At a glance. One lesson of about 60 minutes. It assumes standard form from Year 9 and metric conversions from the Standard course. Students multiply, divide and add in standard form, and convert density and rate units.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students who have a GDC the exponent key, what calculator notation means, and why a result must be rewritten in standard form.
What students will learn
Know
- (a × 10ᵐ)(b × 10ⁿ) = ab × 10ᵐ⁺ⁿ, and the product must be rewritten if ab is 10 or more.
- A length factor k becomes k² for areas and k³ for volumes.
- A compound unit is converted one part at a time.
Understand
- Standard form separates the size of a number from its exponent, so exponents do the heavy lifting.
- A conversion factor is a ratio equal to 1; units are part of the calculation.
- Adding needs equal powers of ten; multiplying does not.
Do
- Multiply, divide and add numbers in standard form and give the result in standard form.
- Convert compound units such as g/cm³ and L per second.
- Choose the form that makes a comparison or calculation clearer.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: calculating in standard form and converting units (questions 1 to 9).
- Criterion B. Investigating patterns: the what-if and the second figure. Students find when a carry happens and generalize it.
- Criterion C. Communicating: writing a conversion as a chain of factors with units, so each step can be checked (worked example 3).
- Criterion D. Applying mathematics in real-life contexts: a river's flow, the mass of a planet and grains of sand, where standard form makes comparison possible (questions 5, 7 and 8).
The one thing to do with the figure
Ask for a prediction before every press. With g/cm³ to kg/m³ selected, ask: “which jump is bigger, the grams or the volume?” before students read the net factor. For the second figure, ask whether the answer will carry before moving a slider.
Then press per km² to per m² and ask why there is only one jump and why it points the other way.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the Sun in hydrogen atoms. Collect predictions of the exponent. |
| 12 min | Both figures with predictions; the key idea. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to the first figure and ask which way each jump points. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 7.2 × 10⁻⁴; 10⁻⁴; 100. |
|---|---|
| Hook | The exponent is 30 − (−27) = 57, and 1 m³ is 1 000 000 cm³. |
| Worked examples | 1.8 × 10⁶; 1.2 × 10⁵⁷ atoms; 7 900 kg/m³. |
| Faded example | 8 × 10⁻¹ m³. |
| 1. A compound unit in one go | C. A converts only the mass; B only the volume; D uses the area factor 10⁴. |
| 2. Multiplying in standard form | 4. 1.28 × 10⁴. |
| 3. A volume factor | 3. (10¹)³ = 10³. |
| 4. Density and volume to mass | 10 800. 4 × 2 700 kg. |
| 5. A rate through several units | 11. 1.8 × 10¹¹ litres per hour. |
| 6. Comparing lengths in different units | A. In metres: 72, 68, 71 and 70. |
| 7. Counting with volumes | 9. 8 × 10⁹ grains. |
| 8. A ratio of standard form numbers | 317. 316.7… to the nearest whole number. |
| 9. Adding in standard form | B. A adds coefficients; C adds the exponents too; D uses the smaller power. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Converted only the mass, only the volume, or used the area factor for a volume. | Ask the student to write kg/m³ as a fraction, convert the top and the bottom separately, and say how many cm³ fit in 1 m³. |
| 3 or 7 in question 2 | Did not carry the coefficient 12.8, or subtracted the exponents. | Ask: is 12.8 × 10³ in standard form? |
| 1 or 2 in question 3 | Used the length factor, or the area factor, for a volume. | Draw a 1 cm cube and ask how many 1 mm cubes fit along each edge, then in total. |
| 10.8 or 10 800 000 in question 4 | Converted only one part of g/cm³. | Write 2.7 g/cm³ in kg/m³ with the two jumps, then multiply. |
| 10 or 8 in question 5 | Left 18 × 10¹⁰ unchanged, or skipped the litres. | Ask for the coefficient; is it between 1 and 10? Did the water change from m³ to litres? |
| B, C or D in question 6 | Compared the coefficients, or converted wrongly. | Convert every length to metres in one column, then compare. |
| 6 or 8 in question 7 | Used the area factor, or slipped a decimal place in 0.5³. | Compute 0.5 × 0.5 × 0.5 in steps; compare 1 m³ with a drawing of 1000 × 1000 × 1000. |
| 32 or 3167 in question 8 | Subtracted the exponents incorrectly. | Write 27 − 24 on the board and have students check it. |
| A, C or D in question 9 | Added coefficients across powers, also added the exponents, or kept the smaller power. | Rewrite 3.0 × 10⁴ as 0.30 × 10⁵ before adding. |
Questions worth asking
- Why is a volume factor the cube of a length factor? Listen for “volume has three lengths, so each length factor is used three times”.
- Why can you multiply 10³ × 10⁻⁶ but not add 10³ + 10⁻⁶ the same way? Multiplying adds exponents; adding needs the same unit size. 10³ + 10⁻⁶ is 1000.000001.
- Is 1.8 × 10⁶ more useful than 1 800 000? For comparing size, yes; for reading an everyday amount, perhaps not. Form depends on purpose.
- Why does the figure show 'per km² to per m²' pointing left? More square metres make each one's share smaller.
Differentiation
A way in for anyone stuck
- Give conversion factors as fractions with units, so units cancel visibly.
- Do the exponent arithmetic on a number line with the figure before calculating by hand.
- Use a 10 cm cube next to a 1 m cube drawing to build the volume factor.
Stretch
- A light year is about 9.46 × 10¹² km. Find, in standard form, the number of metres in 4.2 light years.
- The density of air is 1.2 kg/m³. Give it in mg/cm³.
Students learning in English as an additional language
“Compound unit” means a unit made of two units, such as metres per second. “Per” means “for each”. “Density” is how much mass is in each unit of volume. “Coefficient” here is the number in front of the power of ten.
| Word | Plain meaning | Example |
|---|---|---|
| standard form | a number written as a × 10ⁿ with 1 ≤ a < 10 and n a whole number | 4.2 × 10⁻³ |
| coefficient | the number between 1 and 10 in front of the power of ten | the 6.5 in 6.5 × 10⁸ |
| compound unit | a unit built from two units, one divided by the other | metres per second |
| density | mass divided by volume | the mass in each cm³ |
| conversion factor | a ratio equal to 1 that changes the unit but not the amount | 1 m / 100 cm |
| carry | when the coefficient reaches 10 or more and the exponent goes up by 1 | 40 × 10² becomes 4 × 10³ |
Sentence starters for explaining aloud or in writing:
- I multiply the coefficients and add the exponents, which gives …
- The top unit changes by … and the bottom unit changes by …, so the net factor is …
- This is not in standard form, because …
Two things not to say
“Just move the decimal point.” Say how many places and in which direction, and why.
“1 m is 100 cm, so use 100.” Ask what is being measured: a length, an area or a volume.
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.
- Standard formWork out (6 × 10³)(2 × 10⁴).
1.2 × 10⁸. - UnitsHow many mm² are in 1 m²?
10⁶.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA farm in Kenya uses water at 3.6 × 10³ L per hour. How many m³ is that per day?
3.6 × 10³ × 24 = 86 400 L, which is 86.4 m³. - Earlier work (Lesson 1)Write 0.272727… as a fraction.
27/99 = 3/11.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThe Moon is 3.84 × 10⁸ m away. Light travels at 3.0 × 10⁸ m/s. How long does light take?
3.84 ÷ 3.0 = 1.28 s. - ComparingWhich is greater, 5 × 10⁻³ km or 4.8 × 10³ mm?
5 m is greater than 4.8 m.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.