OR and Complements

Overview

In lesson 8 of 12, students connect equally likely outcomes with addition and complementary events. They apply probability to dice, spinners and LOTTO numbers, then explain decisions using diagrams, mathematical statements and checks. The lesson supports the refreshed New Zealand Curriculum emphasis on reasoning, communicating mathematically and applying learning in meaningful contexts.

Learning intentions

• WALT decide whether two events are mutually exclusive.
• WALT calculate probabilities of mutually exclusive events using addition.
• WALT use complements to solve “not” probability questions.
• WALT communicate and check probability solutions in context.

Success criteria

• I can decide whether two events can occur together.
• I can add probabilities for mutually exclusive events.
• I can use (P(\text{not }A)=1-P(A)).
• I can check that my answer is between 0 and 1 and explain what it means.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making informed decisions from mathematical information.
• Demonstrate mathematical reasoning by using appropriate methods, mathematical statements and representations.
• Use mathematical methods to explore problems related to life in Aotearoa New Zealand or the Pacific, including accurate calculations and communication.
• Refreshed curriculum emphasis: understand, know and do; mathematical and statistical reasoning; communicating ideas clearly and making connections between representations.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Display a six-sided die and ask, “What is the probability of rolling an odd number or a 6?” Open with the hook and retrieval slides and have students give an estimate, then independently recall sample space, favourable outcomes and probability notation. Take two contrasting answers without confirming the result.

2. 8–23 min · Explicit teaching. Model the meaning of “A or B”, using a Venn-style event diagram and a sample-space list. Explain that mutually exclusive events cannot happen on the same trial, so (P(A\text{ or }B)=P(A)+P(B)); introduce the complement rule (P(\text{not }A)=1-P(A)). Students annotate a formula mat on the probability practice worksheet and complete three short examples: rolling a 2 or 5, selecting an even number from 1–10, and not rolling a 6.

3. 23–40 min · Guided representations. Work through dice and spinner examples from the worked-example slides. Students draw event diagrams and sample spaces, then answer: “Can the events occur together?”, “Which outcomes are included?”, and “What does the probability mean in context?” Pause after each example for mini-whiteboard responses. Correct the common error of adding probabilities when an outcome belongs to both events.

4. 40–60 min · Collaborative sorting and practice. In groups of three, students use prepared event statements on the event-sorting and practice pages. They sort each case into “mutually exclusive”, “overlapping”, or “use a complement”, and justify one decision before completing the related calculation. Include dice, a 1–8 spinner and LOTTO numbers 1–40: odd or even, a multiple of 5 or 10, a number from 1–10 or 31–40, and not a multiple of 3. Groups compare one solution with another group and identify any different assumptions.

5. 60–78 min · Independent application and error analysis. Students complete the remaining worksheet questions independently. Questions progress from identifying events to calculating and interpreting probabilities, including a LOTTO-style question such as “What is the probability that one selected number is odd or a multiple of 5?” Students analyse two incorrect solutions, explain the error, and repair the reasoning. Circulate and conference using: “What is the sample space?”, “Can both events happen?”, and “How could you check that answer?”

6. 78–88 min · Extension and discussion. Advanced learners investigate overlapping events, such as rolling a number that is even or greater than 3. They show why simple addition overcounts the shared outcomes and develop (P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)). Other students explain one completed solution to a partner using the sentence frame: “Because the events are/are not mutually exclusive, I…”

7. 88–95 min · Plenary and exit check. Return to the opening question through the plenary and exit-question slide. Students complete a three-part exit response: (a) decide whether “rolling a 1 or an even number” is mutually exclusive, (b) calculate the probability, and (c) state the complement of “a number is a multiple of 4” when choosing from 1–12. Collect responses to identify the starting point for lesson 9.

Resources

• the OR and complementary events slide deck
• the probability practice worksheet
• Six-sided dice, one per group
• Simple paper or digital spinners labelled 1–8
• Mini-whiteboards and pens
• Calculators for checking, not replacing, reasoning
• Highlighters in two colours
• Visualiser or board for modelling

Assessment

• Check mini-whiteboard responses for the distinction between mutually exclusive and overlapping events.
• Question groups during sorting and listen for correct use of “sample space”, “favourable outcomes”, “or”, “not” and “complement”.
• Use the exit response to assess calculation accuracy, representation, reasoning and whether students check that probabilities lie from 0 to 1.

Differentiation

• Support learners with clearly spaced worksheet questions, enlarged print, high-contrast diagrams, colour-coded event regions, a worked example and sentence starters. Offer the worksheet digitally or read questions aloud for dyslexic learners; avoid requiring students to copy dense text.
• Provide a Venn-style event diagram, outcome lists and a formula mat before moving to symbolic notation. Allow students to use physical dice, counters or a spinner to represent outcomes.
• Pair students strategically and assign roles: reader, organiser and explainer. Revoice vocabulary and display “mutually exclusive”, “overlap”, “complement”, “sample space” and “favourable outcome” with concise definitions.
• Extend confident learners through overlapping events, inclusion–exclusion reasoning and a comparison of two methods, requiring them to explain the overcounting and generalise a rule.

Extension

• Design two events involving LOTTO numbers that overlap. Calculate the probability of their union in two ways and explain why direct addition is not valid.
• Investigate whether a claim such as “the probability of odd or a multiple of 5 is the sum of the two probabilities” is always true. Give a counterexample and a corrected statement.