Paddy MacMahon
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Paddy MacMahon
@paddymacmahon.com
1.2K followers 440 following 1.7K posts
Head of Maths at Latymer Upper School
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OCR, Cambridge International and AQA associate, Desmos and Google Colab enthusiast
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#MathsToday #ALevelMaths

Away for half term for the last couple of weeks... but here's today's Monday Area Problem.

Find the exact value of the green area.
The curve y=x² is in red. Two normals, at points in the second quadrant, intersect eachother at a point on the curve in the first quadrant with y-coordinate 3. The area enclosed by the curve and the two normals is shaded green.
November 3, 2025 at 6:26 PM
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#MathsToday #ALevelMaths #FurtherMaths

Find the exact value of the green area.
The graphs of y = a/x³ and y = a - x³ are in red. They share a mutual tangent point in the first quadrant. The area enclosed by the x-axis and the two curves, with no upper limit, is shaded green.
October 13, 2025 at 5:16 PM
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I saw your labels and thought these represented the actual areas (as in the area of region 1 is 1 unit, of region 2 is 2 units, etc...) which got me thinking that that sort of thing might make a nice problem. Impossible to get regions with those areas using a parabola, but I came up with this:
A parabola passing through the origin and a positive value on the x-axis, with a maximum point in the first quadrant. A straight line with positive gradient intersects the parabola at 2 points in the first quadrant. Vertical line segments are drawn from each of these intersection points to the x-axis, so the area enclosed by the parabola and the x-axis is divided into 4 regions. The left hand region has area 13, the right hand region has area 44 and the top region has area 8. The final region is labelled with a question mark.
October 9, 2025 at 6:05 PM
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Yeah, I thought this was at the easier end of the spectrum. Although I needed some pen and paper working...!

Gemini pro made a hash of it...
October 6, 2025 at 2:43 PM
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#ALevelMaths #MathsToday

Monday area problem. Find the exact value of the green area.
The graph of y=x² is in red, together with the point (sqrt3 / 2, 3/4) highlighted and labelled in blue. Also in red is a circle, tangent to the x-axis and touching the curve at the blue point. The region enclosed by the curve, the circle and the x-axis is shaded green.
October 6, 2025 at 5:34 AM
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In #ALevelMaths #MathsToday we started looking at transformations in terms of how you change the coordinates, using @desmos.com. Next time: encoding (some of) them with matrices.

www.desmos.com/calculator/j...
September 30, 2025 at 4:07 PM
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#MathsToday #ALevelMaths

Here's Monday's area problem. Find the exact value of the green area.

( #FurtherMaths really, given it's an improper integral...)
The graphs of y=a/sqrtx, in red, and y=a-x, in blue, are shown. The blue graph is a tangent line to the first at a point, in blue, in the first quadrant. The area enclosed by the y-axis and the two graphs, to the left of the tangent point, is shaded green. This area extends upwards without limit.
September 29, 2025 at 6:36 AM
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In #ALevelMaths #MathsToday we saw that if you make the 'wrong' choice for integration by parts with a logarithm involved, you can still proceed to the correct antiderivative.
Working for finding the integral of x⁵lnx by parts, using u=x⁵ and v'=lnx
September 26, 2025 at 11:11 AM
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Mine are still there, under this little flag thing's menu.
September 25, 2025 at 5:22 PM
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#MathsToday

Dedication from our girls, preparing for the MOG/MCG at 4:30 on a Wednesday afternoon with @heerpal.bsky.social.
September 24, 2025 at 3:30 PM
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If I select 'latest' rather than 'top', I see it.
September 23, 2025 at 8:22 PM
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Feedback, please. What do people think of this as an early task in thinking about parametric equations?

#ALevelMaths #MathsToday
September 23, 2025 at 2:39 PM
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I love lift problems...!

