Identities and proof – Grade 10 math lesson
Lesson 3 of 16 in the Grade 10 math course
US version. US grades, US spelling and dollars ($). · UK Maths Department
A complete 45-minute lesson for grade 10 (ages 15–16) that runs on this page – at school or at home. Every answer is graded right away, with a worked explanation.
- Time
- 45 minutes + 20 min homework
- Grade
- Grade 10 (ages 15–16)
- You need
- Paper, a pencil and this page
- Graded
- 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 graded straight away with a worked explanation, so you never have to grade 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
Learning goal: recognize 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 | Graded by Classfern |
| Teaching input | 10 min | Not assessed, self-checked |
| Worked examples and guided practice | 9 min | Not assessed, graded by Classfern |
| Independent practice | 15 min | Graded by Classfern |
| Closure and exit ticket | 6 min | Graded by Classfern, self-checked |
Starter, "your turn", practice and exit ticket are graded 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 graded instantly.
More practice
More Grade 10 math lessons
- Factoring
- Quadratic equations
- Functions
- Graphing functions
- Exponential functions
- Circles: circumference and area
- Volume and surface area
- The Pythagorean theorem
- Right triangle trigonometry
- Transformations
- Similarity and congruence
- Properties of quadrilaterals
- Coordinate geometry
- Cross sections of 3-D figures
- Geometric constructions
Curriculum
Common Core State Standards HSA.APR.C.4: “Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples.”
Common Core State Standards HSA.SSE.A.2: “Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).”
Our lessons and questions are our own. They are not written or endorsed by any government department, testing agency or exam board.