At a glance. One lesson of about 60 minutes. The scale from 0 to 1, impossible and certain, and one probability written as a fraction, a decimal and a percentage.
Calculator: no calculator. Every number is a fraction with a denominator that divides 100 or can be simplified at sight; the work is placing a chance on the scale.
What students will learn
Know
- 0 is impossible and 1 is certain.
- A probability is never below 0 or above 1.
- P(not A) is the rest of the scale up to 1.
Understand
- A fraction, a decimal and a percentage can name the same probability.
- Two possible results do not make each one half.
Do
- Write a probability from counters as a fraction, decimal and percentage.
- Order probabilities given in different forms.
- Decide whether a stated probability is possible.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: converting between forms and placing chances on the scale (questions 2 to 5).
- Criterion B. Investigating patterns: with the figure, ask students to predict where the marker goes before each move, and to say what happens to the marker when the bag is doubled in size with the same fraction red.
- Criterion C. Communicating: moving between words (unlikely, even chance) and numbers, and the exit ticket's ordering.
- Criterion D. Applying mathematics in real-life contexts: the forecast and raffle questions, and judging whether a probability makes sense.
The one thing to do with the figure
Ask for the hook's mark on the line first, then set 5 red together. Many students put it near the middle because “it's red or not red”. Let the marker land at 1/4 and ask what the bag would need to look like for the marker to be in the middle.
Then ask: what happens to the marker if the bag is doubled to 40 counters with 10 red? Nothing moves, because 10/40 and 5/20 name the same place. That is the bridge to equivalent fractions from Year 7.
A possible order
| 5 min | Warm-up, then answers; dwell on 3/5 = 0.6 = 60%. |
|---|---|
| 10 min | Hook: mark 5 red out of 20 on a blank line; compare marks around the room. |
| 15 min | The figure with predictions; the three names for one position. |
| 10 min | Worked examples; the bag of 25 counters in pairs. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 0.6 and 60%; 0.4; 3/5. |
|---|---|
| Worked examples | Spinner: 3/4 = 0.75 = 75%. Die: 0 and 1. |
| Faded example | 0.4 and 40%. |
| 1. Valid probabilities | C, 1.2. D is a common slip: 100% looks big, but it is exactly 1. |
| 2. Counters as a decimal | 0.7 = 7/10. |
| 3. Tickets as a percentage | 20 (20%). |
| 4. Counters, larger bag | 0.3 = 12/40. |
| 5. Not happening | 0.65. |
| 6. Impossible | B. A is the certain event, at the other end of the scale. |
| 7. Changing the bag | 10 green counters. |
| 8. Comparing chances | 0.54. |
| 9. Ordering chances | B. Writing all three as decimals (0.3, 0.4, 0.6) makes the order clear. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 7 or 10 in question 2 | Gave the count instead of comparing it with the whole. | Ask “out of how many?” and have them write the fraction first. |
| 0.3 in question 2; 0.35 in question 5 | Gave the chance of the opposite event, or stopped before taking it from 1. | Mark both on the scale; they must add to 1. |
| 0.2 or 10 in question 3 | Left the answer as a decimal, or as the count. | Read the question's last words: percentage means out of 100. |
| 5 in question 7 | Forgot that adding counters also changes the total. | Draw the bag: 10 counters plus 5 green is 15, and 5/15 is one third. |
| 3.18 in question 8 | Added 3 instead of multiplying. | Ask whether any probability can exceed 1. |
Questions worth asking
- Can a probability be negative? Greater than 1? No: below 0 would be less likely than impossible. Ask what 1.2 would mean.
- Is a probability of 1/2 the same as 50%? Yes, they name the same place; ask for the decimal as well.
- The bag has red or not red. Why isn't red 1/2? Listen for counting the counters, not the colours.
Differentiation
A way in for anyone stuck
- Use a physical bag of 10 coloured counters; let students pull out a number and write the fraction.
- Keep a strip of paper marked 0, 1/2, 1 so each probability can be placed by hand.
- Start with bags of 10 or 20 where percentages are quick to find.
Stretch
- A bag has 3 red and 2 blue counters. How many more red counters make P(red) = 0.75? (Three more: 6 red out of 8 counters.)
- Can two different events both have probability 1? Give an example (rolling below 7 and rolling a number from 1 to 6).
Students learning in English as an additional language
Pair each word with a position: a hand at the left for impossible, in the middle for even chance, at the right for certain.
| Word | Plain meaning | Example |
|---|---|---|
| probability | a number from 0 to 1 that says how likely an event is | the probability of a red counter is 0.3 |
| impossible | cannot happen, probability 0 | rolling 7 on a standard die |
| certain | must happen, probability 1 | rolling a number below 7 |
| even chance | equally likely to happen or not, probability 1/2 | a fair coin landing on heads |
| event | a result or set of results you are asking about | picking a red counter |
Sentence starters for explaining aloud or in writing:
- The probability of … is … out of …, which is … as a decimal.
- It is likely / unlikely because the probability is more / less than 1/2.
- It cannot be … because a probability is never more than 1.
Two things not to say
“Probability is just a guess.” For equally likely counters it is exactly counted, and the scale is what makes two probabilities comparable.
“It's either red or not, so 50-50.” That is the slip the lesson is built to catch; say how many counters.
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.
- FormsWrite 0.25 as a fraction and a percentage.
1/4 and 25%. - CountersA bag has 5 red and 15 yellow counters. What is P(red) as a decimal?
0.25.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- PercentageA box holds 8 pens and 2 are broken. What is the probability a pen picked is broken, as a percentage?
25%. - Earlier work (fractions)Which is greater, 7/10 or 3/4?
3/4, since 0.75 > 0.7.
3Challenge
Reasoning, generalizing and working backwards.
- Missing countersA bag has 6 red counters and the probability of red is 0.3. How many counters in all?
20. - NotP(A) = 0.62. What is P(not A)?
0.38.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard and by touch; “Show me the steps” gives five cases in words.
On paper: a bag of coloured counters and a line marked 0 to 1 do the same job as the figure.