Best for
Years 7-10 students ready to move from basic chance language into investigation, trial data, and the difference between theoretical and experimental probability.
Pāngarau / Mathematics • Statistics • Years 7-10
Experimental and Theoretical Chance
Predicted probability and actual trial results do not always match perfectly. This investigation helps students compare theory with evidence and explain why short runs can vary.
Free investigation starter, premium adaptation path
This version is ready for tomorrow. If you want a class set with differentiated event structures, local sports or weather contexts, or assessment-ready extension prompts, Te Wānanga can generate those variants quickly.
- Create simplified and extension versions of the same chance trial.
- Swap in culturally relevant local contexts without breaking the mathematics.
- Save adapted copies to My Kete for repeated use.
Kaiako planning snapshot
- Use length: 40-55 minutes.
- Grouping: Pairs or small groups.
- Prep: Coins, dice, or a simple spinner if you want the experiment to be run physically.
- Teaching move: Keep asking students what they expected to happen and why the short-run results may differ from that expectation.
Resources already provided
- Definitions and formula reminder
- Ready-to-run trial options
- Trial-record table
- Comparison and reflection prompts
- Teacher-only curriculum companion
The investigation structure is already built into this page.
Ngā Whāinga Akoranga / Learning Intentions
- We are learning how to calculate theoretical probability for a simple event.
- We are learning how to record experimental results clearly.
- We are learning how to compare prediction with evidence and explain any difference.
Paearu Angitu / Success Criteria
- I can work out the theoretical probability before I start the trial.
- I can collect and organise my results accurately.
- I can compare theoretical and experimental probability using evidence.
Curriculum integration / Te Mātaiaho alignment
This handout is strongest where students are comparing predicted and observed chance, including the language of theoretical probability, experimental probability, and complementary events.
Why this matters in Aotearoa
People use probability when they judge weather forecasts, game strategies, injury risk, traffic, and election predictions. Students need to understand that short runs can vary while still being compatible with the same underlying chance.
That idea matters for science, sport, media claims, and later statistics work. A mātauranga Māori lens also values patient observation over quick overclaiming, which makes this comparison between evidence and expectation especially worth teaching carefully.
Key ideas
Theoretical probability
What we predict from the structure of the event before any trials begin.
Experimental probability
What the results look like after we run the event and count what happened.
Complementary event
The chance that the chosen event does not happen.
Choose one investigation
| Trial option | Theoretical probability of the chosen event | Complementary event |
|---|---|---|
| Flip a fair coin and track heads | ||
| Roll a fair die and track numbers greater than 4 | ||
| Spin a fair spinner with 8 equal sections and track landing on a star when 3 sections are stars |
Record your results
| Trial number | Outcome | Chosen event happened? |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 | ||
| 7 | ||
| 8 | ||
| 9 | ||
| 10 |
If you run more than 10 trials, continue on the back or in your maths book and summarise the totals below.
Compare prediction and evidence
| Question | Your answer |
|---|---|
| How many times did the chosen event happen? | |
| What is the experimental probability? | |
| How close was this to the theoretical probability? |
Explain the difference
Why might your experimental result differ from the theoretical probability, especially with a small number of trials?
Support
Use the coin task first so the event structure is simplest, then rehearse the meaning of complementary event together.
Core
Students complete one investigation and compare theoretical with experimental probability in writing.
Stretch
Ask students how the experimental probability might change if the number of trials rose from 10 to 100.
Neurodiversity and inclusion note
Give students a recording partner or let them mark tallies instead of writing each outcome in full words. Keep the explanation prompt sentence-framed if needed.
Kaiako reminder
Do not rush students into “the experiment was wrong”. Variation in a short run is part of the mathematical story.
Hononga Marautanga · Curriculum Alignment
Mathematics — Pāngarau
Level 3–4: Apply number operations, statistical analysis, and mathematical reasoning to solve real-world problems; represent data using appropriate tools; interpret and communicate mathematical findings clearly.
Social Sciences — Tikanga ā-Iwi
Level 3–4: Understand how mathematical data and statistics are used to describe and analyse social, economic, and environmental patterns; recognise how data can reveal or obscure inequality.
Aronga Mātauranga Māori
Mathematics has always been part of mātauranga Māori — in the navigation of Te Moana-nui-a-Kiwa, in the architectural precision of wharenui, in the sophisticated storage and accounting systems of rua kūmara, and in the patterns of kōwhaiwhai and tukutuku that encode mathematical relationships in visual form. When Māori students engage with mathematics, they are not encountering something foreign: they are meeting a domain of knowledge that their tīpuna practised with extraordinary sophistication. Framing mathematical learning through whakapapa — connecting concepts to real Māori contexts — is not "cultural add-on" but recognition of where much mathematical knowledge lives in this land.
Ngā Rauemi Tautoko · Support Materials
This handout is designed to be used alongside the broader unit resources available at Te Kete Ako handouts library. Related resources from the same unit are linked in the unit planner. All resources are provided — no additional preparation is required to use this handout in your classroom.
📋 Teacher Planning Snapshot
Ngā Whāinga Ako — Learning Intentions
Students will engage with this resource to build pāngarau (mathematical) understanding — developing number sense, pattern recognition, and mathematical reasoning through hands-on, culturally grounded activities that connect to tamariki's world.
Ngā Paearu Angitū — Success Criteria
- ✅ Students can explain their mathematical thinking using words, objects, drawings, or symbols.
- ✅ Students can apply the number or pattern concept in this resource to a real or everyday context.
Differentiation & Inclusion
Scaffold support: Use concrete materials (blocks, counters, fingers) for entry-level engagement before progressing to abstract representations. Offer extension challenges asking students to generalise a pattern, write their own word problem, or explain their strategy to a partner.
ELL / ESOL: Mathematical language is a discipline-specific barrier — pre-teach key terms (e.g., equals, more than, fewer, pattern, factor) using visual representations. Allow students to demonstrate mathematical understanding non-verbally or through drawing. Pair with a bilingual buddy where possible.
Inclusion: Embed choice in how students engage — oral, written, or diagrammatic responses are all valid. Neurodiverse learners benefit from short, chunked task sequences with immediate feedback loops. Avoid timed drills in favour of exploratory tasks that reward curiosity. Make the maths classroom a safe place to be wrong and try again.
Mātauranga Māori lens: Pāngarau is a living tradition in Te Ao Māori — from the geometric precision of tukutuku and kōwhaiwhai patterns to the navigational mathematics of waka hourua, and the seasonal calculations embedded in maramataka. Framing early number sense within these contexts shows tamariki that mathematics is a human, culturally rich endeavour — not a foreign import. Encourage students to see counting, measuring, and patterning as acts of knowing their world.
Prior knowledge: Designed for early learners. No prior formal mathematics knowledge required. Teachers should assess current number knowledge before selecting appropriate entry points.
Curriculum alignment
- Number and Algebra — Number Strategies: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
- Number and Algebra — Patterns and Relationships: Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many.