← Identify tiling pieces and total area · Tile and Cut Figures with Congruent Pieces

Identify tiling pieces and total area · 15 practice problems

4.MD.A.34.OA.A.3

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From the workbook (authentic) — 3

Real practice problems extracted and localized from the source workbook.

Workbook 1

Four kinds of pattern-block pieces were each used several times to build the dog shape on the right: piece R is a rhombus, piece T is a trapezoid, piece H is a hexagon, and piece S is a square. The unit triangle has size $1$ , so a rhombus R has size $2$ , a trapezoid T has size $3$ , and a hexagon H has size $6$ . The square piece S has size about $2$ . About what is the size of the whole dog shape on the right?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A dog shape is built from four kinds of pattern blocks measured against a unit triangle of size 1: rhombus R (size 2), trapezoid T (size 3), hexagon H (size 6), and square S (size about 2). The dog uses 2 rhombi, 2 trapezoids, 3 hexagons, and 2 squares. I need the total size of the whole dog shape.

Givens

The unit triangle has size 1.
Rhombus R has size 2; trapezoid T has size 3; hexagon H has size 6; square S has size about 2.
The dog uses 2 rhombi R (ears), 2 trapezoids T (middle of body), 3 hexagons H (head and two body sections), and 2 squares S (legs).

Unknowns

The total size of the whole dog shape.

Constraints

The whole shape is made only of these four piece types.
Total size = sum over piece types of (count) x (size).

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#1 Draw a Diagram

Break the dog into its four piece types, count each type, multiply each count by that piece's size, then add the four subtotals for the whole size.

3 · Execute5 carry out the plan

1Total the rhombus pieces R (ears)

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

There are 2 rhombus pieces, each of size 2, so the rhombi cover 2 x 2 = 4.

2 \times 2 = 4

Two size-2 pieces add to 4 by multiplying count times size.

2Total the trapezoid pieces T (body middle)

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

There are 2 trapezoid pieces, each of size 3, so the trapezoids cover 3 x 2 = 6.

3 \times 2 = 6

Two size-3 pieces add to 6.

3Total the hexagon pieces H (head and body)

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

There are 3 hexagon pieces, each of size 6, so the hexagons cover 6 x 3 = 18.

6 \times 3 = 18

Three size-6 pieces add to 18 by repeated addition of 6.

4Total the square pieces S (legs)

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

There are 2 square pieces, each of size about 2, so the squares cover about 2 x 2 = 4.

2 \times 2 = 4

Two size-2 pieces add to 4.

5Add all four subtotals

#7 Identify Subproblems 4.OA.A.3

The whole size is 4 + 6 + 18 + 4 = 32.

4 + 6 + 18 + 4 = 32

Combining the four piece-type totals gives the full dog's size.

Answer: about 32

4 · Reviewdoes it hold up?

The three hexagons alone give 18, the biggest chunk, and the smaller pieces (4 + 6 + 4 = 14) bring the total to 32; that is sensible since hexagons are the largest pieces and there are three of them.

Another way: Triangle-unit view (tool 8/15): convert every piece to unit triangles -- 2 rhombi = 4, 2 trapezoids = 6, 3 hexagons = 18, 2 squares = 4 -- and count 32 unit triangles in all, the same answer.

Standardsmin grade 4

4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Finding each piece-type's subtotal by multiplying its count by its size.
4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations — Adding the four piece-type subtotals into the whole dog's size.

💡Takeaway. Multiply each block by its size and add: 4 + 6 + 18 + 4 = about 32 -- just counting and grouping!

Workbook 2

A single pattern-block piece was used several times to build the shape on the right. The piece is a rhombus (the blue pattern-block rhombus). If this rhombus piece has size $1$ , what is the size of the whole shape on the right?

Your answer

Show solution

1 · Understandwhat's really being asked

One rhombus pattern-block piece of size 1 is used over and over to build a figure-eight shape: two equal regular hexagons joined at a single pinch point. Each hexagon is made of 3 of these rhombi. I need the total size of the whole figure-eight shape.

Givens

The piece is a rhombus with size 1.
The shape is two equal regular hexagons joined at one pinch point.
Each hexagon is tiled by 3 of the rhombus pieces.

