Lesson 10 · Faster, or just different?
Two kinds of number today, both from the same two stalls. Prizes won, which is counted, and seconds per turn, which is measured. The idea is that a set can sit lower and be more spread out, and those are two separate things.
| Minutes | What happens |
|---|---|
| 0–5 | Warm-up. Counted or measured? Four quick ones. |
| 5–15 | Launch. Prizes first — a bar chart, same as last lesson. Then the times, and why a bar chart will not do. |
| 15–40 | Explore. Plot the times as dots on the shared line, then the questions. |
| 40–50 | Summarise. Quicker and more spread out are two different findings. |
| 50–55 | Exit ticket. |
Why a dot plot and not a bar chart
Twenty times, and almost every one of them different. A bar chart of the raw values would be a row of bars one high. A dot plot puts one dot per value on a number line, so repeats stack and the shape appears. It is the same information, drawn so you can see it.
This replaces grouping into intervals. Intervals are a real technique and they are the right answer for bigger data sets, but they were costing a whole lesson and burying the comparison. A dot plot gets Year 6 to the same insight in five minutes.
The two findings, kept apart
Where the bulk sits. Treasure Bag turns were quicker — its mode is 24 seconds against 30, and most of its dots are left of most of the Spinner Stall’s.
How spread out they are. Treasure Bag range 15 seconds against 10. Quicker on the whole, but much less consistent, with a tail running out to 36.
Students will find one and stop. The question to ask every time is: and what about the spread?
The trap in question 7
Both sets have a 36 in them. One value in common tells you nothing about either distribution — it is the range error from Lesson 9 wearing a different hat.
Differentiation
Support
- The Spinner Stall dot plot already done, as a model.
- Times listed in order, so plotting is reading not sorting.
- Three questions, with a sentence frame.
- Prizes compared from the printed tally, no chart needed.
Core
- Both dot plots, then all seven questions.
- Range worked out for both sets of times.
Extension
- Unequal group sizes — which comparisons survive and which do not.
- Why a dot plot beats a bar chart for measurements.
What we are learning today
We are learning to compare two sets of numbers — counted ones and measured ones — by where they sit and how spread out they are.
I will know I can do it when…
- I can tell a counted number from a measured one;
- I can put a set of measurements on a dot plot;
- I can say which set sits lower, and which set is more spread out;
- I can explain how one set can be quicker and more spread out at the same time.
Set your goal for today
Read the list above. Which one is the biggest stretch for you? Write it down.
Not sure? The last one. Most people find one difference and stop looking.
How hard is each question?
Every question is labelled so you know what kind of thinking it needs.
SF — simple familiar
Read a mode off a tally, or a range off your dot plot.
CF — complex familiar
Compare the two sets in a sentence, and explain what the spread means.
CU — complex unfamiliar
Decide what a single shared value does and does not prove.
Warm-up · could you have a number in between?
“Today you get two charts that look completely different, even though both are about the same twenty Preps. Before we start, one question about numbers.
Pick any two numbers next to each other. Could there be a number in between them?”
1. Prizes a Prep won: 2 and 3. — Nothing in between. You cannot win 2½ prizes. Counted.
2. Seconds a turn took: 24 and 25. — 24½ is a real time. So is 24.37. Measured.
3. Preps at the stall: 18 and 19. — Nothing in between. Counted.
4. How tall a Prep is: 112 cm and 113 cm. — Of course there is. We just rounded. Measured.
What the warm-up is for
It is not a sorting exercise. It is the reason the second half of the lesson uses a different chart, and students should be able to say that reason before they see the dot plot. If they cannot, the dot plot looks like a new rule to memorise instead of an obvious response to the data.
Number 4 is the one to labour
Students classify by how the number was written down rather than by what it is. A height written as 112 looks counted; it is a measurement rounded to the nearest centimetre, and the true height sits somewhere in between. Today’s times are the same — read off a stopwatch to the nearest second, so “29” really means somewhere near 29.
