To generate cases systematically, change one thing at a time, in order, and record every result in a table. A pattern is almost never visible in a single case, but it often becomes obvious once cases 1, 2, 3, 4 and 5 sit in a column side by side.
This is the first step of any investigation in International Mathematics investigations. It also supports sequences and pattern rules, where the same tables appear.
Why does the order of cases matter?
If you jump from case 1 to case 5, you skip the evidence of how the pattern grows. You also cannot tell which change in the shape caused which change in the count.
A systematic list uses consecutive values of the variable, such as n = 1, 2, 3, 4, 5. It keeps everything else fixed. That makes differences between rows easy to read.
How to generate cases, step by step
- Define the variable. Say what n stands for, for example “the number of squares in the row”.
- Start with the smallest sensible case. This is often n = 1, sometimes n = 0 or n = 2.
- Draw or list the next cases in order. Do not skip values.
- Record each result in a two-column table. Write n on the left and the quantity on the right.
- Look at the structure and the differences. Ask how each case is built from the one before.
- Predict the next case before you build it, then build it to check your prediction.
Worked example
Matchsticks are used to make a row of squares. One square needs 4 matchsticks. How many matchsticks does a row of n squares need?
Step 1, variable: n is the number of squares in the row.
Step 2, cases: draw n = 1, 2, 3, 4 and count sticks carefully, remembering that neighbouring squares share a side.
Step 3, table:
| n | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Matchsticks | 4 | 7 | 10 | 13 | 16 | 19 |
Step 4, structure: the first square needs 4 sticks. Each extra square adds 3 more, because it shares one side with the square before it.
Step 5, prediction: for n = 10 the count is 4 + 9 × 3 = 31. Checking with a second method: 10 squares have 11 vertical sticks and 20 horizontal sticks, and 11 + 20 = 31. The two methods agree.
If the pattern involved repeated percentage change instead, the percentage-base explorer can generate the cases for you. For example, 100 increased by 10% twice gives 110 and then 121.
The mistake to watch for
A common slip is to count every square as if it had its own four sticks.
Mistaken working: 3 squares need 3 × 4 = 12 matchsticks.
This ignores that squares share sides, so the count is too high.
The correction is to draw the case and count the sticks once each. For n = 3 the drawing shows 10 sticks, not 12.
The table then reads 4, 7, 10, and the pattern of adding 3 appears. Always check your structure against one drawn case.
Check yourself
1. Matchsticks make a strip of triangles. One triangle uses 3 sticks and two triangles in a strip use 5 sticks. Complete the table for n = 3, 4 and 5 triangles.
Show answer
Each extra triangle shares one side, so it adds 2 sticks. The counts are 3, 5, 7, 9, 11.
For n = 3, 4, 5 the answers are 7, 9 and 11.
2. Five friends each send one message to every other friend, one message per pair. List the pairs systematically and count them.
Show answer
Name the friends A to E. A pairs with 4 others, B with 3 new, C with 2 new, D with 1 new.
Total 4 + 3 + 2 + 1 = 10 pairs.
3. A student tests n = 1, 2 and 6 only. Give one reason this is a weak way to start.
Show answer
Skipping 3, 4 and 5 hides how the quantity grows from one case to the next, so the pattern and any change in it are harder to see. A full run of consecutive values is better evidence.
Where this leads next
With a clean table, you are ready to move from observed cases to a conjecture. Then try the investigations practice set. The non-calculator working trainer is handy for checking the arithmetic in larger cases.
Some students can build a table but are unsure which quantity to count. That is a good thing to work through with a teacher in online one-to-one Mathematics tuition.