Reasoning · Grade 2-1 Large and Small Shapes
Problem
In (1) the parallels only cut off similar triangles and the bands are trapezoids: 3.
A line parallel to the base always slices off a smaller triangle sharing the same top point, so each parallel line gives exactly one new triangle.
2.G.A.1Make A Systematic ListEach line drawn parallel to the base slices off exactly one new triangle sharing the same apex.
A line parallel to the base never meets it, so it must cut both of the other two sides and close off a triangle below the apex.
One parallel line makes one such triangle and no more, so counting the lines is the same as counting the new triangles.
In (2): two small, two halves, one top, one whole — 6.
Splitting the picture into the left part and the right part (subproblems) makes the small, medium, and whole triangles easy to see and add.
2.G.A.1Identify SubproblemsIn (3) each pair of the four base points gives one triangle: 6.
When all the lines share one top point, each triangle is fixed by which two base points you pick, so listing the base-point pairs counts them with none missed.
2.G.A.1Make A Systematic ListIn (4), counting from the small top one up to the whole: 5.
Adding the lines one at a time and re-checking the picture shows each new triangle without overcounting.
2.G.A.1Make A Systematic ListTogether: 3, 6, 6, 5.
Each figure was counted in small-then-large order, so the four totals are trustworthy.
1.OA.A.2Make A Systematic ListCount the tiny triangles first, then the ones made by joining them, and last the whole outside triangle. Going small-to-big means you never miss one!