Homeschool Guide: These lesson plans are a guide for parents. Content may contain errors — always cross-reference with official exam board specifications.

upper and lower bounds

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4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.

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Lesson Overview

Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA

Learning Objectives

Prerequisites

Materials & Equipment

Lesson 1: Introduction: upper and lower bounds

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about upper and lower bounds. Then check against the key terms: Lower Bound, Upper Bound. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Lower Bound, Upper Bound.
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: upper and lower bounds. 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: Find the bounds for 35 measured to the nearest 5

Plenary (5 minutes)

Check Out

Your student states one thing they learned and one question they still have about upper and lower bounds.

Lesson 2: Core Concepts: upper and lower bounds

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from Lesson 1 on upper and lower bounds. Check them against the notes below.

Main Content (35 minutes)

Lower Bound: The smallest value that would round to the given number.
Upper Bound: The smallest value that would NOT round to the given number (just above the largest value that would round to it).

Practice (10 minutes)

Q: Find the bounds for 35 measured to the nearest 5

Answer: 32.5 ≤ x < 37.5

Plenary (5 minutes)

Explain Back

Your student teaches the key points back to you without looking. Fill any gaps immediately.

Lesson 3: Application: upper and lower bounds

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Lower Bound, Upper Bound. 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: Find the bounds for 35 measured to the nearest 5

Answer: 32.5 ≤ x < 37.5

Q2: A length is 12.4 cm to 1 d.p. Find the error interval.

Answer: 12.35 ≤ x < 12.45

Q3: Two lengths are 20 cm and 15 cm (both nearest cm). Find the minimum perimeter.

Answer: Min perimeter: 19.5 + 14.5 = 34 cm

Q4: A rectangle is 10 cm by 6 cm (both nearest cm). Find the maximum possible area.

Answer: Max area: just below 10.5 × 6.5 = 68.25 cm²

Q5: Distance = 150 km (nearest km), Time = 2 hours (nearest hour). Find the maximum possible speed.

Answer: Max speed: 150.5 ÷ 1.5 = 100.33 km/h

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: upper and lower bounds

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 cuboid has dimensions: length = 10 cm (nearest cm), width = 5 cm (nearest cm), height = 3 cm (nearest cm). (a) Find the upper and lower bounds for each dimension. (b) Calculate the minimum and maximum possible volume. (c) The cuboid is made of gold. Gold costs £45 per cm³. Find the maximum possible cost of the gold used. <div class="

(a) Length: 9.5 ≤ l (b) Min volume = 9.5 × 4.5 × 2.5 = 106.875 cm³. Max volume (just below) = 10.5 × 5.5 × 3.5 = 202.125 cm³. (c) Max cost = 202.125 × £45 = £9,095.63. Mark scheme: 2 marks for (a) all six bounds, 2 marks for (b), 2 marks for (c)

Exam Tips: Always state what the number is rounded to first | Remember: upper bound is NOT included (use < symbol) | For maximum: use largest values in numerator, smallest in denominator | For minimum: use smallest values in numerator, largest in denominator | Write the error interval clearly with correct inequality symbols
Common Errors: Watch Out! 1. Wrong: For max of a ÷ b, use UB(a) ÷ UB(b) Correct: Max of a ÷ b = UB(a) ÷ LB(b) (largest numerator, smallest denominator) 2. Wrong: Upper bound is included in the error interval (using ≤) Correct: Upper bound is NOT included: use 3. Wrong: For "nearest 5", the half-unit is 5 (not 2.5) Correct: Half of 5 = 2.5, so bounds for 40 (nearest 5) are 37.5 ≤ x
AO3 - Reasoning & Interpretation: Reasoning and Interpretation Two students measure a corridor. Student A measures 24 m to the nearest metre. Student B measures 24.0 m to the nearest 0.1 m. (a) Write the error interval for each measurement. (b) A door frame needs to be 24.05 m wide. Which measurement gives more confidence that the corridor is wide enough? Justify your answer. (c) A builder says "It doesn't matter whether you write 24 or 24.0 — they're the same number." Is the builder correct in the context of measurement? Explain. Answers: (a) Student A: 23.5 ≤ x
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how upper and lower bounds connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of upper and lower bounds
  • Critical: "What are the limitations of the models used in upper and lower bounds?"

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

Recommended Resources

🎓 Smart Lesson (Guided)