Mathematics • Year 9 • Unit 1 • Lesson 18
From Atoms to Galaxies
Use scientific notation to compare cosmic and atomic scales, divide huge by tiny, compare orders of magnitude, and round to 3 significant figures. Then write down the calculator key-sequence you would actually press on EE/EXP.
1. Word problems
Each problem uses one or more of the techniques from Lesson 18: multiplying or dividing scientific-notation numbers, comparing orders of magnitude, or rounding to significant figures. Show your working.
1.1 — Atoms across the Sun. The Sun is about $1.5 \times 10^{11}$ m from Earth. A single hydrogen atom is about $5 \times 10^{-11}$ m wide.
(a) How many hydrogen atoms would line up across the Earth–Sun distance? Give your answer in scientific notation.
(b) State the order of magnitude of your answer. 3 marks
1.2 — Electron vs proton. The mass of an electron is $9.11 \times 10^{-31}$ kg. The mass of a proton is $1.67 \times 10^{-27}$ kg.
(a) Which particle is heavier? Justify by comparing the indices.
(b) How many times heavier? Give the answer to 3 significant figures. 3 marks
1.3 — Federal budget per person. Australia's population is about $2.6 \times 10^7$. The federal budget is about $\$6.5 \times 10^{11}$.
(a) Calculate the spending per person, to 3 significant figures.
(b) Write out the exact calculator key-sequence using the EE / EXP key. 3 marks
1.4 — Cubed weight. The mass of the Earth is about $6 \times 10^{24}$ kg. The mass of an average person is about $6 \times 10^1$ kg.
(a) How many people would have the combined mass of the Earth? Give your answer in scientific notation.
(b) State the order of magnitude difference between Earth's mass and a person's mass. 3 marks
1.5 — Salt at dinner. A single grain of salt has a mass of about $5.85 \times 10^{-5}$ g and contains roughly $6.0 \times 10^{18}$ formula units of NaCl. You sprinkle $5.0$ g on dinner.
(a) Estimate how many grains of salt are in $5.0$ g, to 2 significant figures.
(b) Estimate the total number of formula units in those grains, to 2 significant figures. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 Two students argue about which is the bigger number: $1.5 \times 10^9$ or $9 \times 10^8$. Annie says "$9 > 1.5$ so $9 \times 10^8$ is bigger." Ben says "Look at the index — $10^9$ wins." Explain (i) who is correct, (ii) why the index decides for positive coefficients, and (iii) when (if ever) you would compare the coefficients instead. Use the words "order of magnitude" somewhere.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Atoms across the Sun
(a) $N = \dfrac{1.5 \times 10^{11}}{5 \times 10^{-11}} = \dfrac{1.5}{5} \times 10^{11 - (-11)} = 0.3 \times 10^{22} = \mathbf{3 \times 10^{21}}$ atoms (re-normalised).
(b) Order of magnitude $\approx \mathbf{10^{21}}$.
That's about 3 sextillion atoms — a number so big that writing it without scientific notation takes 22 digits.
1.2 — Electron vs proton
(a) The proton is heavier. Comparing the indices: $-27 > -31$, so a $10^{-27}$ number is bigger than a $10^{-31}$ one (less negative means closer to zero, but bigger value).
(b) Ratio $= \dfrac{1.67 \times 10^{-27}}{9.11 \times 10^{-31}} = \dfrac{1.67}{9.11} \times 10^{-27 - (-31)} = 0.1833 \times 10^{4} = 1.833 \times 10^{3} \approx \mathbf{1.83 \times 10^3}$ times heavier (3 s.f.) — about 1830 times.
1.3 — Federal budget per person
(a) Per person $= \dfrac{6.5 \times 10^{11}}{2.6 \times 10^7} = \dfrac{6.5}{2.6} \times 10^{11 - 7} = 2.500 \times 10^4 = \mathbf{\$2.50 \times 10^4}$ $= \$25{,}000$ per person (3 s.f.).
(b) Calculator keys: $\mathbf{6.5 \;\text{EXP}\; 11 \;\div\; 2.6 \;\text{EXP}\; 7 \;=\;}$.
Important: do NOT type "$\times 10$" before the EXP key — EXP already means $\times 10^{?}$.
1.4 — Cubed weight
(a) Number of people $= \dfrac{6 \times 10^{24}}{6 \times 10^1} = \dfrac{6}{6} \times 10^{24 - 1} = 1 \times 10^{23} = \mathbf{1 \times 10^{23}}$ people.
(b) Order of magnitude difference $= 24 - 1 = \mathbf{23}$ orders of magnitude. Earth is roughly $10^{23}$ times heavier than one person.
1.5 — Salt at dinner
(a) Grains $= \dfrac{5.0}{5.85 \times 10^{-5}} = \dfrac{5.0}{5.85} \times 10^{0 - (-5)} \approx 0.8547 \times 10^5 = \mathbf{8.5 \times 10^4}$ grains (2 s.f.) — about 85,000 grains.
(b) Formula units $= (8.5 \times 10^4) \times (6.0 \times 10^{18}) = (8.5 \times 6.0) \times 10^{4 + 18} = 51 \times 10^{22} = \mathbf{5.1 \times 10^{23}}$ formula units (2 s.f.).
That's about half of Avogadro's number — physical chemistry hidden in your dinner.
2.1 — Explain your thinking (sample response)
Ben is correct. When comparing two positive numbers in scientific notation, the index (the power of 10) is what decides, not the coefficient. This is because a jump of 1 in the index multiplies the value by 10 — that's a whole order of magnitude bigger. So $1.5 \times 10^9 = 1{,}500{,}000{,}000$, while $9 \times 10^8 = 900{,}000{,}000$ — even though Annie is right that $9 > 1.5$, the extra factor of 10 from the larger index makes $1.5 \times 10^9$ bigger overall. You should only compare coefficients when the indices are equal; for example, between $7 \times 10^4$ and $3 \times 10^4$, the indices tie so the coefficient $7$ wins.
Marking: 1 mark for naming the correct student (Ben); 1 for explaining why the index dominates (factor of 10 per index); 1 for stating when coefficients matter (equal indices); 1 for using "order of magnitude" correctly.