Unknowns

The total size of the whole figure-eight shape.

Constraints

The whole shape is made only of the rhombus pieces.
Total size = (number of rhombi) x (size of one rhombus).

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#2 Make a Systematic List

Split the figure-eight into its two hexagons, find how many rhombi tile one hexagon, double it to count all the rhombi, then multiply the count by the size of one rhombus.

3 · Execute3 carry out the plan

1Count the rhombi in one hexagon

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

A regular hexagon is tiled by exactly 3 of the rhombus pieces, as shown by the lines inside each hexagon.

3 \text{ rhombi per hexagon}

Three congruent rhombi meeting at the center fill a regular hexagon -- a standard pattern-block fact you can see in the figure.

2Count all the rhombi in the figure-eight

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

There are two hexagons, and they only touch at a single point, so no rhombi are shared. The total number of rhombi is 3 + 3 = 6.

3 \times 2 = 6

Two separate hexagons of 3 rhombi each give 6 rhombi by simple doubling.

3Find the total size

#7 Identify Subproblems 4.MD.A.3

Each of the 6 rhombi has size 1, so the whole shape has size 6 x 1 = 6.

6 \times 1 = 6

Multiplying the count of equal pieces by each piece's size gives the total size.

Answer: 6

4 · Reviewdoes it hold up?

The shape is two hexagons and each hexagon is 3 rhombi, so 6 rhombi of size 1 must total 6. The pinch point is a single shared point (zero area), so nothing is double-counted -- 6 is consistent.

Another way: Pattern view (tool 5): each hexagon repeats the same 3-rhombus pattern, so the size grows by 3 per hexagon; with 2 hexagons the size is 3, 6 -- ending at 6.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Recognizing that a regular hexagon is composed of 3 congruent rhombi.
4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations — Combining the rhombi from both hexagons into one count.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Multiplying the rhombus count by each rhombus's size to get the total size.

💡Takeaway. Each hexagon is just 3 rhombi, so two hexagons are 6 rhombi -- count and multiply, and the size is 6!

Workbook 3

Two pattern-block pieces, piece A and piece B, were each used several times to build the shape on the right. Piece A is an equilateral triangle and piece B is a square. 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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A tall star-topped shape is built from two kinds of pattern blocks: triangle piece A of size 1, and square piece B of size about 2. The shape uses 12 of the triangle pieces and 2 of the square pieces. I need the total size of the whole shape.

Givens

Piece A is a triangle with size 1.
Piece B is a square with size about 2.
The shape is built from 12 triangle pieces A and 2 square pieces B.

Unknowns

The total size of the whole shape.

Constraints

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

2 · Planchoose the strategy

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

Separate the shape into its triangle part and its square part, count each kind of piece, multiply each count by that piece's size, and add the two totals.

3 · Execute3 carry out the plan

1Total the triangle pieces A

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

The star and bowtie parts are made of 12 triangle pieces A, each of size 1, so the triangles cover 12 x 1 = 12.

1 \times 12 = 12

Twelve equal size-1 pieces add to 12 by counting.

2Total the square pieces B

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

Two square pieces B sit at the base, each of size about 2, so the squares cover about 2 x 2 = 4.

2 \times 2 = 4

Two pieces of size 2 each add to 4 by multiplying a count by a size.

3Add the two parts

#7 Identify Subproblems 4.OA.A.3

The whole size is the triangle total plus the square total: 12 + 4 = 16.

12 + 4 = 16

Combining the triangle area and the square area gives the full shape's size.

Answer: about 16

4 · Reviewdoes it hold up?

The triangles give 12 and the two size-2 squares add only 4 more, so a total near 16 makes sense; the triangle part dominates because there are many more triangles.

Another way: Units view (tool 8): measure everything in triangle-units -- each square is worth 2 triangles, so the 2 squares are 4 triangle-units; with 12 triangle-units already, the total is 16 triangle-units.

Standardsmin grade 4

4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Finding each piece-type's total by multiplying its count by its size.
4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations — Adding the triangle total and the square total into the whole shape's size.

💡Takeaway. Twelve size-1 triangles make 12, plus two size-2 squares make 4 -- add them for about 16!

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 triangle pieces and 3 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces.
3 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 6 of the pieces are triangles like piece A and 3 are squares like piece B.

