Counting and measuring

Classroom resources

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Students will be able to

• I can say whether numerical data is discrete (counted) or continuous (measured).
• I can explain why a measurement is always rounded, and choose a sensible precision.

Timing

Predict and Run 6 min · Investigate 10 min · Modify 8 min · Practise and Make 10 min

Before class

Try the Run board first. This short lesson opens the Data unit; the next lesson calculates mean, median, mode and range.

The one idea: counted data takes separate values (discrete); measured data can be any value in a range (continuous). The common mistake is treating any decimal as continuous (shoe sizes, prices); it is planted in Ava's reasoning.

Age and money are worth discussing rather than ruling on: age is measured time recorded in whole years; money is counted in cents but has so many values that it is often treated as continuous.

Running the lesson

1. Predict: take a show of hands; ask each group how they decided.
2. Run: ask whether a 14.35 s time is possible and what would measure it.
3. Investigate: build a two-column list on the board (counted / measured) from students' own examples.
4. Modify: let the age question be a short debate.
5. Make: ask what happens to the six times if rounded to whole seconds.

When students get stuck

Answers to the checks

1. which can be 152.5?: Which of these could have the value $152.5$?
   A height in cm. Heights are measured, so they can be any value (152.5 cm, or 152.47 cm with a better ruler). The others are counted: whole numbers only.
2. add a time: These are 100 m times in seconds. Add a time of $14.3$ s (click the line, or type it in the box). Was your prediction right about measuring?
   
3. Counted or measured: Discrete or continuous?
   Number of pets → Discrete (counted); Mass of a backpack in kg → Continuous (measured); Goals scored in a match → Discrete (counted); Time to solve a puzzle → Continuous (measured); Daily rainfall in mm → Continuous (measured); Number of letters in a name → Discrete (counted)
   Counting gives separate values (discrete); measuring gives any value in a range (continuous).
4. Find the mistake: Ava decides what kind of data shoe size is. Click the line where her reasoning goes wrong.
   line 3
   Line 3: shoe sizes come in fixed steps (7, 7.5, 8, …); there is no size 7.3. Data that can only take separate values is discrete, even with halves in it. A foot length in cm would be continuous.
5. Rounded measurements: A height is recorded as $152$ cm, to the nearest centimetre. What could the real height be?
   Anything from 151.5 cm up to (but not including) 152.5 cm
   Measured values are rounded: 152 cm to the nearest cm means from 151.5 cm up to 152.5 cm.
6. Same topic, two variables: Same topic, two variables: the number of apples in a bag, and the mass of the bag. Which is which?
   Apples: discrete. Mass: continuous.
   Count the apples (discrete); weigh the bag (continuous).
7. Money: Prices in dollars and cents, such as $\$3.95$. Discrete or continuous?
   Discrete: prices go in steps of 1 cent
   Money is counted in cents, so it is discrete (though with so many possible values it is often treated like continuous data).
8. Ratings: A survey asks students to rate a movie from 1 to 5 stars (whole stars). Discrete or continuous?
   Discrete
   Only five separate values: discrete.
9. Age: Edge case: age. You are 12 years old, but you are not exactly 12. What kind of data is age?
   Continuous (time is measured), though we often record it as whole years
   Age is a length of time, so continuous; recording it in whole years rounds it, which is why it looks discrete.
10. Continuous: Which is continuous data?
    The temperature at noon each day
    Temperature is measured.
11. Discrete: Which is discrete data?
    The number of text messages sent in a day
    Messages are counted.
12. Precision: To compare 100 m times in a race, which precision makes most sense?
    To the nearest 0.01 s
    Runners are often a fraction of a second apart, so record hundredths of a second.
13. Review: decimals: Review: which is bigger, $14.3$ or $14.25$? Type the bigger one.
    14.3
    14.30 > 14.25: compare the tenths first.
14. your own measured data: Make a data set of 6 swim times, all between 13 s and 14 s.
    
    Values such as 13.1, 13.3, 13.4, 13.6, 13.8, 13.9: measured data needs decimals to tell them apart. Rounded to whole seconds, they would all be 13 or 14.

Solutions

One possible solution for each task. Other correct programs are fine.

your own measured data

Open task: any program that does what the step asks.

Stretch and support

Stretch: Find a variable that could be recorded either way, depending on how you collect it (time in whole minutes, or measured with a stopwatch).

Support: Ask 'Could you have half of one?' and 'Is it counted or measured?' for every variable.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Counting and measuring".

The idea: Distinguishing discrete and continuous numerical variables: I can Students distinguish discrete from continuous numerical variables and understand how this affects data acquisition.

What understanding looks like:
- Correctly classifies given variables as discrete or continuous.
- Explains the key distinction (countable vs measurable).
- Identifies appropriate units and precision for a continuous variable.

Mistakes students often make here:
- A student calls any decimal data continuous (Ava's mistake).
- A student calls large counts continuous.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Counting and measuring".

Exit ticket

Classify: (a) number of apps on a phone, (b) battery temperature, (c) screen size in inches. Explain one.

(a) discrete (counted), (b) continuous (measured), (c) continuous (measured, though sold in common sizes).

What each step is for

• which can be 152.5?: Committing to which variable can take the value 152.5 shows whether they think in terms of counting versus measuring.
• add a time: Placing a value between whole numbers on a measured scale shows that continuous data can fall anywhere, limited only by how precisely it is measured.
• counted or measured: Counted gives discrete data, measured gives continuous; and measured values are always rounded. Ava's mistake thinks any decimal means continuous.
• one change at a time: One change at a time: the same topic counted or measured, money, a rating scale, then the edge case of age, which is measured but usually recorded as a count of years.
• kinds of data: Classifying variables from contexts, choosing a precision, and spotting rounding, mixed; the review keeps decimals fresh.
• your own measured data: Building a data set whose values all lie between two whole numbers shows that continuous data needs decimals, and that more precision separates values that rounding would merge.

Where it leads

Next: Mean, median, mode and range (node 2), then choosing the most useful measure (node 3).

Where it leads: histograms for continuous data and sampling (Year 8 statistics).