Whole Numbers
FreeGrade 7
1. Place Value and Ordering
In Grade 7 we work with whole numbers including large values into the billions. We use place value to read, write, and compare numbers. To order numbers we compare digit by digit from the left, starting with the highest place value. We also revise rounding to various place values.
Place value positions
| Billions | Hundred millions | Ten millions | Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Units |
|---|---|---|---|---|---|---|---|---|---|
| 1 000 000 000 | 100 000 000 | 10 000 000 | 1 000 000 | 100 000 | 10 000 | 1 000 | 100 | 10 | 1 |
Ordering and rounding rules
Ordering
First compare the number of digits — more digits means a larger number. If the digit count is equal, compare the digits one by one from the left until they differ.
Rounding
Identify the place you are rounding to. Look at the digit one place to the right. If it is 5 or more, round up. If it is less than 5, round down (keep the digit the same). Replace all digits to the right with zeros.
Reading large numbers
Group digits in threes from the right to read large numbers: 4 523 100 is read as "four million, five hundred and twenty-three thousand, one hundred." Using spaces as separators (not commas) is the South African convention.
Worked Examples
- 1Compare digit count: 45 200 000 has 8 digits. Both 4 523 100 and 4 519 800 have 7 digits. More digits means a larger number — so 45 200 000 is the largest.
- 2Compare the two 7-digit numbers digit by digit: Both start with 4 — same. Next digit: 5 vs 5 — same. Next: 2 vs 1 — here they differ. Since 1 < 2, we know 4 519 800 < 4 523 100.
- 3Order (smallest to biggest): 4 519 800, 4 523 100, 45 200 000
- 1Identify the hundred thousands digit in 38 462 719: the digit is 4 (in the hundred thousands position).
- 2Look one place to the right — the ten thousands digit is 6. Since 6 ≥ 5, we round up.
- 3Round the 4 up to 5 and replace all digits to the right with zeros. Answer: 38 500 000
Diagram
Place value chart showing 38 462 719 with the hundred thousands digit highlighted blue, the ten thousands digit highlighted orange, and the rounding decision highlighted green
2. Operations with Large Numbers
We apply addition, subtraction, multiplication, and division to large whole numbers using the same methods learned in earlier grades, now with bigger values and more complex multi-step problems. Estimating first helps verify whether the final answer is reasonable.
Operation strategies
Addition & Subtraction
Align digits in columns. Work right to left. Carry (addition) or borrow (subtraction) as needed.
Multiplication
Use long multiplication or break into parts. Estimate first by rounding to check your answer is reasonable.
Division
Use long division. Check your answer by multiplying the quotient by the divisor.
Estimation
Round each number to its highest place value before calculating. Compare the estimate to your answer to spot errors.
Multi-step problems: work in the right order
When a problem has more than one operation, underline the intermediate result after each step. This makes it easy to spot errors and follow your working. Write the final answer clearly at the end.
Worked Examples
- 1Identify the operation: a year has 12 months, so we multiply: 24 580 × 12.
- 2Estimate first: 25 000 × 12 = 300 000. Our answer should be close to 300 000.
- 3Calculate: 24 580 × 12
= 24 580 × 10 + 24 580 × 2
= 245 800 + 49 160
= 294 960 - 4Check against estimate: 294 960 ≈ 300 000 ✓ — the answer is reasonable. Answer: 294 960 units.
- 1This is a two-step problem. First subtract the amount spent, then add the amount earned.
- 2Step 1 — subtract: 450 000 − 128 750 = 321 250
- 3Step 2 — add: 321 250 + 67 300 = 388 550
- 4Answer: Lerato has R388 550.
Diagram
Column layout showing 24 580 multiplied by 12 with intermediate steps highlighted orange and the final answer highlighted green
3. Exponents — Squares and Cubes
An exponent shows how many times a number is multiplied by itself. A squared number is multiplied by itself twice, written with a small 2 — for example 5² = 5 × 5 = 25. A cubed number is multiplied by itself three times, written with a small 3 — for example 3³ = 3 × 3 × 3 = 27. The square root of a number is the value that, when squared, gives that number. The cube root works the same way for cubes.
Key terms
Base number
The number being multiplied by itself. Written in normal size.
Exponent (index)
The small raised number showing how many times the base is multiplied by itself.
Square root (√)
The value that, when squared, gives the original number. √64 = 8 because 8² = 64.
Cube root (∛)
The value that, when cubed, gives the original number. ∛27 = 3 because 3³ = 27.
Perfect squares to know
Perfect cubes to know
Roots and powers are inverses
Squaring and taking the square root undo each other — just like multiplication and division. If you know your perfect squares and cubes by heart, you can read off square and cube roots instantly.
Worked Examples
- 1Calculate 8²: The exponent is 2, so we multiply the base by itself twice: 8 × 8 = 64
- 2Find √64: We need the value that, when squared, gives 64. Ask: "__ × __ = 64?" Since 8 × 8 = 64, the square root of 64 is 8.
- 3Summary: 8² = 64 and √64 = 8 ✓
- 1Calculate 4³: The exponent is 3, so we multiply the base by itself three times: 4 × 4 × 4 = 16 × 4 = 64
- 2Find ∛27: We need the value that, when cubed, gives 27. Ask: "__ × __ × __ = 27?" Since 3 × 3 × 3 = 27, the cube root of 27 is 3.