If they are stuck
Ask what half of it would be. Half a prize is nothing; half a second is a real amount of time. It is a weaker test than the in-between question, so use it as a prompt rather than the rule.
Part one · prizes won
Same two stalls as last lesson. Each Prep had five turns. This is how many prizes they won.
| Prizes won | The Spinner Stall | The Treasure Bag |
|---|---|---|
| 0 | 2 | 7 |
| 1 | 5 | 6 |
| 2 | 6 | 4 |
| 3 | 4 | 2 |
| 4 | 2 | 1 |
| 5 | 1 | 0 |
| Preps altogether | 20 | 20 |
“Counted, so a bar chart works, exactly like last lesson. Mode of each? … 2 and 0.
Seven of the twenty Preps at the Treasure Bag won nothing at all. Two did at the Spinner Stall.”
Part two · how long a turn took
“Now the times, in seconds. Here are the Spinner Stall’s twenty.”
26 27 27 28 28 29 29 29 30 30 30 30 31 31 32 32 33 34 35 36
“Measured, so from the warm-up you already know what happens. Try a bar chart of those. One bar for 26, one for 27… nearly every bar is one high. It tells you nothing.
So we put one dot above the number line for each time. Repeats stack up. That is a dot plot, and the shape appears straight away.”
Build the broken bar chart for real
Open Compare Two Sets, press Seconds per turn, and look at the top chart. Nearly every bar is one or two high. Say out loud that this is what measurements always do, then switch to the dot plots on paper. Thirty seconds, and it earns the technique.
Then let them plot the second one
The Treasure Bag’s times are on the worksheet, already in order. Plotting is the quick part; the comparison is the lesson.
Plot the times, then the questions
Prizes won: what is the mode for each stall?
How many Preps won no prizes at all at each stall?
Write one sentence comparing the prizes won at the two stalls.
Now your two dot plots. Which stall’s turns were generally quicker? How can you tell by looking?
What is the range of each set of times? What does the difference between those two ranges tell you?
The Treasure Bag was quicker and more spread out. Explain how both of those can be true at the same time.
One Prep at the Treasure Bag took 36 seconds, which is exactly the same as the slowest turn at the Spinner Stall. Does that mean the two stalls were equally slow? Explain.
2. Two at the Spinner Stall, seven at the Treasure Bag. Both out of twenty, so the counts compare directly.
3. Looking for the Lesson 9 structure: Preps won more at the Spinner Stall — its mode was 2 prizes against 0, and only 2 of its 20 Preps won nothing, against 7 at the Treasure Bag.
4. The Treasure Bag. Its dots sit further left on the same line — most of them between 21 and 27, while most of the Spinner Stall’s are between 28 and 32. Accept any answer that points at where the dots are rather than at one number.
5. Spinner Stall 36 minus 26 = 10 seconds. Treasure Bag 36 minus 21 = 15 seconds. The Treasure Bag’s turns were less consistent — more variety in how long a turn took, even though it was quicker on the whole.
6. The idea of the lesson. They are two different questions. Quicker is about where the bulk of the dots sit, and the Treasure Bag’s bulk is further left. More spread out is about how far the dots reach, and the Treasure Bag’s stretch from 21 all the way to 36. Most of its turns were fast, and a few were very slow. A student who thinks these must contradict each other is reading both as the same measurement.
7. No. One value in common says nothing about the twenty around it. Both sets reach 36, but the Spinner Stall has most of its turns up near 30 while the Treasure Bag has most of its down near 24 — the 36 is an outlier there and typical here. This is the Lesson 9 range error again: judging a whole set from one or two extreme values.
Choose your path
Prizes won: which number of prizes happened most often at each stall?
Plot the Treasure Bag’s times on the second number line. Then say which stall has more dots on the left-hand side.
Finish the sentence on your sheet to say which stall’s turns were quicker.
