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Count distinct shapes built by adding congruent tiles · 3 practice problems

4.G.A.2

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Real practice problems extracted and localized from the source workbook.

Workbook 1

Using $4$ identical equilateral-triangle tiles, how many different shapes can you make? (You must join the tiles edge to edge, full side against full side, and two shapes that match after rotating or flipping count as one and the same shape.)

Your answer

Show solution

1 · Understandwhat's really being asked

I have 4 identical equilateral triangles. I join them edge to edge (a full side touching a full side) into one connected shape, and I count how many genuinely different shapes are possible, treating two shapes as the same when one becomes the other by turning or flipping.

Givens

There are 4 congruent equilateral-triangle tiles.
Tiles must be joined edge to edge, full side against full side.
Shapes that match after rotation or reflection count as one and the same shape.

Unknowns

The number of different shapes that can be built from the 4 equilateral triangles.

Constraints

Build the shapes step by step, adding one triangle at a time onto an open edge.
Throw out any arrangement that is just a rotation or a flip of one already listed.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram#6 Guess and Check

This is a small 'how many shapes' question, so the safe method is to draw every way the triangles can fit together and then cross out the ones that are the same after turning or flipping, leaving only the truly different outlines.

3 · Execute3 carry out the plan

1Build up to 3 triangles first

#1 Draw a Diagram 4.G.A.2

Two triangles joined on an edge make one shape: a rhombus (the two triangles share a full side). Adding a third triangle to the rhombus gives only one new shape up to turning and flipping: a longer half-strip (a trapezoid made of 3 triangles). So after 3 tiles there is essentially one shape to grow.

2 \text{ tiles} \Rightarrow 1 \text{ shape}, \quad 3 \text{ tiles} \Rightarrow 1 \text{ shape}

Equilateral triangles only fit one way up to turning, so the early steps each give a single shape.

2Add the 4th triangle and list the distinct results

#2 Make a Systematic List 4.G.A.2

Attach the 4th triangle onto the open edges of the 3-triangle piece and remove repeats. Exactly three different outlines survive: (1) a BIG TRIANGLE of side 2 (three triangles pointing up with one upside-down triangle filling the middle); (2) a STRAIGHT STRIP of 4 triangles in a row, which forms a long trapezoid; and (3) a BENT STRIP (a chevron) where the row of triangles turns a corner.

\text{big triangle} + \text{straight strip} + \text{bent strip} = 3

Listing every place the last tile can go and crossing out flips and turns leaves three different outlines.

3Confirm the three shapes are all different

#6 Guess and Check 4.G.A.2

Check each pair: the big triangle has three equal sides, the straight strip is a long trapezoid, and the bent strip turns a corner, so none can become another by turning or flipping. No duplicates remain, so the count is 3.

\text{distinct shapes} = 3

Turning or flipping never merges these three outlines, so each one is genuinely different.

Answer: 3 shapes

4 · Reviewdoes it hold up?

Four small triangles can only form a compact triangle or a strip that is either straight or bent, so a small count like 3 is reasonable; counting flips and turns separately would wrongly inflate it.

Another way: Cut out 4 paper triangles (tool 10) and physically slide the last triangle to every open edge, sorting the results into piles that look the same after turning or flipping -- you end with 3 piles.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Recognizing the equilateral-triangle tiles and judging when two built shapes are the same after rotation or reflection.

💡Takeaway. List every way the last triangle can attach, toss out the flips and turns, and only 3 truly different shapes are left!

Workbook 2

Using $4$ identical square tiles, how many different shapes can you make? (You must join the tiles edge to edge, full side against full side, and two shapes that match after rotating or flipping count as one and the same shape.)

Your answer

Show solution

1 · Understandwhat's really being asked

I have 4 identical squares. I join them edge to edge (a full side touching a full side) into one connected shape, and I count how many genuinely different shapes are possible, treating two shapes as the same when one becomes the other by turning or flipping.

Givens

There are 4 congruent square tiles.
Tiles must be joined edge to edge, full side against full side.
Shapes that match after rotation or reflection count as one and the same shape.

Unknowns

The number of different shapes that can be built from the 4 squares.

Constraints

Build the shapes by adding one square at a time onto an open edge.
Throw out any arrangement that is just a rotation or a flip of one already listed.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram#6 Guess and Check

This is a small 'how many shapes' question, so the safe method is to draw every way the four squares can fit together edge to edge and then cross out the ones that are the same after turning or flipping, leaving only the truly different outlines.

3 · Execute3 carry out the plan

1Build up to 3 squares first

#1 Draw a Diagram 4.G.A.2

Two squares joined on an edge make one shape (a 1-by-2 rectangle). Adding a third square gives only two different shapes up to turning and flipping: a STRAIGHT row of 3 squares, and an L of 3 squares. These are the two pieces I grow with the fourth square.

2 \text{ squares} \Rightarrow 1 \text{ shape}, \quad 3 \text{ squares} \Rightarrow 2 \text{ shapes}

Starting smaller keeps the listing organized, so I do not miss or double-count a shape.

