At a glance. One lesson of about 60 minutes. It assumes lessons 2, 3 and 5. Students turn a difference in readings into a ratio and a ratio into a difference, use the decibel formula, and see why equal sounds add intensities, not levels.
Calculator: scientific calculator. A scientific calculator is used for the decibel formula and for non-whole differences, because the point is the reading of the scale. Most numbers are chosen so that the logarithms are exact.
What students will learn
Know
- R = log10 (amount) on a base 10 scale, so a difference d is a ratio of 10d.
- A lower pH is a larger concentration of acid.
- L = 10 log10 (I ÷ I0) in decibels, with each 10 dB a factor of 10.
Understand
- Why a step on the scale multiplies the amount, rather than adding to it.
- Why equal sounds add intensities and not levels.
- Why scientists choose a logarithmic scale for quantities with a very wide range.
Do
- Convert between a difference in readings and a ratio of amounts.
- Evaluate a decibel level from a ratio or an intensity.
- Combine two equal sounds.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the scale relations (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say what stays the same when both readings go up by 1.
- Criterion C. Communicating: explaining why a reading of 8 is not a third more than 6.
- Criterion D. Applying mathematics in real-life contexts: acidity, earthquakes and noise (hook and questions 2, 3, 4 and 7).
The one thing to do with the figure
Ask for the prediction first. Then set A and B to neighbouring readings and move them together along the scale, and ask what the ratio is each time.
Ask: “Why does the straight line crowd the small amounts together?” The largest amount sets the scale, and 1, 10 and 100 are tiny beside 100 000.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the news report about the two earthquakes; collect predictions. |
| 12 min | Figure, key idea and warning card; fill a table of readings against amounts. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask whether more acid means a higher or a lower pH. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 3; 1 000 000; 0.30. |
|---|---|
| Hook | The readings differ by 2, so the amplitude is 102 = 100 times larger, far more than “a third bigger”. |
| Worked examples | 100 000 times; about 32 times; 86.0 dB. |
| Faded example | 60 dB. |
| 1. From a ratio to a pH reading | C. A goes the wrong way; B subtracts the ratio; D divides the reading. |
| 2. Ratio from a difference in pH | 100. 102. |
| 3. Ratio of intensities from decibels | 1000. 30 dB is three factors of 10. |
| 4. From a ratio to a reading | 9.5. 3.5 + 6. |
| 5. Decibels from a ratio | 50. L = 10 × 5. |
| 6. Decibels with a calculator | 75.1. 10 × 7.505. |
| 7. Combining two sounds | 83. 80 + 3.01. |
| 8. Pairs of readings with a given ratio | B. A counts the digits; C uses the decibel step of 10; D adds the ratio. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Moved the pH the wrong way, subtracted the ratio, or divided the reading. | Ask: is the more acidic water higher or lower on the pH scale, and how many factors of 10 are in 1000? |
| 2 or 10 in question 2 | Gave the number of steps, or one factor of 10. | Make a short table of pH 7, 8, 9 with the acidity multiplying by 10 at each step. |
| 30 or 1.75 in question 3 | Gave the difference in dB, or divided the levels. | Ask: how many decibels is a factor of 10? |
| 63.5 or 21 in question 4 | Used the decibel step of 10, or multiplied the reading by the number of steps. | Ask what a factor of 10 does to the reading, then a factor of 1 000 000. |
| 5 or 15 in question 5 | Left out the multiplier 10, or added it. | Check the formula with a ratio of 10: it should give 10 dB. |
| 7.5 or −44.9 in question 6 | Left out the multiplier 10, or did not divide by I0. | Write the ratio on its own line before taking the logarithm. |
| 160 or 90 in question 7 | Added the levels, or added 10. | Ask: what happens to intensity when a second machine starts, and what does the formula do with 2? |
| A, C or D in question 8 | Counted the digits of the ratio, used the decibel step of 10, or added the ratio. | Write 10 000 as a power of 10, then subtract each pair. |
Questions worth asking
- What does a reading of 0 mean on a base 10 scale? Listen for: an amount of 1 unit, since 10 to the power 0 is 1.
- Why are two equal sounds not twice as loud in decibels? Listen for: intensities add, and the scale is logarithmic.
- Why can a lower pH be a bigger quantity? Listen for: pH is minus the logarithm, so it falls as the concentration rises.
- What is the advantage of a logarithmic scale? Listen for: a huge range of amounts fits in a few readings.
Differentiation
A way in for anyone stuck
- Give a table with readings 0 to 6 and a column for the amount 10 to the reading, to fill in.
- Provide cards with a reading difference on one side and the ratio on the other.
- Give the frame: “The readings differ by …, so the ratio is 10 to the power …, which is ….”
Stretch
- Show that adding 3 dB to a sound level roughly doubles its intensity, using log10 2 = 0.301.
- A magnitude scale is defined so that each 1 is a factor of 10 in amplitude. If energy goes as amplitude to the power 1.5, find the energy ratio for a difference of 1.
Students learning in English as an additional language
“Scale” here means a way to report a measurement. “Acidic” describes a liquid with a low pH. “Intensity” is how much sound energy arrives. “Amplitude” is the size of the ground movement. “Ratio” means how many times larger.
| Word | Plain meaning | Example |
|---|---|---|
| logarithmic scale | a scale whose readings are the logarithms of the amounts | pH or decibels |
| pH | a scale of acidity, lower for stronger acids | a lemon is acidic |
| magnitude | the number used to report the size of an earthquake | a magnitude 5.9 earthquake |
| decibel | a unit of sound level, ten times the logarithm of a ratio of intensities | dB |
| intensity | the power of a sound per unit area | W/m2 |
Sentence starters for explaining aloud or in writing:
- The readings differ by … so the ratio is …
- Each step on this scale means …
- I added the intensities, not the …
Two things not to say
“The scale is just a number line.” Say instead: equal steps on the scale are equal factors in the amount.
“Twice the decibels is twice as loud.” Say instead: the decibel level is a logarithm, so a doubling of level is a huge change in intensity.
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.
- FindHow many times more acidic is pH 3 than pH 5?
100. - FindA 90 dB sound is how many times more intense than a 70 dB sound?
100.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextAn earthquake of magnitude 6.7 has an amplitude 100 times that of another. What is the other's magnitude?
4.7. - Earlier work (Year 10)Write 0.001 as a power of 10.
10−3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThree identical sounds at 60 dB each play together. Find the combined level to the nearest decibel.
10 log 3 = 4.77, so about 65 dB. - Working backwardsA sound level rises by 25 dB. By what factor does the intensity rise, to 3 significant figures?
102.5 = 316.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
A scientific calculator is needed for questions 6 and 7.