Mathematics • Year 9 • Unit 1 • Lesson 17
Operations in Scientific Notation
Build fluency with $\times$, $\div$, $+$ and $-$ on numbers in $a \times 10^n$ form. Multiply coefficients and add indices, divide coefficients and subtract indices, match powers before adding or subtracting, and re-standardise the coefficient back into $[1, 10)$ when it drifts.
1. I do, fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Calculate $(5 \times 10^3) \times (4 \times 10^2)$. Give your answer in scientific notation.
Step 1, Spot the operation.
Two numbers in $a \times 10^n$ form being multiplied. So: $\times$ on the coefficients, $+$ on the indices.
Reason: $(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}$, the product rule on $10^m \cdot 10^n$ adds indices.
Step 2, Multiply the coefficients.
$5 \times 4 = 20$
Reason: the coefficients are just ordinary numbers, handle them with ordinary arithmetic.
Step 3, Add the indices.
$10^3 \times 10^2 = 10^{3+2} = 10^5$
Reason: product rule on powers of the same base, add the indices.
Step 4, Combine.
$20 \times 10^5$
Reason: $20$ is outside $[1, 10)$, this is right value, wrong form. Re-standardise.
Step 5, Re-standardise.
$20 \times 10^5 = 2.0 \times 10^6$ (decimal shifts one left $\to$ index goes up by 1)
Reason: shift the decimal one place left, bump the power of 10 up by 1, so the value is unchanged.
Answer: $\mathbf{2 \times 10^6}$.
2. We do, fill in the missing steps
Same structure as Section 1, with the working faded. Fill in each blank. 4 marks
Problem. Calculate $3 \times 10^4 + 2 \times 10^3$. Give your answer in scientific notation.
Step 1, Spot the operation: addition, so the powers of $10$ must __________________ first.
Step 2, Rewrite the smaller-power term to share the larger power $10^4$:
$2 \times 10^3 = \_\_\_\_ \times 10^4$
Step 3, Now both terms share $10^4$. Add the coefficients:
$3 + \_\_\_\_ = \_\_\_\_$
Step 4, Write the result:
$3 \times 10^4 + 2 \times 10^3 = \_\_\_\_\_\_\_ \times 10^4$
Step 5, Check the form: Is the coefficient in $[1, 10)$? __________ (yes / no)
3. You do, independent practice
Show your working under each problem. The first four are foundation (single operation, clean numbers). The middle two are standard (need a re-standardise or negative index). The last two are extension (real context or extra step).
Foundation, single operation
3.1 Calculate $(4 \times 10^3) \times (2 \times 10^5)$. 1 mark
3.2 Calculate $\dfrac{9 \times 10^8}{3 \times 10^3}$. 1 mark
3.3 Calculate $\dfrac{6 \times 10^9}{3 \times 10^4}$. 1 mark
3.4 Calculate $4 \times 10^5 + 3 \times 10^4$. 1 mark
Standard, re-standardise or negatives
3.5 Calculate $(6 \times 10^5) \times (5 \times 10^{-2})$. Give your answer in scientific notation. 2 marks
3.6 Calculate $5 \times 10^{-3} - 2 \times 10^{-4}$. Give your answer in scientific notation. 2 marks
Extension, push your thinking
3.7 Calculate $(8 \times 10^4) \times (5 \times 10^3)$, giving the answer in scientific notation. (Hint: the coefficient won't stay in $[1, 10)$ at first.) 2 marks
3.8 A student writes "$3 \times 10^4 + 2 \times 10^3 = 5 \times 10^7$." Explain in one sentence what they did wrong, then give the correct answer in scientific notation. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (faded $3 \times 10^4 + 2 \times 10^3$)
Step 1: powers of 10 must match first.
Step 2: $2 \times 10^3 = \mathbf{0.2} \times 10^4$.
Step 3: $3 + \mathbf{0.2} = \mathbf{3.2}$.
Step 4: $3 \times 10^4 + 2 \times 10^3 = \mathbf{3.2} \times 10^4$.
Step 5: $3.2$ is in $[1, 10)$, yes, already in standard form.
3.1, $(4 \times 10^3) \times (2 \times 10^5)$
Coefficients: $4 \times 2 = 8$. Indices: $10^{3+5} = 10^8$. Answer: $\mathbf{8 \times 10^8}$.
3.2, $\dfrac{9 \times 10^8}{3 \times 10^3}$
Coefficients: $9 / 3 = 3$. Indices: $10^{8 - 3} = 10^5$. Answer: $\mathbf{3 \times 10^5}$.
3.3, $\dfrac{6 \times 10^9}{3 \times 10^4}$
Coefficients: $6 / 3 = 2$. Indices: $10^{9 - 4} = 10^5$. Answer: $\mathbf{2 \times 10^5}$.
3.4, $4 \times 10^5 + 3 \times 10^4$
Match powers: $3 \times 10^4 = 0.3 \times 10^5$. Add coefficients: $4 + 0.3 = 4.3$. Answer: $\mathbf{4.3 \times 10^5}$.
3.5, $(6 \times 10^5) \times (5 \times 10^{-2})$
Coefficients: $6 \times 5 = 30$. Indices: $10^{5 + (-2)} = 10^3$. Current: $30 \times 10^3$, coefficient out of $[1, 10)$. Re-standardise: $\mathbf{3 \times 10^4}$.
3.6, $5 \times 10^{-3} - 2 \times 10^{-4}$
Match powers to the larger index ($-3$): $2 \times 10^{-4} = 0.2 \times 10^{-3}$. Subtract: $5 - 0.2 = 4.8$. Answer: $\mathbf{4.8 \times 10^{-3}}$.
3.7, $(8 \times 10^4) \times (5 \times 10^3)$
Coefficients: $8 \times 5 = 40$. Indices: $10^{4+3} = 10^7$. Current: $40 \times 10^7$, $40$ is outside $[1, 10)$. Re-standardise: $40 = 4.0 \times 10^1$, so $40 \times 10^7 = 4 \times 10^8$. Answer: $\mathbf{4 \times 10^8}$.
3.8, Find the mistake in $3 \times 10^4 + 2 \times 10^3$
The student added the coefficients and the indices together ($3 + 2 = 5$ and $4 + 3 = 7$), but addition of scientific-notation terms requires the powers of $10$ to match first you never add indices when adding terms.
Correct working: rewrite $2 \times 10^3 = 0.2 \times 10^4$, then $(3 + 0.2) \times 10^4 = \mathbf{3.2 \times 10^4}$.