Math lesson plan · 11th grade (Year 12 in England) · 30 minutes
Remainder and factor theorems
Algebra
CCSS HSA.APR.B.2CCSS HSA.APR.B.3
Objective and success criteria
Students will: Remainder and factor theorems.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Add, subtract and multiply polynomials; Function notation
Leads to: Polynomial functions: zeros and end behaviour; Divide a polynomial by (x − a); Use the factor theorem to find an unknown coefficient
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Add, subtract and multiply polynomials; Function notation). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Simplify: (x² − 8x − 9) + (−x² − x − 1) → -9x-10Prompt: Remove the brackets.
- f(x) = 2x² + 6x − 5. Find f(3). → 31Prompt: Replace every x with 3.
- Expand and simplify: (−3x − 1)(x² + 4) → -3x^3-x^2-12x-4Prompt: Multiply each term of the first bracket by every term of the second.
Hook · 3 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Remainder and factor theorems. Read the success criteria together.
I do: model it · 7 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Find the remainder when x³ − 3x² − 7x − 6 is divided by (x + 1).
- Step 1: f(−1) = −1 − 3 + 7 − 6
- Step 2: = −3
- Answer: -3
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Find the remainder when x³ + 6x² + 9x − 7 is divided by (x − 1). → 9Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
- (x + 3) is a factor of x³ + 7x² + 7x − 15. Solve x³ + 7x² + 7x − 15 = 0. Give all three solutions, separated by commas. → x = -5 or x = -3 or x = 1Prompt: Divide by the known factor to get a quadratic.
You do: independent practice · 7 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Find the remainder when x³ − 5x² + 4x + 6 is divided by (x + 3). → -78Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
- (x − 4) is a factor of x³ − 5x² − 2x + 24. Solve x³ − 5x² − 2x + 24 = 0. Give all three solutions, separated by commas. → x = -2 or x = 3 or x = 4Prompt: Divide by the known factor to get a quadratic.
Watch out for
- Doing an operation to one side of an equation but not the other.
- Reading 3x as 3 + x, or thinking x always stands for one particular number.
Support and stretch
Support
- Review first: Add, subtract and multiply polynomials.
- Use concrete materials: balance scales or a balance drawing.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Polynomial functions: zeros and end behaviour; Divide a polynomial by (x − a); Use the factor theorem to find an unknown coefficient.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-11.
Free lesson plan from eurekami.com/lesson-plan-builder