The shape of the lesson
By the end of the lesson, learners can
- Building the curve: In the lab, add up the frequencies as you go to fill the cumulative frequency column.
- Median, quartiles and IQR: Every reading is the same move: start on the cumulative frequency axis, go across to the curve, then down to the horizontal axis.
- Percentiles and how many questions: The \(p\)th percentile is read across at \(p\%\) of \(n\).
- Table: add the frequencies as you go; the last total is \(n\).
- Plot each total at the upper class boundary, as a cross, and join with a smooth curve.
- Read across at \(\dfrac{n}{4}\), \(\dfrac{n}{2}\), \(\dfrac{3n}{4}\) (or \(p\%\) of \(n\)), then down.
- IQR = UQ \(-\) LQ, both read from the horizontal axis.
- More than: total \(-\) reading.
One picture every idea returns to
Cumulative frequency is a running total. Plot each total at the UPPER boundary of its class and join with a smooth curve. For \(n\) values, read across at \(\dfrac{n}{4}\), \(\dfrac{n}{2}\) and \(\dfrac{3n}{4}\) for the lower quartile, median and upper quartile.
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, Running totals, smooth curve.. Extended. 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: Building the curve | In the lab, add up the frequencies as you go to fill the cumulative frequency column. Project tab 1 of the lab (Cumulative frequency and histograms lab, cumulative frequency) and let a student drive it while the class predicts. Model the worked example From a frequency table to a curve on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Median, quartiles and IQR | Every reading is the same move: start on the cumulative frequency axis, go across to the curve, then down to the horizontal axis. Model the worked example Read the curve on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Percentiles and how many questions | The \(p\)th percentile is read across at \(p\%\) of \(n\). Model the worked example Percentile on the board, one line at a time. Your turn: A cumulative frequency curve for the marks of 200 students gives 64 marks at the 80th percentile. How many students scored more than 64? | Jigsaw |
| 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, 15 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 Percentiles and how many questions, 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.
Jigsaw
Split the content into expert pieces. Expert groups master one piece each, then regroup so every home team has one expert per piece, and each expert teaches their piece to the team.
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 |
|---|---|
| Plotting at the midpoints (5, 15, 25,...) shifts the whole curve 5 minutes to the left and every reading is wrong. | The cf belongs to the upper boundary. |
| The IQR is the difference of two TIMES read from the horizontal axis (\(38 - 20\)), not the difference of the cumulative frequencies (\(60 - 20 = 40\)). | Return to Median, quartiles and IQR: ask a pair to state the correct version and show it on the lab before moving on. |
| 'How many took MORE than 45 minutes' is the total minus the reading. | Writing 71, the number who took 45 minutes or less, is the most common lost mark on this topic. |
Cumulative frequency diagrams: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
Complete the cumulative frequency table for the masses of 50 parcels.
| Mass \(m\) (kg) | \(0 < m \le 20\) | \(20 < m \le 40\) | \(40 < m \le 60\) | \(60 < m \le 80\) | \(80 < m \le 100\) |
|---|---|---|---|---|---|
| Frequency | 5 | 13 | 22 | 7 | 3 |
Write down the coordinates of the points you would plot for a cumulative frequency curve of the parcels in the previous question.
A cumulative frequency curve for the heights of 120 plants passes through these points.
| Height (cm) | 10 | 15 | 20 | 25 | 30 | 35 | 40 |
|---|---|---|---|---|---|---|---|
| Cumulative frequency | 0 | 12 | 30 | 60 | 90 | 108 | 120 |
Find the median and the interquartile range.
For the plants in the previous question, find (a) the 90th percentile (b) the number of plants taller than 35 cm.
A second batch of plants, grown in shade, has median 28 cm and IQR 16 cm. Compare the two batches, using the results from the earlier question (median 25 cm, IQR 10 cm).
The cumulative frequency curve for the masses of 200 apples passes through (100, 15), (120, 62), (140, 130), (160, 178) and (180, 200), where the first number is the mass in grams.
Apples heavier than 140 g are graded 'large'.
(a) How many apples are large? (b) What percentage of the apples are large? (c) How many apples have a mass between 120 g and 160 g?
Answer key and marking notes
- Cumulative frequencies: 5, 18, 40, 47, 50.
- Check: the last value equals the total, 50.
- (0, 0), (20, 5), (40, 18), (60, 40), (80, 47), (100, 50).
- Each total goes at the upper boundary of its class.
- Median: read at \(\dfrac{120}{2} = 60\), giving 25 cm.
