Identify tiling pieces and total area

4.G.A.24.MD.A.3 · adapt · grade 4

Archetype: Tile and Cut Figures with Congruent Pieces · step in a 5-type progression

Piece A and piece B were each used several times to build the shape on the right. If piece A has size $1$ and piece B has size about $2$ , about what is the size of the whole shape on the right?

Show solution

Understand

A shape is built only from two kinds of pattern blocks: piece A is a triangle of size 1, and piece B is a square of size about 2. The shape has 6 of piece A meeting in the center to form a hexagon, plus 4 of piece B around the outside. I need the total size (area) of the whole shape.

Givens

Piece A is a triangle with size 1.
Piece B is a square with size about 2.
The center uses 6 of piece A to form a regular hexagon.
Around the outside, 4 of piece B are attached.

Unknowns

The total size (area) of the whole built shape.

Constraints

The whole shape is made only of these triangle and square pieces.
Total size = (number of A) x (size of A) + (number of B) x (size of B).

Plan

#7 Identify Subproblems · also uses: #2 Make a Systematic List#8 Analyze the Units

Break the area into the part made of triangles and the part made of squares, count how many of each piece is used, multiply each count by that piece's size, and add the two totals.

Execute

#2 Make a Systematic List 4.MD.A.3

Six triangle pieces A meet in the center to make the hexagon. Each piece A has size 1, so the triangles together cover 6 x 1 = 6.

6 \times 1 = 6

Six equal pieces of size 1 add to 6 by simple repeated addition.

#2 Make a Systematic List 4.MD.A.3

Four square pieces B are attached around the hexagon. Each piece B has size about 2, so the squares together cover about 4 x 2 = 8.

4 \times 2 = 8

Four pieces of size 2 each add to 8, again just multiplying a count by a size.

#7 Identify Subproblems 4.OA.A.3

The whole size is the triangle total plus the square total: 6 + 8 = 14.

6 + 8 = 14

Combining the area from triangles and squares gives the full shape's size.

Answer: about 14

Review

The answer 14 is built from sensible parts: 6 triangles of size 1 (=6) and 4 squares of size 2 (=8). Since piece B (size 2) is twice piece A (size 1), the 4 squares should outweigh the 6 triangles only slightly, and 8 vs 6 matches that.

Units view (tool 8): measure everything in 'triangle units' -- each square is worth 2 triangles, so the squares are 4 x 2 = 8 triangle-units and the hexagon is 6 triangle-units, total 14 triangle-units, the same answer.

Standards · min grade 4

4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Finding each piece-type's total area by multiplying count times piece size.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Combining the triangle total and square total into the whole shape's size.

💡 Add up the size of every block: six size-1 triangles plus four size-2 squares makes about 14 -- just counting and multiplying!