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By the end of Key Stage 2 you should know all your times tables up to 12 x 12 and be able to multiply and divide large numbers using written methods. You will also learn about prime numbers, square and cube numbers, and the order of operations.
You need to know all multiplication facts up to 12 x 12 instantly. Here are some patterns that help:
All times table answers up to 12 x 12 should come to you without needing to count.
Short multiplication is used to multiply a large number by a single digit. Work from right to left, carrying as needed.
Example: 3,456 x 7
3 4 5 6
x 7
-------
Long multiplication multiplies a number by a 2-digit (or larger) number.
Example: 324 x 26
Break 26 into 20 + 6:
Grid method (useful to see the steps):
| x | 300 | 20 | 4 |
|---|---|---|---|
| 20 | 6,000 | 400 | 80 |
| 6 | 1,800 | 120 | 24 |
Add all cells: 6,000 + 400 + 80 + 1,800 + 120 + 24 = 8,424 ✓
Short division (the "bus stop" method) divides step by step from left to right.
Example: 952 / 4
2 3 8
4 | 9 5 2
Long division handles larger divisors.
Example: 7,248 / 12
How many times does 12 go into each part?
Remainders: When a division does not work out exactly, you will have a remainder.
Factor: a number that divides exactly into another number.
Multiple: the result of multiplying a number by a whole number.
Prime number: a number with exactly two factors — 1 and itself.
Composite number: a number with more than two factors.
Square numbers are made by multiplying a number by itself.
1x1=1, 2x2=4, 3x3=9, 4x4=16, 5x5=25,
6x6=36, 7x7=49, 8x8=64, 9x9=81, 10x10=100,
11x11=121, 12x12=144
We write "n squared" as n^2. For example, 7^2 = 49.
Cube numbers are made by multiplying a number by itself twice.
1x1x1=1, 2x2x2=8, 3x3x3=27, 4x4x4=64, 5x5x5=125
We write "n cubed" as n^3. For example, 4^3 = 64.
When a calculation has several operations, you must do them in the right order.
BODMAS / BIDMAS priority order. Work top to bottom; operations on the same row are equal priority and resolved left to right:
flowchart TD
A["1. Brackets<br/>e.g. 3+4"] --> B["2. Orders / Indices<br/>squares, cubes, roots"]
B --> C["3. Division and Multiplication<br/>left to right"]
C --> D["4. Addition and Subtraction<br/>left to right"]
D --> E[Final answer]
BODMAS tells you the order:
Division and multiplication have equal priority — do them left to right. Addition and subtraction have equal priority — do them left to right.
Example: 3 + 4 x 2
Example: (3 + 4) x 2
Example: 20 - 4 x 3 + 1
A school orders 24 boxes of pencils. Each box contains 48 pencils. The pencils are then shared equally between 16 classes. How many pencils does each class get? Are there any pencils left over?
Step 1 — Total number of pencils (long multiplication). Multiply 24 x 48 using the formal column method.
48
x 24
----
192 (48 x 4 = 192)
960 (48 x 20 = 960; written with a placeholder zero)
----
1152
So 24 x 48 = 1,152 pencils.
Check using the grid method:
| x | 40 | 8 |
|---|---|---|
| 20 | 800 | 160 |
| 4 | 160 | 32 |
Total: 800 + 160 + 160 + 32 = 1,152 ✓
Step 2 — Share between 16 classes (long division). Calculate 1,152 / 16 using the formal compact method.
How many 16s in 11? Zero. How many 16s in 115? 7 (because 7 x 16 = 112). Remainder 3. Bring down the 2 to make 32. How many 16s in 32? Exactly 2 (because 2 x 16 = 32). Remainder 0.
Answer: **72 pencils per class**, with **0 remainder**.
Step 3 — Verify with multiplication. 72 x 16 should equal 1,152.
72
x 16
----
432 (72 x 6 = 432)
720 (72 x 10 = 720)
----
1152 ✓
Step 4 — A new scenario with a remainder. Suppose only 17 classes share the pencils instead of 16. Calculate 1,152 / 17.
How many 17s in 115? 6 (6 x 17 = 102). Remainder 13. Bring down the 2 to make 132. How many 17s in 132? 7 (7 x 17 = 119). Remainder 13.
Answer: **67 pencils per class, remainder 13.**
Each class gets 67 pencils and 13 pencils are left over for the school office.
Step 5 — Express the remainder as a fraction or decimal.
13 / 17 ≈ 0.76 (rounded to 2 d.p.)
So 1,152 / 17 ≈ 67.76 pencils per class — but in real life you cannot split a pencil, so the integer-with-remainder form is more useful.
Step 6 — Apply the order of operations. A teacher writes the calculation: 1,152 / 16 + 8 x 3 - 5.
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