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The prompt
The prompt
Mathematical inquiry processes: Generate examples and counter-examples; conjecture, generalise and prove; analyse structure. Conceptual field of inquiry: Formulae linked to arrays; algebraic expressions.
The prompt is suitable for all classes in secondary school, with the amount of a teacher's guidance and instruction contingent on the students' level of independent inquiry skills and mathematical knowledge.
The inquiry starts with questions and statements about the prompt:
What do l, h and n stand for?
The formula works because 62 + 32 = 45 and there are 45 lines.
The length is twice the height.
Does the formula hold true if the ratio of l:h is different?
Is there a separate formula for square-shaped arrays?
Does the formula work for other rectangular arrays?
Are there separate formulas for the numbers of inside and outside lines?
Could n stand for the number of gaps between the lines? (Can we count an 'outside' gap as the same as an 'inside' gap?)
Does the formula have something to do with Pythagoras' Theorem where n stands for the square on the hypotenuse?
We should draw another rectangle to check the formula works.
After the initial stage the teacher might define the variables, at least at the start of the inquiry, as follows: l - length of the rectangle, h - height of the rectangle, n - number of lines or sticks.
The prompt intrigues students and they search enthusiastically for other arrays with lengths and widths that satisfy the formula. They quickly realise that the formula does not work in most cases.
Conjectures
So what is it about the rectangle in the prompt that is special? Could it be that the length of the array is twice the height? Or do both the length and height have to be multiples of three? The teacher should share students' conjectures as they arise in the inquiry.
The formula works for the array in the prompt because the length and height are consecutive triangular numbers. If students do not 'discover' this themselves, the teacher can lead them towards the idea using the examples they have (see the Exploration section below).
There are a variety of lines of inquiry at different levels of challenge that follow from this first phase. Three are outlined in these slides designed for a structured inquiry.
Inquiry Maths workshop
Inquiry Maths workshop
Dr Andrew Blair ran an Inquiry Maths workshop for trainee teachers at St Mary's University (Twickenham, UK) in September 2026. The trainees explored the formulae prompt. The question, notice, and wonder phase led to speculation about whether the formula generalises to other cases.
After realising that they could calculate the number of sticks in the array by using 7 x 3 + 4 x 6, the trainees devised the general form: l(h + 1) + h(l + 1). Most then equated the two formulae l2 + h2 = l (h + 1) + h(l + 1) and followed an algebraic approach.
During presentations at the end of the inquiry one pair of trainees demonstrated how the equation could be rearranged to give (l - h)2 = l + h. Thus, the formula in the prompt works for arrays when the square of the difference between the length and height equals their sum. Below we present two other lines of inquiry.
Pietro Tozzi, the PGCE course lead, organised the workshop.
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Read more reports of inquiries into the formulae prompt during Inquiry Maths workshops.
The quadratic formula
The quadratic formula
Peter Haasjes and Jack Hardman rearranged l2 + h2 = l (h + 1) + h(l + 1) to create a quadratic equation in x and y. (See their working above, left and right respectively.) By solving the equation with the quadratic formula, they expressed x in terms of y. When y is a triangular number greater than one, the formula gives the previous and next triangular numbers. For example, when y = 6, x = 3 or 10. Moreover, when the discriminant (8y + 1) is a perfect square, the values of x and y will be consecutive triangular numbers.
Connection to the circle
Connection to the circle
Dariya Ainekova and Joe Haldenby noticed that the formula in the prompt is in the same form as the equation of a circle with centre (0,0) and radius √(n). The graph connects different lines of inquiry. It shows concentric circles with radii of √10, √45, √136, √325, and √666 - that is, the square root of the sum of the squares of two consecutive triangular numbers. By plotting the implicit equation (x - y)2 = x + y, the points of intersection represent the dimensions of rectangular lattices that satisfy the formula in the prompt.
Lines of inquiry
Lines of inquiry
- Exploration
If the class explores systematically, students soon find the smallest case that satisfies the formula (length 3, height 1). This leads to speculation that the next one after the 3-by-1 and 6-by-3 arrays is 9-by-6 by using multiples of three.
That proves to be close but not quite right. The teacher encourages students to change the length or height by one and check.
With the first three results involving 1, 3, 6, and 10 the teacher focuses students' attention on the sequence.
With the first three results involving 1, 3, 6, and 10 the teacher focuses students' attention on the sequence.
If the class does not recognise the triangular numbers, then the teacher reminds them of the sequence - perhaps supporting an explanation with diagrams.
If the class does not recognise the triangular numbers, then the teacher reminds them of the sequence - perhaps supporting an explanation with diagrams.
Another period of exploration might follow when the teacher asks if the triangular numbers have to be consecutive.
2. Deduce a formula and prove
2. Deduce a formula and prove
An analysis of the structure of the array helps students develop another formula for the number of lines.
