Pyramid addition inquiry
The prompt
Mathematical inquiry processes: Notice properties; generate examples; conjecture, generalise and prove. Conceptual field of inquiry: Sum of integers and other types of numbers; collect like terms; Yang Hui's (Pascal's) Triangle.
The prompt was designed by Dan Walker (a mathematics teacher in north London, UK). He designed a series of lessons in which the prompt is part of a structured investigation. As the stimulus for inquiry, the prompt can lead into many different lines of inquiry. Students start by using addition to sum the numbers in two adjacent cells to make the number in the cell above. Normally they do this instinctively, but the teacher might have to direct a class towards addition.
Students invariably notice that 23 comes from the two and three in the bottom row. The teacher might emphasise that the '2' is two tens and the '3' stands for three units, which underpins the move into a general representation later in the inquiry. The consecutive numbers in the bottom row become n and n + 1, while the top cell now contains 10n + n + 1 or 11n + 1.
A period of exploration might follow initial questions and observations as students try to find numbers that 'fit' the two cells in the bottom row. The teacher might introduce variables into the inquiry at an early stage. If the two cells are filled by a and b, then it can be proved that a + b = 6. The proof is accessible to students in lower secondary and even upper primary.
Changes to the prompt
Students have developed their own lines of inquiry by making the following changes to the prompt:
(1) Change the starting numbers
Retain a difference of one - for example, 3 and 4, 4 and 5, etc.
Make the numbers the same - for example, 2 and 2, 3 and 3, etc.
Reverse them numbers - for example, 3 and 2, 4 and 3, etc.
(2) Change the difference between the two starting numbers - for example 3 and 5, 4 and 6, etc. In general, if the bottom row of the pyramid is n, a, b, n + k (where n is the first number, a and b are the single digit numbers to be found and k is the difference between the two starting numbers), then 11n + k = 3a + 3b + 2n + k. This simplifies to a + b = 3n and, therefore, the sum of the two numbers to be found in the bottom row is three times the first number.
Change 2: A year 8 student is close to proving that a + b = 3n when the difference between the starting numbers (in the first and fourth cells of the bottom row) is 2.
(3) Change the position of the numbers in the bottom row.
Change 3: The relationship between a and b when 2 and 3 are in different cells of the bottom row.
Change 3: A year 10 student proves that 3y + x = 14 when 2 and 3 appear in the second and fourth cell of the bottom row respectively.
(4) Change the number of rows (see resource sheet below).
(5) Change the operation and use multiplication or subtraction.
Questions and conjectures
Students in a year 8 mixed attainment class started their inquiry by posing questions and making conjectures (see below). After the conjecture about the sum of the two missing numbers on the bottom row was confirmed using different types of cases (including decimals and negative numbers), the teacher introduced the variables a and b. Students learnt how to prove that a + b = 6.
A student proves a + b = 6 when 2 and 3 appear in the first and fourth cell of the bottom row respectively.
Focusing on the regulatory card Change the prompt at the start of the second lesson, the teacher asked the class to suggest changes they could make to the prompt. Each student then selected one change and developed a line of inquiry, which involved making a conjecture, testing different cases and attempting to prove a generalisation.
Classroom inquiry
Links to Yang Hui's (Pascal's) triangle
Students can make links to Yang Hui's (Pascal's) triangle during the inquiry (see below), particularly when they change the number of rows. If, for example, there are five rows, the entry in the top cell would be a + 4b + 6c + 4d + e. Students could use this property of the pyramid to speed up their search for numbers that satisfy the original prompt. When the bottom row of the pyramid is made up of, in order, 2, a, b and 3, the top number (23) equals (1 x 2) + 3a + 3b + (1 x 3).
Resources
Acknowledgement
Richard Mills, Amy Flood and Michael Joseph, all mathematics teachers at Haverstock School (Camden, London), trialled the prompt in their classrooms in June and July 2018.