#MathsToday #ALevelMaths
A picture of a question on a whiteboard: A man is holding a parcel of mass 3.4kg by a piece of string. He's standing on a set of scales which is travelling upwards and decelerating at 1.3m/s². The scales read 85kg. What is the man's mass? What's the tension in the string?
September 23, 2025 at 12:16 PM
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Yes, and it's a result that I don't think is taught (here in the UK).

If the self-intersection point is at an x- or y- extreme I think it's quite easy to explain (see my attempt below - it's the area between two functions of x), but needs more 'pieces' when that's not the case.
September 22, 2025 at 9:06 PM
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#MathsToday #ALevelMaths

Here's this week's Monday area problem:

Find the exact value of the green area.
The graph with parametric equations x = t² - 2t, y = at - t³ is shown in red. This contains a closed off loop, the interior of which is shaded green. The point where the curve intersects itself is in the 4th quadrant and is shown with a blue dot. The tangent to the curve at this point is horizontal, and is a blue dashed line.
September 22, 2025 at 10:03 AM
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Using spreadsheets to work out the APR of a loan in #CoreMaths #MathsToday.
A screenshot of a loan repayment calculator: £10000 is borrowed over 30 months. The monthly repayments are £360.78 and the total amount payable is £10823.40. The APR for the loan has been obscured.
September 22, 2025 at 8:50 AM
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In #MathsToday these arrived, so ready to put theory into practice with our roulettes...
September 18, 2025 at 3:24 PM
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Also in #ALevelMaths #FurtherMaths #MathsToday, I used this as a starter for U6.

Finding the coordinates of S is fun!
The curve with parametric equations x=t^2-2t-2, y=9t-t^3 is shown in purple. This forms a loop, intersects the x-axis at the origin, a negative value (labelled A) and a positive value (labelled B). The curve has a local maximum point (Q) in the second quadrant and a local minimum point (R) in the fourth quadrant. The extreme left-most point (P) lies in the second quadrant. The curve intersects itself at a point S in fourth quadrant, to the left of the minimum point. The instruction, in red, is to find the coordinates of points, A, B, P, Q, R and S.
September 17, 2025 at 9:43 AM
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September 15, 2025 at 9:19 PM
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#MathsToday #ALevelMaths

We're looking at parametric equations today, so pupils are trying to animate roulettes using @desmos.com.

I'll try to make a guided worksheet for this...

This sort of thing is the desired outcome: www.desmos.com/calculator/9...
September 15, 2025 at 11:36 AM
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In #CoreMaths #MathsToday pupils have built a mortgage calculator, and they're using it to answer these questions.
Some mortgage questions: Use your mortgage calculator spreadsheet to answer these questions: If I borrow £350000 for 20 years at 4.2% p.a., what are my monthly payments? If I borrow £500000 at 3.7% p.a. and pay £3200 each month, how long will it take to pay off my mortgage? If I want to have monthly payments of £1800 and pay for 25 years, what’s the most I can borrow if interest rates are 4.5% ; 2.9% ? I can afford to pay £2050 a month, and my mortgage interest rate is 5.3%. What’s the most I can borrow if the term of my mortgage is 20 years; 25 years; 40 years? I want to get a £450000 mortgage over 18 years. What interest rate should I look for if I want my monthly payments to be £2450; £1800?
September 15, 2025 at 9:19 AM
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Meanwhile, on the other side of the world, ginkgo leaves coming to the end of their lives.
A branch of a gingko tree with yellowing leaves against a blue sky.
September 15, 2025 at 8:11 AM
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#MathsToday #ALevelMaths

Monday's area problem.

Find the exact value of the green area.
The graph of y²=x²(a-x) is shown in red. It's symmetrical about the x-axis and has a closed loop to the right of the y-axis, and passes through the origin twice. The normal at the origin, in blue, passes through a stationary point on the graph in the first quadrant. The interior of the loop below the normal is shaded green.
September 15, 2025 at 5:42 AM
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This one has perimeter 38. I'm confident that x is transcendental...
September 13, 2025 at 7:25 AM
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...and even with rational sides, the area isn't unique:
September 12, 2025 at 8:18 PM