Scientific notation, measurement error and logarithmic scales
Writing and calculating with very large and very small numbers, quantifying measurement error, and reading logarithmic scales like Richter and pH
About four lessons of 45 to 60 minutes
Numbers too big, too small, or too uncertain for ordinary digits
The distance from Earth to the Sun is about 150,000,000,000 metres. The width of a virus is about 0.00000009 metres. Neither number is comfortable to read, write or calculate with in full, so scientists write both in scientific notation instead: a single digit, a decimal point, and a power of 10 doing the heavy lifting.
No measurement is ever perfectly exact either, a ruler marked in millimetres cannot report a length more precisely than that, which is why every measurement carries some error. And when a quantity ranges from tiny to enormous, such as earthquake energy or sound intensity, scientists switch to a logarithmic scale, where each equal step means an equal MULTIPLY rather than an equal add.
- Distance to the Sunabout 1.5 x 10^11 metres, far easier to write in scientific notation
- Width of a virusabout 9 x 10^-8 metres, a very small number written the same way
- A measured length versus the true lengthpercentage error compares the two
- The Richter scale for earthquakeseach whole step up is about 10 times stronger
What students will be able to do
Students will convert numbers between standard and scientific notation, multiply expressions written in scientific notation, calculate the absolute and percentage error of a measurement compared to a true value, explain why every measurement carries uncertainty, and interpret logarithmic scales such as the Richter and pH scales.
- I can write a very large or very small number in scientific notation.
- I can convert a number in scientific notation back to an ordinary number.
- I can multiply two numbers written in scientific notation, renormalising the mantissa if needed.
- I can calculate the absolute error and the percentage error of a measurement.
- I can explain why a logarithmic scale is used, and read what a given number of steps on one means.
Standards this unit teaches
- AC9M9M02Australian Curriculum v9 (ACARA)Scientific notation
Solve problems involving very small and very large measurements, time scales and intervals expressed in scientific notation.
- AC9M9M04Australian Curriculum v9 (ACARA)Measurement error
Calculate and interpret absolute, relative and percentage errors in measurements, recognising that all measurements have a degree of uncertainty.
- AC9M10M02Australian Curriculum v9 (ACARA)Logarithmic scales
Interpret and use logarithmic scales in applied contexts involving small and large quantities and change.
- AC9M10M04Australian Curriculum v9 (ACARA)Impact of measurement error
Identify the impact of measurement errors on the accuracy of results in practical contexts.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 8 rational numbers, exponents & roots teaching unitthe exponent rules this unit's scientific-notation arithmetic relies on
- Grade 8 scientific notation teaching unitthe Grade 8 foundation this unit extends with error and logarithmic scales
- Percentage in the glossarya refresher before percentage error is introduced
Words to teach and display
- Scientific notation
- a way to write a number as a x 10^n, where the leading number a is between 1 and 10
- Exponent
- the power of 10 in scientific notation, telling how many places the decimal point moves
- Absolute error
- the size of the gap between a measured value and the true value, in the original units
- Percentage error
- the absolute error compared to the true value, written as a percentage
- Logarithmic scale
- a scale where each equal step represents multiplying by the same factor, not adding the same amount
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 very large and very small numbers in scientific notation
ConcreteScientific notation writes a number as a x 10^n, where a is between 1 and 10. For a large number, count how many places the decimal point moves left to sit after the first non-zero digit; that count is a positive exponent. For a very small number, the decimal point moves right instead, giving a negative exponent.
The width of a virus, about 0.00000009 metres, becomes 9 x 10^-8 metres: the decimal point moves 8 places to the right to reach 9. The negative exponent shows the number is small, it does not make the number itself negative.
Write 78,000,000 in scientific notation.
- Place the decimal point after the first non-zero digit: 7.8.
- Count how many places the decimal point moved from its original position: 7 places.
- Write it as a x 10^n: 7.8 x 10^7.
Answer: 7.8 x 10^7.
