Ruling Out Every Case

The big idea

Sometimes a problem has a small, complete set of possibilities. A learner can prove a conclusion by naming every possible case and ruling out the cases that do not work. The organization matters: the list explains why there is no hidden fifth case, and each rejection needs a specific reason. When only one case remains, it is not a guess. It is forced by the complete check.

Worked example

A two-digit number uses 2 and 5 exactly once. The complete possibilities are 25 and 52. Suppose the number must be even. Twenty-five is not even because its last digit is 5. Fifty-two is even because its last digit is 2. Since 25 and 52 are the only cases, 52 is the only possible answer.

How children may show it

Children may use a short list, a table with a check or cross beside each row, cards that can be moved into “works” and “does not work” groups, or a sentence for each case. Ask, “Why are these all the cases?” before asking whether each one works. That separates completeness from checking.

Common mix-up

A child may rule out several tempting choices and then stop, even though another possibility has not been considered. Ask the child to describe the rule that generated the list. Another mix-up is rejecting a case with “it looks wrong.” Ask which exact condition it fails and where that failure can be seen.

Try it together

Choose one shirt from red or blue and one hat from black or white. List all four outfits. Add the rule “the shirt and hat cannot both be dark.” Check each outfit against that one rule, cross out the failing case, and explain why the remaining list is complete.