PRACTISE THE CONVERSATION
Four useful habits.
Room to think.
- 01Listen
Understand the question. Ask if unsure.
- 02Start
Try an example, a diagram or a calculation.
- 03Explain
Say why your approach makes sense.
- 04Respond
Use a hint. Check or refine the idea.
11+ & 13+: Maths discussion and genuine interests.
16+: Mathematical arguments and Physics reasoning.
I allow thinking time, offer a small prompt when needed, then try a related question with less support.
What schools assess & what these lessons cover
St Paul’s describes numeracy, source material, wider interests and conversation at 13+. Westminster 11+ discusses genuine interests; its current entry booklet also includes classroom activities. The exact format depends on the school and entry route.
My offer covers Maths, Physics where relevant, and general conversation practice. English teaching and specialist verbal or non-verbal reasoning need separate consideration. We agree the focus before a paid lesson.
The exercises below are original discussion activities, not past questions or predictions. Raphael Education is independent and is not endorsed by the schools, ISEB or UKMT. Follow your school’s guidance and invitation.
Try a short academic conversation.
Read the question, allow a first attempt and ask “What are you thinking?” Open a hint only when it helps. The extension is optional: understanding the first part matters more than rushing to the hardest one.
How many ways can you make 45?
Write 45 as the sum of at least two consecutive positive whole numbers. Can you find another way? How could you know that your list is complete?
A first hint
Check what “consecutive” means. Start with two numbers next to each other, then try three. If three consecutive numbers total 45, what must the middle number be?
How I would guide the thinking
After a first example, I would suggest organising attempts by the number of terms. A table helps us avoid repeating the same attempt or missing a possibility.
The five ways are 22 + 23; 14 + 15 + 16; 7 + 8 + 9 + 10 + 11; 5 + 6 + 7 + 8 + 9 + 10; and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.
Ten or more terms would total at least 1 + 2 + … + 10 = 55. So it is enough to check lengths two through nine. Lengths four, seven and eight do not give a positive whole-number starting value.
What I would notice: whether your child can explain how the list was organised, even if they need a prompt to finish it.
Take the idea further
Could two consecutive positive whole numbers add to an even number?
Reasoning: one is odd and the other even, so their sum is always odd.
Does the headline follow?
A school has 200 pupils. Twenty answer a survey: twelve say they cycle to school and eight say they do not. A headline reads, “60% of the school cycles.”
What is correct about the calculation? What has the headline assumed? What would you want to find out?
A first hint
What group does 12 out of 20 describe? What do we know about the other 180 pupils?
How I would guide the thinking
These dots represent the respondents. The other 180 pupils’ travel habits are unknown.
The calculation describes respondents; it does not prove that 60% of all pupils cycle.
I would ask how the respondents were chosen. A well-chosen random sample could support an estimate, with uncertainty. Asking only the cycling club would give a biased picture.
What I would notice: whether your child separates what the figures show from what the headline assumes. A plain spoken explanation is enough.
Take the idea further
If the responses are truthful, what are the smallest and largest possible numbers of cyclists in the whole school?
Reasoning: at least 12 cycle. At most 192 cycle: the twelve known cyclists plus all 180 pupils who did not respond. Eight pupils are already known not to cycle.
A minimum without calculus.
For a positive real number x, how small can x + 4/x be? Explain why your answer works for every positive x.
A first hint
Try x = 1, 2 and 4. What do those examples suggest? Can you rewrite x + 4/x − 4 as one fraction?
How I would guide the thinking
Examples suggest a minimum of 4 at x = 2. To prove it, write x + 4/x − 4 = (x − 2)²/x.
The numerator is non-negative and x is positive. Therefore the expression is at least 4, with equality at x = 2.
What I would notice: whether your child moves from examples to a reason that covers every allowed value, and explains why “positive” matters.
Take the idea further
What changes if x is negative?
Reasoning: x + 4/x + 4 = (x + 2)²/x is now non-positive, so the expression is at most −4, attained at x = −2. It has no minimum: increasingly large negative values of x make it arbitrarily negative. At x = 0, it is undefined.
Where did the energy go?
In a hypothetical experiment, a 420 W heater runs for 100 seconds in 0.50 kg of water. The water’s specific heat capacity is 4200 J kg⁻¹ °C⁻¹. Assume no change of state. If all electrical energy heats the water, what temperature rise would you expect? The measured rise is 14°C. What might explain the difference?
A first hint
Find the energy supplied using E = Pt. Connect energy to temperature rise using E = mcΔT. State the assumptions before calculating.
How I would guide the thinking
The input is 420 × 100 = 42,000 J. The ideal temperature rise is 42,000 ÷ (0.50 × 4200) = 20°C.
A rise of 14°C corresponds to 0.50 × 4200 × 14 = 29,400 J gained by the water, or 70% of the stated electrical input. Some energy may warm the vessel or surroundings. Measurement uncertainty or a different actual heater power could also matter.
What I would notice: whether your child distinguishes a plausible explanation from something the data prove. One result cannot tell us exactly where every joule went.
Take the idea further
What if we double the water mass but keep the same energy input?
Reasoning: the ideal rise halves to 10°C. This assumes the same specific heat capacity, no phase change and all input energy going to the water. Real heat losses may differ, so we cannot automatically halve the measured rise.
Feedback they can use next time.
Did they identify the question and find a useful first step — a diagram, an example or a calculation?
Could they explain why a step works, use a hint and check an assumption, unit or answer?
I identify one thing that worked and one specific next step. We keep track of what your child can do independently and what still needs a prompt. Thinking time, a quiet manner or a change of approach are not failures.
Will my child rehearse personal answers?
We can practise a natural conversation about genuine interests: what they enjoy, something they have learned and a question they are curious about. I use follow-up questions so they can develop their own explanation. I do not supply an invented personality or a speech to memorise.
Do schools recommend interview tuition?
Each school gives its own advice. Westminster’s 11+ FAQ discourages outside tuition and values spontaneous enthusiasm. Read the school’s guidance first. Our practice focuses on understanding, listening and explaining ideas, with no claim that tuition is necessary or guarantees an offer.
Can we focus on an interview that is coming up soon?
Share the invitation, confirmed date, entry route and subjects on the parent call. We can then discuss whether the time and availability allow useful practice, and agree a focused lesson plan. Fees and arrangements are confirmed before booking paid tuition.
Start with the official information.
Links checked 20 October 2026. Schools can update requirements; follow the information for your child’s entry year.