At a glance. One lesson of about 60 minutes. It assumes the median and quartiles of a short list and running totals. Students tabulate and plot cumulative frequency, read the median and quartiles by interpolating, and find the interquartile range.
Calculator: no calculator. No calculator is needed: the interpolation fractions are chosen to be simple, and the reasoning about the graph is the lesson.
What students will learn
Know
- Cumulative frequency is a running total, plotted at the upper class boundary.
- Quartiles are read at n/4, n/2 and 3n/4.
- Interquartile range = upper quartile minus lower quartile.
Understand
- A count belongs on the vertical axis and a value on the horizontal axis.
- Between two plotted points, a fraction of the count gives the same fraction of the class width.
- The interquartile range ignores the extremes, so it is steadier than the range.
Do
- Build a cumulative frequency table and plot it.
- Read the median, quartiles and the number of values above or below a point.
- Compare the range with the interquartile range.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: running totals, plotting positions, quartiles and interpolation (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to predict how the range and the interquartile range change as the last class stretches.
- Criterion C. Communicating: showing the count, the fraction along the class and the value (the worked example and question 4).
- Criterion D. Applying mathematics in real-life contexts: waiting times, museum visitors, offcuts; deciding which spread measure to quote.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “At which count will you read the median?” and have them write the number before moving the slider. Then ask which stretch of the curve it falls on.
In the second figure, ask what will happen to the interquartile range as the last class is stretched, and take a show of hands before pressing anything.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the ferry queue. Collect predictions about where the answer would be on a graph of running totals. |
| 10 min | First figure and key idea: plot at the upper boundary, then read across and down. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose A, B or D, return to what a running total counts. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 7.5; the third class; 35. |
|---|---|
| Hook | No raw times are needed: the running total at each class end is enough. The answer is read at a count of 10 (a quarter of 40) on the vertical axis and then down to the horizontal axis; for the first figure that is 17.5 minutes. |
| Worked example | Median 40; lower quartile 25, upper quartile 55; interquartile range 30. |
| Faded example | Median 27 cm (count 40 lies between the points (20, 26) and (30, 46)). |
| 1. Which point to plot | C. A plots at the start; B at the midpoint; D plots the class frequency, not the running total. |
| 2. A running total | 26. 6 + 9 + 11. |
| 3. Which count gives the lower quartile | 20. 80 ÷ 4. |
| 4. Interpolating along a segment | 22.5. 20 + 10 × 5 ÷ 20. |
| 5. Interquartile range from two quartiles | 17. 51 − 34. |
| 6. Interquartile range from a cumulative table | 47.5. 70 − 22.5. |
| 7. Values above a given point | 4. 60 − 56. |
| 8. Reading the lower quartile | B. A is the midpoint; C reads the count as the value; D reads the count at the left-hand point. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Plots at the start or the midpoint of the class, or plots the class frequency instead of the running total. | Ask: has the total reached 24 at the start of the class? |
| 11, 30 or 15 in question 2 | Gave one class, the whole total, or stopped early. | Write the running total row under the frequency row. |
| 40, 60 or 25 in question 3 | Used the median count, the upper quartile count, or the percentage. | Ask: what fraction of 80 is below the lower quartile? |
| 25, 27.5 or 17 in question 4 | Used the class midpoint, measured from the wrong end, or gave the count. | Sketch the line segment and mark 17 on the vertical side. |
| 85, 60 or 42.5 in question 5 | Added the quartiles, gave the range, or gave the halfway value. | Shade the middle half on a number line. |
| 55, 27.5 or 92.5 in question 6 | Used the count as a value, used the median minus the lower quartile, or added. | Make students label every reading as a count or a value. |
| 56, 8 or 52 in question 7 | Gave the number below, the whole last class, or the count at 80. | Draw the curve between 80 and 100 and mark 90. |
| A, C or D in question 8 | Used the midpoint, the count, or the count at the left point. | Do the proportion on the segment out loud. |
Questions worth asking
- Why is a running total plotted at the end of a class? Listen for “only at the end of the class are all its values counted”.
- Why does the interquartile range not change when the last class stretches? The quartiles are read from the middle of the curve, and the stretched class lies beyond the upper quartile.
- Why do we assume a straight line between points? The values inside a class are assumed to be spread evenly; it is an estimate.
- When would you choose the range, and when the interquartile range? The range for the full spread of a clean data set; the interquartile range when extreme values might mislead.
Differentiation
A way in for anyone stuck
- Give a partly completed running-total row and a printed grid with the points already plotted.
- Use a checklist: count or value? vertical or horizontal? which two points is it between?
- Mark the quartile counts on the vertical axis before students use the curve.
Stretch
- Show that a curve that is a straight line from (0, 0) to (100, n) has its median at 50 and an interquartile range of 50.
- Two curves cross. What does the crossing point mean, and what does it say about the two groups either side of it?
Students learning in English as an additional language
“Cumulative” means adding up as you go. “Running total” is the same idea. “Quartile” comes from “quarter”: a fourth part. “Interpolate” means to find a value between two known values.
| Word | Plain meaning | Example |
|---|---|---|
| cumulative frequency | a running total of the frequencies up to the end of a class | how many values are at or below a boundary |
| upper boundary | the top end of a class, where its running total is plotted | the right-hand end of the interval |
| lower quartile | the value with a quarter of the data below it | read at a count of n divided by 4 |
| upper quartile | the value with three quarters of the data below it | read at a count of 3n divided by 4 |
| interquartile range | the upper quartile minus the lower quartile; the spread of the middle half | the width of the box in a box plot |
| interpolate | find a value between two plotted points in proportion | a count a third of the way up gives a value a third of the way along |
Sentence starters for explaining aloud or in writing:
- The count I read at is … because ….
- The curve reaches that count between … and ….
- The interquartile range tells me ….
Two things not to say
“Read the number off the side.” The side shows counts; the value is found by going across and down.
“The range is the spread.” The range depends on two values only; the interquartile range describes the middle half.
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.
- Running totalFrequencies 3, 7, 9 and 5. Write the running totals.
3, 10, 19, 24. - Quartile countA curve has 200 values. At what counts are the three quartiles read?
50, 100 and 150.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextOn a bus route in Lagos, 50 journeys have a cumulative frequency curve through (10, 15) and (20, 40). Estimate the value at a count of 25.
10 + 10 × 10 ÷ 25 = 14. - Earlier work (Lesson 1)A table has 3 values in 0 to under 10 and 7 in 10 to under 20. Estimate the mean.
(5 × 3 + 15 × 7) ÷ 10 = 12.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA curve for 100 values has a lower quartile of 18 and an upper quartile of 42. The largest value is changed from 80 to 140. What happens to the range and the interquartile range?
The range goes up by 60. The interquartile range stays 24. - Working backwardsA curve is a straight line from (0, 0) to (60, 120). At what value is the upper quartile?
At count 90, so value 45.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use squared paper and a ruler: plot the six points and read the quartiles with a pencil line across and down.