2Add the 4th square and list every distinct outline

#2 Make a Systematic List 4.G.A.2

Attach the 4th square to the open edges of the 3-square pieces and remove repeats. Exactly five different outlines survive: (1) a STRAIGHT row of 4 squares (the I); (2) a 2-by-2 SQUARE (the O); (3) a T made of a row of 3 with one square on the middle's side; (4) an L made of a row of 3 with one square on an end's side; and (5) an S/Z step shape (two squares, then a shift, then two more). The S and Z look the same after a flip, and the L and its mirror also match after a flip, so each counts once.

\text{I} + \text{O} + \text{T} + \text{L} + \text{S} = 5

These are exactly the five 'tetromino' shapes, the same pieces seen in falling-block puzzles.

3Confirm the five shapes are all different

#6 Guess and Check 4.G.A.2

Check each pair: I is a long bar, O is a fat square block, T has a bump in the middle, L has a bump at an end, and S is a slanted step. None can become another by turning or flipping, so no duplicates remain and the count is 5.

\text{distinct shapes} = 5

Turning or flipping never merges these five outlines, so each one is genuinely different.

Answer: 5 shapes

4 · Reviewdoes it hold up?

These are the well-known tetromino pieces, and there are exactly 5 of them, which matches the careful list of bar, block, T, L, and S.

Another way: Cut out 4 paper squares (tool 10) and physically build each arrangement, sorting them into piles that look the same after turning or flipping -- you end with 5 piles.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Recognizing the square tiles and judging when two built shapes are the same after rotation or reflection.

💡Takeaway. Four squares make exactly the 5 puzzle-block shapes (bar, block, T, L, and S) -- once you toss out the flips and turns!

Workbook 3

Using $2$ equilateral-triangle tiles and $1$ square tile, all with the same side length, how many different shapes can you make? (You must join the tiles edge to edge, full side against full side, and two shapes that match after rotating or flipping count as one and the same shape.)

Your answer

Show solution

1 · Understandwhat's really being asked

I have 2 identical equilateral triangles and 1 square, all with the same side length. I join them edge to edge (a full side touching a full side) into one connected shape, and I count how many genuinely different shapes are possible, treating two shapes as the same when one becomes the other by turning or flipping.

Givens

There are 2 congruent equilateral-triangle tiles and 1 square tile, all with the same side length.
Tiles must be joined edge to edge, full side against full side.
Shapes that match after rotation or reflection count as one and the same shape.

Unknowns

The number of different shapes that can be built from the 2 triangles and 1 square.

Constraints

Organize the count by how the pieces connect: either each triangle touches the square, or the two triangles touch each other.
Throw out any arrangement that is just a rotation or a flip of one already listed.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #1 Draw a Diagram#17 Visualize Spatial Relationships

Because the side lengths all match, every join is full-side to full-side, so I can split the count into clear cases by which pieces are touching, draw each case, and cross out rotations and flips.

3 · Execute3 carry out the plan

1Case A: both triangles attach to the square

#17 Visualize Spatial Relationships 4.G.A.2

Put a triangle on a side of the square; the square is symmetric, so where I start does not matter. The second triangle then goes on a different side of the square, and that side is either NEXT TO the first triangle (two sides meeting at a corner) or ACROSS FROM it (two parallel sides). Those give two different shapes: triangles on two ADJACENT sides, and triangles on two OPPOSITE sides. Putting both triangles on the same side is not allowed (only one full side fits one tile).

\text{adjacent sides} + \text{opposite sides} = 2 \text{ shapes}

The square only cares whether the two triangles sit on sides that meet or on sides that are parallel, so this case gives exactly two outlines.

2Case B: the two triangles attach to each other

#1 Draw a Diagram 4.G.A.2

Join the two triangles along a full side; they form a RHOMBUS. Now attach the square to one side of that rhombus. By turning and flipping, every choice of which rhombus side the square goes on gives the same single shape: a rhombus with a square stuck on one edge. So this case gives 1 shape.

\text{rhombus} + \text{square} = 1 \text{ shape}

The rhombus made from the two triangles is symmetric, so the square lands the same way up to turning and flipping.

3Add up the cases and confirm no repeats

#2 Make a Systematic List 4.G.A.2

Case A gives 2 shapes (triangles on adjacent sides, triangles on opposite sides) and Case B gives 1 shape (rhombus plus a square). A Case A shape always has the square showing two free corners between the triangles, while the Case B shape has the two triangles fused into one rhombus, so no Case A shape equals the Case B shape. The total is 2 + 1 = 3.

2 + 1 = 3

The cases describe different connection patterns, so shapes from different cases can never be the same after turning or flipping.

Answer: 3 shapes

4 · Reviewdoes it hold up?

There are only a few ways three matching-edge tiles can connect -- triangles on adjacent sides, triangles on opposite sides, or triangles fused into a rhombus with the square attached -- so a small count like 3 is reasonable.

Another way: Cut out 2 paper triangles and 1 paper square of the same side length (tool 10) and physically build every arrangement, sorting them into piles that look the same after turning or flipping -- you end with 3 piles.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Recognizing the triangle and square tiles and judging when two built shapes are the same after rotation or reflection.

💡Takeaway. Sort the joins into cases -- triangles on adjacent sides, on opposite sides, or fused into a rhombus -- and only 3 different shapes appear!

Where next?

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