At a glance. One lesson of about 60 minutes, following the estimated mean. Students build the running totals, plot at upper boundaries, read and interpolate quartiles and percentiles, and apply the 1.5 times IQR fences.
Calculator: no calculator. The arithmetic is short, and the reading is by interpolation between plotted points. No calculator is needed.
What students will learn
Know
- Plot the running total at the upper boundary of each class.
- A quartile or percentile is read at a count: n ÷ 4, n ÷ 2, 3n ÷ 4 or P ÷ 100 of n.
- An outlier is beyond Q₁ − 1.5 × IQR or Q₃ + 1.5 × IQR.
Understand
- The curve holds the whole distribution, so any percentile can be read from it.
- A steep section means many values in a short range.
- Interpolating assumes values are spread evenly inside a class, so the reading is an estimate.
Do
- Build a cumulative frequency table and curve.
- Estimate the median, quartiles, percentiles and the percentage above or below a value.
- Test for outliers with the fences.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: running totals, reading and interpolating percentiles, the outlier fences (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if. Students link the steepness of the curve to the IQR and generalize.
- Criterion C. Communicating: labelled axes, points plotted at upper boundaries, and dashed construction lines in the figure and in student work.
- Criterion D. Applying mathematics in real-life contexts: the rainfall hook, reading the value reached on one day in ten.
The one thing to do with the figure
Ask for the count before the slide. At 50%, ask: “How many of the 80 days is that, and which piece of the curve will it fall on?” Only then let a student move the slider. The habit to build is share, then count, then read across.
Then ask why 25% and 75% land exactly on a plotted boundary but 50% does not, and what that says about how reliable a read-off between points is.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the rain gauge. Collect predictions of where the curve is steepest. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, return to the plotting rule. |
| 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 | 30; 12; 28. |
|---|---|
| Hook | Steepest in the class with the most days; a value reached on one day in ten is the 90th percentile. |
| Worked examples | Q₁ ≈ 25.9, median ≈ 33.3, Q₃ ≈ 39.6; 90th percentile ≈ 47.3 cm; fences about 5 and 60, so no outliers. |
| Faded example | 31.3. |
| 1. Point plotted for a class | B. A uses the midpoint; C uses the class frequency; D uses the lower boundary. |
| 2. Class frequency from running totals | 16. 27 − 11. |
| 3. Count at the lower quartile | 15. 60 ÷ 4. |
| 4. Interpolating a median | 41.7. 40 + 2 ÷ 12 × 10. |
| 5. Percentage above a value | 36.8. (50 − 31.6) ÷ 50. |
| 6. Upper fence for outliers | 66.5. 41 + 1.5 × 17. |
| 7. Values between two percentiles | 48. 40% of 120. |
| 8. Deciding which value is an outlier | A. B uses 1.5 × Q₃; C uses 1 × IQR; D adds 1.5. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Plotted at the midpoint, plotted the frequency, or plotted at the lower boundary. | Ask: when has the whole class been counted? |
| 27 or 38 in question 2 | Gave the running total, or added two running totals. | Draw a number line with the two boundaries and shade the class. |
| 25 or 45 in question 3 | Used the percentage as a count, or the upper quartile. | Always write the count: 60 ÷ 4. |
| 45 or 42 in question 4 | Gave the class midpoint, or treated values as units of x. | Write the fraction of the climb, then apply it to the width. |
| 63.2 or 31.6 in question 5 | Gave the percentage below, or the count below. | Underline 'greater than'; subtract from n. |
| 25.5, 42.5 or 61.5 in question 6 | Left out Q₃, added 1.5, or multiplied Q₃. | IQR first, in its own line. |
| 40 or 22 in question 7 | Used the percentage as a count, or the difference in x. | Which axis answers 'how many'? |
| B, C or D in question 8 | Multiplied Q₃, left out the 1.5, or added 1.5. | Write IQR, then 1.5 × IQR, then the fence, each on its own line. |
Questions worth asking
- Why must the first point be a count of 0 at the lower boundary? Because no values are below the smallest possible value; the curve must start at zero.
- Can two different data sets have the same cumulative frequency curve? Yes, if they have the same classes and frequencies, but only the grouped values are known.
- Why can the 1.5 × IQR rule flag a value that is not an error? It flags values far from the middle half; they may be genuine and informative.
- Why is a percentile read from a graph only an estimate? The curve assumes the values are spread evenly inside each class.
Differentiation
A way in for anyone stuck
- Give a pre-drawn grid with the upper boundaries marked on the horizontal axis.
- Have students write the count to look for before touching the graph.
- Provide a three-line frame for interpolation: lower point, upper point, how far along.
Stretch
- Show that for a straight piece from (a, c) to (b, d) the value at count k is a + (k − c) ÷ (d − c) × (b − a).
- Compare the box plot drawn from the curve with one drawn from the raw data and say where they differ.
Students learning in English as an additional language
“Cumulative” means adding up as you go, a running total. “Percentile” means the value below which that percentage of the data lies. “Fence” here is a boundary beyond which a value is an outlier.
| Word | Plain meaning | Example |
|---|---|---|
| cumulative frequency | the running total of frequencies up to a class boundary | a total of 7 up to the first boundary |
| upper boundary | the boundary at the top end of a class, where its running total is plotted | 40 for the class 30 ≤ x < 40 |
| percentile | the value below which a given percentage of the data lie | 12 s, if 30% of the times are under 12 s (the 30th percentile) |
| interquartile range | the upper quartile minus the lower quartile | IQR = Q3 − Q1 |
| outlier | a value beyond the fences, far from the rest of the data | above Q3 + 1.5 × IQR |
| interpolate | estimate a value between two known points along a straight line | halfway between two plotted points |
Sentence starters for explaining aloud or in writing:
- The count I need is … because …
- The point is plotted at … because …
- The value is an estimate because …
Two things not to say
“Read the percentage on the vertical axis.” The vertical axis is a count unless the axis is labelled as a percentage.
“Plot the point in the middle of the class.” The running total is known only at the upper boundary.
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.
- ReadingA curve has n = 120. At which count do you read the upper quartile?
90. Three quarters of 120. - TableRunning totals at boundaries 10, 20 and 30 are 4, 9 and 15. How many values are in the class 20 to under 30?
6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA river gauge in Zambia records 150 daily levels. The 10th percentile is 1.2 m and the 90th is 3.1 m. How many days had levels between them?
120. 80% of 150. - Earlier work (Lesson 1)A table of 20 values has estimated mean 12 and every class 4 wide. Between what limits is the true mean?
Between 10 and 14.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningQ₁ = 12 and Q₃ = 20. A value of 35 is recorded. Is it an outlier?
The fence is 20 + 1.5 × 8 = 32, and 35 is above it: yes. - GeneralizeShow that for the outlier fences, adding 5 to every value moves both fences by 5 but leaves their distance apart unchanged.
Q₁ and Q₃ both rise by 5, IQR is unchanged, so both fences rise by 5 and the gap 2 × 1.5 × IQR + IQR stays the same.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is required; a ruler helps students who read from the printed graph.