The shape of the lesson
By the end of the lesson, learners can
- Relative frequency: In the lab, pick an outcome to count and add trials 1, 10, 100 or 1000 at a time.
- Fair, biased and random: These three words come up in almost every question on experiments.
- Expected frequency: In the lab, set a probability and a number of trials, see the expected number, then press Try it once for real.
- Relative frequency: times it happened \(\div\) number of trials.
- Reliability: more trials, better estimate.
- Combining: add the successes and add the trials; never average the fractions.
- Fair or biased: compare the relative frequency with the fair probability.
- Expected frequency: probability \(\times\) number of trials.
One picture every idea returns to
Relative frequency \(= \dfrac{\text{number of times it happened}}{\text{number of trials}}\), an estimate of the probability that gets more reliable with more trials. Expected frequency \(= \text{probability} \times \text{number of trials}\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Experiments and expectations.. When outcomes are not equally likely, you estimate a probability by experiment. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Relative frequency | In the lab, pick an outcome to count and add trials 1, 10, 100 or 1000 at a time. Project tab 2 of the lab (Probability lab, relative frequency) and let a student drive it while the class predicts. Model the worked example A drawing pin on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Fair, biased and random | These three words come up in almost every question on experiments. Model the worked example Combining results on the board, one line at a time. Your turn: Group A spins a spinner 30 times and gets 12 reds. Group B spins it 120 times and gets 33 reds. Find the best estimate of P(red). | Pairs |
| 21 to 30 min | Teach and check: Expected frequency | In the lab, set a probability and a number of trials, see the expected number, then press Try it once for real. Project tab 3 of the lab (Probability lab, expected frequency) and let a student drive it while the class predicts. Model the worked example A fair dice on the board, one line at a time. Your turn: P(red) = 0.24 for a biased spinner. It is spun 75 times. How many reds do you expect? | Mix-Pair-Share |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 12 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Expected frequency, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Mix-Pair-Share
Students mix around the room until you call pair; they pair with the nearest classmate, discuss the prompt, then mix again. Three prompts of rising difficulty, three different partners.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| A relative frequency is an ESTIMATE. | With only 10 throws (Fig. 1) a fair coin gave 0.7. Never decide a dice is biased from a handful of trials. |
| Do not average two relative frequencies: \(\tfrac{1}{2}(0.4 + 0.275)\) gives the small group as much weight as the big one. | Add the counts and the trials instead. |
| Expected frequency is probability TIMES trials, not trials divided by the probability. \(240 \div 0.35 = 686\) is bigger than the number of spins, which is impossible. | Return to Expected frequency: ask a pair to state the correct version and show it on the lab before moving on. |
Relative and expected frequencies: worksheet
6 questions, 12 marks. Written in the style of the papers; no past paper question is reproduced.
A coin is thrown 80 times and lands heads 32 times. Find the relative frequency of heads.
Explain what is meant by a biased spinner.
The probability that a light bulb is faulty is 0.04. A shop receives 2500 bulbs. How many would you expect to be faulty?
A four-sided spinner is spun 160 times. The results are: 1 came up 35 times, 2 came up 42 times, 3 came up 38 times and 4 came up 45 times.
(a) Find the relative frequency of 4. (b) Estimate how many 4s there would be in 800 spins.
Tom throws a dice 50 times and gets 9 sixes. Sara throws the same dice 250 times and gets 46 sixes.
(a) Whose results are more reliable? Give a reason. (b) Find the best estimate of the probability of a six. (c) Does the dice appear to be fair?
A random sample of 120 students at a school is asked how they travel to school. 42 of them walk.
The school has 1400 students. Estimate how many of them walk to school, and state one reason why this is only an estimate.
Answer key and marking notes
\(32 \div 80 = 0.4\)
The outcomes are not all equally likely: some sections come up more often than others.
\(0.04 \times 2500 = 100\)
- (a) \(45 \div 160 = 0.28125\) (0.281)
- (b) \(0.28125 \times 800 = 225\)
- (a) Sara's, because she did more trials.
- (b) \(\dfrac{9 + 46}{50 + 250} = \dfrac{55}{300} = 0.183\) (3 s.f.)
- (c) A fair dice gives \(\tfrac{1}{6} = 0.167\); 0.183 is close, so the dice appears fair.
- Relative frequency \(= 42 \div 120 = 0.35\).
- Estimate \(= 0.35 \times 1400 = 490\) students.
- It is only an estimate because a different sample of 120 could give a different number.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A four-coloured spinner is spun 160 times. It lands on blue 44 times. Find the relative frequency of blue.
- A0.275
- B0.379
- C0.25
- D0.725
The probability that a bus is late is 0.08. Over 225 days, how many times would you expect the bus to be late?
- A18
- B2812.5
- C207
- D8
A fair six-sided dice is rolled 450 times. How many times would you expect a score greater than 4?
- A150
- B75
- C225
- D300
Mia throws a coin 10 times and gets 8 heads. Which conclusion is best?
- A10 throws is too few to decide; she should throw it many more times
- BThe coin is definitely biased towards heads
- CThe probability of heads for this coin is exactly 0.8
- DThe next throw is more likely to be tails, to even things out
Group A drops a cup 60 times and it lands upright 18 times. Group B drops the same cup 240 times and it lands upright 75 times. Find the best estimate of the probability that the cup lands upright.
- A0.31
- B0.306
- C0.3125
- D0.3
A spinner is spun 200 times. The results are shown in the table.
| Colour | Red | Blue | Green | Yellow |
|---|---|---|---|---|
| Frequency | 46 | 59 | 39 | 56 |
- A196
- B175
- C504
- D2500
Quiz key
Relative frequency \(= \dfrac{44}{160} = 0.275\). The tempting 0.379 divides by the 116 spins that were NOT blue instead of by all 160 trials. 0.25 assumes the spinner is fair and ignores the experiment, and 0.725 is the relative frequency of not blue.
Expected number \(= 0.08 \times 225 = 18\) days. The tempting 2812.5 divides by the probability, giving more late days than there are days. 207 is the expected number of days NOT late, and 8 treats 0.08 as 8 out of every 100 days but forgets there are 225 days.
Only 5 and 6 are greater than 4, so the probability is \(\tfrac{2}{6} = \tfrac{1}{3}\). Expected number \(= \tfrac{1}{3} \times 450 = 150\). The tempting 75 uses \(\tfrac{1}{6}\), counting only one score. 225 treats it as an even chance, and 300 is the expected number of scores of 4 or less.
A relative frequency is only an estimate, and with 10 throws even a fair coin often gives 7 or 8 heads. More trials give a more reliable estimate. The tempting "definitely biased" reads too much into a tiny experiment. 0.8 is an estimate, not the exact probability, and a coin has no memory, so the next throw is not more likely to be tails.
Best estimate \(= \dfrac{18 + 75}{60 + 240} = \dfrac{93}{300} = 0.31\). The tempting 0.306 averages the two relative frequencies \(\tfrac{1}{2}(0.3 + 0.3125)\), which gives the 60 trials as much weight as the 240. 0.3125 uses only group B and 0.3 only group A; all 300 trials together give the best estimate.
Relative frequency of yellow \(= \dfrac{56}{200} = 0.28\). Expected number \(= 0.28 \times 700 = 196\). The tempting 175 assumes the four colours are equally likely, \(\tfrac{1}{4} \times 700\), ignoring the experiment. 504 is the estimate for NOT yellow, and 2500 divides 700 by 0.28 instead of multiplying.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.