The shape of the lesson
By the end of the lesson, learners can
- Mean, median and mode from a table: Open the lab's frequency table of goals scored.
- Grouped data: an estimate: Extended only.
- Working backwards with totals: Many exam questions give the mean and ask for something else.
- Mean: \(\dfrac{\sum fx}{\sum f}\).
- Median: position \(\dfrac{n+1}{2}\), then count down the running totals.
- Mode: the value (or class) with the highest frequency.
- Grouped (Extended): use midpoints, and say 'estimate'.
- Backwards: total = mean \(\times\) number; add totals, never means.
One picture every idea returns to
Mean \(= \dfrac{\sum fx}{\sum f}\): multiply each value by its frequency, add, then divide by the TOTAL frequency. For grouped data (Extended) use the midpoint of each class as \(x\), and call the answer an estimate.
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, Every row counts.. When data come in a frequency table, each value stands for several people. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Mean, median and mode from a table | Open the lab's frequency table of goals scored. Project tab 2 of the lab (Averages and charts lab, frequency tables) and let a student drive it while the class predicts. Model the worked example All four from a table on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Grouped data: an estimate | Extended only. Model the worked example Estimate the mean (Extended) on the board, one line at a time. Your turn: (Extended) Ages: \(10 < a \le 20\): 3 people, \(20 < a \le 30\): 7 people, \(30 < a \le 50\): 10 people. Estimate the mean age. | Pairs |
| 21 to 30 min | Teach and check: Working backwards with totals | Many exam questions give the mean and ask for something else. Model the worked example Combining two groups on the board, one line at a time. | Round Table |
| 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 Working backwards with totals, 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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| Divide \(\sum fx\) by the total FREQUENCY (25 matches), not by the number of rows (5). | And the mode is the value 1, not its frequency 8. |
| Use the MIDPOINT, not the upper bound of each class. | Using 10, 20, 30, 40 in the example gives 26 minutes, which is too big. Note also that the last class here is 20 wide, so its midpoint is 40, not 35. |
| The mean of the two means, \(\dfrac{64 + 59}{2} = 61.5\), is wrong because the classes are different sizes. | Combine TOTALS, then divide by the total number of students. |
Averages from frequency tables: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
The table shows the numbers of goals scored by a team in 20 matches.
| Goals | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 4 | 6 | 5 | 3 | 1 | 1 |
Find (a) the mode (b) the median (c) the range.
Calculate the mean number of goals for the table in the previous question.
(Extended) The masses, \(m\) kg, of 40 parcels are shown.
| Mass | \(0 < m \le 20\) | \(20 < m \le 40\) | \(40 < m \le 60\) | \(60 < m \le 100\) |
|---|---|---|---|---|
| Frequency | 6 | 14 | 16 | 4 |
Calculate an estimate of the mean mass.
(Extended) For the parcels in the previous question, write down (a) the modal class (b) the class that contains the median.
| \(x\) | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| \(f\) | 6 | \(x\) | 8 | 4 |
The mean of this distribution is 3.4. Show that the missing frequency is 7.
A class has 30 students. The 18 girls do a mean of 6.5 hours of homework a week. The mean for the whole class is 6.1 hours. Find the mean for the boys.
Answer key and marking notes
- (a) Highest frequency 6, so the mode is 1 goal.
- (b) \(n = 20\): median halfway between the 10th and 11th values. Running totals 4, 10, 15: the 10th is 1 and the 11th is 2, so the median is 1.5.
- (c) Range \(= 5 - 0 = 5\).
- \(\sum fx = 0 + 6 + 10 + 9 + 4 + 5 = 34\).
- Mean \(= \dfrac{34}{20} = 1.7\) goals.
- Midpoints: 10, 30, 50, 80 (the last class is 40 kg wide).
- \(6 \times 10 + 14 \times 30 + 16 \times 50 + 4 \times 80 = 60 + 420 + 800 + 320 = 1600\).
- Estimated mean \(= \dfrac{1600}{40} = 40\) kg.
- (a) Highest frequency 16: \(40 < m \le 60\). (Its width, 20 kg, matches the first three classes.)
- (b) Median position \(\dfrac{40+1}{2} = 20.5\). Running totals 6, 20, 36, so the 20.5th value is in \(40 < m \le 60\).
- \(\sum fx = 12 + 3x + 32 + 20 = 64 + 3x\) and \(\sum f = 18 + x\).
- \(64 + 3x = 3.4(18 + x) = 61.2 + 3.4x\).
- \(2.8 = 0.4x\), so \(x = 7\). Check: \(\dfrac{85}{25} = 3.4\).
