Logarithmic scales: a week of ready-to-teach maths

Five lessons for Year 10 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus twelve size cards to match, twelve comparison cards to sort, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair. Cut the 12 size cards before Day 3 and the 12 comparison cards before Day 4, and keep each set in an envelope.
4. Calculators with a log button are used on Day 2; a ruler marked in centimetres is used on Days 1 and 2.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A linear scale adds: each step along it adds the same amount. A logarithmic scale multiplies: each step along it multiplies by the same factor, usually ten. That lets one scale hold a virus and the Earth, or a whisper and a jet engine, without the small values vanishing into the first millimetre. The price is that reading it takes multiplying thinking: halfway between 1 and 10 is about 3.16, not 5.5, and two steps up a scale of change, such as earthquake magnitude, mean a hundred times as much, not twice.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these

Powers of ten are written with a raised exponent: 10⁻³, 10⁰, 10⁵. On a log scale the distance between two marks stands for a factor, so we say “ten times” or “a hundred times”, never “two more”. Sizes are given in metres, to the nearest order of magnitude.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Days 1 and 2 build the log scale and learn to read it; Day 3 uses it for sizes; Day 4 for the scales of change used in science; and Day 5 for graphs, where exponential growth turns into a straight line. It builds on Year 9, where students wrote very small and very large measurements in scientific notation, and it prepares the logarithms of senior mathematics and the scales of science.

Materials for the week (one trip)

• From the classroom: calculators with a log button, centimetre rulers, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 size cards (Day 3).
• Cut-out sheet 2, one per pair: 12 comparison cards (Day 4).
• If you have them: a spreadsheet that can set a chart axis to a log scale, for Day 5. No other equipment to buy.

Words for the word wall

linear scale, logarithmic scale, power of ten, order of magnitude, decade, factor, magnitude, decibel, pH, acidic, semi-log graph, exponential growth

Dear families

This week in maths, Year 10 is learning to read logarithmic scales, where each step means ten times as much. They are the scales behind earthquake magnitudes, decibels and pH, and they let one ruler hold both a virus and the whole Earth.

Try this at home

• Ask: how many times longer is a kilometre than a metre? (1000 times: three powers of ten.)
• Ask: is a magnitude 7 earthquake twice as strong as a magnitude 5? (No: a hundred times the shaking.)
• Ask: a person is about 1 m tall and the Earth about 10 000 000 m across. How many powers of ten apart? (Seven.)
• Ask: which is more acidic, pH 3 or pH 5, and by how much? (pH 3, a hundred times.)

My powers of ten week

Fill one row a day. Write a very big or very small quantity you met, its nearest power of ten, and a thing of the same size.

Two rulers

A linear ruler adds and a log ruler multiplies. Students place 1, 10, 100 and 1000 on both, see how the log ruler spreads them evenly, and write numbers as powers of ten.

We are learning to

• compare a linear scale with a logarithmic scale,
• place powers of ten on a log scale at equal distances,
• write numbers, large and small, as powers of ten.

Success criteria

• I can explain why 1, 10 and 100 crowd together on a linear ruler but spread out on a log ruler.
• I can write 0.001 as 10⁻³.

You need

Centimetre rulers; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Spacing a log ruler by adding: 1, 10, 20, 30 is linear; a log ruler goes 1, 10, 100, 1000.
• “10⁰ = 0.” Any number to the power 0 is 1, which sits at the start of the log ruler.
• Reading 10⁻³ as a negative number: it is 0.001, small but positive.

Answers

• Part A: 1: 0.01 cm on the linear ruler, 0 cm on the log ruler; 10: 0.15 cm on the linear ruler, 5 cm on the log ruler; 100: 1.50 cm on the linear ruler, 10 cm on the log ruler; 1 000: 15.00 cm on the linear ruler, 15 cm on the log ruler.
• Part B: 3, 30 and 300 sit 5 cm apart, the same gap as 1, 10 and 100: each is ten times the one before.
• Part C: 100 000 = 10⁵; 0.001 = 10⁻³; 1 = 10⁰; 10 000 000 = 10⁷; 0.1 = 10⁻¹.
• Part D: 1 000 000 = 10⁶ sits 30 cm along; 0.1 = 10⁻¹ sits 5 cm before the 1. Zero cannot go on a log ruler at all: no power of ten equals 0.

Two rulers

A. Where do they go?

The linear ruler runs from 0 to 1 000 over 15 cm. The log ruler puts each power of ten 5 cm further on, starting with 1 at 0 cm. The first row is done for you.

B. The log ruler

Here is a log ruler. Mark 3, 30 and 300 on it as well as you can.

C. Powers of ten

D. Stretch: off the end

Between the marks

On a log scale the gap between two marks stands for a factor, so the values between them are not where a linear eye expects. Students place 2, 3, 5 and 8 on one cycle with the log button, find the value halfway between two marks, and see that equal steps multiply.