- LQ: read at 30, giving 20 cm. UQ: read at 90, giving 30 cm.
- IQR \(= 30 - 20 = 10\) cm.
- (a) 90% of 120 = 108, which the curve reaches at 35 cm.
- (b) 108 plants are 35 cm or shorter, so \(120 - 108 = 12\) are taller.
- The shade plants are taller on average (median 28 cm against 25 cm).
- The first batch is more consistent in height (IQR 10 cm against 16 cm).
- (a) 130 apples are 140 g or less, so \(200 - 130 = 70\) are large.
- (b) \(\dfrac{70}{200} \times 100 = 35\%\).
- (c) \(178 - 62 = 116\) apples.
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.
The cumulative frequency diagram shows the masses of 120 eggs. Use it to find the interquartile range of the masses.
- A20 g
- B60 g
- C50 g
- D40 g
Use the cumulative frequency diagram of the masses of 120 eggs to find how many eggs have a mass of more than 70 g.
- A8
- B112
- C22
- D10
In a grouped frequency table of egg masses, the class \(40 < m \le 50\) has frequency 30 and cumulative frequency 60. Which point should be plotted for this class on a cumulative frequency diagram?
- A(50, 60)
- B(45, 60)
- C(50, 30)
- D(40, 60)
A cumulative frequency curve for the finishing times of 200 runners passes through these points.
| Time (min) | 20 | 25 | 30 | 35 | 40 | 45 |
|---|---|---|---|---|---|---|
| Cumulative frequency | 12 | 50 | 120 | 170 | 190 | 200 |
- A35 minutes
- B170 minutes
- C30 minutes
- D40 minutes
The cumulative frequency diagram shows the masses of 120 eggs from Farm A. Eggs from Farm B have median 54 g and interquartile range 12 g. Which statement is correct?
- AFarm B's eggs are heavier on average and more consistent in mass.
- BFarm A's eggs are heavier on average, but Farm B's are more consistent.
- CFarm B's eggs are heavier on average, but Farm A's are more consistent.
- DFarm A's eggs are more consistent, because their IQR is larger.
Use the cumulative frequency diagram of the masses of 120 eggs to find how many eggs have a mass between 40 g and 60 g.
- A60
- B90
- C30
- D20
Quiz key
\(n = 120\), so read across at \(\dfrac{120}{4} = 30\) and \(\dfrac{3 \times 120}{4} = 90\). Going down from the curve: LQ = 40 g and UQ = 60 g, so IQR \(= 60 - 40 = 20\) g. The tempting 60 is \(90 - 30\), the difference of the cumulative frequencies instead of the masses. 50 g is the median (read at 60), and 40 g is only the lower quartile.
Up from 70 g to the curve, then across: the cumulative frequency is 112, so 112 eggs have a mass of 70 g or less. More than 70 g: \(120 - 112 = 8\) eggs. The tempting 112 is the number at or BELOW 70 g. 22 is the number between 60 g and 70 g (\(112 - 90\)), and 10 is \(80 - 70\), a difference of masses, not a number of eggs.
60 eggs have a mass of 50 g or less, so the cumulative frequency 60 is plotted at the upper class boundary, 50: the point (50, 60). The tempting (45, 60) uses the midpoint, which shifts the whole curve 5 g to the left. (50, 30) plots the frequency instead of the running total, and (40, 60) uses the lower boundary.
85% of 200 \(= 0.85 \times 200 = 170\). The curve reaches a cumulative frequency of 170 at 35 minutes, so the 85th percentile is 35 minutes: 85% of the runners finished within 35 minutes. The tempting 170 is the cumulative frequency, not the time. 30 minutes is where the curve reaches 120 (the 60th percentile), and 40 minutes is where it reaches 190 (the 95th percentile).
From the curve, Farm A has median 50 g, LQ 40 g and UQ 60 g, so its IQR is 20 g. Average: \(54 > 50\), so Farm B's eggs are heavier on average. Spread: \(12 < 20\), so Farm B's masses are more consistent. The tempting third option thinks a larger IQR means more consistent; it is the opposite, since a larger IQR means the middle half of the masses is more spread out.
At 60 g the cumulative frequency is 90, and at 40 g it is 30. So \(90 - 30 = 60\) eggs have a mass between 40 g and 60 g (this is the middle half, between the quartiles). The tempting 90 is the number of eggs of 60 g or less, which includes those lighter than 40 g. 30 is only the number up to 40 g, and 20 is \(60 - 40\), a difference of masses.
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.