Mathematical inquiry processes: make connections; generate examples and counter-examples; conjecture, generalise and prove. 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. Younger students can substitute numbers into the formula and check whether it satisfies arrays of different sizes; older students can develop other formulae and extend the inquiry into series. 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 h:l is different?
Is there a separate formula for square-shaped arrays?
Does the formula work for other 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.
This prompt can intrigue and frustrate in equal measure. Students want the neat formula to 'work' in all cases, although they can realise very quickly that it does not. The formula in the prompt works for the array presented, but only because the length and height are consecutive triangular numbers. For other types of array, the formula does not give the total number of lines. However, this does not stop students from speculating about the conditions under which the formula works. Could it be when the length of the array is twice the height or both length and height are multiples of three? Only after a period of exploration might they acknowledge that the formula does not work far more often than it does, and start the search for an alternative.
Deducing a formula for any rectangular array
At this point, students might suggest (or be guided towards) collecting their results in a table, from which they attempt to induce (or 'discover') a formula. Even when students are successful in this type of enterprise, however, very few can link the formula derived from a numerical pattern to the structure of the array. It is incumbent on the inquiry teacher to draw out an explanation or give an exposition (see below) on how to deduce the formula from the array in the prompt.
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).
Lines of inquiry
Extending the inquiry
1. Considering consecutive triangular numbers
The inquiry teacher might have to direct students towards triangular numbers. It seems more intuitive to speculate about the ratio of l:h as 2:1 or both l and h being multiples of 3. Once these possibilities are exhausted, students might come to triangular numbers if they find that a 3-by-1 array also satisfies the formula or in response to a teacher's prompt about sequences that contain 3 and 6. Otherwise, the inquiry teacher points out that both numbers are triangular and invites further speculation: do the triangular numbers have to be consecutive or not? Through a short episode of exploration students soon come to the realisation that rectangles with side lengths 1 and 3, 6 and 10, 10 and 15, and so on (see table) meet the conditions in the formula.
The table shows that for consecutive triangular numbers, both formulae provide 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).
2. Triangular and hexagonal arrays
The method to deduce a formula can be generalised to arrays of other shapes - for example, equilateral triangles and hexagons. See mathematical notes 2 for formulae that give the total number of lines in these shapes.
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 fulfil 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's second question takes the inquiry into new territory and we are delighted that she sent the proof to the inquiry maths website.
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 squares in the rectangle.
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 two comments can result in the development of formulae related to the number of squares in a rectangular array. The other comment can lead to students extending the array to form a sequence of arrays all with a height of three. An expression for the nth term of the sequence is 7n + 3. A year 10 class decided to find as many sequences of arrays as possible that include the one in the prompt.
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."
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.
ATM conference 2016
The formulae prompt was used in the Inquiry Maths workshop at the 2016 conference of the Association of Teachers of Mathematics. Initial questions included:
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, we paused to consider the regulatory cards. The majority of the 30 participants 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, colleagues fed back their findings. Dave showed how l2 + h2 = l(h + 1) + h(l + 1) simplified 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 is shown below. The approach he presented to the session follows.
(l - h)2 = l + h
Making l - h = x [A], then l + h = x2 [B]
[A] + [B] gives 2l = x2 + x and l = x(x + 1)/2
[A] - [B] gives 2h = x2 - x and h = x(x - 1)/2
x(x + 1)/2 and x(x - 1)/2 are expressions for consecutive triangular numbers.
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 are shown above.
As would be expected of some of the UK's foremost maths educators, the responses to the prompt could take the inquiry in multiple directions. Mike Ollerton categorised the gaps between the sticks as "2, 3- or 4- hole". The ratio of squares to lines (sticks) is a novel idea. The participants at the session decided either to follow their own strands of inquiry or to focus on the questions in the box at the top of the sheet.
Kate Bell presented her approach to the other participants. She deduced that the sum of l and h equals a square number, specifically the square of their difference.
l2 + h2 = l(h + 1) + h(l + 1)
l2 + h2 = 2lh + h + l
l2 - 2lh + h2 = l + h
(l - h)2 = l + h
Kate was reminded of a problem involving this diagram (below), 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."
During the Inquiry Maths session at BCME 9, Becky Warren (NRICH) and Michael Anderson (National STEM Learning Centre) arrived at:
l2 + h2 = l + (2l + 1)h
Rearranging gave them: h2 - (2l + 1)h + (l2 - l) = 0
Using the quadratic formula, a = 1, b = -(2l + 1), c = l2 - l
As b2 - 4ac > 0, [-(2l + 1)]2 - (4)(1)(l2 - l) > 0
Which gives 8l + 1 > 0
As h is an integer, 8l + 1 is a square number.
Therefore 8l + 1 = x2
The values of l that satisfy the equation are 1, 3, 6, 10, etc.
In a workshop for teacher trainees at London Metropolitan University (January 2018), Tom Mee and Eduardo Abend 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. You can see their initial responses to the prompt here and their subsequent reasoning here.
Tom and Eduardo presented their reasoning to the rest of the teacher trainees.
Mixed Attainment Maths
At the Mixed Attainment Maths conference at Sheffield Hallam university (June 2017), one participant of 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.
Early professional development
At an Early Professional Development Day (June 2014), maths teachers 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 matchsticks, which she shows is always a triangular number. Alice presents her full working here.
Ranelagh Maths Network's critical reflection on this prompt during an Inquiry Maths session 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.