Learning intention
We are learning to place a probability on a scale and write it three different ways.
Success criteria
- I can put an event on a 0–1 scale and say why it sits there.
- I can write the same probability as a fraction, a decimal and a percentage.
- I can explain what 0 and 1 mean on the scale.
- I can tell the difference between an event I can calculate and one I can only estimate.
Curriculum
AC9M6P01 — recognise that probabilities lie on numerical scales of 0–1 or 0%–100% and use estimation to assign probabilities that events occur in a given context, using common fractions, percentages and decimals.
Shape of the lesson
| Phase | Time | What happens |
|---|---|---|
| Warm-up | 5 min | Open question on "even chance" |
| Launch | 12 min | Human number line, then relabelled as percentages |
| Explore | 22 min | Placing events on a line and recording three ways |
| Summarise | 8 min | What 0 and 1 mean; calculate versus estimate |
| Review | 5 min | Exit ticket |
What we are learning today
We are learning to place a probability on a scale and write it three different ways.
I will know I can do it when…
- I can put an event on a 0–1 scale and say why it sits there.
- I can write the same probability as a fraction, a decimal and a percentage.
- I can explain what 0 and 1 mean on the scale.
- I can tell the difference between an event I can calculate and one I can only estimate.
Set your goal for today
Read the success criteria above. Which one is the biggest stretch for you? Write one thing you are going to focus on. You will come back to it at the end.
Not sure? Converting between fractions, decimals and percentages is the one most people find hardest today.
How hard is each question?
Every question in this unit is labelled so you know what kind of thinking it needs. The assessment has questions at all three levels.
SF — simple familiar
You have seen this kind of question before and it is clear what to do. One step, friendly numbers.
CF — complex familiar
You know the method, but there is more than one step, or you have to pick which method to use.
CU — complex unfamiliar
Nobody tells you the method. You work out what to do, and explain why you did it.
Warm-up · 5 minutes
An even chance of what?
Think on your own first, then share with a partner. There are lots of right answers.
Or instead · 15 minutes
Diagnostic pre-test
If the pre-test is running in this lesson, it replaces the warm-up. 10 items, 15 minutes, no calculator.
Read the items aloud one at a time for students who need it. Untimed for those students, and it does not gate their entry to the lesson. Analyse against the teacher analysis sheet before Lesson 2.
Launch · 12 minutes
The human number line
Clear a line across the room. 0 at one wall, 1 at the other, and a blank card at the halfway point.
Events to call out
Events 3 and 4 land in the same place. Students find this surprising — one has two outcomes, the other six. First chance to say probability is about the proportion of favourable outcomes, not the number of them.
Event 6, the rain. Somebody will say 50% because "either it rains or it doesn't". This is the equiprobability bias and the most durable misconception in the cycle. Ask "by that reasoning, what is the chance the sun rises tomorrow?" and let the room do the work.
Then relabel the line
Explore · 22 minutes
Part A · Put the events on the line
Write the letter of each event where it belongs, then work out its probability three ways. Everything here is on the worksheet in the same order, so you can work on screen or on paper.
Part B · Write each probability three ways
Fraction, then decimal, then percentage. Use the conversion mat if you need it.
A tossed coin lands heads.
An ordinary six-sided die shows a 6.
An ordinary six-sided die shows an even number.
An ordinary six-sided die shows a 7.
A four-section spinner, one section shaded, lands on shaded.
A bag holds 8 counters: 2 red, 6 blue. You draw a red one.
The same bag. You draw a counter that is not red.
An eight-section spinner has 3 shaded sections. It lands on shaded.
A bag holds 5 counters: 1 red, 4 green. You draw a green one.
The next car that drives past the school gate is white.
You cannot work this one out exactly. Estimate it, then explain what you based your estimate on.
F 2/8 = 1/4 = 0.25 = 25% · G 6/8 = 3/4 = 0.75 = 75% · H 3/8 = 0.375 = 37.5% · I 4/5 = 0.8 = 80%
J No single answer. Any sensible estimate with a reason behind it.
Part C · Show your thinking
Events C and A ended up in the same place on the line. Explain why, even though a die has six outcomes and a coin has two.
Look at events F and G. Add your two answers together. What do you get? Why does that happen?
Event J — the next car past the gate is white. Write your estimate, then explain what you based it on.
2 1/4 + 3/4 = 1. Red and not-red cover every possible outcome, and all the probabilities in a sample space add to 1.
