Complex numbers – Grade 12 math lesson
Lesson 1 of 5 in the Grade 12 math course
US version. US grades, US spelling and dollars ($). · UK Maths Department
A complete 45-minute lesson for grade 12 (ages 17–18) 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 12 (ages 17–18)
- 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: “What happens to a complex number when you multiply it by i? Try it on the plane.”
- Later that day or the next day: the homework (20 minutes), then a practice test.
Watch out for
- Leaving i² in an answer. Replace i² with −1 straight away.
- Adding a real part to an imaginary part. Keep them separate, like unlike terms in algebra.
Tip for home: Talk about why mathematicians invented numbers whose square is negative.
What your child will learn
Learning goal: add, subtract and multiply complex numbers and use i² = −1.
Success looks like
- I can add and subtract complex numbers.
- I can multiply complex numbers.
- I can use the conjugate.
Key words
- imaginary unit
- i, where i² = −1
- complex number
- a number a + bi
- real part
- the a in a + bi
- conjugate
- a + bi and a − bi
What to have ready
- Paper and a pencil
- This page on a computer, tablet or phone
- For the hands-on part: Plot complex numbers as points on a plane (real across, imaginary up) and see what adding does.
The big ideas (for the adult)
The number i is defined so that i² = −1. A complex number has the form a + bi, with a real part a and an imaginary part b.
Add and subtract complex numbers by combining the real parts and the imaginary parts separately.
Multiply like expanding parentheses, then replace i² with −1. The conjugate of a + bi is a − bi; their product is a real number.
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 12 math lessons
Curriculum
Common Core State Standards HSN.CN.C.9: “(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.”
Common Core State Standards HSN.CN.A.2: “Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.”
Our lessons and questions are our own. They are not written or endorsed by any government department, testing agency or exam board.