Math lesson plan · 11th grade (Year 12 in England) · 60 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 · 6 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: (4x² − 8x + 6) − (x² + 5x − 1) → 3x^2-13x+7Prompt: Subtracting a bracket changes the sign of every term inside it.
- f(x) = 2x² − 2x − 7. Find f(1). → -7Prompt: Replace every x with 1.
- Expand and simplify: (2x − 5)(x² − 2x − 5) → 2x^3-9x^2+25Prompt: Multiply each term of the first bracket by every term of the second.
Hook · 6 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 · 12 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 · 12 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 + 8x − 11 is divided by (x − 3). → 40Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
- (x + 4) is a factor of x³ + 2x² − 23x − 60. Solve x³ + 2x² − 23x − 60 = 0. Give all three solutions, separated by commas. → x = -4 or x = -3 or x = 5Prompt: Divide by the known factor to get a quadratic.
You do: independent practice · 17 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 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- (x + 5) is a factor of x³ − 4x² − 25x + 100. Solve x³ − 4x² − 25x + 100 = 0. Give all three solutions, separated by commas. → x = -5 or x = 4 or x = 5Prompt: Divide by the known factor to get a quadratic.
- Find the remainder when x³ + 5x² + 7x + 4 is divided by (x + 2). → 2Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
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