We are learning to

• use log10 to place a value between two powers of ten,
• find the value halfway between two marks as the square root of their product,
• describe equal steps on a log scale as equal factors.

Success criteria

• I can place 2 at about 3.0 cm along a 10 cm cycle.
• I can explain why halfway between 1 and 10 on a log scale is about 3.16.

You need

Calculators with a log button; centimetre rulers; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Spacing 2 to 9 evenly inside a cycle: the marks crowd together towards the 10.
• “Halfway is the average.” Halfway on a log scale is the square root of the product.
• Reading the log button as the value: it gives the distance along the cycle, as a fraction of the cycle.

Answers

• Part A: 2: log 0.301, so 3.01 cm; 3: log 0.477, so 4.77 cm; 5: log 0.699, so 6.99 cm; 8: log 0.903, so 9.03 cm.
• Part B: halfway between 1 and 10 is √(1 × 10) = 3.16; halfway between 10 and 1 000 is √(10 × 1 000) = 100.00; halfway between 1 and 100 is √(1 × 100) = 10.00.
• Part C: one step up from 3 is 30, two steps 300, three steps 3 000: each equal step multiplies by 10.
• Part D: 2 sits at 3.01 cm and 5 at 6.99 cm; together 10.00 cm, the place of 10 = 2 × 5. Adding distances on a log scale multiplies the values: the idea behind the slide rule.

Between the marks

A. One cycle

One cycle of a log scale runs from 1 to 10 over 10 cm. A value sits at log(value) × 10 cm. The first row is done for you.

Mark your four values on this cycle, which has its 2 to 9 marks drawn in.

B. Halfway

The value halfway between two marks is the square root of their product. Check one with the log button.

C. Equal steps

D. Stretch: adding distances

The tiny and the vast

Sizes from a virus to the Earth span fourteen powers of ten. Students write each size in metres as a power of ten, place six things on one log scale, match six things to their orders of magnitude, and compare sizes as factors.

We are learning to

• write a size in metres to the nearest power of ten,
• place very small and very large sizes on one log scale,
• compare two sizes by how many powers of ten apart they are.

Success criteria

• I can place a red blood cell and the Earth on the same scale.
• I can say that the Earth is about 10⁷ times as wide as a person is tall.

You need

The size cards (cut-out sheet 1), one set per pair; calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Rounding 8 × 10⁻⁶ m down to 10⁻⁶: on a log scale 8 is nearer 10 than 1, so it is about 10⁻⁵.
• Mixing units: change every size to metres first.
• Miscounting for small numbers: 0.0001 = 10⁻⁴.

Answers

• Part A: a virus: 10⁻⁷ m, about 10⁻⁷ m; a red blood cell: 8 × 10⁻⁶ m, about 10⁻⁵ m; an ant: 5 × 10⁻³ m, about 10⁻² m; a person: 1.7 × 10⁰ m, about 10⁰ m; a humpback whale: 1.5 × 10¹ m, about 10¹ m; the Earth: 1.27 × 10⁷ m, about 10⁷ m.
• Part B: from a person to the Earth is 7 powers of ten; from a virus to a person 7. A person is about as many times bigger than a virus as the Earth is bigger than a person.
• Part C: the card key is with the mini-check answers.
• Part D: 14 powers of ten. On a linear ruler 1 m long with the Earth at the end, the virus would sit about 10⁻¹⁴ m from 0, far smaller than an atom.

The tiny and the vast

A. Sizes in metres

Write each size in metres in scientific notation, then to the nearest power of ten. The first row is done for you.

B. One scale for all

Mark the six things on this log scale, in metres.

C. Match the sizes

Match each thing card to its order of magnitude card. Write the letters.

D. Stretch: one ordinary ruler

Scales of change

Earthquake magnitude, decibels and pH are log scales of change: one whole step means ten times as much. Students compare quakes, sounds and acids, turn steps into factors and back, and sort twelve comparisons by their factor.

We are learning to

• read one step on a log scale of change as a factor of ten,
• turn a difference in steps into a factor, and a factor back into steps,
• explain why a small difference on these scales can be a large difference in reality.

Success criteria

• I can say that a magnitude 7 earthquake shakes 100 times as hard as a magnitude 5.
• I can find how many times as intense a 70 dB sound is as a 40 dB sound.

You need

The comparison cards (cut-out sheet 2), one set per pair; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Subtracting instead of multiplying: two magnitudes apart is 10 × 10 = 100 times.
• Treating every decibel as a factor of 10: it is every 10 dB.
• Reading pH upwards: a lower pH is the more acidic.

Answers

• Part A: Earthquakes, magnitude 7 against magnitude 5: 2 steps, so 100 times; Sound, 70 dB against 40 dB: 3 steps, so 1 000 times; Acids, pH 3 against pH 6: 3 steps, so 1 000 times (the lower pH is the more acidic).
• Part B: a sound with this many times the intensity (100) is 20 dB louder; a quake with this many times the shaking (1 000) is 3 magnitudes bigger; an acid this many times as acidic (10 000) is 4 pH units lower.
• Part C: the card key is with the mini-check answers.
• Part D: magnitude 6.5 against 5 is 1.5 steps: 10^1.5 ≈ 31.6 times. 63 dB against 60 dB is 0.3 of a step: 10^0.3 ≈ 2.0, so doubling the intensity adds about 3 dB.