- Would the exponent be positive or negative for a number smaller than 1?
- Why must the leading number a always be between 1 and 10?
2. Converting back to an ordinary number, and multiplying in scientific notation
PictorialTo convert back, move the decimal point the number of places shown by the exponent (right for positive, left for negative). To multiply two numbers in scientific notation, multiply the leading numbers and add the exponents, then renormalise if the result is 10 or more.
Simplify (3 x 10^4) x (5 x 10^6), giving the answer in scientific notation.
- Multiply the leading numbers: 3 x 5 = 15.
- Add the exponents: 4 + 6 = 10.
- 15 x 10^10 is not in proper form, since 15 is 10 or more. Rewrite 15 as 1.5 x 10^1, and add that extra exponent: 1.5 x 10^11.
Answer: 1.5 x 10^11.
- Why is 15 x 10^10 not considered proper scientific notation?
- What operation do you use on the exponents when multiplying, addition or multiplication?
3. Absolute, relative and percentage error
AbstractNo measuring tool is perfectly precise, so every measurement is really an estimate of the true value. Absolute error is the size of the gap between the measured and true values, in the original units. Percentage error compares that gap to the true value, so measurements of very different sizes can be compared fairly.
A length is measured as 44 cm, but the true length is 40 cm. Find the absolute error and the percentage error.
- Absolute error = |measured - true| = |44 - 40| = 4 cm.
- Relative error = absolute error / true value = 4 / 40 = 0.1.
- Percentage error = relative error x 100 = 0.1 x 100 = 10%.
Answer: Absolute error is 4 cm; percentage error is 10%.
- Why does percentage error use the TRUE value on the bottom, not the measured value?
- Could two measurements have the same absolute error but different percentage errors?
4. Reading a logarithmic scale: Richter, decibel and pH
AbstractSome quantities, such as earthquake energy or sound intensity, range from tiny to enormous. A normal scale would be useless for them, so scientists use a logarithmic scale instead, where each equal step means an equal MULTIPLY (usually by 10), not an equal add.
An earthquake measures 4 on the Richter scale. A second earthquake measures 6. About how many times stronger is the second earthquake?
- Each whole step on the Richter scale is about 10 times stronger.
- From magnitude 4 to magnitude 6 is 2 steps.
- 2 steps means 10 x 10 = 100 times stronger.
Answer: About 100 times stronger.
- Why would a normal (linear) scale be impractical for earthquake energy?
- How many times stronger would a 3-step difference on the Richter scale represent?
Common misconceptions and how to address them
MisconceptionTreating a negative exponent as making the number itself negative, e.g. thinking 9 x 10^-8 is a negative number.
Why it happens: The minus sign sits right next to the exponent, so it is easy to read it as describing the whole number rather than just the size of the shift.
How to address it: Show that 9 x 10^-8 = 0.00000009, a very small but still positive number. A negative exponent means 'very small', never 'negative'.
MisconceptionForgetting to renormalise the leading number after multiplying two scientific-notation numbers, leaving an answer like 15 x 10^10.
Why it happens: The multiplication step is complete once the numbers are multiplied and the exponents are added, so the extra renormalising step is easy to skip.
How to address it: Build in a final check every time: is the leading number between 1 and 10? If it is 10 or more, move the decimal point one place left and add 1 to the exponent.
MisconceptionConfusing absolute error with percentage error, e.g. reporting '4' as if it were already a percentage.
Why it happens: Both numbers come from the same subtraction step, so it is easy to stop before the final division and multiplication.
How to address it: Keep the units attached: absolute error always keeps the original unit (cm, g, and so on); percentage error is unitless and always ends in a % sign, from dividing by the true value and multiplying by 100.
MisconceptionBelieving a measurement can be made with zero error, if only the person measuring is careful enough.
Why it happens: Errors feel like mistakes that a careful person should be able to avoid entirely.