- Class total \(= 30 \times 6.1 = 183\) hours. Girls' total \(= 18 \times 6.5 = 117\) hours.
- Boys' total \(= 183 - 117 = 66\) hours, shared by \(30 - 18 = 12\) boys.
- Boys' mean \(= \dfrac{66}{12} = 5.5\) hours.
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 table shows the number of pets owned by 30 students.
| Pets \(x\) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Frequency \(f\) | 3 | 7 | 9 | 6 | 5 |
- A3.1
- B18.6
- C3
- D6.2
The numbers of books read by 28 students in a month are shown.
| Books | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 5 | 8 | 6 | 4 | 3 | 2 |
- A2
- B2.5
- C14.5
- D1
| Score | 10 | 11 | 12 | 13 | 14 |
|---|---|---|---|---|---|
| Frequency | 0 | 4 | 7 | 5 | 2 |
- A3
- B4
- C7
- D18
In a club, 12 juniors have a mean time of 15 minutes for a run and 18 seniors have a mean time of 20 minutes. Find the mean time of all 30 runners.
- A18 minutes
- B17.5 minutes
- C17 minutes
- D270 minutes
(Extended) The lengths, \(l\) cm, of 40 fish are shown.
| Length | Frequency |
|---|---|
| \(0 < l \le 10\) | 8 |
| \(10 < l \le 20\) | 14 |
| \(20 < l \le 30\) | 10 |
| \(30 < l \le 50\) | 8 |
- A20.5 cm
- B26.5 cm
- C19.5 cm
- D205 cm
| \(x\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(f\) | 5 | 9 | \(k\) | 4 |
- A7
- B2.1
- C4.2
- D3
Quiz key
\(\sum fx = 1 \times 3 + 2 \times 7 + 3 \times 9 + 4 \times 6 + 5 \times 5 = 3 + 14 + 27 + 24 + 25 = 93\) and \(\sum f = 30\), so the mean is \(\dfrac{93}{30} = 3.1\) pets. The tempting 18.6 divides by the 5 columns instead of the 30 students. 3 is the mean of the values 1 to 5, ignoring the frequencies, and 6.2 divides by \(1 + 2 + 3 + 4 + 5 = 15\).
\(n = 28\), so the median is at position \(\dfrac{29}{2} = 14.5\), halfway between the 14th and 15th values. Running totals: 5 (0 books), 13 (up to 1 book), 19 (up to 2 books). The 14th and 15th values are both 2, so the median is 2. The tempting 2.5 is the middle of the row of values 0 to 5, ignoring how often each occurs. 14.5 is the position, not the value, and 1 is the mode.
Nobody scored 10 (frequency 0), so the smallest score is 11. The largest is 14. Range \(= 14 - 11 = 3\). The tempting 4 uses \(14 - 10\), but 10 is only a heading in the table, not a score anyone got. 7 is the largest frequency minus the smallest (\(7 - 0\)), and 18 is the number of people.
Juniors' total \(= 12 \times 15 = 180\) minutes. Seniors' total \(= 18 \times 20 = 360\) minutes. Combined mean \(= \dfrac{180 + 360}{30} = \dfrac{540}{30} = 18\) minutes. The tempting 17.5 is the mean of the two means, \(\dfrac{15 + 20}{2}\), which is wrong because the groups are different sizes. 17 pairs each group size with the other group's mean, and 270 divides the total by 2 instead of 30.
Midpoints: 5, 15, 25 and 40 (the last class is 20 cm wide). \(\sum f \times \text{midpoint} = 8 \times 5 + 14 \times 15 + 10 \times 25 + 8 \times 40 = 40 + 210 + 250 + 320 = 820\). Estimated mean \(= \dfrac{820}{40} = 20.5\) cm. The tempting 19.5 cm uses 35 as the last midpoint, as if every class were 10 cm wide. 26.5 cm uses the upper bounds, and 205 cm divides by the 4 classes instead of the 40 fish.
\(\sum fx = 5 + 18 + 3k + 16 = 39 + 3k\) and \(\sum f = 18 + k\). So \(39 + 3k = 2.4(18 + k) = 43.2 + 2.4k\), giving \(0.6k = 4.2\) and \(k = 7\). Check: \(\dfrac{60}{25} = 2.4\). The tempting 2.1 comes from multiplying out the bracket as \(43.2 + k\), forgetting to multiply \(k\) by 2.4. 4.2 stops before dividing by 0.6, and 3 is the value of \(x\) above the missing frequency.
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.