Scale of the Universe
Students compare the mass of a virus to the mass of a whale and the distance to the moon to the width of a human hair, using scientific notation to make sense of extremes.
Common Core standards covered
- 8.EE.A.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
- 8.EE.A.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Express and operate on very large and very small quantities in scientific notation, and compare them by orders of magnitude.
Student target: “I can write very large and very small quantities in scientific notation, compare them by orders of magnitude, and multiply and divide numbers in scientific notation using the exponent laws.”
Success criteria: I can convert between standard and scientific notation in both directions, including small decimals. · I can say how many TIMES larger one quantity is than another using powers of ten. · I can multiply and divide in scientific notation, fixing the result into proper form. · I can choose a sensible unit for a measurement instead of drowning it in zeros.
Essential question
A virus and the Sun cannot fit on the same number line — or can they, if the line counts ZEROS instead of units?
Learning target: I can write very large and very small quantities in scientific notation, compare them by orders of magnitude, and multiply and divide numbers in scientific notation using the exponent laws.
Success criteria
- I can convert between standard and scientific notation in both directions, including small decimals.
- I can say how many TIMES larger one quantity is than another using powers of ten.
- I can multiply and divide in scientific notation, fixing the result into proper form.
- I can choose a sensible unit for a measurement instead of drowning it in zeros.
Why this matters
Science runs on numbers no one can picture — a virus at a ten-millionth of a meter, the Sun at 150 billion meters away — and scientific notation is the only sane way to write, compare and compute with them. The skill under the skill is yesterday's exponent laws: multiplying measurements is the product rule wearing units. And unit choice (millimeters per year for a spreading seafloor) is the difference between a number a reader feels and one they skim.
Students should already be able to
- The exponent laws, especially products and quotients of powers — yesterday's lab.
- Multiply and divide by 10, 100, 1000 by moving the decimal point (5.NBT.A.2).
- Round and estimate with decimals.
Vocabulary to define explicitly
- Scientific notation
A number written as a × 10ⁿ where a is at least 1 and less than 10.
Example: 38 000 000 m to the Moon region… properly: 3.8 × 10⁸ m.
Not an example: 45 × 10³ breaks the rule — the front number must be under 10. Proper form: 4.5 × 10⁴.
🔒 3 more terms with examples and non-examples — Order of magnitude, Negative power of ten, Proper form — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The warm-up comparison — exponent-first reasoning, or expansion?
- Inventory ordering during curation: the red-blood-cell/hair adjacency is the diagnostic.
- Plaque computations: proper-form fix-ups observed in the wild.
Summative — the graded evidence: The exit ticket (conversion, product, comparison, unit choice), scored with the four-criterion rubric.
The task students work on
Your team curates the museum's new Scale of the Universe exhibit. Eight measurement cards — from a virus to the Earth–Sun distance — must be written in proper scientific notation, hung in order on the magnitude line, and connected by comparison plaques ('a hair is 1 000 viruses wide'). The exhibit's rule: every plaque needs the computation behind it.
Exhibit inventory — convert, then order from smallest to largest.
| Object | Measurement (m) | Scientific notation | Order (smallest = 1) |
|---|---|---|---|
| Virus (diameter) | 0.0000001 | ||
| Red blood cell | 0.000008 | ||
| Human hair (width) | 0.0001 | ||
| Human (height) | 2 | ||
| Earth (diameter) | 13 000 000 | ||
| Earth to Moon | 380 000 000 | ||
| Sun (diameter) | 1 400 000 000 | ||
| Earth to Sun | 150 000 000 000 |
Anticipated student strategies
Convert, then sort, then compare — each strategy consumes the previous one's output. The divide-to-compare step is where yesterday's lab pays its rent, and it should be said out loud.
- Walk the decimal point. The student converts 13 000 000 by walking the point seven places to land at 1.3, writing 1.3 × 10⁷ — and walks right for the small numbers.
The mechanical workhorse. Share first, with the check: does the power count the walk?
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
2 × 10⁶ wins: one more order of magnitude is a factor of ten, and no front number under 10 can make up for it. Name the reading order: exponent first, front number for ties.
Listen for: The 9 multiplies a smaller power. 9 × 10⁵ is 900 000; 2 × 10⁶ is 2 000 000 — the exponent carries more weight than the front.
Then the form rule via the trap: is 45 × 10³ scientific notation? No — 45 is too big; fix to 4.5 × 10⁴, same number, proper name. End with the reading: 10⁻⁷ is not negative, it is tiny.
- Hops left = positive exponent (big number); hops right = negative (small number).
- Proper form is what makes exponent-first comparison legal.
Listen for: Same number, different names. Only the proper name lets you compare by exponent at a glance.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “10³ = 30.” 10³ = 10 · 10 · 10 = 1000. The exponent counts factors of ten, it does not multiply by 3.
- “10⁻⁶ is bigger than 10⁻⁴ because 6 beats 4.” — the teacher response is in the full lesson.
- “45 × 10³ is fine scientific notation.” — the teacher response is in the full lesson.
- “To multiply, multiply the exponents: 10⁴ · 10⁵ = 10²⁰.” — the teacher response is in the full lesson.
- “One order of magnitude bigger means 'a bit more'.” — the teacher response is in the full lesson.
- “A negative exponent makes the number negative.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 4 independent.
- I doWrite 45 000 in scientific notation.
🔒 5 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- Write 0.00052 in scientific notation, and 3.1 × 10⁵ in standard form.
🔒 3 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Review converting between standard and scientific notation using powers of ten. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Teams order eight real measurements from smallest to largest on a magnitude strip. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Students calculate how many times larger one quantity is than another. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Each team presents one comparison using an everyday analogy. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Provide conversion scaffolds for support; extend with division and mixed decimal-scientific problems.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: Students multiply two numbers in scientific notation and explain how many orders of magnitude separate two quantities.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.