We are given 15 capital letters A, C, D, E, F, L, M, N, O, P, R, S, T, V, Z. For each of three criteria every letter goes into exactly one of two groups, has it or has it not. The criteria are straight strokes, enclosed regions, and folding in half exactly. Find the two groups for each criterion.
Your answer
Watch outD looks like a curves-only letter and R looks like it has a loop. D has a straight vertical and R is drawn open.
How to solve
StrategyMake a Systematic List — There is a small, fixed set of 15 letters, so we go through them one by one (a systematic list) and check each property. Picturing or lightly sketching each letter's shape (a diagram) helps us decide whether it has a straight line, an enclosed region, or a fold-line of symmetry.
1STEP 1
Criterion 1 — straight line or not
Only C, O, S are all curve; the other twelve carry a straight stroke.
straight = {A, D, E, F, L, M, N, P, R, T, V, Z}, no straight = {C, O, S}
Sorting by 'is there a straight edge?' is a defining-attribute question — exactly the kind of attribute first graders use to describe shapes.
1.G.A.1Make A Systematic List
2STEP 2
Criterion 2 — enclosed region or not
The letters that close a region right off are A, D, O, P.
enclosed = {A, D, O, P}, not enclosed = {C, E, F, L, M, N, R, S, T, V, Z}
Deciding whether a shape encloses a space is a visual attribute young learners can see directly by looking for a closed loop.
1.G.A.1Draw A Diagram
3STEP 3
Criterion 3 — folds in half exactly (line symmetry)
Folding across or down, the ones that match are A, C, D, E, M, O, T, V.
matches = {A, C, D, E, M, O, T, V}, does not = {F, L, N, P, R, S, Z}
Folding a letter in half to see if the halves land on top of each other is the hands-on meaning of a line of symmetry.
4.G.A.1Draw A Diagram
A letter has a line of symmetry exactly when folding it along that line lands the two halves on top of each other.
Why?
Folding sends every point of one half straight across the crease to a point the same distance away on the other side, so the crease is the mirror.
🧱A fold is a mirror lineFolding puts a point onto its landing spot, so the crease sits the same distance from both and crosses the line joining them square on.
Why?
Because the fold does not stretch or shrink anything, the two halves can only match if they were genuinely the same shape and size to begin with.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Answer
no straight = C, O, S / encloses = A, D, O, P / folds exactly = A, C, D, E, M, O, T, V
Each of the three groupings uses all 15 letters with none left out and none counted twice: 12+3=15, 4+11=15, and 8+7=15. The problem's own example letters all land where expected: A and D have a straight line, R has no enclosed region, and C folds in half exactly. That agreement gives confidence in the rest of each grouping.
Takeaway
You can sort the same letters lots of ways just by changing the question you ask about them!
Criterion 1 — straight line or not
Criterion 2 — enclosed region or not
Criterion 3 — folds in half exactly (line symmetry)