Scales of change

A. Steps into factors

On each scale one whole step means ten times as much: one magnitude, 10 dB, or one pH unit (lower pH is more acidic). The first row is done for you.

B. Factors into steps

C. Ten, a hundred or a thousand?

Sort the twelve comparison cards by how many times as much each one is, then write the letters.

D. Stretch: half steps

Graphs on a log axis

Exponential growth curves upwards on ordinary axes but runs straight on a log axis. Students plot doubling bacteria both ways, read the straight line to predict, and finish with the self-check quiz.

We are learning to

• plot data on a grid with a logarithmic axis,
• recognise exponential growth as a straight line on a log axis,
• read and extend the line to make a prediction.

Success criteria

• I can plot 100, 200, 400, ... on a log axis and see a straight line.
• I can use the line to say when the count first passes 10 000.

You need

Rulers; the worksheet, one per student; a spreadsheet that can set a log axis, if you have one; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; do Parts C and D and the quiz next time.

Watch for these ideas

• Spacing a log axis by adding: its main lines go 100, 1000, 10 000, with 200 to 900 crowding towards each one.
• Reading between the gridlines as if they were evenly spaced.
• Extending a curve by eye on linear axes: the log axis turns it into a line that is easy to extend.

Answers

• Part A: 100, 200, 400, 800, 1 600, 3 200, 6 400.
• Part B: on the linear grid the points curve upwards ever more steeply; on the log grid they lie on a straight line, rising 0.30 of a power of ten each hour.
• Part C: the line passes 10 000 between 6 and 7 hours; the count first passes it at 7 hours (12 800).
• Part D: a straight line on a log axis means the same factor each step, so a falling straight line means decay by the same factor, such as halving each hour.

Graphs on a log axis

A. Doubling bacteria

A colony starts with 100 bacteria and doubles every hour. Fill the table. The first two are done for you.

B. Two graphs

Plot the counts on both grids: an ordinary axis on the left, a log axis on the right.

C. Extend the line

D. Stretch: a falling line

Things and their orders of magnitude

For Day 3, one set per pair. Six cards name a thing; six give a size in metres to the nearest power of ten. Match each thing to its size.

Teacher note: the key is with the mini-check answers.

How many times as much?

For Day 4, one set per pair. Each card compares two quantities on a log scale. Sort the cards into ten times, a hundred times and a thousand times as much.

Teacher note: the key is with the mini-check answers.

What we know: Logarithmic scales

Work on your own. A calculator is allowed. Show your working.

1. On a log scale with 2 cm for each power of ten and 1 at 0 cm, where do 1, 10, 100, 1 000, 10 000 go? ______________________
2. On a log scale, which value is halfway between 10 and 100? ______________________
3. Write 0.000001 m and 3 000 000 m to the nearest power of ten. ______________________
4. How many times the shaking is a magnitude 8 earthquake compared with a magnitude 6? ______________________
5. One sound is 30 dB louder than another. How many times the intensity is it? ______________________
6. On a log scale, equal distances mean, circle one:   equal factors  ·  equal differences  ·  equal sums
7. Which is more acidic, pH 2 or pH 6, and how many times as acidic? ______________________
8. A count starts at 10 and is multiplied by 3 every day. On a log axis its points make a straight line. After how many days does it first pass 10 000? ______________________

Answers and marking guide

Answers with working

1. 1 at 0 cm, 10 at 2 cm, 100 at 4 cm, 1 000 at 6 cm, 10 000 at 8 cm.
2. √(10 × 100) = √1000 ≈ 31.62, not 55.
3. 0.000001 = 10⁻⁶ m; 3 000 000 = 3 × 10⁶, nearest power of ten 10⁶ m (log 6.48).
4. 2 steps: 100 times.
5. 3 steps of 10 dB: 1 000 times.
6. equal factors.
7. pH 2: 4 steps, so 10 000 times as acidic.
8. 10 × 3⁶ = 7 290, 10 × 3⁷ = 21 870: after 7 days.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, and speaks of factors (“a hundred times”) rather than differences on a log scale.

Card keys

Size cards (Day 3): A virus: G and A (10⁻⁷ m); A red blood cell: D and J (10⁻⁵ m); A grain of fine sand: E and L (10⁻⁴ m); A person: B and H (10⁰ m); A football field: K and F (10² m); The width of the Earth: I and C (10⁷ m). Comparison cards (Day 4): 10 times: M, P, R, U, X; 100 times: N, Q, T, W; 1000 times: O, S, V.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: two rulers, reading between the marks, sizes, scales of change, and graphs on a log axis.