Math lesson plan · 11th grade (Year 12 in England) · 30 minutes
Use the factor theorem to find an unknown coefficient
Algebra
CCSS HSA.APR.B.2England: A-level Pure Mathematics
Objective and success criteria
Students will: Use the factor theorem to find an unknown coefficient.
- 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: Remainder and factor theorems
Leads to: the next skill in this area.
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 (Remainder and factor theorems). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find the remainder when x³ − 6x² + x − 4 is divided by (x + 1). → -12Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
- (x + 4) is a factor of x³ + 6x² − 7x − 60. Solve x³ + 6x² − 7x − 60 = 0. Give all three solutions, separated by commas. → x = -5 or x = -4 or x = 3Prompt: Divide by the known factor to get a quadratic.
- Find the remainder when x³ − 4x² + 4x + 5 is divided by (x − 3). → 8Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
Hook · 3 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Use the factor theorem to find an unknown coefficient. 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: (x − 2) is a factor of x³ + kx² − 3x − 10. Find k.
- Step 1: 2³ + k × 2² − 3 × 2 − 10 = 0
- Step 2: 4k − 8 = 0
- Step 3: k = 2
- Answer: 2
- 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.
- (x + 1) is a factor of x³ + kx² − 5x − 8. Find k. → 4Prompt: If (x + 1) is a factor, then f(−1) = 0.
- When 2x³ + 3x² + kx − 78 is divided by (x − 3), the remainder is 0. Find k. → -1Prompt: Remainder 0 means x = 3 makes the polynomial zero.
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.
- When 3x³ + 6x² + kx + 6 is divided by (x + 3), the remainder is 0. Find k. → -7Prompt: Remainder 0 means x = −3 makes the polynomial zero.
- (x − 1) is a factor of x³ + kx² + 3x + 1. Find k. → -5Prompt: If (x − 1) is a factor, then f(1) = 0.
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: Remainder and factor theorems.
- 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
- 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