experimental vs theoretical
4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.
Lesson Overview
Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Learning Objectives
- Explain the key ideas of experimental vs theoretical
- Apply experimental vs theoretical to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for experimental vs theoretical first (GCSE Revise summary)
- Key vocab to pre-teach: key terms from experimental vs theoretical
- Basic skills: reading the summary notes and answering the practice questions there
Materials & Equipment
- Exercise book, pencil, ruler
- Scientific calculator
- Pair of compasses and protractor (geometry topics)
- Graph paper
- Printed revision notes (link below)
Lesson 1: Introduction: experimental vs theoretical
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about experimental vs theoretical. Then check against the key terms: key terms from experimental vs theoretical. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: key terms from experimental vs theoretical.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Before lesson: Read the script below. Pre-teach key vocab: key terms from experimental vs theoretical.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: experimental vs theoretical. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
Mins 0-5 - Hook: "Today: experimental vs theoretical. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look
Start with the revision notes summary, then attempt: A fair dice is rolled 60 times. How many times would you expect to roll an even number?
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about experimental vs theoretical.
Lesson 2: Core Concepts: experimental vs theoretical
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on experimental vs theoretical. Check them against the notes below.
Main Content (35 minutes)
Key Idea: As the number of trials increases, the relative frequency (experimental probability) gets closer to the theoretical probability.
| Term | Meaning | Example |
|---|---|---|
| Theoretical Probability | Expected probability based on equally likely outcomes | Calculation |
| Experimental Probability | Probability based on actual results | Trials/experiments |
| Relative Frequency | Same as experimental probability | Counting outcomes |
| 10 | 3 | 3/10 = 0.300 |
| 100 | 19 | 19/100 = 0.190 |
| 1000 | 168 | 168/1000 = 0.168 |
| 10000 | 1662 | 1662/10000 = 0.1662 |
Practice (10 minutes)
Q: A fair dice is rolled 60 times. How many times would you expect to roll an even number?
Answer: P(even) = 3/6 = 1/2. Expected = 1/2 × 60 = 30 times
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: experimental vs theoretical
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: key terms from experimental vs theoretical. Define each in one sentence.
Main Content (35 minutes)
Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.
Q1: A fair dice is rolled 60 times. How many times would you expect to roll an even number?
Answer: P(even) = 3/6 = 1/2. Expected = 1/2 × 60 = 30 times
Q2: A spinner is spun 150 times. It lands on red 45 times. Estimate P(red).
Answer: P(red) = 45/150 = 3/10 = 0.3
Q3: A coin is flipped 100 times with 55 heads. Is this what you'd expect? Explain.
Answer: 55/100 = 0.55 is close to 0.5, so this is expected variation for a fair coin
Q4: A bag contains coloured counters. In 40 draws, 16 were blue. Estimate how many blue counters there would be if there are 200 in total.
Answer: 16/40 = 0.4. Expected blue in 200 = 0.4 × 200 = 80 blue counters
Q5: A biased dice has P(6) = 0.25. How many sixes would you expect in 400 rolls?
Answer: Expected sixes = 0.25 × 400 = 100 sixes
Plenary (5 minutes)
Error Review
Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.
Lesson 4: Exam Practice: experimental vs theoretical
Duration: 50 minutes
Starter Activity (5 minutes)
Command Words
Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).
Main Content (35 minutes)
Extended Answer
Extended question: Extended Answer 6 marks: A company tests a biased coin. In 50 flips, heads appears 35 times. (a) Find the experimental probability of heads. (b) Use this to estimate the number of heads in 200 flips. (c) Another test of 500 flips gives 310 heads. Calculate the new experimental probability and comment on which estimate is more reliable. <div class="
(a) Experimental P(H) = 35/50 = 0.7. (b) Expected heads in 200 = 0.7 × 200 = 140. (c) New experimental P(H) = 310/500 = 0.62. The second estimate (0.62) is more reliable because it is based on 500 trials rather than 50. With more trials, the relative frequency gives a better estimate of the true probability. The true probability is likely between 0.62 and 0.7, and closer to 0.62. Mark scheme: M1 for 35/50, A1 for 0.7, M1 for 0.7×200, A1 for 140, M1 for 310/500=0.62, A1 for stating second estimate more reliable with reason (more trials)
Exam Tips: More trials = more accurate estimate of true probability | Expected frequency = probability × number of trials | Small samples can give misleading results - need many trials | Use experimental probability when you can't calculate theoretical | If experimental ≈ theoretical over many trials, the object is likely fair
Common Errors: Watch Out! 1. Wrong: Saying a coin is biased after only 10 flips because you got 7 heads Correct: With small samples, results vary a lot — you need many trials (hundreds) before concluding bias 2. Wrong: Calculating expected frequency as probability + number of trials instead of multiplying Correct: Expected frequency = probability × number of trials (multiply, not add) 3. Wrong: Saying "the relative frequency will eventually equal the theoretical probability" Correct: It tends towards (gets closer to) the theoretical probability, but rarely equals it exactly
AO3 - Reasoning & Interpretation: Reasoning and Interpretation Aisha rolls a fair dice 60 times and gets 15 sixes. Ben rolls the same dice 600 times and gets 108 sixes. (a) Calculate the relative frequency of sixes for each person. (b) Whose result is closer to the theoretical probability? Explain why. (c) Aisha says "The dice must be biased because I got way more sixes than expected." Evaluate her claim. Answers: (a) Aisha: 15/60 = 0.25. Ben: 108/600 = 0.18. (b) Ben's result (0.18) is closer to theoretical P(6) = 1/6 ≈ 0.167 because he did more trials. (c) Aisha's claim is not well supported. With only 60 rolls, getting 15 sixes (expected 10) is not unusual variation. Ben's much larger sample shows 0.18, much closer to 0.16
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how experimental vs theoretical connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of experimental vs theoretical
- Critical: "What are the limitations of the models used in experimental vs theoretical?"
Plenary (5 minutes)
Assessment Criteria
- Got it: Confident explanation + correct worked examples
- Getting there: Main points OK, needs support with detail
- Not yet: Confused on key concepts - re-run Lesson 2
Homework & Consolidation
- Consolidation: Re-answer any Lesson 3 practice questions answered incorrectly (20 mins)
- Retrieval: Write flashcards for the key terms: key terms from experimental vs theoretical (10 mins)
- Exam practice: One past-paper question on experimental vs theoretical from the board websites (15 mins)
- Extension: Explain experimental vs theoretical to someone else in your own words (10 mins)