Approaches to the formulae prompt

On this page we present approaches to the formulae prompt from Inquiry Maths workshops. The approaches involve a wide array of mathematical ideas, concepts, and techniques.

The reports illustrate how the same prompt can give rise to different lines of inquiry and attest to the idea of mathematics as a creative human endeavour.

Participants at the workshops have included trainee teachers on postgraduate courses, teachers in schools, and attendees at conferences.

Using sequences to derive a formula

During an Inquiry Maths workshop at the University of Sussex (UK) in October 2025, trainee teachers studying on the secondary PGCE course looked at the formulae prompt.

Kirsty Hoad and Harry Armstrong developed a novel approach using expressions for the nth term of sequences created by increasing the height systematically. The general expression for the number of sticks with height h is (2h + 1)n + h (where n represents the pattern number). As the pattern number corresponds to the length of the lattice (l), we can replace n with l. By rearranging and factorising, we show that the sum of the length and the height equals the square of their difference.

The course tutor James Bashford organised the workshop.

Different approaches

In an Inquiry Maths workshop at St Mary's University (Twickenham, UK) in June 2025, trainee teachers on the PGCE course took different approaches to the formulae prompt.

Tom Cates and Paul Hackworth used an algebraic approach (pictured left) to deduce that the difference between l and h equals the square root of the sum of l and h. In their working, the sum is given by n2 and the difference by n. They developed their idea further to devise two expressions using n2 and n (where n > 1) that give all the pairs of l and h that satisfy the formula in the prompt.

Jaspreet Kaur Rashid developed a systematic approach to find all the rectangular arrays that satisfy the formula (pictured top right). She explained to the other participants that for the first set of arrays (with a height of 1), there are no more examples after the 3 x 1 array because 'n actual', which increases in a linear sequence, can never catch 'n formal ', which increases in a quadratic sequence. Jaspreet's approach also proves that there is not an array of height 2 that satisfies the formula.

Francis D took a third approach (pictured bottom right). He used expressions for the nth term of linear sequences to devise an alternative formula for the total number of sticks in any rectangular array.

Pietro Tozzi, the PGCE course lead, organised the workshop. He reported that the Inquiry Maths model impressed the trainees and they gave excellent feedback on the workshop.

Algebraic proof

In January 2025 Tymoteusz Zdunek organised an Inquiry Maths workshop with Dr Andrew Blair at Akademeia High School in Warsaw (Poland). The head of mathematics, Dr Peter Kowalski, and the rest of the department participated in the workshop.

In one session, the teachers analysed the formulae inquiry. Monika Klimczak-Gromska devised a proof (pictured) that consecutive triangular numbers satisfy the condition in the prompt.

Later in the day a class of year 10 students inquired into the difference of two cubes prompt while the teachers observed. The workshop ended with each teacher presenting a plan for an inquiry with one of their classes.

ATM conference 2016

During the Inquiry Maths workshop at the Association of Teachers of Mathematics conference 2016, the thirty participants inquired into the formulae prompt.

Question, notice, and wonder

They posed the following questions during the first phase of the inquiry:

What are we trying to find?
Is there a link between the diagram and formula?
What rectangles satisfy the formula?
Can we generalise?
If we vary l and h, does n give a sequence or interesting set of numbers?
Is there a formula for rectangles that don't satisfy the given formula?

After agreeing on definitions of the terms l, h and n, the participants paused to consider the regulatory cards. The majority selected either Decide what the problem is or Try to find some more examples, although a strong argument was made for Think about the structure in order to avoid an unsystematic search for more cases.

After a period of working on the prompt, participants fed back their findings. Dave showed how l2 + h2 = l(h + 1) + h(l + 1) simplifies to (l - h)2 = l + h.

Then Sabrina used the result to show how she had found more rectangles that satisfied the formula using this table:

Luke described how he had used the same result to develop an exhaustive method for finding rectangles.

Finally, Julian explained how he and Jo had developed their approach using simultaneous equations to deduce that l and h had to be consecutive triangular numbers. Julian's note sheet (illustrated) shows their reasoning, which is carried on below.

BCME 2014

The formulae prompt was the starting point for an inquiry in the Inquiry Maths session at the British Congress of Mathematics Education (April 2014). Some of the questions and comments about the prompt are shown in the picture (right).

As would be expected from some of the UK's foremost maths educators, the responses to the prompt could take the inquiry in multiple directions. Mike Ollerton, for example, categorised the gaps between the sticks as "2, 3- or 4- hole" - an observation that leads into a novel line of inquiry.

The participants at the session decided either to follow their own lines of inquiry or to focus on the questions in the box at the top of the sheet.

Kate Bell presented her approach. She deduced that the sum of l and h equals the square of their difference:

Kate was reminded of a problem involving the diagram (right), which shows that this is a property of two consecutive triangular numbers.

After the session, Dominic Penney said that the key point he would take away was that Inquiry Maths is not about unsupported individual exploration. Rather, the teacher participates in directing the inquiry while giving the students as much responsibility as possible. The class develops the inquiry together, discussing its direction and content at regular points. Dominic described it this way: "We are all on the same platform and then we jump off together."

Postscript

During the Inquiry Maths session at BCME 9 (2018), Becky Warren (nrich) and Michael Anderson (National STEM Learning Centre) arrived at an expression for the value of l that satisfies the formula:

Other approaches

Simulataneous equations

In a workshop for trainee teachers at London Metropolitan University in January 2018, Eduardo Abend and Tom Mee took a novel approach that moved from the general to the particular.

Taking their starting point as l2 + h2 = l(h +1) + h(l +1), they derived (l - h)2 = l + h and deduced that l + h must be a square number. From this they were able to set up simultaneous equations to find values for l and h that satisfied the condition in the prompt. For example, when l + h = 25, l - h = 5 and l = 16, h = 9.

Eduardo and Tom presented their reasoning to the rest of the teacher trainees.

A triangular number of sticks

At an Early Professional Development Day (June 2014), teachers of mathematics in their first and second years of teaching in Brighton and Hove (UK) looked at the formulae prompt.

Alice-Rae Gilbert (from Dorothy Stringer secondary school) noticed a link between the length of the longest side of the array and the total number of sticks, which she shows is always a triangular number. Alice presents her full working here.

A graphical representation

At the Mixed Attainment Maths conference in Sheffield Hallam university (June 2017), one participant in the Inquiry Maths workshop graphed the following equations:

l + h + 2hl = 45 and l2 + h2 = 45

She was able to show the solutions of l = 6, h = 3 and l = 3, h = 6 at the intersections of the circle and curve (see picture).

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