The shape of the lesson
By the end of the lesson, learners can
- Frequency density: In the lab's histogram tab, tick 'show the mistake: height = frequency' and see how the wide 30 to 50 class towers over the others.
- Reading a histogram: area: To get information out of a histogram, turn heights back into frequencies: frequency = frequency density \(\times\) class width.
- When the scale is hidden: Harder questions give a histogram with no numbers on the frequency density axis, plus the frequency of one class.
- Width = upper \(-\) lower boundary of each class.
- Draw: frequency density = frequency \(\div\) width.
- Read: frequency = frequency density \(\times\) width (the area).
- Part of a class: use only the width inside your range.
- Hidden scale: use a known class to find what one unit of height is worth.
One picture every idea returns to
Frequency density \(= \dfrac{\text{frequency}}{\text{class width}}\). Plot frequency density on the vertical axis; then frequency \(=\) frequency density \(\times\) class width, which is the area of the bar.
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, Area is frequency.. 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: Frequency density | In the lab's histogram tab, tick 'show the mistake: height = frequency' and see how the wide 30 to 50 class towers over the others. Project tab 2 of the lab (Cumulative frequency and histograms lab, histograms) and let a student drive it while the class predicts. Model the worked example Draw a histogram on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Reading a histogram: area | To get information out of a histogram, turn heights back into frequencies: frequency = frequency density \(\times\) class width. Model the worked example A frequency from a bar on the board, one line at a time. Your turn: Use Fig. 1 to estimate the number of parcels heavier than 60 kg. | Pairs |
| 21 to 30 min | Teach and check: When the scale is hidden | Harder questions give a histogram with no numbers on the frequency density axis, plus the frequency of one class. Model the worked example Find the scale on the board, one line at a time. | Rally Coach |
| 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, 18 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 When the scale is hidden, 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.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
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 |
|---|---|
| The 30 to 50 class has the biggest frequency (24) but NOT the tallest bar: its density is only 1.2. | Plotting frequency instead of frequency density is the mistake examiners look for first. |
| Using the whole class when only part is wanted overcounts: 25 to 40 kg is NOT \(22 + 24 = 46\). | Use only the widths 5 and 10 inside the range. |
| Comparing bar heights alone gives \(\dfrac{6}{4.5} \times 30 = 40\), which ignores that the second class is only half as wide. | Always multiply by the class width. |
Histograms with unequal class widths: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
The times, \(t\) seconds, taken by 65 students to solve a puzzle are shown. Calculate the frequency density of each class.
| Time | \(0 < t \le 20\) | \(20 < t \le 30\) | \(30 < t \le 35\) | \(35 < t \le 40\) | \(40 < t \le 60\) |
|---|---|---|---|---|---|
| Frequency | 12 | 18 | 14 | 11 | 10 |
For the puzzle times in the previous question, which is the modal class? Explain why it is not \(20 < t \le 30\).
Using the puzzle times, estimate the number of students who took between 25 and 38 seconds.
A histogram has bars for \(0 < d \le 5\) with frequency density 4 and \(5 < d \le 15\) with frequency density 2.5. How many values are there in these two classes altogether?
Calculate an estimate of the mean time for the 65 students in the first question.
A histogram shows the ages of people at a concert, but the frequency density axis has no scale. The bar for \(20 < a \le 30\) represents 60 people and is 4 cm tall. The bar for \(30 < a \le 50\) is 3 cm tall, and the bar for \(50 < a \le 80\) is 0.8 cm tall.
(a) How many people are aged \(30 < a \le 50\)? (b) How many are aged \(50 < a \le 80\)? (c) Estimate how many people are aged between 40 and 60.
Answer key and marking notes
- Widths: 20, 10, 5, 5, 20.
- Frequency densities: \(12 \div 20 = 0.6\), \(18 \div 10 = 1.8\), \(14 \div 5 = 2.8\), \(11 \div 5 = 2.2\), \(10 \div 20 = 0.5\).
- The modal class has the highest frequency DENSITY: \(30 < t \le 35\) (2.8).
- \(20 < t \le 30\) has the highest frequency, 18, but it is twice as wide, so its values are less crowded.
- 25 to 30 s: \(5 \times 1.8 = 9\).
- 30 to 35 s: the whole class, 14. 35 to 38 s: \(3 \times 2.2 = 6.6\).
- Total \(= 9 + 14 + 6.6 = 29.6\), so about 30 students.
- \(4 \times 5 = 20\) and \(2.5 \times 10 = 25\).
- Total \(= 20 + 25 = 45\).
- Midpoints: 10, 25, 32.5, 37.5, 50.
- \(12 \times 10 + 18 \times 25 + 14 \times 32.5 + 11 \times 37.5 + 10 \times 50 = 120 + 450 + 455 + 412.5 + 500 = 1937.5\).
- Mean \(\approx \dfrac{1937.5}{65} = 29.8\) seconds (3 s.f.)
