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# Stem-and-leaf plots

Data and chance · Data

<!-- Speaker note: Students build ordered stem-and-leaf plots (tens as stems, units as leaves, every stem included, with a key) and read values, the median, mode and range from them (AC9M7ST02, node 1). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 8 min). -->

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## Today

**We are learning to** a stem-and-leaf plot keeps every value and shows the shape: the tens are the stems, the units are the leaves, written in order. Include every stem, even with no leaves, and add a key.

- I can build an ordered stem-and-leaf plot with a key.
- I can read values, the median, the mode and the range from a stem-and-leaf plot.

<!-- Speaker note: Before class: Do the Run board first. Students need Mean, median, mode and range. This is a hinge: describing and comparing distributions (next lesson) reads shape from these plots. The common mistake is leaving out a stem with no leaves, which hides gaps; it is planted in Jo's working. Others: unordered leaves, and reading a leaf as a value on its own (the board's key helps). The board shows the dot plot and the stem-and-leaf plot together in Run and Make, so students can see they show the same data. -->

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## Why it matters

> Thirty homework times in a list tell you very little. Sort them into a stem-and-leaf plot and the shape jumps out, and you can still read every value.

<!-- Speaker note: A **stem-and-leaf plot** is a display that keeps every value. For two-digit numbers: - the **stem** is the tens digit, written once down the left; - the **leaves** are the units digits, written in order along each stem's row; - a **key** says how to read it: $3 \mid 5$ means $35$. Turned on its side, it looks like a bar graph, so you can see the shape of the data and still read every value. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**In a stem-and-leaf plot, a row reads $3 \mid 1\ 5\ 5\ 8$. Which values does it show?**

- 31, 35, 35 and 38
- The number 31 558
- 3, 1, 5, 5 and 8
- 3, 15 and 58

<!-- Speaker note: What happens: 31, 35, 35 and 38. The stem 3 is the tens; each leaf is the units of one value. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands; students who chose 3, 1, 5, 5, 8 read digits, not values. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fdata%2Fstem-and-leaf%2Fstep-2&code=widget%3A+data%0Avalues%3A+23%2C+31%2C+35%2C+35%2C+38%2C+42%2C+47%2C+51%0Aview%3A+both%0Aedit%3A+on%0Ashow%3A+none%0Alabel%3A+Minutes+spent+on+homework%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/data/stem-and-leaf/?full=step-2#step-2)

<!-- Speaker note: The dot plot and the stem-and-leaf plot show the same data. Which is easier to read exact values from? -->

<!-- Speaker note: Teacher: ask where 29 goes and why it goes at the end of its row. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 31, 35, 35 and 38. The stem 3 is the tens; each leaf is the units of one value.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Build, then read

Here is how to build one, and read from it.

Now read the median of the plot in the picture.

$$23, 31, 35, 35, 38, 42, 47, 51$$

1) $\text{stems: } 2, 3, 4, 5$ — The tens digits, from smallest to largest, none missed out.
2) $3 \mid 1\ 5\ 5\ 8$ — Each value's units digit goes on its stem's row, in order.
3) $\text{Key: } 3 \mid 5 = 35$ — A key says how to read the plot.
4) $\text{median} = \tfrac{35 + 38}{2} = 36.5$ — 8 values: count in to the 4th and 5th leaves.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 23, 31, 35, 35, 38, 42, 47, 51 and say "Sort the values first (or sort the leaves at the end)." Line 1: write stems: 2, 3, 4, 5 and say "The tens digits, from smallest to largest, none missed out." Line 2: write 3 1558 and say "Each value's units digit goes on its stem's row, in order." Line 3: write Key: 3 5 = 35 and say "A key says how to read the plot." Line 4: write median = 35 + 38 ÷ 2 = 36.5 and say "8 values: count in to the 4th and 5th leaves." -->

<!-- Speaker note: Teacher: build a plot of the class's own data (arm spans, say) on the board. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Build, then read

**What is the median of the data in this plot?**

* Answer: 27

<!-- Speaker note: 11 values, so the median is the 6th: 14, 18, 21, 23, 23, 27. -->

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## Find the mistake

**Jo draws a stem-and-leaf plot. Click the line where her working goes wrong.**

```
 1  Values: 12, 15, 21, 27, 43, 48.
 2  The stems are the tens digits.
 3  Write the stems that have values: 1, 2, 4.
 4  Leaves: 1 | 2 5, 2 | 1 7, 4 | 3 8.
```

* Answer: line 3

<!-- Speaker note: Line 3: every stem from the smallest to the largest goes in, even with no leaves: 1, 2, 3, 4. The empty 3 row shows the gap between the 20s and the 40s. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Range from the plot

**What is the range of the data?**

* Answer: 31

<!-- Speaker note: Highest 45, lowest 14: 45 − 14 = 31. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Build one:** Each task changes one thing.
- **The mode:** 
- **Count the values:** 
- **Use the key:** 
- **Three digits:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/data/stem-and-leaf/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: build a plot on the board, read the mode from a different plot, count the values, use the key, then the edge case of three-digit values. Hints, in order: Type the values in the box: 8, 15, 17, 22, 26, 26, 31. For students who need more support: Sort the values first, then write them in rows by tens before making the plot. For early finishers: Draw a back-to-back stem-and-leaf plot for two classes' scores (leaves going left for one, right for the other). When students get stuck: A student leaves out an empty stem (Jo's mistake). Ask: Are there any tens between your smallest and largest stems with no values? Fix: Write every stem in order; leave its row empty. / A student reads a leaf as a value. Ask: What does the key say 3 | 5 means? Fix: Stem and leaf together make one value. / A student writes leaves out of order. Ask: Can you find the median easily from this row? Fix: Sort the values first, or the leaves at the end. -->

<!-- Speaker note: Teacher: before the three-digit task, ask what the stems should be for 105. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Median

**What is the median?**

* Answer: 63

<!-- Speaker note: 5th and 6th values are both 63, so the median is 63. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Range

**What is the range?**

* Answer: 27

<!-- Speaker note: 79 − 52 = 27. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Which stems

**Values $41, 44, 62, 65$. Which stems should the plot have?**

- 4 and 6
- 1, 2, 4 and 5
- 4, 5 and 6 (5 has no leaves)

* Answer: 4, 5 and 6 (5 has no leaves)

<!-- Speaker note: Every stem from 4 to 6, so the empty 5 row shows the gap. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: mean

**Review: find the mean of $23, 31, 35, 35$.**

* Answer: 31

<!-- Speaker note: 124 ÷ 4 = 31. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a plot with a given median

Then: could you do it with no value of 35 at all?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/data/stem-and-leaf/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building eight two-digit values with a median of 35 reverses reading: the 4th and 5th leaves must average 35. Hints, in order: 8 values: the median is halfway between the 4th and 5th. / Make the 4th and 5th values 34 and 36, or 35 and 35. -->

<!-- Speaker note: Teacher: collect answers; ask whether 35 has to appear. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Draw a stem-and-leaf plot of 17, 25, 22, 41, 18, 29, 26 and find its median.**

* 1 | 7 8, 2 | 2 5 6 9, 3 | (empty), 4 | 1. Median: the 4th of 7 values, 25.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: Comparing data sets (AC9M7ST02 nodes 2 and 3). Where it leads: back-to-back plots, histograms and grouped data (Year 8 and 9). -->