Scientific notation
Writing very large and very small numbers in scientific notation, and adding, multiplying and dividing them
About three lessons of 45 to 60 minutes
How do scientists write a number with 24 zeros without losing their mind?
The mass of the Earth is about 5,970,000,000,000,000,000,000,000 kg. Typing all those zeros correctly, let alone multiplying two numbers like it, is a disaster waiting to happen. Scientific notation solves this: the same number becomes 5.97 x 10^24, one digit, a decimal, and a power of ten showing exactly how many places the decimal point has moved.
The real payoff comes when you calculate with these numbers: multiplying or dividing two numbers in scientific notation only means multiplying the small decimal parts and adding or subtracting the exponents, no matter how enormous or tiny the original numbers were.
- The mass of the Earth5.97 x 10^24 kg, instead of 25 digits
- The width of a bacteriumabout 1 x 10^-6 m, a number too small to write comfortably in decimal form
- The distance to the nearest starabout 4 x 10^16 m, used constantly in astronomy
- A computer chip's transistor sizemeasured in nanometres, a scientific-notation-sized number
What students will be able to do
Students will convert between standard form and scientific notation for very large and very small numbers, and add, multiply and divide numbers written in scientific notation, interpreting the results in context.
- I can write a large number in scientific notation, as a single digit times a power of ten.
- I can write a small decimal number in scientific notation, using a negative exponent.
- I can convert a number in scientific notation back to standard form.
- I can multiply and divide two numbers in scientific notation.
- I can add two numbers in scientific notation, including when their exponents differ.
Standards this unit teaches
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Scientific notation
- a way of writing a number as a single digit (1 to 9), a decimal part, times a power of ten, such as 5.97 x 10^24
- Coefficient
- the decimal part of a number in scientific notation, always at least 1 and less than 10
- Standard form
- a number written out in full, the ordinary way, such as 45,000,000
- Order of magnitude
- the power of ten in a number's scientific notation, a rough sense of how large or small it is
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Writing numbers in scientific notation
ConcreteTo write a large number in scientific notation, move the decimal point left until only one nonzero digit remains before it; the number of places you moved becomes a POSITIVE exponent. For a small decimal, move the decimal point right instead, and the exponent is NEGATIVE.
45,000,000 becomes 4.5 x 10^7: the decimal point moves 7 places left (from after the last 0 to between the 4 and the 5). The small number 0.000621 becomes 6.21 x 10^-4: the decimal point moves 4 places right (from before the 6 to after it), and moving right always gives a negative exponent.
Write 45,000,000 and 0.000621 in scientific notation.
- 45,000,000: move the decimal point left until one digit remains before it: 4.5, moved 7 places. So 45,000,000 = 4.5 x 10^7.
- 0.000621: move the decimal point right until one digit remains before it: 6.21, moved 4 places. Moving right gives a negative exponent, so 0.000621 = 6.21 x 10^-4.
Answer: 45,000,000 = 4.5 x 10^7. 0.000621 = 6.21 x 10^-4.
- Why does moving the decimal point LEFT give a POSITIVE exponent?
- How many places would the decimal point move to write 0.05 in scientific notation, and which direction?
2. Multiplying and dividing in scientific notation
PictorialMultiplying two numbers in scientific notation multiplies their coefficients and ADDS their exponents (the product rule from exponent laws). Dividing multiplies... divides the coefficients and SUBTRACTS the exponents.
(3 x 10^4)(2 x 10^3) = (3 x 2) x 10^(4+3) = 6 x 10^7. Sometimes the coefficients multiply to 10 or more, and the result needs renormalising: (5 x 10^6)(4 x 10^3) = 20 x 10^9, but 20 is not a single digit, so shift it: 20 x 10^9 = 2.0 x 10^10.
Multiply (5 x 10^6)(4 x 10^3), then divide (8 x 10^9) / (2 x 10^3).
- Multiply the coefficients: 5 x 4 = 20. Add the exponents: 6 + 3 = 9. Result so far: 20 x 10^9.
- 20 is not between 1 and 10, so renormalise: 20 x 10^9 = 2.0 x 10^1 x 10^9 = 2.0 x 10^10.
- Divide the coefficients: 8 / 2 = 4. Subtract the exponents: 9 - 3 = 6. Result: 4 x 10^6.
Answer: (5 x 10^6)(4 x 10^3) = 2.0 x 10^10. (8 x 10^9) / (2 x 10^3) = 4 x 10^6.
- Why must a coefficient in scientific notation always be renormalised back to between 1 and 10?
- What operation do you do to the exponents when dividing, and why?
3. Adding numbers in scientific notation
AbstractMultiplying and dividing work on any two numbers in scientific notation right away, but adding (or subtracting) only works directly when the exponents MATCH, because you can only add coefficients that describe the same size of place value.
(3 x 10^5) + (4 x 10^5) = (3 + 4) x 10^5 = 7 x 10^5, since both terms already share the exponent 5. If the exponents differ, rewrite one so they match first: (2 x 10^6) + (3 x 10^5) = (2 x 10^6) + (0.3 x 10^6) = 2.3 x 10^6.
Add (2 x 10^6) + (3 x 10^5).
- The exponents differ (6 and 5), so rewrite 3 x 10^5 with exponent 6: 3 x 10^5 = 0.3 x 10^6.