3 No correct number. A strong answer names what the estimate is based on — "about 1/4, because white is one of the common car colours and I see a lot of them."
Talk it through with a partner
Show your partner your answer to question 2. Your partner picks one of these to ask you:
- Would that always happen, or just for this bag?
- Can you show me where the 1 comes from on the number line?
- What would the two answers add to if the bag had three colours?
Write down the question you were asked and what you said back.
Choose your path
Pick the one that matches where you are right now. You can change your mind.
Do events A, B, C, D and E only.
A bag holds 10 counters: 5 red, 5 blue. You draw a red one. Write the probability three ways.
A bag holds 4 counters: 3 red, 1 blue. You draw a red one. Write the probability three ways.
Events G and I are close together. Which is more likely? Say how you know without working either one out.
Invent an event with a probability of exactly 5/8. Write it as a decimal and a percentage too.
Is there an event with a probability of exactly 1 that is not about the sun or the calendar?
★2 Any device with 5 favourable outcomes in 8. 5/8 = 0.625 = 62.5%.
★3 Yes — anything certain by definition. "I roll a number less than 7 on an ordinary die."
Summarise · 8 minutes
Pulling it together
Three things to land
- A probability is a number on a scale. 0 to 1, or 0% to 100%. Nothing sits outside it.
- 0 and 1 are not "very unlikely" and "very likely". They mean impossible and certain.
- The same probability can be written three ways. ½, 0.5 and 50% are the same number wearing different clothes.
Vocabulary
Frayer boards today: probability, outcome, impossible, certain, even chance.
Review · 5 minutes
Exit ticket
Write 0.2 as a fraction and as a percentage.
Draw a spinner, a bag of counters or a die that would give this probability. Label it so someone else can see why it works.
Mark 0.2 on the number line with an arrow.
Sort into three piles as you read, and do not write on them.
Pile 1 — all three correct. Core or Extension for Lesson 2.
Pile 2 — correct fraction and percentage, drawing does not match. Very common; the student can convert but has not connected the number to a physical sample space, which is what Lesson 2 does.
Pile 3 — conversion errors, or 0.2 placed past the midpoint. Conversion mat permanently on the desk.
The most useful single signal is question 3. A student who marks 0.2 to the right of ½ is reading the decimal as a whole number, and that will wreck Lesson 4 unless it is caught here.
Think back
Your goal
Go back to the Intentions tab and read the goal you set. Did you meet it?
Nothing you type here is saved. Copy it into your book before you close the page.
What you need
- resources/l1-01-worksheet-core.pdf — 3 pages, most of the class
- resources/l1-02-worksheet-support.pdf — 3 pages, six events
- resources/l1-04-diagnostic-pre-test.pdf and teacher-only-NOT-for-github/l1-diagnostic-teacher-analysis.pdf (teacher folder — not on the website)
- resources/l1-03-event-cards.pdf — 34pt, for the human number line
- shared/conversion-mat.pdf — used every lesson this cycle
- shared/frayer-boards.pdf — probability, outcome, impossible, certain, even chance
Lesson-specific adjustments
Standing adjustments are on the linked Personalised Adjustments document.
- Pre-test read aloud item by item. Untimed for students who need it.
- Floor positions marked with tape at 0, ¼, ½, ¾ and 1.
- Event set reduced to six for the support tier — the "Need a hand" path on Explore.
- Students may observe the Launch from a seat and place events on the printed line instead.
Misconceptions to expect today
"Either it happens or it doesn't, so it's 50-50"
The equiprobability bias. Returns in Lessons 4 and 6.
Response. Apply the student's own reasoning to an event where it obviously fails — "what is the chance the sun rises tomorrow, then?"
Reading 0.25 as larger than 0.5
The student reads the digits as a whole number. It shows as a decimal placed past the midpoint.
Response. Go back to the fraction. If ¼ is one part of four and ½ is one part of two, which piece is bigger?
"Even chance" meaning "even number"
Predictable, and not only among EAL/D students.
Response. Name both meanings on the everyday-versus-mathematical chart. It is a reasonable reading of the English, not an error.
Treating 0 as "very unlikely"
Students place impossible events slightly above 0.
Response. Rolling a 7 on a six-sided die. There is no 7 on the die.
What comes next
Lesson 2 — Sample space and spinners. Students build spinners and meet the numerator and denominator properly. Lesson 2 also opens the question this lesson leaves alone.