At a glance. One lesson of about 60 minutes. It uses Lessons 3 and 5 (the product and quotient laws, negative exponents) and Lesson 6 (writing numbers in scientific notation). Students multiply, divide and add numbers in scientific notation, handle the carry, and use a scientific calculator once, to check an estimate.
Calculator: scientific calculator. A scientific calculator is used once, to enter numbers with the EXP key and check an estimate. The estimate comes first, so that a slipped exponent is caught, and the calculator is not trusted blindly.
What students will learn
Know
- Multiply the front numbers, add the exponents; divide the front numbers, subtract the exponents.
- If the front number reaches 10 or falls below 1, move a factor of ten into the exponent.
- To add or subtract, match the powers of ten first.
Understand
- Multiplication regroups into front numbers and powers of ten; addition does not.
- The power of ten is the unit when adding, so it stays the same.
- An answer's accuracy is limited by the data.
Do
- Multiply and divide numbers in scientific notation, with and without a calculator.
- Add and subtract numbers in scientific notation with different powers.
- Solve real-life problems with very large and very small numbers to a sensible accuracy.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the four operations in scientific notation (questions 1 to 7).
- Criterion B. Investigating patterns: the figures. Ask students to find states where the multiplication needs a carry and say what the front numbers have in common.
- Criterion C. Communicating: explaining why the exponents are not added when numbers are added, with an example in ordinary numbers (question 9).
- Criterion D. Applying mathematics in real-life contexts: the speed of light and transistor sizes (questions 7 and 8).
The one thing to do with the figure
Ask for a prediction first. “(3 × 108) × (4 × 10−3): what do you get, and is it finished?” Take 12 × 105 from the room and ask whether it is in scientific notation.
In the second figure, set the second exponent three below the first and ask students what happens to the second number's contribution. Then set the difference to 0 and ask why the powers need no matching.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, predictions first. Include a carry and a negative exponent. |
| 10 min | Key idea; the second figure; the warning on adding exponents. |
| 10 min | Worked examples; the faded (6 × 10−4) × (5 × 107). |
| 5 min | Question 1, the check; then the calculator note: estimate first, then press. |
| 10 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 10 min | Exit ticket, question 9; then ask which of the four operations needed a carry or a match. |
The answers
| Warm-up | 4.9 × 10−3; 103; 2 × 10−3. |
|---|---|
| Worked examples | 2.8 × 109; 3.2 × 109. |
| Faded example | 3 × 104. |
| 1. Multiplying in scientific notation | A. B multiplies exponents; C adds front numbers; D subtracts exponents. |
| 2. A carry on multiplying | 9. 2 × 109. |
| 3. Dividing in scientific notation | 6. 4 × 106. |
| 4. Adding with unequal powers | 6.8. 6.8 × 105. |
| 5. Multiplying with negative exponents | −5. 6 × 10−5. |
| 6. Subtracting with unequal powers | 4.6. 4.6 × 106. |
| 7. Distance from speed and time, to two significant figures | 1.4. 1.44 × 1011 m, rounded to two significant figures. |
| 8. How many fit | 50000. 5 × 104. |
| 9. Adding, not multiplying | B. A adds exponents; C multiplies; D ignores the unequal powers. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Multiplied exponents, added front numbers, or subtracted exponents. | Write the numbers out in factors: 30000 × 200000. |
| 8 or 15 in question 2 | Did not carry, or multiplied exponents. | Ask whether the front number is below 10 after multiplying. |
| 12 or 3 in question 3 | Added or divided the exponents. | Remind: dividing powers subtracts exponents (Lesson 3). |
| 14 or 6.08 in question 4 | Added the front numbers without matching, or shifted wrongly. | Write both numbers in full: 600000 + 80000. |
| 5 or 6 in question 5 | Lost the sign, or multiplied negatives. | Add −3 and −2 on a number line. |
| 1 or 5.4 in question 6 | Subtracted the front numbers without matching, or added instead of subtracting. | Write both numbers in full: 5000000 − 400000. |
| 14, 1.44 or 1.5 in question 7 | Did not carry, gave too many significant figures, or rounded the data first. | Estimate 3 × 5 first, then multiply the data as given and round at the end. |
| 100000 or 5000 in question 8 | Forgot to divide the front numbers, or miscounted the exponent. | Subtracting a negative is adding: write −3 + 8. |
Questions worth asking
- Why do exponents add when you multiply but not when you add? Multiplying joins runs of tens; adding combines amounts of the same unit. Listen for “unit”.
- When does multiplying need a carry? When the front numbers multiply to 10 or more. Listen for “10 or more”.
- Why is the Sun's distance given only to two significant figures? The data is only that accurate. Listen for “the data limits the answer”.
- What would 2 × 103 + 2 × 103 be? 4 × 103. Listen for “the power stays”.
Differentiation
A way in for anyone stuck
- Give a card with the four rules and a one-line example of each; keep it for the unit.
- Use whole-number front numbers (2, 3, 4) before decimals.
- Ask students to write each number in full beside the notation for the first two questions.
Stretch
- Question 8, then: how many transistors fit in a chip 1.5 × 10−2 m across, in one row? (7.5 × 105.)
- Find two numbers in scientific notation whose product is exactly 1 × 106 but neither front number is 1. (2 × 103 and 5 × 102.)
Students learning in English as an additional language
“Carry” here means moving a factor of ten into the exponent. “Significant figures” are the digits you trust. The E on a calculator display means “times ten to the power”.
| Word | Plain meaning | Example |
|---|---|---|
| front number | the number from 1 up to 10 in scientific notation | 2.5 in 2.5 × 106 |
| carry | move a factor of ten into the exponent | 10 × 108 = 1 × 109 |
| EXP key | the calculator key for “times ten to the power” | 2.5 EXP 6 |
| estimate | a rough answer worked out first | 10 × 108 |
| significant figures | the digits that carry accuracy | 2.5 has 2 significant figures |
| match | write with the same power of ten | 3.2 × 104 = 0.032 × 106 |
Sentence starters for explaining aloud or in writing:
- I multiplied the front numbers and added …
- The front number is 10 or more, so I carried …
- I matched the powers of ten first because …
Two things not to say
“Add the exponents.” Say which operation: they add when multiplying, they stay when adding.
“The calculator shows it, so it is right.” Always have an estimate first.
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.
- MultiplyCalculate (3 × 105) × (2 × 103).
6 × 108. - DivideCalculate (9 × 106) ÷ (3 × 102).
3 × 104.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA computer does 2 × 109 operations a second. How many in 5 × 102 seconds?
10 × 1011 = 1 × 1012. - Earlier work (Lesson 6)Write 6.8 × 10−4 as a decimal.
0.00068.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy can't 3 × 104 + 2 × 105 be done by adding the exponents?
3 × 104 = 0.3 × 105, so the sum is 2.3 × 105. - SquaringCalculate (4 × 10−3)2.
16 × 10−6 = 1.6 × 10−5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All eight sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
A scientific calculator is used once. No calculator? Write the numbers in full for the estimates.