Reasoning · Grade 4-1 Joining and Cutting Figures
Problem
Fix the rule: same after a turn or flip means same shape.
Turning and flipping never stretches or breaks the shape, so the piece really is unchanged — a child can settle every duplicate question by physically laying one cut-out on top of another.
8.G.A.2Create A Physical RepresentationOne square gives 1, two give 1, three give 2.
Practising the growing-and-crossing-off routine on cases small enough to check by eye makes it trustworthy before it is used on the harder ones.
1.G.A.2Solve An Easier Related ProblemAdd one square to each tromino to grow the tetrominoes.
Every 4-square shape must contain a 3-square shape inside it, so growing from the complete list of trominoes cannot possibly miss anything.
1.G.A.2Make A Systematic ListEvery four-square shape contains a three-square shape inside it, so growing from the complete list of trominoes cannot miss anything.
Remove any one square from the edge of a four-square shape and what is left is still a connected three-square shape, so it must appear on the smaller list.
Sorting the new shapes by which tromino they grew from covers every shape and lets duplicates be crossed off once and for all.
Removing the flip-duplicates leaves 5.
Looking for a line of symmetry tells you in advance whether flipping a shape will give something new, which is quicker than redrawing it and comparing.
4.G.A.3Create A Physical RepresentationSorting by bounding box confirms the same 5.
Sorting by the box a shape fits into is a totally different rule from growing shapes one square at a time, so agreement between the two counts is real evidence, not the same mistake made twice.
1.G.A.2Make A Systematic ListThe same growing gives 12 pentominoes.
Grouping by the rectangle a shape sits in splits one long scary list into four short lists, and each short list is small enough to check square by square.
1.G.A.2Make A Systematic ListNothing missing or doubled, so the answer is 5 and 12.
Physically stacking the cut-outs settles the two dangers of any counting problem — counting something twice and leaving something out — without needing any formula.
8.G.A.2Create A Physical RepresentationGrow shapes one square at a time and test every new one by turning and flipping a paper cut-out — that is how you get all 5 tetrominoes and all 12 pentominoes with none missed and none doubled!