Students zoom in on √2, √3, π and 1/7 one digit at a time, and sort numbers into rational (fractions, decimals that stop or repeat) and irrational (AC9M8N01, node 1).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 8 min · Investigate 10 min · Modify 8 min · Practise and Make 9 min
Do the Run board yourself first. Students need recurring decimals (previous unit) and square roots (Number Foundations).
The one idea: rational numbers are fractions, so their decimals stop or repeat; irrational numbers never stop and never repeat. The common mistake is calling every square root irrational (Ana's mistake).
The board checks each zoom with squares for square roots, and with "too big, too small" for π.
Proving √2 is irrational is beyond Year 8; the board is evidence, not proof (say so). Wider: historical approximations of π (Egyptian, Chinese, Indian, Greek), the golden ratio.
Believing a decimal can go on forever without repeating. Let the zooming carry on past where students expect it to settle.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student calls a long calculator decimal irrational. | Does it repeat? Is it a fraction? | Repeating means rational; only endless and non-repeating is irrational. |
| A student calls every root irrational. | What is √16? | Perfect squares have rational roots. |
| A student thinks π is 22/7. | What is 22 ÷ 7 to 4 places? | 3.1428… against 3.1415…: close, not equal. |
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Stretch: Find how ancient Egyptian, Chinese and Indian mathematicians approximated π. Which approximation was closest?
Support: A card with two columns: 'stops or repeats: rational' and 'never stops, never repeats: irrational'.
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 8 student in Australia. Help me understand ONE idea only, from the lesson "Irrational numbers". The idea: Irrational numbers as non-terminating, non-recurring decimals: I can Students understand that irrational numbers cannot be expressed as fractions and have decimal expansions that neither terminate nor recur. What understanding looks like: - States that irrational numbers cannot be expressed as fractions and have non-terminating non-recurring decimal expansions. - Correctly classifies given numbers as rational or irrational. - Does not confuse 'irrational' with 'very large' or 'impossible to use in calculation'. Mistakes students often make here: - A student calls a long calculator decimal irrational. - A student calls every root irrational. - A student thinks π is 22/7. 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 "Irrational numbers".
Sort √20, √25, 0.4 and 0.454545… into rational and irrational.
Rational: √25 = 5, 0.4, 0.454545… (= 5/11). Irrational: √20.
This starts AC9M8N01.
Where it leads: placing irrational numbers on the number line (next lesson), Pythagoras' exact answers, and surds (Year 10).