- First bar: frequency density \(= 60 \div 10 = 6\), shown by 4 cm, so 1 cm represents 1.5.
- (a) Density \(= 3 \times 1.5 = 4.5\); frequency \(= 4.5 \times 20 = 90\) people.
- (b) Density \(= 0.8 \times 1.5 = 1.2\); frequency \(= 1.2 \times 30 = 36\) people.
- (c) 40 to 50: \(10 \times 4.5 = 45\). 50 to 60: \(10 \times 1.2 = 12\). Total \(\approx 45 + 12 = 57\) people.
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 histogram shows the waiting times of 100 customers at a bank. How many customers waited more than 20 minutes?
- A60
- B4.5
- C45
- D75
Use the histogram of the waiting times of 100 customers to estimate how many customers waited between 12 and 25 minutes.
- A44
- B65
- C10
- D59
The class \(15 < x \le 40\) has frequency 60. What is the height of its bar in a histogram?
- A2.4
- B1.5
- C1500
- D4
The table shows the distances, \(d\) km, that 69 people travel to work.
| Distance | \(0 < d \le 10\) | \(10 < d \le 30\) | \(30 < d \le 35\) | \(35 < d \le 50\) |
|---|---|---|---|---|
| Frequency | 12 | 30 | 9 | 18 |
- A\(30 < d \le 35\)
- B\(10 < d \le 30\)
- C\(35 < d \le 50\)
- D\(0 < d \le 10\)
A histogram of the ages of people at a festival has no scale on the frequency density axis. The bar for \(10 < a \le 20\) represents 40 people and is 5 cm tall. The bar for \(20 < a \le 50\) is 2 cm tall. How many people does it represent?
- A48
- B16
- C32
- D60
Use the histogram of the waiting times of 100 customers to calculate an estimate of the mean waiting time.
- A25.1 minutes
- B30.8 minutes
- C20.4 minutes
- D502.5 minutes
Quiz key
The bar from 20 to 30 minutes has height 3 and width 10: \(3 \times 10 = 30\). The bar from 30 to 50 minutes has height 1.5 and width 20: \(1.5 \times 20 = 30\). Total \(= 30 + 30 = 60\) customers. The tempting 45 treats the last bar as 10 wide, like the others; 75 uses 30 as its width (from 20 to 50). 4.5 adds the heights, which are frequency densities, not frequencies.
12 to 15 min: width 3 of the bar with height 3, so \(3 \times 3 = 9\). 15 to 20 min: the whole bar, \(5 \times 4 = 20\). 20 to 25 min: width 5 of the bar with height 3, so \(5 \times 3 = 15\). Total \(\approx 9 + 20 + 15 = 44\) customers. The tempting 65 adds the whole of the three classes (\(15 + 20 + 30\)), counting people outside the range. 59 uses all of the 20 to 30 class, and 10 adds the heights 3, 4 and 3.
Class width \(= 40 - 15 = 25\), so frequency density \(= \dfrac{60}{25} = 2.4\). The tempting 1.5 divides by the upper boundary, 40, instead of the width. 4 divides by the lower boundary, 15, and 1500 multiplies the frequency by the width instead of dividing.
Frequency densities: \(12 \div 10 = 1.2\), \(30 \div 20 = 1.5\), \(9 \div 5 = 1.8\), \(18 \div 15 = 1.2\). The highest is 1.8, so the modal class is \(30 < d \le 35\): it would be the tallest bar on a histogram. The tempting \(10 < d \le 30\) has the highest FREQUENCY, but it is four times as wide, so its values are less crowded. The other two classes both have density 1.2.
First bar: frequency density \(= 40 \div 10 = 4\), shown by 5 cm, so 1 cm stands for \(4 \div 5 = 0.8\). Second bar: frequency density \(= 2 \times 0.8 = 1.6\), and its width is \(50 - 20 = 30\), so the frequency is \(1.6 \times 30 = 48\) people. The tempting 16 compares heights only (\(\dfrac{2}{5} \times 40\)), ignoring that the second class is three times as wide. 32 multiplies by 20 instead of the width 30, and 60 uses the height in cm as the frequency density (\(2 \times 30\)).
Frequencies from the areas: \(0.5 \times 10 = 5\), \(3 \times 5 = 15\), \(4 \times 5 = 20\), \(3 \times 10 = 30\), \(1.5 \times 20 = 30\) (total 100). Midpoints: 5, 12.5, 17.5, 25, 40. \(\sum f \times \text{midpoint} = 25 + 187.5 + 350 + 750 + 1200 = 2512.5\). Mean \(\approx \dfrac{2512.5}{100} = 25.1\) minutes (3 s.f.). The tempting 30.8 uses the upper class boundaries instead of the midpoints. 20.4 uses the frequency densities as if they were frequencies, and 502.5 divides by the 5 classes instead of the 100 customers.
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.