, then the next biggest, and so on: 7530.
7 5 3 0Biggest digit first.
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Step 2
Make the smallest.
Smallest digit first, but a number can’t start with 0. So 3 goes first, then the 0: 3057.
3 0 5 7Not 0357: that’s just 357.
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Step 3
Take away.
Count up from 3057: 3 to 3060, 940 to 4000, 3530 to 7530. That’s 3 + 940 + 3530 = 4473. Or use column subtraction, as at school: 7530 − 3057 = 4473.
Two mistakes to spot
| The mistake | Do this instead |
|---|---|
| Putting the 0 first. 0357 is just 357, so you’d get 7530 − 357 = 7173. | A 4-digit number can’t start with 0. Put it second. |
| Stopping at 7530. That’s the largest number, but the question wants the difference. | Read the last line again: largest take away smallest. |
So it’s 4473. Check: 4473 + 3057 = 7530.
Remember: Biggest first. Never 0 first.
Two more exam questions
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The gap across 0
It’s −6°C at night and 9°C by day. How many degrees warmer? 6 up to 0, then 9 more: 15. Trap: 3, from 9 − 6.
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Prime factors
Write 84 as primes multiplied together. Split it into two factors, and keep splitting until every end is prime: 84 = 2 × 2 × 3 × 7. Trap: stopping at 4 or 21.
Chapter 6
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Name the columns from the right.
- Read the digit’s column.
- Check by saying it.
Remember: Ones first, then count left.
For parents.
More on place value and number, and what parents ask us.
More on place value and number
Number lines and thermometers
Shape Facts draws a number line or a thermometer with an arrow and asks what it shows. Before reading, work out what one small step is worth: the gap between two labels divided by the number of steps between them. From 5 to 6 in five steps, each step is 0.2, so an arrow one step before 7 shows 6.8. About four in ten of these questions offer the nearest label (7 here) for a child who rounds to it instead of counting steps.
Questions parents ask
What prime numbers should you know for the 11+?
Know the primes under 20 by heart: 2, 3, 5, 7, 11, 13, 17 and 19. The national curriculum expects these in Year 5, along with deciding whether a number up to 100 is prime. To decide, try 2, 3, 5 and 7: if none of them divides the number, it is prime. There are 25 primes under 100.
How do you round to the nearest 10, 100 or 1,000?
Find the column you are rounding to and look at the digit one place to its right. 5 or more rounds up, 4 or less rounds down, and every column after it becomes 0. So 2524 to the nearest 100 looks at the tens digit, 2, and rounds down to 2500.
Do you need to know Roman numerals for the 11+?
Reading them is on the Year 5 national curriculum, up to M (1,000) and including years, so a paper can ask about them. A smaller letter before a bigger one is taken away: IX is 9 and XC is 90.
What is the quickest way to check for multiples of 3, 4, 6 and 9?
The digit sum handles 3 and 9, and a multiple of 6 is an even number that passes the 3 test. For 4, look only at the last two digits: 1,396 ends in 96, which is 4 × 24, so 1,396 is a multiple of 4.
Rush: 11+ Practice & KS2 SATsPractise place value and number in Rush.
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