6\text{ triangles},\quad 3\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 6 of them come to 6.

6 \times 1 = 6

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 3 of them come to about 6.

3 \times 2 = 6

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 6 + 6 = about 12.

6 + 6 = 12

Areas of pieces that do not overlap simply add.

Answer: about 12

4 · Reviewdoes it hold up?

There are 9 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 9 and 18. About 12 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 6 + 2 x 3 = 12 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 2 easy

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 triangle pieces and 2 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces.
2 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 6 of the pieces are triangles like piece A and 2 are squares like piece B.

6\text{ triangles},\quad 2\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 6 of them come to 6.

6 \times 1 = 6

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 2 of them come to about 4.

2 \times 2 = 4

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 6 + 4 = about 10.

6 + 4 = 10

Areas of pieces that do not overlap simply add.

Answer: about 10

4 · Reviewdoes it hold up?

There are 8 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 8 and 16. About 10 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 6 + 2 x 2 = 10 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 3 easy

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 triangle pieces and 6 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces.
6 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 6 of the pieces are triangles like piece A and 6 are squares like piece B.

6\text{ triangles},\quad 6\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 6 of them come to 6.

6 \times 1 = 6

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 6 of them come to about 12.

6 \times 2 = 12

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 6 + 12 = about 18.

6 + 12 = 18

Areas of pieces that do not overlap simply add.

Answer: about 18

4 · Reviewdoes it hold up?

There are 12 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 12 and 24. About 18 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 6 + 2 x 6 = 18 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 4 medium

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 triangle pieces and 5 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces.
5 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 6 of the pieces are triangles like piece A and 5 are squares like piece B.

6\text{ triangles},\quad 5\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 6 of them come to 6.

6 \times 1 = 6

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 5 of them come to about 10.

5 \times 2 = 10

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 6 + 10 = about 16.

6 + 10 = 16

Areas of pieces that do not overlap simply add.

Answer: about 16

4 · Reviewdoes it hold up?

There are 11 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 11 and 22. About 16 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 6 + 2 x 5 = 16 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 5 medium

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 triangle pieces and 1 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces.
1 square piece are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 6 of the pieces are triangles like piece A and 1 are squares like piece B.

6\text{ triangles},\quad 1\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 6 of them come to 6.

6 \times 1 = 6

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 1 of them come to about 2.

1 \times 2 = 2

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 6 + 2 = about 8.

6 + 2 = 8

Areas of pieces that do not overlap simply add.

Answer: about 8

4 · Reviewdoes it hold up?

There are 7 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 7 and 14. About 8 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 6 + 2 x 1 = 8 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 6 easy

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 triangle pieces and 4 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces.
4 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 6 of the pieces are triangles like piece A and 4 are squares like piece B.

6\text{ triangles},\quad 4\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 6 of them come to 6.

6 \times 1 = 6

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 4 of them come to about 8.

4 \times 2 = 8

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 6 + 8 = about 14.

6 + 8 = 14

Areas of pieces that do not overlap simply add.

Answer: about 14

4 · Reviewdoes it hold up?

There are 10 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 10 and 20. About 14 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 6 + 2 x 4 = 14 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 7 hard

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 9 triangle pieces and 3 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces plus 3 mores around the outside.
3 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 9 of the pieces are triangles like piece A and 3 are squares like piece B.

9\text{ triangles},\quad 3\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 9 of them come to 9.

9 \times 1 = 9

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 3 of them come to about 6.

3 \times 2 = 6

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 9 + 6 = about 15.

9 + 6 = 15

Areas of pieces that do not overlap simply add.

Answer: about 15

4 · Reviewdoes it hold up?

There are 12 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 12 and 24. About 15 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 9 + 2 x 3 = 15 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 8 medium

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 10 triangle pieces and 2 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces plus 4 mores around the outside.
2 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 10 of the pieces are triangles like piece A and 2 are squares like piece B.

10\text{ triangles},\quad 2\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 10 of them come to 10.

10 \times 1 = 10

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 2 of them come to about 4.

2 \times 2 = 4

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 10 + 4 = about 14.

10 + 4 = 14

Areas of pieces that do not overlap simply add.

