Systematic Listing

Activities to try

1. 1

   Pair and light the grid: building every meal in order

   Why it works: Working in a fixed order — fix one choice, run through every option of the other — guarantees no combination is missed or repeated, and the complete set fills a grid whose total is rows × columns, the product of the options.

   1. Start with 3 starters and 4 mains and first PREDICT how many different meals you think are possible before listing any — write down your prediction and say whether you think it is 3 + 4 or 3 × 4.
   2. Begin by pairing starters and mains in any order you like, recording each meal you make in a two-way grid (starters down the side as rows, mains across the top as columns) by marking the matching cell.
   3. Look at the grid: check whether random pairing has left scattered marked cells with empty gaps, or whether you tried to mark any meal twice. Ask yourself: have I found them all? Can I be sure?
   4. Now switch to a fixed order: hold starter 1 and pair it with every main in turn (filling a whole ROW), then starter 2's row, then starter 3's row, until every cell is marked with no gaps.
   5. Count the completed grid as rows × columns = 3 × 4 = 12, and compare this with your earlier prediction.
2. 2

   The tree-diagram route to the same complete set

   Why it works: A tree diagram is a second systematic structure that enumerates the same complete set; the repeated identical sub-branching is the product 3 × 4 made visible a different way.

   1. Draw a tree: one branch for each starter, then from each starter branch draw a sub-branch for every main.
   2. Read each complete path from left to right as one meal (e.g. starter 1 → main 3).
   3. Count the end-points of the tree and check it matches the grid total of 12.
   4. Explain why each starter branch must sprout the same 4 sub-branches, so the total is 3 lots of 4 = 3 × 4.
3. 3

   Check and extend: how many if a category grows?

   Why it works: The product structure lets you predict the count when a category grows without enumerating every case, confirming the skill transfers beyond the original numbers.

   1. Answer 'How many meals are there altogether?' and justify whether it is 3 + 4 or 3 × 4.
   2. Fill in a deliberately missing combination given a partly-completed grid or list.
   3. Predict, WITHOUT listing them all, how many meals there would be if the café added a 4th starter (4 × 4 = 16), explaining your reasoning from the product structure.
   4. Match each listed meal to its grid cell to confirm the set is complete and non-repeating.