- 3Summary: 4³ = 64 and ∛27 = 3 ✓
Diagram
Visual diagram showing 8 squared as a square with 8 rows and 8 columns equalling 64, and 3 cubed as a cube with 3 layers each 3 by 3 equalling 27, with base numbers blue, exponents orange, and answers green
4. Properties of Operations and Order of Operations
The four operations follow rules that let you rearrange a calculation into an easier one. Knowing these properties is what turns slow arithmetic into fast mental arithmetic — and they are the same rules that make algebra work later.
The commutative property — order
a + b = b + a and a × b = b × a
Only + and × are commutative. 12 − 5 is not 5 − 12, and 12 ÷ 4 is not 4 ÷ 12 — so you may never reorder a subtraction or a division.
The associative property — grouping
(a + b) + c = a + (b + c)
Regroup to make friendly numbers: 47 + 25 + 75 is much easier as 47 + (25 + 75) = 47 + 100 = 147.
The distributive property — splitting
a(b + c) = ab + ac
This is how you multiply mentally: 6 × 47 = 6 × (40 + 7) = 240 + 42 = 282. It is also the rule behind every long multiplication you have ever done.
Identity and inverse
Adding 0 leaves a number unchanged, and multiplying by 1 leaves it unchanged — these are the identity elements. Every number also has an inverse: adding 7 is undone by subtracting 7, and multiplying by 5 is undone by dividing by 5. That is precisely how equations get solved.
Order of operations
When several operations appear in one expression the order is fixed: brackets, then exponents and roots, then × and ÷ left to right, then + and − left to right. So 5 + 3 × 4 is 5 + 12 = 17, not 8 × 4 = 32.
Common mistake: assuming × must always come before ÷. They have equal priority and are done left to right: 24 ÷ 6 × 2 is 4 × 2 = 8, not 24 ÷ 12 = 2.
Worked Examples
- 1Regroup so the friendly pair is together: 47 + (25 + 75)
- 225 + 75 = 100
- 347 + 100 = 147 ✓
- 1Split the 47: 6 × (40 + 7)
- 26 × 40 = 240
- 36 × 7 = 42
- 4240 + 42 = 282 ✓
- 1× and ÷ first, left to right: 3 × 4 = 12 and 6 ÷ 2 = 3
- 2The expression becomes 5 + 12 − 3
- 3Now + and − left to right: 5 + 12 = 17, then 17 − 3 = 14
- 4Answer: 14 ✓
- 1÷ and × have equal priority, so work strictly left to right
- 224 ÷ 6 = 4
- 34 × 2 = 8
- 4Doing the × first would give 24 ÷ 12 = 2, which breaks the left-to-right rule ✓
- 1Left: 25 ÷ 5 = 5, then 100 ÷ 5 = 20
- 2Right: 100 ÷ 25 = 4, then 4 ÷ 5 = 0,8
- 3Division is not associative, so the brackets change the answer
- 4Answer: no — grouping matters for division ✓
5. Rounding and Estimation
Rounding replaces a number with a nearby, simpler one. Estimation uses rounded numbers to work out roughly what an answer should be — which is how you catch a calculation that has gone badly wrong before you hand it in.
The rounding rule
Look at the digit after the place you are rounding to.
If it is 5 or more, round up. If it is 4 or less, leave the digit as it is. Everything after that place becomes zero.
Rounding to different places
| Number | to 10 | to 100 | to 1 000 |
|---|---|---|---|
| 4 762 | 4 760 | 4 800 | 5 000 |
| 2 349 | 2 350 | 2 300 | 2 000 |
Notice 2 349 rounds up to 2 350 but down to 2 300 — the place you are rounding to changes which digit you look at.
Round once, from the original
To round 4 649 to the nearest thousand, look at the 6 in the hundreds place: it is 5 or more, so the answer is 5 000. Do not round to 4 650, then 4 700, then 5 000 in stages — that chain can push a number the wrong way.
Estimating a calculation
Round every number to one easy figure, then calculate. 387 × 21 is roughly 400 × 20 = 8 000. The exact answer is 8 127, so the estimate was excellent — and if your calculator had shown 81 270 you would know instantly that a digit had slipped.
When to round up regardless
Context sometimes beats the rule. If 127 learners need buses that seat 50, then 127 ÷ 50 = 2,54 — but you cannot hire 0,54 of a bus, so the answer is 3 buses. Always ask whether a part of a thing makes sense before you round in the usual way.
Common mistake: giving a rounded answer where an exact one was asked for. Round only when the question says to — or at the very end of a calculation, never in the middle, because rounding early makes every later step wrong.
Worked Examples
- 1The hundreds digit is 7; look at the digit after it, which is 6
- 26 is 5 or more, so round up: the 7 becomes 8
- 3Everything after becomes zero
- 4Answer: 4 800 ✓
- 1The thousands digit is 4; look at the hundreds digit, which is 6
- 26 is 5 or more, so round up: the 4 becomes 5
- 3Do not round in stages — work from the original number
- 4Answer: 5 000 ✓
- 1Round each number: 387 → 400 and 21 → 20
- 2400 × 20 = 8 000
- 3The exact value is 8 127, so the estimate is within about 2%
- 4Answer: 8 000 — close enough to check a calculation ✓
- 1127 ÷ 50 = 2,54
- 2The ordinary rounding rule would give 3, but check the context anyway
- 3Two buses seat only 100, leaving 27 learners behind
- 4Answer: 3 buses — you must always round up ✓