How to address it: Every measuring tool has a limit to its precision, for example a ruler marked in millimetres cannot report a length more precisely than that, no matter how carefully it is read. The true value can be approached but never captured exactly.
MisconceptionReading a logarithmic scale as if the steps were equal amounts, so a jump from magnitude 5 to magnitude 7 feels like it should be roughly double.
Why it happens: Every other number line encountered before this has equal steps meaning equal ADDS, so the habit carries over.
How to address it: Count the number of steps first, then multiply by the scale's factor that many times (usually x10 per step), rather than adding.
Guided practice (with answers)
1. Write 5,200,000 in scientific notation.
Answer: 5.2 x 10^6, because the decimal point moves 6 places to sit after the 5.
2. Write 0.00043 in scientific notation.
Answer: 4.3 x 10^-4, because the decimal point moves 4 places to the right to sit after the 4.
3. Write 6.1 x 10^5 as an ordinary number.
Answer: 610,000, because the decimal point moves 5 places to the right.
4. Simplify (2 x 10^3) x (4 x 10^5).
Answer: 8 x 10^8, because 2 x 4 = 8 (already between 1 and 10) and 3 + 5 = 8.
5. A recipe calls for 250 g of flour. A kitchen scale measures 235 g. Find the percentage error.
Answer: 6%, because |235 - 250| / 250 x 100 = 15 / 250 x 100 = 6%.
6. The pH scale is logarithmic. A solution with pH 3 is how many times more acidic than one with pH 5?
Answer: About 100 times more acidic, because 2 steps means 10 x 10 = 100.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong.
Differentiation
- Start with positive exponents (large numbers) only, before introducing negative exponents for small numbers.
- Give the renormalising rule ('is the leading number 10 or more?') as a checklist to work through after every multiplication.
- Use whole-number percentage errors before decimal ones.
- Talk through the Richter example with physical stacking (10, then 10 lots of 10, then 10 lots of that) before naming it 'logarithmic'.
- Work with very small measurements, such as nanometres or microseconds, using negative exponents throughout.
- Compare the percentage error of several repeated measurements of the same quantity and discuss which is most reliable.
- Research a real logarithmic scale (decibel, Richter, pH, or stellar magnitude) and explain in writing why a normal scale would not work for it.
- Investigate how an error in a measured length affects a calculated area or volume, where the error is effectively squared or cubed.
Assessment: exit ticket
A short exit ticket covering scientific notation, percentage error and reading a logarithmic scale.
1. Write 0.0000091 in scientific notation.
Answer: 9.1 x 10^-6, because the decimal point moves 6 places to the right to sit after the 9.
2. A wall is measured at 3.2 m, but its true length is 3.0 m. Find the percentage error, to 1 decimal place.
Answer: 6.7%, because |3.2 - 3.0| / 3.0 x 100 = 0.2 / 3.0 x 100 ≈ 6.7%.
3. Using the idea that equal steps on a logarithmic scale mean equal multiplies, explain why a magnitude 8 earthquake is far more than 'a bit worse' than a magnitude 5 earthquake.
Answer: Magnitude 8 is 3 steps above magnitude 5. Three steps means 10 x 10 x 10 = 1000 times stronger, not just moderately worse.
Teacher notes and timings
- Rough timing: Lesson 1 scientific notation (section 1), Lesson 2 converting and multiplying (section 2), Lesson 3 measurement error (section 3), Lesson 4 logarithmic scales and the exit ticket (section 4 and assessment).
- This unit deliberately spans two ACARA year levels, Year 9 (scientific notation, measurement error) and Year 10 (logarithmic scales, the impact of error), because all four descriptors build a single narrative about quantifying scale and uncertainty; splitting them would break that story.
- AC9M9M02, AC9M9M04, AC9M10M02 and AC9M10M04 previously had no dedicated lesson; the closest existing unit, Grade 8 scientific notation, explicitly names error and logarithmic scales as 'a step beyond this unit's scope'.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.