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# Counting and measuring

Data and chance · Data

<!-- Speaker note: Students sort numerical variables into discrete (counted) and continuous (measured), place a measured value between whole numbers on a dot plot, and see that measurements are always rounded to some precision (AC9M7ST01, node 1). About 35 minutes (Predict and Run 6 min · Investigate 10 min · Modify 8 min · Practise and Make 10 min). -->

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## Today

**We are learning to** numerical data is discrete when it is counted (siblings: 0, 1, 2, …) and continuous when it is measured (a 100 m time can be any value, such as 14.37 s). Measured values are always rounded to some precision.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> How many siblings do you have? How long does it take you to run 100 m? One answer is always a whole number; the other could be anything in between.

<!-- Speaker note: Statistics starts with **data**: values collected to answer a question. Numerical data comes in two kinds: - **Discrete** data is **counted**: number of siblings, goals scored, cars in a car park. It takes separate values, usually whole numbers. - **Continuous** data is **measured**: height, time, mass, temperature. It can take any value in a range; we round it when we record it. The kind of data changes how you collect it, and how you display it later in this unit. -->

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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

**Which of these could have the value $152.5$?**

- A person's height in cm
- The number of pages in a book
- The number of goals in a season
- The number of students in a school

<!-- Speaker note: What happens: 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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands; ask each group how they decided. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fdata%2Fkinds-of-data%2Fstep-2&code=widget%3A+data%0Avalues%3A+13.2%2C+13.8%2C+14.1%2C+14.6%2C+15.0%2C+15.4%2C+16.2%0Arange%3A+12%2C+17%0Astep%3A+0.1%0Aedit%3A+on%0Ashow%3A+none%0Alabel%3A+Time+to+run+100+m+%28s%29%0A&fresh=1}}

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

<!-- Speaker note: Times can land between the marks. Could you add 14.35 s? What would you need to measure it? -->

<!-- Speaker note: Teacher: ask whether a 14.35 s time is possible and what would measure it. -->

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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?

* 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.

<!-- 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>

## Counted or measured

Here is how to tell the kinds apart.

Sort each one: discrete or continuous?

$$\text{siblings: } 0, 1, 2, 3$$

1) $\text{height: } 152.5\text{ cm}$ — Measured: any value in a range. **Continuous.**
2) $152.5 \approx 152.47\ldots$ — A measurement is rounded to the precision of the instrument (here 0.1 cm).
3) $\text{shoe size: } 7, 7.5, 8$ — Fixed steps with nothing in between: discrete, even with halves.

<!-- 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 siblings: 0, 1, 2, 3 and say "Counted: separate values, no values in between. Discrete." Line 1: write height: 152.5 cm and say "Measured: any value in a range. Continuous." Line 2: write 152.5 152.47 and say "A measurement is rounded to the precision of the instrument (here 0.1 cm)." Line 3: write shoe size: 7, 7.5, 8 and say "Fixed steps with nothing in between: discrete, even with halves." -->

<!-- Speaker note: Teacher: build a two-column list on the board (counted / measured) from students' own examples. -->

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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>

## Counted or measured

**Discrete or continuous?**

Groups: Discrete (counted) · Continuous (measured)

Number of pets · Mass of a backpack in kg · Goals scored in a match · Time to solve a puzzle · Daily rainfall in mm · Number of letters in a name

* Answer: 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)

<!-- Speaker note: Counting gives separate values (discrete); measuring gives any value in a range (continuous). -->

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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>

## Find the mistake

**Ava decides what kind of data shoe size is. Click the line where her reasoning goes wrong.**

```
 1  Shoe sizes in our class: 7, 7.5, 8, 8.5, 9.
 2  Shoe size has decimals in it.
 3  So shoe size is continuous data.
```

* Answer: line 3

<!-- Speaker note: 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. -->

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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>

## Rounded measurements

**A height is recorded as $152$ cm, to the nearest centimetre. What could the real height be?**

- Exactly 152 cm
- Anything from 151.5 cm up to (but not including) 152.5 cm
- Anything from 152 cm to 153 cm

* Answer: Anything from 151.5 cm up to (but not including) 152.5 cm

<!-- Speaker note: Measured values are rounded: 152 cm to the nearest cm means from 151.5 cm up to 152.5 cm. -->

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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



- **Same topic, two variables:** Each task changes one thing.
- **Money:** 
- **Ratings:** 
- **Age:** 

<!-- Speaker note: What this step is for: 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. For students who need more support: Ask 'Could you have half of one?' and 'Is it counted or measured?' for every variable. For early finishers: Find a variable that could be recorded either way, depending on how you collect it (time in whole minutes, or measured with a stopwatch). When students get stuck: A student calls any decimal data continuous (Ava's mistake). Ask: What values are possible between 7 and 7.5 for a shoe size? Fix: Fixed steps with nothing in between: discrete. / A student calls large counts continuous. Ask: Is it counted or measured? Fix: Counted, however big: discrete. -->

<!-- Speaker note: Teacher: let the age question be a short debate. -->

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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>

## Continuous

**Which is continuous data?**

- The number of people on a bus
- The number of rainy days in a month
- The temperature at noon each day

* Answer: The temperature at noon each day

<!-- Speaker note: Temperature is measured. -->

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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>

## Discrete

**Which is discrete data?**

- The length of a phone call
- The number of text messages sent in a day
- The mass of a phone

* Answer: The number of text messages sent in a day

<!-- Speaker note: Messages are counted. -->

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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>

## Precision

**To compare 100 m times in a race, which precision makes most sense?**

- To the nearest minute
- To the nearest 0.01 s
- To the nearest 10 s

* Answer: To the nearest 0.01 s

<!-- Speaker note: Runners are often a fraction of a second apart, so record hundredths of a second. -->

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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: decimals

**Review: which is bigger, $14.3$ or $14.25$? Type the bigger one.**

* Answer: 14.3

<!-- Speaker note: 14.30 > 14.25: compare the tenths first. -->

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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: your own measured data

Then: if you rounded your times to the nearest second, how many different values would be left?

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

<!-- Speaker note: What this step is for: 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. Hints, in order: Every value must be more than 13 and less than 14. / Type them in the box, such as 13.2, 13.5, … -->

<!-- Speaker note: Teacher: ask what happens to the six times if rounded to whole seconds. -->

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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

**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).

<!-- 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: 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). -->