By colouring the lines in the way shown in the illustration, students can understand how to develop a formula from the structure of the array. There are 4 x 6 red lines, where 6 is the length and 4 is one more than the height. Similarly, there are 7 x 3 green lines, where 3 is the height and 7 is one more than the length.
As any rectangular array can be represented by horizontal and vertical lines in this way, the formula for the number of lines in any rectangular array is: l(h + 1) + h(l + 1).
Students can complete a table (the one below is from a year 10 student) to verify that, for consecutive triangular numbers, both formulae give the same answer.
See mathematical notes 1 for a proof that if l and h are consecutive triangular numbers, then l2 + h2 = l(h + 1) + h(l + 1).
The proof below comes from Amelia (a year 10 student at Longhill High School, Brighton, UK). After being given the general expressions for two consecutive triangular numbers, she produced the proof independently.
If-and-only-if proof
If-and-only-if proof
Would any two consecutive triangular numbers satisfy the formula? Or must they be consecutive?
Dr Daniela Vasile (Head of Mathematics at South Island School, Hong Kong) gave the formulae prompt to some of her year 11 students who are doing the Cambridge Additional Mathematics qualification.
Daniela explains how the inquiry developed:
The students worked through and found the pattern - sides of the rectangle being consecutive triangular numbers.
Then I asked, are we sure that all pairs of consecutive triangular numbers fulfill the condition? It was not difficult for them to prove it true, but my next question was: Are we sure that there are no other pairs verifying the condition? They struggled with this proof, so we did it together. The prompt turned into a very nice if-and-only-if one - I really enjoyed it!
Daniela sent the proof to the Inquiry Maths website.
Other lines of inquiry
Other lines of inquiry
1. Arrays in other shapes
1. Arrays in other shapes
The method to deduce a formula for the number of sticks can be generalised to arrays of other shapes - for example, equilateral triangle, rhombus, and hexagon. (See mathematical notes 2.)
2. Gaps between the lines
2. Gaps between the lines
Is there a formula for the number of gaps between the lines? Are there formulae for the different types of gaps? In a rectangle, there will '2-gaps' where two lines meet, '3-gaps' where three lines meet and '4-gaps' where four lines meet.
In the prompt there are 4 2-gaps, 14 3-gaps, and 10 4-gaps. Students might collect more results and attempt to find a generalisation by spotting patterns. They will notice that there are always 4 2-gaps (in the corners), but struggle to find expressions for 3- and 4-gaps.
To find those, an analysis of structure is better. The 3-gaps are on the perimeter. The total is the sum of two lots of one less than the length and two lots of one less than the height. The 4-gaps are in the centre. The total of 4-gaps is the product of one less than the length and one less than the height.
3. The array only
3. The array only
Although the inquiry teacher might wish to insist that students consider the array and the formula together, comments that relate to the array separately can lead the inquiry down different pathways. They include the following:
There are 18 1-by-1 squares, 10 2-by-2 squares and 4 3-by-3 squares, making 32 in total.
If you add another column, you need seven more lines.
The first line of inquiry leads to the question 'How many squares are there in a rectangular array? Again a structural approach can lead to formulae for the different sized squares and one for the total number of squares.
The second line of inquiry can lead to students extending the array to form a sequence of arrays all with a height of three.
If a 1-by-3 array is taken as the first in the sequence, the nth term of the number of lines is 7n + 3 (where the +3 are the first three lines on the left of the array).
4. Computer programming to develop inquiry
4. Computer programming to develop inquiry
During an Inquiry Maths workshop at Mulberry Academy (Shoreditch, London) in June 2019, Duygu Gumus deduced the relationship l - h = √(l + h). In the interval, Lauren Gillott wrote a programme in Python to generate all the pairs of l and h that satisfy the equation up to 1000. Amy Flood (Head of Mathematics) talked of her plans to incorporate Inquiry Maths into the department's schemes of learning, "We are excited to begin creating inquiry lessons and inquiry classrooms in order to develop students' reasoning."
import math
for i in range(1000):
for j in range(1000):
if (i - j) == math.sqrt(i + j):
print(str(i) + ' ' + str(j))
Lauren Gillott is 2 i/c in the mathematics and computer science department; Duygu Gumus is a teacher of mathematics and deputy director of sixth form.
5. Using GeoGebra to explore
5. Using GeoGebra to explore
During an Inquiry Maths workshop at the Swiss Mathematics Conference for secondary teachers in Geneva (February 2018), Dan Pearcy designed a tool on Geogebra that allows students to explore the prompt and find other cases that are consistent with the formula.
Resources
Resources
Acknowledgement
Ranelagh Maths Network's critical reflection on this prompt during an Inquiry Maths workshop in November 2013 was invaluable for the development of the inquiry. The group of maths teachers from Bracknell Forest, Reading and the Royal Borough of Windsor and Maidenhead (UK) meet for professional development activities over the school year in order to keep up to date with developments to the curriculum, assessment and pedagogy. Their organiser is Yvonne Scott from Ranelagh School, Bracknell.
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