45-minute lesson
Secondary 3 Mathematics: Scientific notation and significant figures
Lesson objective
- Write very large and very small numbers in scientific notation.
- Convert numbers in scientific notation back to ordinary (decimal) form.
- Round numbers to a given number of significant figures.
Success criteria:
- I can write 4 300 000 as 4.3 × 10⁶ and 0.000 52 as 5.2 × 10⁻⁴.
- I can write 3.07 × 10⁵ as an ordinary number.
- I can round a number such as 0.004 718 to 2 significant figures.
Materials
- Scientific calculator
- Paper and pencil
- A place-value chart (optional)
Introduction
5 minIntroduce today's words:
- Scientific notation: A way of writing a number as a number from 1 up to (but not including) 10, multiplied by a power of 10, such as 4.3 × 10⁶.
- Power of 10: 10 raised to an index, such as 10³ = 1000 or 10⁻² = 0.01.
- Negative index: An index below zero, used for small numbers: 10⁻ⁿ = 1 ÷ 10ⁿ, so 10⁻³ = 0.001.
- Ordinary form: A number written out in full with ordinary digits, such as 307 000.
- Significant figures: The digits in a number that carry meaning about its size, starting from the first non-zero digit.
Ask your child what they already know about scientific notation and significant figures.
Explanation
9 minScientists work with huge numbers (the distance to the Sun is about 150 000 000 km) and tiny ones (a bacterium is about 0.000 002 m long). Writing all those zeros is slow and easy to get wrong, so we use scientific notation: a number from 1 up to (but not including) 10, multiplied by a power of 10.
Powers of 10 build on place value: 10¹ = 10, 10² = 100, 10³ = 1000, and so on. For small numbers we use negative indices: 10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001.
To write a large number in scientific notation:
- Place the decimal point after the first non-zero digit: 4 300 000 → 4.3.
- Count how many places the decimal point moved to get there (6 places), and use that as a positive index: 4.3 × 10⁶.
For a small number, the decimal point moves to the right, so the index is negative. 0.000 52 → 5.2, moving 4 places right, so 0.000 52 = 5.2 × 10⁻⁴. A quick check: a negative index means the number is less than 1.
To convert back to ordinary form, move the decimal point the number of places shown by the index — right for a positive index (the number gets bigger) and left for a negative index (the number gets smaller). Fill gaps with zeros.
Significant figures (s.f.) count from the first non-zero digit. Zeros at the start of a decimal (like the zeros in 0.0047) are not significant; zeros between other digits are; zeros at the end of a decimal are (0.4050 has 4 s.f.). To round to a number of significant figures, look at the next digit: 5 or more rounds up. When rounding a whole number, keep its size by filling with zeros: 86 451 to 3 s.f. is 86 500, not 865.
Calculators often show scientific notation with an E, so 3.2E8 means 3.2 × 10⁸.
Worked examples
7 minLarge and small numbers
Write (a) 4 300 000 and (b) 0.000 52 in scientific notation.
Step 1: (a) Put the point after the 4: 4.3. The point moves 6 places to the left, so the index is 6: 4.3 × 10⁶.
Step 2: (b) Put the point after the 5: 5.2. The point moves 4 places to the right, so the index is −4: 5.2 × 10⁻⁴.
Answer: (a) 4.3 × 10⁶; (b) 5.2 × 10⁻⁴
Back to ordinary form
Write (a) 3.07 × 10⁵ and (b) 8.1 × 10⁻³ as ordinary numbers.
Step 1: (a) Positive index: move the point 5 places right. 3.07 → 307 000.
Step 2: (b) Negative index: move the point 3 places left. 8.1 → 0.0081.
Answer: (a) 307 000; (b) 0.0081
Significant figures
Round (a) 0.004 718 to 2 significant figures and (b) 86 451 to 3 significant figures.
Step 1: (a) The first significant figure is 4. The first two are 4 and 7; the next digit is 1, so round down: 0.0047.
Step 2: (b) The first three significant figures are 8, 6 and 4; the next digit is 5, so round up to 865, then fill with zeros: 86 500.
Answer: (a) 0.0047; (b) 86 500
Guided practice (do together)
9 min1. Write 52 000 in scientific notation.
(a) 52 × 10³ (b) 5.2 × 10⁴ (c) 5.2 × 10³ (d) 0.52 × 10⁵
2. Write 0.0009 in scientific notation.
(a) 9 × 10⁻⁴ (b) 9 × 10⁴ (c) 9 × 10⁻³ (d) 0.9 × 10⁻³
3. Write 6.1 × 10⁻² as an ordinary number.
(a) 0.61 (b) 0.061 (c) 0.0061 (d) 610
4. Round 3.14159 to 3 significant figures.
(a) 3.14 (b) 3.142 (c) 3.1 (d) 3.15
Independent practice
11 min5. Write 7.25 × 10³ as an ordinary number.
6. Write 640 000 in scientific notation.
7. Round 0.006 382 to 2 significant figures.
8. How many significant figures does 0.04050 have?
Questions to check understanding
- Can you write 4 300 000 as 4.3 × 10⁶ and 0.000 52 as 5.2 × 10⁻⁴?
- Can you write 3.07 × 10⁵ as an ordinary number?
- Can you round a number such as 0.004 718 to 2 significant figures?
- What was the trickiest part today?
Answer guide
1. 5.2 × 10⁴ — The first part must be from 1 to less than 10, so use 5.2. The point moves 4 places: 5.2 × 10⁴. (52 × 10³ and 0.52 × 10⁵ have the right value but are not in scientific notation.)
2. 9 × 10⁻⁴ — The point moves 4 places to the right to give 9, so the index is −4.
3. 0.061 — A negative index of −2 means move the point 2 places left: 0.061.
4. 3.14 — The first three significant figures are 3, 1 and 4. The next digit is 1, so round down: 3.14.
5. 7250 — Move the point 3 places right: 7.25 → 7250.
6. 6.4 × 10⁵ — Put the point after the 6 to get 6.4. The point moves 5 places, so 6.4 × 10⁵.
7. 0.0064 — The first two significant figures are 6 and 3. The next digit is 8, so round up to 64: 0.0064.
8. 4 — Start at the first non-zero digit (4). The digits 4, 0, 5 and the final 0 are all significant: 4 significant figures.
Review
4 minRecap the success criteria together. Watch for these common misconceptions:
- Writing a first part that is 10 or more (such as 52 × 10³). The number in front must be at least 1 and less than 10.
- Mixing up the sign of the index. Remind the learner: big numbers have positive indices; numbers between 0 and 1 have negative indices.
- Dropping place-holder zeros when rounding whole numbers, such as writing 86 451 to 3 s.f. as 865. The rounded number must be about the same size as the original.
Extension activities
- Make a “powers of ten” poster showing objects from the size of an atom (about 10⁻¹⁰ m) to the size of a galaxy.
- Multiply numbers in scientific notation by multiplying the front numbers and adding the indices, such as (2 × 10³) × (4 × 10⁵) = 8 × 10⁸.
- Research how many significant figures a kitchen scale or a measuring tape can reliably give, and why.
Suggested follow-up
- Revisit scientific notation and significant figures tomorrow with two or three quick questions from memory.
- Try the online practice check for this topic and look at any questions that need another go.
Tips for parents
- Ask the learner to say the number aloud after converting (“three hundred and seven thousand”) to check it is sensible.
- Look up real measurements together — the mass of the Earth, the width of a human hair — and convert them both ways.
- Show how your calculator displays very large results (for example 99 999 × 99 999 999) and what the E means.