Higher November 2019 Paper 6 Q10
10 Nine consecutive numbers are written on a 3-by-3 grid.
They are arranged, in ascending order, in a spiral as shown.
(a) Karen writes the numbers 3 to 11 on her grid.
| 3 | 4 | 5 |
| 10 | 11 | 6 |
| 9 | 8 | 7 |
The total of the first column is \(3 + 10 + 9 = 22\).
Karen says
The total of the first column is one less than the total of the second column.
Show that this is correct for Karen’s grid. [1]
(b) Victor says
If any nine consecutive numbers are arranged in ascending order in this spiral on a 3-by-3 grid, the total of the first column will always be one less than the total of the second column.
Prove that Victor is correct. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4 + 11 + 8 = 23 seen | 1 | Accept written as a sum in a column | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| e.g. First column: \(n + (n + 7) + (n + 6) = 3n + 13\) Second column: \((n + 1) + (n + 8) + (n + 5) = 3n + 14\) \((3n + 14) - (3n + 13) = 1\) | 5 | B2 for consistent algebraic terms for at least first two columns of the grid or B1 for at least 3 algebraic terms for consecutive numbers seen AND M1 for algebraic sum of first or second column shown M1 for algebraic sum of first and second columns shown and correctly simplified A1 for sum of second column – sum of first column = 1 calculated or explained from correct working or M1 for difference of one pair of algebraic terms from first and second column shown M1 for differences of two further pairs of algebraic terms from first and second column, with all three pairs correctly simplified A1 for each difference found as +1 or –1 oe and summed/explained to a difference of +1. Correct algebra and reasoning throughout If 0 scored, allow SC1 for a correct numerical or descriptive example using either method and stating an overall difference of 1 | e.g. \(n\), \((n + 7)\), \((n + 6)\) and \((n + 1)\), \((n + 8)\), \((n + 5)\) e.g. \(n\), \((n + 1)\), \((n + 2)\) e.g. \(n + (n + 7) + (n + 6)\) or in column e.g. \(n + (n + 7) + (n + 6) = 3n + 13\) A1 for e.g. \(3n + 14\) and \(3n + 13\) and “second column is 1 more than the first” but A0 for e.g. \(3n + 14\) and \(3n + 13\) and “difference of 1” or for \((3n + 14) - 3n + 13 = 1\) e.g. “the difference between \(n + 1\) and \(n\) is 1” e.g. “\(n + 1\) is 1 more than \(n\)” Condone poor use of brackets for both M marks |