Stem-and-leaf plots
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.
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.
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: means .
Turned on its side, it looks like a bar graph, so you can see the shape of the data and still read every value.
- Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
- Try: Try it on the board and compare the result with your prediction.
- Investigate: Work out why it happens, one step at a time.
- Modify: Change the problem on purpose and see what happens.
- Practise: A few on your own, to check you have it.
- Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
Read a row
How many values are in that row?
Add two values
The dot plot and the stem-and-leaf plot show the same data. Which is easier to read exact values from?
Add 44 and 29 to the data (type them in the box). Where does each go? Was your prediction right about reading a row?
Use the board. This ticks when you get there.
Build and read
Here is how to build one, and read from it.
- Sort the values first (or sort the leaves at the end).
- The tens digits, from smallest to largest, none missed out.
- Each value's units digit goes on its stem's row, in order.
- A key says how to read the plot.
- 8 values: count in to the 4th and 5th leaves.
Now read the median of the plot in the picture.
Read Jo's working. Which stem is missing, and why does it matter?
What is the median of the data in this plot?
Jo draws a stem-and-leaf plot. Click the line where her working goes wrong.
What is the range of the data?
One change at a time
Each task changes one thing.
Need a hint?
Type the values in the box: 8, 15, 17, 22, 26, 26, 31.
Build the stem-and-leaf plot for these quiz scores: .
Use the board. This ticks when you get there.
What is the mode?
How many values are in this plot?
Read the plot
What is the median?
What is the range?
Review: find the mean of .
A plot with a given median
Then: could you do it with no value of 35 at all?
Need a hint?
8 values: the median is halfway between the 4th and 5th.
Another hint
Make the 4th and 5th values 34 and 36, or 35 and 35.
Make a stem-and-leaf plot of 8 two-digit values with a median of 35.
Use the board. This ticks when you get there.
For teachers
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.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- 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.
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.
Running the lesson
- Predict: take a show of hands; students who chose 3, 1, 5, 5, 8 read digits, not values.
- Run: ask where 29 goes and why it goes at the end of its row.
- Investigate: build a plot of the class's own data (arm spans, say) on the board.
- Modify: before the three-digit task, ask what the stems should be for 105.
- Make: collect answers; ask whether 35 has to appear.
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.
Stretch and support
Stretch: Draw a back-to-back stem-and-leaf plot for two classes' scores (leaves going left for one, right for the other).
Support: Sort the values first, then write them in rows by tens before making the plot.
Where it 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).
What each step is for
- read a row: Committing to what a row means before building one tests whether they combine the stem and each leaf into one value.
- add two values: Adding values and watching them land in the right row, in order, shows the plot's rule: the tens choose the row, the units the place.
- build and read: Stems in order with none missed, leaves in order, a key; then the median, mode and range read straight off. Jo's mistake leaves out a stem with no leaves, which hides a gap.
- one change at a time: 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.
- read the plot: Reading the median, range and number of values from plots, mixed; the missing-stem slip in a question; the review keeps the mean fresh.
- a plot with a given median: Building eight two-digit values with a median of 35 reverses reading: the 4th and 5th leaves must average 35.
- Build an ordered stem-and-leaf plot with a key.
- Read values, the median, the mode and the range from a stem-and-leaf plot.