Math lesson plan · 10th grade (Year 11 in England) · 45 minutes
Rationalise denominators like a + √b
Number sense
CCSS HSN.RN.B.3England: GCSE Higher Number
Objective and success criteria
Students will: Rationalise denominators like a + √b.
- 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: Rationalise the denominator; Add and subtract surds
Leads to: the next skill in this area.
Materials: Place value chart, Base-ten blocks or a number line, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Rationalise the denominator; Add and subtract surds). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Rationalise: 30/√5 = k√5. What is k? → 6Prompt: Multiply top and bottom by √5.
- Simplify: 8√3 + 9√3 − 7√3 → 10√3Prompt: Treat √3 like a letter: these are like terms.
- Rationalise 8/(3 + √5). The result is p − q√5. What is p? → 6Prompt: Multiply top and bottom by the conjugate 3 − √5.
Hook · 5 min
- Show a number in the news or a price tag and ask what each digit is worth.
- Share the objective: Rationalise denominators like a + √b. 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: Rationalise 1/(4 − √6). What is the denominator after rationalising?
- Step 1: (4 + √6)(4 − √6) = 16 − 6
- Step 2: = 10
- Answer: 10
- 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.
- Rationalise 1/(1 − √2). What is the denominator after rationalising? → -1Prompt: Multiply top and bottom by the conjugate, 1 + √2.
- Rationalise and simplify: 6/(2 + √7) (4 + 2√7 / −4 − 2√7 / 12 − 6√7 / −4 + 2√7) → −4 + 2√7Prompt: Multiply top and bottom by the conjugate.
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.
- Rationalise and simplify: 18/(1 − √7) (−3 + 3√7 / −3 − 3√7 / 18 + 18√7 / 3 − 3√7) → −3 − 3√7Prompt: Multiply top and bottom by the conjugate.
- Rationalise 1/(3 + √10). What is the denominator after rationalising? → -1Prompt: Multiply top and bottom by the conjugate, 3 − √10.
Watch out for
- Reading 1,045 as one thousand four hundred five: point to each place and say its value.
- Thinking a longer number is always bigger, even with decimals (0.45 vs 0.5).
Support and stretch
Support
- Review first: Rationalise the denominator.
- Use concrete materials: place value chart.
- 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-10.
Free lesson plan from eurekami.com/lesson-plan-builder