Problems we wrote
The organised list: count every case, miss none
Many young problem solvers find some answers and stop. An organised list fixes the order in which possibilities are written, so the child knows when every case has been found.
Fix the first choice
Write every option that starts with the first item, then every option starting with the second, and so on. Order is what makes a list organised.
Never go backwards
When pairing friends, A with B already covers B with A. Only pair each person with the people after them in the list.
Count the pattern
Once a list is in order, its shape often shows a pattern, such as 3, then 2, then 1, that makes the total easy to check.
Original Year 4 counting problems with the organised list written out| Problem | Organised list | Answer |
|---|
| Using the digits 2, 4 and 7 once each, how many two-digit numbers can you make? | 24, 27; 42, 47; 72, 74 | 6 |
| A cafe offers chocolate, mango or vanilla ice cream, in a cone or a cup. How many choices are there? | Chocolate cone, chocolate cup; mango cone, mango cup; vanilla cone, vanilla cup | 6 |
| Four friends each shake hands once with everyone else. How many handshakes? | A with B, C, D; B with C, D; C with D | 6 |
| How many ways can 10 be made by adding three different counting numbers? (Order does not matter.) | 1 + 2 + 7, 1 + 3 + 6, 1 + 4 + 5, 2 + 3 + 5 | 4 |
| A strip of 3 squares in a row. How many rectangles can you find (count squares too)? | Length 1: three; length 2: two; length 3: one | 6 |
| Using 1, 2 and 3 once each, how many three-digit numbers are there, and which is the largest even one? | 123, 132; 213, 231; 312, 321 | 6 numbers; 312 is the largest even one |
Several answers here are 6, and that is no accident. Three things arranged in order, three ice cream flavours each with two containers, and four people paired up all happen to give six cases. Spotting that different-looking problems share a structure is exactly the kind of reasoning a maths section rewards.
Try the handshake problem with a child and watch the list grow. Ava shakes hands with Ben, Cal and Dev: three handshakes. Ben has already met Ava, so he only needs Cal and Dev: two more. Cal still has Dev: one more. Dev has met everyone. Three, then two, then one makes six. Ask what happens with five friends, and a child who has understood the list will predict four, three, two and one, which is ten, before drawing anything.
The fourth problem shows why order matters. A child who writes sums at random may list 1 + 2 + 7 and 2 + 1 + 7 as different, or miss 2 + 3 + 5 entirely. Starting with the smallest first number and keeping each list increasing guarantees every set appears exactly once.
