At a glance. One lesson of about 60 minutes. It uses negative powers of 10 from Lesson 5 and place value from Year 7, Unit 1. Students slide the point on eight numbers, write large and small numbers in scientific notation, convert back, and order numbers. Lesson 7 calculates with them.
Calculator: no calculator. Nothing here needs a calculator. The point is the structure of the notation, and a calculator's E display is met in the what-if and in Lesson 7.
What students will learn
Know
- a × 10n with 1 ≤ a < 10 and n an integer.
- A number of 10 or more has a positive n; a number between 0 and 1 has a negative n.
- Compare by the exponent first, then by a.
Understand
- The exponent counts places moved, not zeros.
- The rule on a makes each number's scientific notation unique, which is what lets exponents order numbers.
- A negative exponent means small, not negative.
Do
- Write a large or small number in scientific notation.
- Convert from scientific notation to an ordinary number.
- Order a list of numbers in scientific notation.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the form, conversions, and ordering (questions 1 to 7).
- Criterion B. Investigating patterns: the figure. Ask students to find how the exponent relates to the number of digits for numbers above 10, and to the position of the first non-zero digit for numbers below 1.
- Criterion C. Communicating: explaining why 0.52 × 10−3 is not scientific notation (question 9).
- Criterion D. Applying mathematics in real-life contexts: the Earth to Sun distance and a red blood cell (the hook), and the sizes in the extra questions.
The one thing to do with the figure
Ask for a prediction first. “Write 150000000 as a number from 1 to 10 times a power of 10.” Take 1.5 × 108 and 15 × 107 from the room and let the rule settle which is right.
Use 0.0305 (three zeros written, 2 places moved) and 120 (one zero, 2 places moved) to show that the exponent is not the number of zeros. Ask students to say which direction the point moves before they use the slider.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, predictions first. Include 0.0305 and 120. |
| 10 min | Key idea; the warning on a outside the range; the worked example on large and small. |
| 10 min | Ordering by exponent first; the faded 7.1 × 106. |
| 5 min | Question 1, the check. If many choose B, go back to counting the places. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 0.001; 450; 106. |
|---|---|
| Worked examples | 3.8 × 105 and 4.6 × 10−6; 8.1 × 10−5, 3.2 × 10−3, 4.7 × 10−3. |
| Faded example | 7100000. |
| 1. Writing a small number | A. B counts five places; C has a outside the range; D has the wrong sign. |
| 2. The exponent of a large number | 7. 5.4 × 107. |
| 3. The exponent of a small number | −5. 3.2 × 10−5. |
| 4. A negative exponent to a decimal | 0.0037. 3.7 ÷ 1000. |
| 5. Digits in a large number | 10. 3400000000. |
| 6. Ordering numbers in scientific notation | A. B orders by a only; C is descending; D swaps the two with exponent 5. |
| 7. Moving a out of range | 6. 4.56 × 106. |
| 8. Which are in scientific notation | 3. 3.5, 6 and 9.99. |
| 9. The value is right but the form is not | B. A accepts a front number below 1; C moves the exponent the wrong way; D has a front number above 10. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Miscounted, left a outside the range, or used the wrong sign. | Move the point one place at a time and count aloud. |
| 8 or 6 in question 2 | Counted digits or zeros. | Count places moved, not zeros or digits. |
| 5 or −4 in question 3 | Gave a small number a positive exponent, or counted zeros. | Ask: is the number above 10 or below 1? |
| 0.00037, 0.037 or −3700 in question 4 | Moved the point the wrong number of places, or made it negative. | Move one place at a time: 3.7, 0.37, 0.037, 0.0037. |
| 9 or 11 in question 5 | Gave the exponent, or one too many. | Write the number out and count. |
| B, C or D in question 6 | Ordered by a only, descending, or swapped ties. | Exponents first, then the front numbers. |
| 5 or 7 in question 7 | Forgot to raise the exponent, or raised it too far. | Write 45.6 as 4.56 × 10 and combine the tens. |
| 2 or 4 in question 8 | Rejected exponent 0, or accepted 10 × 105. | Test each front number against 1 and 10 only. |
Questions worth asking
- Why must a be less than 10? So each number has one name. Listen for “unique” or “one form”.
- Which is bigger: 9.9 × 103 or 1.1 × 104? 1.1 × 104; the exponent decides. Listen for “exponent first”.
- Is 105 in scientific notation? Yes, as 1 × 105. Listen for “a is 1”.
- Can zero be written in scientific notation? No; there is no non-zero first digit. Stretch only.
Differentiation
A way in for anyone stuck
- Give a place-value chart from 10−6 to 106, and let students write digits on it and read off the exponent.
- Use numbers with one significant digit first (5000, 0.003), then two.
- Ask students to say “the point moves … places …” before writing the exponent.
Stretch
- Question 8, then: write 25 × 10−3 in scientific notation. (2.5 × 10−2.)
- Which is larger, 9.9 × 10−3 or 1.0 × 10−2? (1.0 × 10−2 = 0.01.)
Students learning in English as an additional language
“Scientific notation” is also called standard form in some countries. “Significant digit” is a digit that carries value, not a zero used only to hold places. Say “times ten to the power of” for the notation.
| Word | Plain meaning | Example |
|---|---|---|
| scientific notation | a number written as a times a power of ten, with a from 1 up to 10 | 4.5 × 104 |
| decimal point | the dot that separates whole numbers from parts | the point in 0.0305 |
| exponent | the power of ten in scientific notation | −4 in 7.2 × 10−4 |
| significant digit | a digit that is part of the value, not a place-holder zero | 4 and 5 in 45000 |
| ordinary number | a number written in the usual way, with no power of ten | 45000 |
| order of magnitude | roughly how many powers of ten a number has | 105 is larger than 103 |
Sentence starters for explaining aloud or in writing:
- The point moves … places …, so the exponent is …
- The exponent is negative because …
- I compared the exponents first because …
Two things not to say
“Count the zeros.” It fails for 0.0305, 120 and 6.3 × 105.
“Negative exponent, negative number.” It is a small positive number.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- LargeWrite 82000 in scientific notation.
8.2 × 104. - SmallWrite 4.1 × 10−2 as a decimal.
0.041.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA hair is 0.00008 m wide. Write that in scientific notation.
8 × 10−5 m. - Earlier work (Lesson 5)Evaluate 10−2.
1/100 = 0.01.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhich is larger, 6.2 × 10−4 or 8.5 × 10−5? Explain.
6.2 × 10−4: the exponent −4 is greater than −5. - CorrectingWrite 0.00036 × 108 in scientific notation.
3.6 × 104.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard (tab to it, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write each number on squared paper, one digit per square, and draw the point's journey with an arrow.