Algebraic proof and identities – Year 11 maths lesson
Lesson 16 of 17 in the Year 11 maths course
UK version. UK school years, UK spelling and pounds (£). · US Math Department
A complete 45-minute lesson for Year 11 (ages 15–16) that runs on this page – at school or at home. Every answer is marked straight away, with a worked explanation.
- Time
- 45 minutes + 20 min homework
- Year
- Year 11 (ages 15–16)
- You need
- Paper, a pencil and this page
- Marked
- Instantly, with explanations
For parents: how to use this lesson
You don't need to know this topic yourself. The lesson explains each step on screen, one tidy step at a time. Every answer is marked straight away with a worked explanation, so you never have to mark anything.
- Read what your child will learn and the big ideas together.
- Press Start the lesson below and sit with your child for the teaching and the worked examples (about 19 minutes).
- Let them do the practice on their own. They choose Getting started, On track or Challenge – start on On track and move up after a few right in a row.
- Finish with the exit ticket and the talk question: “Why can checking ten examples never prove a statement for all numbers?”
- Later that day or the next day: the homework (20 minutes), then a practice test.
Watch out for
- Thinking examples that work prove a statement. Examples can only show it is false; a proof needs algebra.
- Using the same letter for two different numbers (n and n for two odd numbers). Use different letters: 2n + 1 and 2m + 1.
Tip for home: Share number tricks ("think of a number…") and work out why they always work.
What your child will learn
We are learning to recognise identities, find counterexamples and use algebra to show statements are true.
Success looks like
- I can tell an identity from an equation.
- I can find a counterexample.
- I can write a simple algebraic argument.
Key words
- identity
- true for all values (≡)
- counterexample
- one example that shows a statement is false
- proof
- an argument that shows something is always true
- consecutive
- following one after another
What to have ready
- Paper and a pencil
- This page on a computer, tablet or phone
- For the hands-on part: Test a statement such as "the sum of three consecutive numbers is a multiple of 3" with numbers, then prove it.
The big ideas (for the adult)
An identity is true for every value of the letter, such as (x + 1)² ≡ x² + 2x + 1. An equation is only true for some values.
One counterexample is enough to show a statement is false.
To prove something about any whole number, use letters: an even number is 2n and an odd number is 2n + 1.
The lesson, step by step
| Part | Time | How it is checked |
|---|---|---|
| Starter: quick recall | 5 min | Marked by Classfern |
| Teaching input | 10 min | Not assessed, self-checked |
| Worked examples and guided practice | 9 min | Not assessed, marked by Classfern |
| Independent practice | 15 min | Marked by Classfern |
| Plenary and exit ticket | 6 min | Marked by Classfern, self-checked |
Starter, "your turn", practice and exit ticket are marked by Classfern. Worked examples and teaching are not assessed. The hands-on task and the reflection are self-assessed. Explanations in talk are judged by the parent and not recorded.
The lesson
It runs here on the page. No sign-up and no names – progress is kept only in this browser.
Homework and practice at home
20 minutes: 10 practice questions, 3 multiple-choice questions and 2 to remember from earlier work, then a short reflection. New questions every time, all marked instantly.
More practice
More Year 11 maths lessons
Curriculum
National Curriculum for England, Year 11: “know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs}”
GCSE subject content A6: “know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs”
Contains public sector information licensed under the Open Government Licence v3.0.
Our lessons and questions are our own. They are not written or endorsed by any government department, testing agency or exam board.