Frame: “The ______ turns were quicker, because most of its dots are between ______ and ______, while most of the other stall’s are between ______ and ______.”
Stuck on 6? Answer two separate questions: where are most of the dots, and how far do they reach?
Both stalls had twenty Preps today. Suppose the Treasure Bag had only ten. Which of your comparisons would still be fair, and which would stop being fair? Explain.
Why is a dot plot a better choice than a bar chart for the times, when a bar chart was fine for the prizes? What is different about the two kinds of number?
Two findings, not one
“Somebody tell me which stall was quicker. … Treasure Bag. Good.
Now tell me which stall was more consistent. … Spinner Stall. Also good — and that is a different question with a different answer.”
| Question | What you look at | Here |
|---|---|---|
| Which sits lower? | Where the bulk of the dots is. | Treasure Bag — mode 24 seconds against 30. |
| Which is more spread out? | How far the dots reach. | Treasure Bag — range 15 seconds against 10. |
| Which is more consistent? | The same thing, said the other way round. | Spinner Stall. Its turns cluster tightly around 30. |
Which stall would you rather run?
Worth thirty seconds if you have them. A stall where every turn takes about thirty seconds is easy to queue; one where turns run from 21 to 36 is not, even though it is quicker on average. The maths has a practical consequence, which is the point of doing it on real-looking data.
Keep the dot plots
Lesson 11 is about displays that mislead. Students who have drawn an honest one by hand are much harder to fool.
Exit ticket
Two more stalls timed their turns, in seconds. Ten Preps at each.
| Stall | The ten times |
|---|---|
| Rocket | 18, 19, 20, 20, 21, 21, 22, 23, 24, 32 |
| Comet | 20, 21, 22, 23, 23, 24, 24, 25, 26, 27 |
What is the range of each set?
Which stall was more consistent? Explain how you can tell.
Rocket has the quickest turn and the slowest turn of all twenty. What does that tell you about using the fastest time to judge a stall?
The answers
1. Rocket 32 minus 18 = 14. Comet 27 minus 20 = 7.
2. Comet. Its ten times sit between 20 and 27 with nothing unusual; Rocket’s nine quick times are followed by a single 32 that drags its range out to 14. Students who answer “Rocket, because it is faster” have answered a different question — consistency is about spread, not speed.
3. The fastest time is one Prep out of ten, and the same stall produced the slowest. One value, at either end, tells you nothing about the rest. The answer worth an A says you should look at where most of the times sit instead.
Back to your goal
Read what you wrote on the Intentions tab. Did you do it? One sentence.
What you need for this lesson
| File | Who gets it |
|---|---|
| l10-01-worksheet-core.pdf | One per student. Prize table, both number lines at the same scale, and the seven questions. |
| l10-02-worksheet-support.pdf | Spinner Stall dot plot already done, times pre-sorted, three questions with a frame. |
| Compare Two Sets | On the board. Press Prizes won, then Seconds per turn to show why a bar chart fails for measurements. |
| Games Day — what to print | The collection sheet and the rating card, and how to run the day. Games Day sits between this lesson and Lesson 11. |
Where this sits in the achievement standard
AC9M6ST01 names discrete and continuous numerical variables. Prizes won is discrete; seconds per turn is continuous. With Lesson 9’s ordinal categorical ratings, the three types named in the achievement standard are all covered.
The comparison is still in terms of mode, range and shape, which is the wording of the content descriptor. Nothing beyond that is required at Year 6.
For anyone who asks about the middle
Somebody usually wants to add the times up and divide. That is the mean, it belongs to Year 7, and it is a reasonable instinct. Name it, say they will meet it next year, and point out that it would not have answered question 6 — the mean tells you where the middle is and nothing at all about the spread.
Australian Curriculum content descriptions and achievement standard extracts © ACARA 2010–present, licensed under CC BY 4.0. ACARA does not endorse this material.