Is this continuous data?

What this sequence teaches

One feature: whether the quantity being recorded can take any value in between two of its possible values. Everything else is held still. Every example is numerical data, so the sequence never asks whether something counts as data in the first place — that question belongs to the card at the bottom of this page, and to the atom before this one.

Students arrive with shortcuts that work most of the time and fail at exactly the moments that matter: that counting means discrete and measuring means continuous, that whole numbers mean discrete, and that the words “number of” settle it. The sequence is built to break all three.

The teaching sequence

Example 1 — the number of leaves on each sunflower (not continuous). A count, and the only example on the card that everyone will already sort correctly. It exists to set up Example 2.

Example 1 → Example 2. The same sunflowers, and the same four words: “the number of”. Only the end of the sentence changes, and it changes on screen. Students who have quietly learned that “number of” means discrete meet the contradiction in the first transition rather than three weeks later.

Example 2 — the number of centimetres each sunflower has grown (continuous). A sunflower could have grown 12 cm, or 12.4 cm, or 12.47 cm. Expect a class to object that you count centimetres on a ruler — that objection is the misconception surfacing, and it is better out than in.

Example 2 → Example 3. A clean fade: a new example, not a transformation.

Example 3 — each person’s age (continuous). The whole-number example. Everyone says their age as a whole number, and age is still continuous, because what is changing is how we write it down, not what an age could be. This is the single most useful frame on the card.

Example 3 → Example 4. A clean fade.

Example 4 — each child’s foot length (continuous). A second measurement in a new context, positioned to make the last example possible.

Example 4 → Example 5. Same children, same feet. Only the end of the sentence changes.

Example 5 — each child’s shoe size (not continuous). Nothing is being counted, so the counted-versus-measured shortcut gives the wrong answer here. The rule doesn’t.

Why the sequence is shaped this way

• “The number of” survives the opening boundary. Both of the first two examples begin the same way, so the phrase cannot be what decides the verdict. The only thing that changes is whether anything sits between two possible values.
• Units decide nothing either. Across the five examples, one positive has a unit on the stage and two don’t; neither negative has one. A student looking for “has a unit” as a shortcut finds it doesn’t work.
• The two animated pairs are the lesson. Both keep the context and the first half of the sentence fixed and change only the quantity, and in both the verdict flips.
• The middle three are as different as the design allows — plants, people and children; growth, age and body measurement.

What this sequence doesn’t address

Mass, temperature and speed never appear. Three positives can’t carry every kind of quantity; they are probed on the testing card instead.

A count with an enormous range — where the belief “continuous means lots of possible values” lives — is also on the testing card.

Data that isn’t numbers, money, and data worked out from other data are outside this sequence’s scope. They sit on the expansion card.

The rule and the script

The rule, exactly as the Show rule button displays it: For the following examples: if there is always another possible value in between, then it is continuous data. “For the following examples” matters: the expansion card is where students meet the edge of it.

The spoken justification for each example:

• Leaves — “This is not continuous data. How do I know? Because there isn’t always another possible value in between — a sunflower can have 7 leaves or 8 leaves, but nothing in between.”
• Centimetres grown — “This is continuous data. How do I know? Because there is always another possible value in between — a sunflower could have grown 12 cm, or 12.4 cm, or 12.47 cm.”
• Age — “… someone could be 14, or 14 and a half, or 14 years and 63 days.”
• Foot length — “… a foot could be 23 cm, or 23.7 cm, or 23.68 cm.”
• Shoe size — “This is not continuous data. How do I know? Because there isn’t always another possible value in between — a child can wear size 7 or size 7½, but nothing in between.”

Every justification names two neighbouring values. That is the move worth making habitual: to show something isn’t continuous, name two values with nothing between them; to show it is, name a value in between.

Whether students see the words is your call. For some classes the rule and the justifications help; for others the examples should do the talking. Nothing appears unless you click. Show rule puts the rule above the stage — you can turn it on before Example 1 — and it stays until you turn it off. Why? appears after each reveal: it shows the justification, and at the same moment the deciding phrase pulses and is underlined. On the two fades (Examples 3 and 4), where no animation has pointed at anything, it is worth asking “What decided it?” before you click.

Vocabulary. The word discrete is not on this page anywhere. Introduce it once the distinction is secure: “The kind that isn’t continuous has a name — discrete — and we’ll use it from now on.” The same applies to qualitative and quantitative.

Where this atom sits

Before it: telling numerical data from data that isn’t numbers. This atom assumes it and holds it constant — every example here is numbers.

This atom: that the category depends on what values the quantity could take, not on how the values are written down and not on whether anything is being counted.

After it: the expansion card below, where the rule meets its edge, and grouped frequency tables, which are how continuous data is usually recorded.

Running the sequence

Two clicks per example: the first reveals the tick or cross, the second moves on. The pause before the reveal is the prediction, and it is where the learning happens.

The Replay button appears on Examples 2 and 5, where a single change is the lesson. It doesn’t appear on Examples 3 and 4: those are new examples, not transformations.