Answer: about 14

4 · Reviewdoes it hold up?

There are 12 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 12 and 24. About 14 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 10 + 2 x 2 = 14 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 9 medium

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 8 triangle pieces and 3 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces plus 2 mores around the outside.
3 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 8 of the pieces are triangles like piece A and 3 are squares like piece B.

8\text{ triangles},\quad 3\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 8 of them come to 8.

8 \times 1 = 8

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 3 of them come to about 6.

3 \times 2 = 6

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 8 + 6 = about 14.

8 + 6 = 14

Areas of pieces that do not overlap simply add.

Answer: about 14

4 · Reviewdoes it hold up?

There are 11 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 11 and 22. About 14 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 8 + 2 x 3 = 14 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 10 hard

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 11 triangle pieces and 1 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces plus 5 mores around the outside.
1 square piece are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 11 of the pieces are triangles like piece A and 1 are squares like piece B.

11\text{ triangles},\quad 1\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 11 of them come to 11.

11 \times 1 = 11

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 1 of them come to about 2.

1 \times 2 = 2

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 11 + 2 = about 13.

11 + 2 = 13

Areas of pieces that do not overlap simply add.

Answer: about 13

4 · Reviewdoes it hold up?

There are 12 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 12 and 24. About 13 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 11 + 2 x 1 = 13 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 11 hard

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 9 triangle pieces and 2 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces plus 3 mores around the outside.
2 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 9 of the pieces are triangles like piece A and 2 are squares like piece B.

9\text{ triangles},\quad 2\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 9 of them come to 9.

9 \times 1 = 9

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 2 of them come to about 4.

2 \times 2 = 4

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 9 + 4 = about 13.

9 + 4 = 13

Areas of pieces that do not overlap simply add.

Answer: about 13

4 · Reviewdoes it hold up?

There are 11 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 11 and 22. About 13 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 9 + 2 x 2 = 13 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Variant 12 hard

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?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A shape is built from 8 triangle pieces and 4 square pieces. A triangle is worth 1 and a square about 2. I want the size of the whole shape.

Givens

Piece A, the triangle, has size 1.
Piece B, the square, has size about 2.
The hexagon in the middle is 6 triangle pieces plus 2 mores around the outside.
4 square pieces are attached around the hexagon.

Unknowns

The size of the whole shape.

Constraints

The shape uses only these two pieces, with no gaps and no overlaps.
Because piece B's size is only approximate, the total is approximate too.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #2 Make a Systematic List#17 Visualize Spatial Relationships

Sort the pieces by shape before counting anything. Each kind is then a count times a size, and the two parts add to the whole -- which is easier and safer than trying to measure the outline.

3 · Execute4 carry out the plan

1Sort the pieces by shape

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

Going round the figure, 8 of the pieces are triangles like piece A and 4 are squares like piece B.

8\text{ triangles},\quad 4\text{ squares}

Telling the two shapes apart is the whole first half of the job.

2Total the triangle pieces

#7 Identify Subproblems 4.MD.A.3

Each triangle is worth 1, so 8 of them come to 8.

8 \times 1 = 8

With a size of 1, the count is the total.

3Total the square pieces

#7 Identify Subproblems 4.MD.A.3

Each square is worth about 2, so 4 of them come to about 8.

4 \times 2 = 8

A square covers about two of the triangles.

4Add the two parts

#7 Identify Subproblems 4.MD.A.3

The whole shape is the triangle part plus the square part: 8 + 8 = about 16.

8 + 8 = 16

Areas of pieces that do not overlap simply add.

Answer: about 16

4 · Reviewdoes it hold up?

There are 12 pieces altogether and none is worth less than 1 or more than 2, so the total has to land between 12 and 24. About 16 sits inside that range.

Another way: Swap each square for two triangles in your head and count triangles only: 8 + 2 x 4 = 16 triangles' worth, the same answer reached without ever adding two different units.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size — Telling the triangle pieces from the square pieces by shape.
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems — Adding the area of the parts to get the area of the whole.

💡Takeaway. Count each kind of piece, multiply by what one is worth, and add. Shapes made of pieces are easier to measure than shapes that are not.

Where next?

← Back to the problem: Identify tiling pieces and total area