Reasoning · Grade 5-2 Line-Symmetric and Point-Symmetric Figures
Problem
Give the eight triangles labels.
Composing bigger shapes out of small identical triangles is something children do with pattern blocks; naming the pieces first is what turns a picture into something you can talk about and list.
1.G.A.2Draw A DiagramA mirror can only be vertical, horizontal or slanted.
A fold line has to fold drawn edges onto drawn edges - if it did not, the folded picture would have lines in places where the original has none, and the two halves could not match.
4.G.A.3Organize Information In More WaysOnly a few directions can hold a mirror line, so the search splits into that many separate cases.
A mirror line must send the whole figure onto itself, so it has to run along a direction the figure already respects.
Each symmetric figure has a definite mirror direction, so the cases never overlap and their counts can simply be added.
A vertical mirror gives 3 small ones first.
Starting from the one-triangle case makes the problem easy to enter, and every later figure is just one of these with more triangles stuck on - so the small cases teach you what to look for.
4.G.A.3Solve An Easier Related ProblemLarger ones add 5 more.
Once the mirror line is fixed as a particular vertical line, checking a figure is easy: the triangles on the left of that line must be the exact reflections of the triangles on the right, and you can see that at a glance.
4.G.A.3Make A Systematic ListVertical mirrors give 8 in all.
Adding up a short list of counts is Grade 4 arithmetic; the useful part is that the list was built in size order, so the total is trustworthy.
4.OA.A.3Make A Systematic ListA horizontal mirror adds none.
Deciding that two figures are 'the same' means checking that one can be moved onto the other by a turn or a flip - the rhombus found here is the very same rhombus as before, seen with a different fold line.
8.G.A.2Organize Information In More WaysA slanted mirror adds 1.
It is easy to forget slanted mirror lines because school pictures usually stand up straight, but on a triangle grid the slanted directions are just as natural as the upright one, so they have to be checked.
4.G.A.3Make A Systematic ListAltogether that is 9.
Because the three cases were chosen so that every possible mirror direction falls into exactly one of them, adding the three answers is safe - no figure is counted twice and none is left out.
4.OA.A.3Make A Systematic ListSort by where the fold line could go - upright, flat, or slanted - and then count up by size; the slanted fold is the one everybody forgets, and it is the ninth answer.