If a class struggles with Example 5, go back to Example 4 and replay the change: “What’s stayed the same? What changed?”

About the testing sequence

Ten items in a new random order each time: five continuous, five not. Every item is numerical data, so the rule decides every one. There is no Show rule button here — students apply the rule without it on screen. Why? works as in the teaching sequence, after each answer.

The items are static. The animations in the teaching sequence mark the change between two examples; they don’t carry the meaning of continuous, so a still sentence is a fair test.

What each item is diagnosing

A runner’s time, to the nearest second. The belief that whole recorded numbers mean discrete. The rounding is stated on the stage on purpose.

The number of litres a car takes, and the number of nights a patient stays. A matched pair: identical wording, opposite verdicts. A student working from the phrase gets exactly one of them right and can’t say why.

Dress size, a mark out of 20, and pizza sizes of 9, 12 or 16 inches. The belief that anything not being counted must be continuous. The pizza item is the hard one: inches are a real measuring unit, and the data is still discrete because the shop sells three sizes.

The number of grains in a bag of rice. The belief that an enormous range of possible values means continuous.

The width of a hair. A magnitude no textbook exercise would use, so the item can’t be answered by recognising a familiar example.

The mass of a parcel and the speed of a car. Quantity types the teaching sequence never showed, checking that the rule travels.

Common confusions to watch for

“It says ‘number of’, so it’s counted” — exposed by the matched pair of litres and nights.

“The numbers written down are whole, so it’s discrete” — exposed by the runner’s time.

“Nothing is being counted, so it must be continuous” — survives most of the pool and then fails on dress sizes and pizza sizes. Worth naming once it has failed, rather than before.

Discussion prompts

1. “Say the rule. Now use it on the pizza item.”
2. “Two items begin ‘we record the number of…’. One is continuous and one isn’t — which, and what does the rule say about each?”
3. “Name two values that show the dress size item isn’t continuous.”
4. “For the runner’s time: what would have to be true for the answer to be no?”

Reading the summary

At the end, all ten items are shown together. Each cell carries two separate signals: the tick or cross in the corner says what the item actually was; the green or red tint says whether the student answered correctly.

A red cell with a tick is a continuous item answered “no” — usually the whole-numbers belief. A red cell with a cross is a discrete item answered “yes” — usually the not-being-counted belief. The pattern across the grid tells you which of the two to teach next.

About the expansion sequence

Six items that sit outside what the rule covers, to see how far students’ understanding transfers. It works exactly like the testing sequence: random order, Continuous or Not continuous, feedback, Why?, and a summary.

The rule has been scoped from the beginning — “For the following examples”. This card is where students meet the edge of it.

Two kinds of item

The rule still works on an unfamiliar surface. Shirt numbers are numbers that measure nothing, and the rule handles them without modification. A country’s mean family size is the one to spend time on: every family contains a whole number of people, and the mean of those whole numbers doesn’t have to be whole.

Applying the rule mechanically gives the wrong answer. Colours shade into one another, so the rule appears to pass them; the answer is still no, because they aren’t numbers, so there is nothing in between to look for. Percentages look as though they can be anything, and in a class of 30 they can only be multiples of a thirtieth. And heights recorded in groups look discrete, because nothing sits between one group and the next, when in fact grouping is a way of writing heights down, not a limit on what a height could be.

The grouped heights item has the longest future of anything on this page. Grouped frequency tables are the next thing these students will meet, and the data in them is continuous.

Who it’s for

Students who have been through the teaching and testing cards. The percentage item needs fluency with percentages of a class of 30; the mean item needs averages. Skip either if the arithmetic will be the obstacle rather than the idea.

The justifications

Here the Why? reasons go beyond the rule and name what’s new, because the rule alone can’t settle these:

• Shirt numbers — “… a player can wear 7 or 8, but nothing in between. The rule still works here, even though the numbers aren’t measuring anything.”
• Colour — “… these aren’t numbers at all, so there is nothing in between to look for. Colours do shade into one another, but the rule is about values a number could take.”
• Grouped heights — “… a child could be 148 cm, or 148.4 cm, or 148.37 cm. Putting heights into groups changes how they are written down, not what a height could be.”

As on the other cards, the deciding phrase pulses when Why? opens. On the items where the rule fails that is deliberate: it points at what the rule looks at, while the words say why it doesn’t settle the question.

A note on the price item

The rule decides it: a price can be £2.40 or £2.41 with nothing in between, so it isn’t continuous. You will find books and colleagues who treat money as continuous, because the steps are so small it makes no practical difference. The justification says so. It is the only place on this page where students are told that people disagree, and that is deliberate — the rule is a tool for thinking, not a fact about the world.

Discussion prompts

1. “Which of these could you decide by using the rule, and which needed something else?”
2. “Name two values that show the percentage item isn’t continuous.”
3. “The heights are recorded in groups. What could a child’s actual height be?”

Reading the summary

The same two channels as the testing card: the corner marker for what the item was, the tint for whether it was answered correctly.

Expect lower scores here than on the testing card, and read them as information about how far the idea has travelled rather than as a mark.