Math lesson plan · 11th grade (Year 12 in England) · 45 minutes
Divide a polynomial by (x − a)
Algebra
CCSS HSA.APR.D.6CCSS HSA.APR.B.2England: A-level Pure Mathematics
Objective and success criteria
Students will: Divide a polynomial by (x − a).
- 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: Solve a cubic equation when one root is known
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 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 − 8x − 3 is divided by (x + 2). → 5Prompt: 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.
- Find the remainder when x³ + 3x² − 9x + 9 is divided by (x − 2). → 11Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
Hook · 5 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Divide a polynomial by (x − a). Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Divide x³ + 7x² + 18x + 24 by (x + 4). The remainder is 0. Type the quotient.
- Step 1: x³ + 7x² + 18x + 24 = (x + 4)(x² + 3x + 6)
- Step 2: x² + 3x + 6
- Answer: x^2+3x+6
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Divide x³ + 3x² − 8x − 4 by (x − 2). The remainder is 0. Type the quotient. → x^2+5x+2Prompt: Use long division or synthetic division with 2.
- Divide x³ − x² − 18x − 11 by (x + 3). Type the quotient (ignore the remainder). → x^2-4x-6Prompt: Bring down the leading coefficient, multiply by −3, add, and repeat.
You do: independent practice · 10 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 · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Divide x³ + 7x² + 15x + 12 by (x + 4). The remainder is 0. Type the quotient. → x^2+3x+3Prompt: Use long division or synthetic division with −4.
- Divide x³ + 7x² + 4x − 21 by (x + 2). Type the quotient (ignore the remainder). → x^2+5x-6Prompt: Bring down the leading coefficient, multiply by −2, add, and repeat.
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
- Move on to: Solve a cubic equation when one root is known.
- 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