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The prompt
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.
The class starts by using addition to sum the numbers in two adjacent cells to make the number in the cell above. Although students can do this instinctively, the teacher might have to explain the rule.
For example, if the bottom row is 2 1 4 3, then the second and third rows are 3 5 7 and 8 12 respectively. That gives a top number of 20, which is not the target number.
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.
June 2018
Lines of inquiry
Lines of inquiry
Students have developed their own lines of inquiry by making changes to the prompt.
(1) Change the numbers
Use other numbers with a difference of one - for example, 3, 4, and 34 or 4, 5, and 45. What pairs of numbers in the bottom row lead to the target number at the top of the pyramid?
Reverse the numbers - for example, 3, 2 and 32. Is there a connection between the pair of missing numbers in the prompt and in the 'reverse pyramid'?
(2) Change the difference between the numbers
The students might work systematically to explore examples with a difference of two (for example 3, 5, and 35 or 4, 6, and 46), then a difference of three, four and so on.
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. The illustration shows a year 8 student's proof when the difference between the starting numbers is 2.
(3) Change the position of the numbers
Another change to the prompt involves putting the 2 and 3 in different positions in the bottom row. However, the teacher should point out that the numbers have to be in the same order to give 23 in the top cell.
In the pyramid below a year 10 student has moved the 2 in the bottom row to the second cell and goes on to show that the relationship between x and y is 3x + y =14.
There are six possible arrangements of 2 and 3 in that order along the bottom row (see the illustration). The relationship between a and b in each case is shown to the right.
Students are often intrigued when they notice that two expressions for the relationship between a and b appear twice. Why do they repeat? Why are the two that are different not a repeat of each other? The teacher can use Yang Hui's (Pascal's) triangle (see below) to help the students explain.
(4) Change the number of rows
When students increase the number of rows, they test whether the sum of the numbers in the bottom row is still 11. As they explore, they soon realise that the sum must be smaller because the number in the top cell is always greater than 23 when they use 11.
One solution for five rows is 2 2 1 1 3. Another is 2 0.75 2 0.75 3. Students notice that the sums of the numbers are not the same. They begin to understand that the numbers are weighted in the their contribution to the target of 23. Again the teacher can use Yang Hui's (Pascal's) triangle to help the students explain.
Links to Yang Hui's (Pascal's) triangle
Another line of inquiry develops when students, guided by the teacher, make links to Yang Hui's (Pascal's) triangle. The illustration shows that the number in the top cell is the sum of each outside number in the bottom row and three lots of each middle number.
Students can 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).
The triangle becomes even more useful when students increase the number of rows. If there are five rows, the entry in the top cell would be a + 4b + 6c + 4d + e.
Questions and conjectures
Questions and conjectures
Students in Andrew Blair's year 8 mixed attainment class responded to the prompt by posing questions and making conjectures (see the illustration).
The inquiry started with students creating more examples to test the conjecture that the sum of the numbers in the bottom row is always 11. After considering different types of cases, including decimal and negative numbers, the class decided that the conjecture is true.
However, Andrew explained that testing more and more cases did not prove the sum had to be 11 in all cases.
The students had shown that the sum of the bottom row was 11 only in the examples they had created.
To prove the conjecture is always true, Andrew introduced the variables a and b in the empty cells on the bottom row. Some of the students went on to construct an algebraic proof that a + b = 6 (see illustration), which they explained to the class.
Focusing on the regulatory card Change the prompt at the start of the second lesson, the class suggested 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
Classroom inquiry
Resources
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.
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