Students choose events on a dice and two dice, show their complements, and use P(not A) = 1 − P(A), including for 'at least one' (AC9M8P01, nodes 1 and 2).
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 12 min · Practise and Make 10 min
Do the Run board yourself first. Students need sample spaces and probability as a fraction (Year 7 Chance).
The one idea: an event and its complement cover every outcome exactly once, so their probabilities add to 1. The common mistake is calling any two events that can't happen together complementary (Ben's mistake); complementary events must also cover everything. "Mutually exclusive" is the formal word for can't-happen-together; it is used in the teacher notes, not required of students yet.
'At least one' is the main use: one block to count (none) instead of many cases.
Wider: the chance of at least one six in four rolls of a dice, 1 − (5/6)⁴, a famous gambling question (needs multiplying, Year 10).
Telling complementary from 'can't happen together'. Ask: does every outcome land in one of the two?
| What you will see | A question to ask | The fix |
|---|---|---|
| A student says a heart and a spade are complementary. | Where do clubs go? | Complementary events cover every outcome. |
| A student counts 12 outcomes with at least one six. | Which outcome did you count twice? | (6, 6) is in both rows; or count no six (25) and subtract from 36. |
| A student writes P(not A) = P(A). | Which outcomes are not in A? | Count everything else, or subtract from 1. |
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: Two dice are rolled. Which is more likely: at least one six, or a sum of 7 or more? Use complements.
Support: A sample space printed as a grid, with two colours of highlighter.
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 the ideas in ONE lesson only, from the lesson "Complementary events". The ideas: - Recognising complementary events: I can Students recognise complementary events as pairs that together cover all outcomes, and state that P(A) + P(not A) = 1. - Using the complement relationship to calculate probabilities: I can Students use P(not A) = 1 − P(A) to calculate probabilities in applied contexts, including cases where finding the complement is more efficient than direct calculation. What understanding looks like: - Correctly identifies the complement of a given event. - States P(A) + P(not A) = 1 and verifies it for a specific case. - Does not confuse complementary events with mutually exclusive events. - Correctly calculates P(not A) given P(A) and vice versa. - Identifies at least one context where finding the complement is more efficient. - Applies the complement relationship correctly to a practical problem. Mistakes students often make here: - A student says a heart and a spade are complementary. - A student counts 12 outcomes with at least one six. - A student writes P(not A) = P(A). 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 "Complementary events".
P(it snows tomorrow) = 0.08. What is P(it does not snow)? Then: two coins are tossed; what is P(at least one tail)?
0.92. And 1 − P(HH) = 1 − 1/4 = 3/4.
This covers AC9M8P01.
Where it leads: two-way tables and Venn diagrams (next), and 'and', 'or' and 'not' in compound events.