- Now add the coefficients, since the exponents match: 2 + 0.3 = 2.3.
- Result: 2.3 x 10^6.
Answer: (2 x 10^6) + (3 x 10^5) = 2.3 x 10^6.
- Why can you not just add 2 and 3 directly in (2 x 10^6) + (3 x 10^5)?
- Once the exponents match, what part of each term do you actually add?
Common misconceptions and how to address them
MisconceptionA small decimal like 0.00003 converts to a POSITIVE exponent, since the original number needed a lot of zeros.
Why it happens: Students associate 'lots of digits' with 'big exponent' and lose track of which direction the decimal point actually moved.
How to address it: The sign of the exponent depends on the DIRECTION the decimal point moves, not the size of the number: moving right (to make small numbers into a single digit) always gives a NEGATIVE exponent. 0.00003 = 3 x 10^-5, not 3 x 10^5.
MisconceptionAfter multiplying two numbers in scientific notation, the coefficient can stay however large it comes out, like 20 x 10^9.
Why it happens: Students correctly multiply the coefficients and add the exponents, then stop, forgetting scientific notation requires the coefficient to be a single digit before the decimal.
How to address it: Always check the final coefficient is between 1 and 10. If it is not, shift the decimal point and adjust the exponent to match: 20 x 10^9 becomes 2.0 x 10^10, not left as is.
MisconceptionWhen adding two numbers in scientific notation, you add the exponents the same way you do when multiplying.
Why it happens: Students over-apply the 'add the exponents' rule from multiplication to every operation involving scientific notation.
How to address it: Exponents are only added when MULTIPLYING. When ADDING two numbers in scientific notation, the exponents must first be made equal (by rewriting one term), and then only the coefficients are added; the exponent stays the same.
MisconceptionThe coefficient in scientific notation must be a whole number.
Why it happens: Students round 6.21 to 6, assuming decimals are not allowed in the coefficient.
How to address it: The coefficient only needs to be at least 1 and less than 10; it is very often a decimal, like 6.21 x 10^-4. Rounding it changes the value and should only be done deliberately, for an estimate.
Guided practice (with answers)
1. Write 6,300,000 in scientific notation.
Answer: 6.3 x 10^6, because the decimal point moves 6 places left to leave a single digit before it.
2. Write 0.00042 in scientific notation.
Answer: 4.2 x 10^-4, because the decimal point moves 4 places right, giving a negative exponent.
3. Multiply (2 x 10^5)(3 x 10^2).
Answer: 6 x 10^7, because 2 x 3 = 6 and 5 + 2 = 7.
4. Divide (9 x 10^8) / (3 x 10^5).
Answer: 3 x 10^3, because 9 / 3 = 3 and 8 - 5 = 3.
5. Add (4 x 10^4) + (5 x 10^3).
Answer: 4.5 x 10^4, because 5 x 10^3 = 0.5 x 10^4, and 4 + 0.5 = 4.5.
Independent practice worksheets
Practise writing and calculating with scientific notation, with computed, never-wrong answer keys.
Differentiation
- Practise counting decimal-point moves on whole numbers with an obvious place value first (450, 4,500, 45,000) before tackling small decimals.
- Provide a 'left is positive, right is negative' arrow reference card next to every conversion problem while the rule is new.
- Keep multiplication and division problems to already-normalised results at first (coefficient under 10) before introducing renormalising.
- For adding, always show the rewritten equal-exponent form explicitly as its own line, never combine it with the addition step.
- Estimate real-world quantities (population, distances in space, sizes of cells) using scientific notation and compare orders of magnitude.
- Explore what happens when you add two scientific-notation numbers whose result needs renormalising, such as (9 x 10^5) + (5 x 10^5).
- Investigate how many times bigger one order of magnitude is than the one below it, and why that makes scientific notation so efficient for comparing sizes.
- Combine multiplication, division and renormalising in a single multi-step real-world problem, such as finding the total mass of a large number of identical tiny particles.
Assessment: exit ticket
A three-question exit ticket sampling writing, multiplying, and adding in scientific notation.
1. Write 0.0091 in scientific notation.
Answer: 9.1 x 10^-3, because the decimal point moves 3 places right.
2. Multiply (4 x 10^3)(2 x 10^6).
Answer: 8 x 10^9, because 4 x 2 = 8 and 3 + 6 = 9.
3. Add (7 x 10^5) + (2 x 10^4).
Answer: 7.2 x 10^5, because 2 x 10^4 = 0.2 x 10^5, and 7 + 0.2 = 7.2.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 writing scientific notation both directions (section 1), Lesson 2 multiplying and dividing (section 2), Lesson 3 adding plus the exit ticket (section 3 and assessment).
- This unit assumes comfort with the product and quotient exponent rules from the Grade 8 rational numbers, exponents and roots unit. Revisit that first if combining exponents feels shaky.
- Language to repeat: left is positive, right is negative (for the direction the decimal point moves); multiplying and dividing always work directly, adding needs matching exponents first.
- Curriculum note: 8.EE.A.3 (Common Core) covers writing and estimating with scientific notation; 8.EE.A.4 covers operating on numbers already in scientific notation. Both are grouped in this one unit since they are taught and practised together in almost every classroom sequence.
- Present and print both work: use the Print button for a clean handout, or work a real-world scientific-notation fact (a planet's mass, a virus's size) live on the board as motivation before the practice problems.