Number Pattern Relationships

Activities to try

1. 1

   Generate, align, and compare two sequences

   Why it works: The two sequences are related by a constant factor (×2) at corresponding positions. 'B is 6 more' does not describe the relationship, because the difference between corresponding terms is not constant but grows (0, 3, 6, 9, …) — so no single 'add a number' rule fits every column. 'B is twice A' holds in every column instead, and here this works because the step 6 is twice 3 and both sequences start at 0.

   1. Write out the first six terms of Rule A (start at 0, add 3): 0, 3, 6, 9, 12, 15
   2. Write out the first six terms of Rule B (start at 0, add 6): 0, 6, 12, 18, 24, 30
   3. Line the two sequences up in a two-row table so position 1 sits above position 1, position 2 above position 2, and so on
   4. For each column, compare Rule B's term with Rule A's term in the SAME position (vertical comparison across the two rows, not along one row)
   5. State the relationship multiplicatively: 'B is twice (×2) A in every column' and explain why — Rule B adds 6, which is twice the 3 Rule A adds, and both start at 0
2. 2

   Make ordered pairs and plot them

   Why it works: Corresponding terms of two proportional sequences form ordered pairs whose points lie on a straight line through the origin; the constant ×2 ratio is the same thing as the equation y = 2x and the slope of that ray.

   1. Turn each column into an ordered pair as (Rule-A term, Rule-B term): (0,0), (3,6), (6,12), (9,18)…
   2. For each pair, use the first number to move ACROSS (x, Rule A) and the second number to move UP (y, Rule B)
   3. Plot the first four pairs in the first quadrant of a coordinate grid
   4. Describe the shape the points make, and notice they lie on a straight line through the origin
   5. Connect this to y = 2x, linking the table, the pairs, and the line as one object
3. 3

   Test whether the ×2 depends on the starting number

   Why it works: The multiplicative relationship between corresponding terms comes from the steps AND the starting numbers; in the original rules both start at 0. When A starts at 1 but B starts at 0, 'twice' no longer holds in every column even though one step is still twice the other — so pupils should not assume 'twice' from the steps alone.

   1. Keep Rule B as 'start at 0, add 6' but change Rule A to 'start at 1, add 3' (1, 4, 7, 10, …)
   2. Generate both sequences and line them up by position
   3. Check each column: is B still exactly twice A now?
   4. Explain what broke and why — the clean ×2 relationship relied on the starting numbers as well as the steps (in the original rules, both started at 0), not on the steps alone