How often Number patterns and generalisation is asked

Where it was asked

• Supp 202340 marks
• Oct/Nov 202332 marks
• Oct/Nov 202224 marks

The questions

1. Supp 2023, Q1.1.19 marks

   Three diagrams show stacks of cubes with discs balanced on top of some of the cubes (note that some discs are hidden from view), and the stacks form a growing pattern: diagram 1 is a single cube, diagram 2 is a taller arrangement of cubes, and diagram 3 is a still taller arrangement of cubes. A table records, for each diagram number, the number of discs, the number of cubes, and the sum of the discs and cubes in that diagram. The table already gives: for diagram 1 - 1 disc, 1 cube, sum 2; for diagram 2 - 6 discs, 8 cubes, sum 14; for diagram 3 - 21 discs, 27 cubes, sum 48; and for diagram 4 - sum 116 (with the discs and cubes entries for diagram 4 left blank, as are all entries for diagrams 5, 6, 7 and 23). Using the pattern shown by the diagrams of cubes with discs on top (where diagram 1 has 1 disc and 1 cube, diagram 2 has 6 discs and 8 cubes, diagram 3 has 21 discs and 27 cubes, and the sums for diagrams 1 to 4 are 2, 14, 48 and 116), complete the missing entries in the table for diagrams 5, 6 and 7, giving the number of discs, the number of cubes, and the sum of discs and cubes for each of these three diagrams.

2. Supp 2023, Q1.1.26 marks

   Three diagrams show stacks of cubes with discs balanced on top of some of the cubes (note that some discs are hidden from view), and the stacks form a growing pattern: diagram 1 is a single cube, diagram 2 is a taller arrangement of cubes, and diagram 3 is a still taller arrangement of cubes. A table records, for each diagram number, the number of discs, the number of cubes, and the sum of the discs and cubes in that diagram. The table already gives: for diagram 1 - 1 disc, 1 cube, sum 2; for diagram 2 - 6 discs, 8 cubes, sum 14; for diagram 3 - 21 discs, 27 cubes, sum 48; and for diagram 4 - sum 116 (with the discs and cubes entries for diagram 4 left blank, as are all entries for diagrams 5, 6, 7 and 23). Using the number of discs recorded for diagrams 1, 2 and 3 (1, 6 and 21 discs respectively) and any further values you found, determine the algebraic rule, in terms of n, for the number of discs in diagram n. Show all your working.

3. Supp 2023, Q1.1.35 marks

   Three diagrams show stacks of cubes with discs balanced on top of some of the cubes (note that some discs are hidden from view), and the stacks form a growing pattern: diagram 1 is a single cube, diagram 2 is a taller arrangement of cubes, and diagram 3 is a still taller arrangement of cubes. A table records, for each diagram number, the number of discs, the number of cubes, and the sum of the discs and cubes in that diagram. The table already gives: for diagram 1 - 1 disc, 1 cube, sum 2; for diagram 2 - 6 discs, 8 cubes, sum 14; for diagram 3 - 21 discs, 27 cubes, sum 48; and for diagram 4 - sum 116 (with the discs and cubes entries for diagram 4 left blank, as are all entries for diagrams 5, 6, 7 and 23). Given that diagram 1 has 1 cube, diagram 2 has 8 cubes and diagram 3 has 27 cubes, show that the number of cubes in diagram n is equivalent to n cubed. Explain your reasoning as you would to a Grade 6 learner, making sure your explanation shows the relationship between the diagram number and the number of cubes in each diagram.

4. Supp 2023, Q1.1.46 marks

   Three diagrams show stacks of cubes with discs balanced on top of some of the cubes (note that some discs are hidden from view), and the stacks form a growing pattern: diagram 1 is a single cube, diagram 2 is a taller arrangement of cubes, and diagram 3 is a still taller arrangement of cubes. A table records, for each diagram number, the number of discs, the number of cubes, and the sum of the discs and cubes in that diagram. The table already gives: for diagram 1 - 1 disc, 1 cube, sum 2; for diagram 2 - 6 discs, 8 cubes, sum 14; for diagram 3 - 21 discs, 27 cubes, sum 48; and for diagram 4 - sum 116 (with the discs and cubes entries for diagram 4 left blank, as are all entries for diagrams 5, 6, 7 and 23). Complete the column of the table for diagram 23, giving the number of discs, the number of cubes, and the sum of discs and cubes for diagram 23.

5. Supp 2023, Q1.2.11 mark

   Three diagrams show stacks of cubes with discs balanced on top of some of the cubes (note that some discs are hidden from view), and the stacks form a growing pattern: diagram 1 is a single cube, diagram 2 is a taller arrangement of cubes, and diagram 3 is a still taller arrangement of cubes. A table records, for each diagram number, the number of discs, the number of cubes, and the sum of the discs and cubes in that diagram. The table already gives: for diagram 1 - 1 disc, 1 cube, sum 2; for diagram 2 - 6 discs, 8 cubes, sum 14; for diagram 3 - 21 discs, 27 cubes, sum 48; and for diagram 4 - sum 116 (with the discs and cubes entries for diagram 4 left blank, as are all entries for diagrams 5, 6, 7 and 23). Consider the pattern: 2 squared minus 0 squared equals 4; 3 squared minus 1 squared equals 8; 4 squared minus 2 squared equals 12. If this pattern continues, what is the next equation in the sequence?