Practice · Perimeter plus tiles needed to build it · Sensim Math

Variant 1 easy answer: 6 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $2\text{ cm}$ long. To make a rectangle with a perimeter of $16\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $2\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 2 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 16 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 2 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 2 cm by 2 cm square.
The finished rectangle must have a perimeter of 16 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 2 cm because the building square is 2 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 2 cm, the two triangles snap into a square that is 2 cm wide and 2 cm tall.

2 \text{ triangles} = 1 \text{ square } (2\text{ cm} \times 2\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 2 cm squares side by side in one row. If there are n squares, the rectangle is 2 cm tall and 2 times n cm long.

\text{length} = 2 \times n, \quad \text{height} = 2

Laying equal 2 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 16 cm and find n: 2 times (2n + 2) = 16, so 2n + 2 = 8, so 2n = 6, so n = 3. We get a 6 cm by 2 cm rectangle.

2\,(2n + 2) = 16 \;\Rightarrow\; 2n + 2 = 8 \;\Rightarrow\; n = 3

Try row lengths until the trip around the rectangle measures 16 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 3 squares is made from 2 triangles, so the rectangle needs 3 times 2 = 6 triangles.

3 \times 2 = 6

3 squares, 2 triangles each, is 6 triangles.

Answer: 6 triangles

4 · Reviewdoes it hold up?

The rectangle is 6 cm by 2 cm, and its perimeter is 6 + 2 + 6 + 2 = 16 cm, exactly as required. 6 right triangles with legs of 2 cm cover an area of 6 times (2 times 2 divided by 2) = 12 square cm, which equals the 6 by 2 rectangle's area of 12 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 2 cm with perimeter 16 cm must be 6 cm long, which is 3 squares, hence 6 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 16 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 2 easy answer: 4 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $3\text{ cm}$ long. To make a rectangle with a perimeter of $18\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $3\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 3 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 18 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 3 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 3 cm by 3 cm square.
The finished rectangle must have a perimeter of 18 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 3 cm because the building square is 3 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 3 cm, the two triangles snap into a square that is 3 cm wide and 3 cm tall.

2 \text{ triangles} = 1 \text{ square } (3\text{ cm} \times 3\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 3 cm squares side by side in one row. If there are n squares, the rectangle is 3 cm tall and 3 times n cm long.

\text{length} = 3 \times n, \quad \text{height} = 3

Laying equal 3 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 18 cm and find n: 2 times (3n + 3) = 18, so 3n + 3 = 9, so 3n = 6, so n = 2. We get a 6 cm by 3 cm rectangle.

2\,(3n + 3) = 18 \;\Rightarrow\; 3n + 3 = 9 \;\Rightarrow\; n = 2

Try row lengths until the trip around the rectangle measures 18 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 2 squares is made from 2 triangles, so the rectangle needs 2 times 2 = 4 triangles.

2 \times 2 = 4

2 squares, 2 triangles each, is 4 triangles.

Answer: 4 triangles

4 · Reviewdoes it hold up?

The rectangle is 6 cm by 3 cm, and its perimeter is 6 + 3 + 6 + 3 = 18 cm, exactly as required. 4 right triangles with legs of 3 cm cover an area of 4 times (3 times 3 divided by 2) = 18 square cm, which equals the 6 by 3 rectangle's area of 18 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 3 cm with perimeter 18 cm must be 6 cm long, which is 2 squares, hence 4 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 18 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 3 easy answer: 8 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $2\text{ cm}$ long. To make a rectangle with a perimeter of $20\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $2\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 2 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 20 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 2 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 2 cm by 2 cm square.
The finished rectangle must have a perimeter of 20 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 2 cm because the building square is 2 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 2 cm, the two triangles snap into a square that is 2 cm wide and 2 cm tall.

2 \text{ triangles} = 1 \text{ square } (2\text{ cm} \times 2\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 2 cm squares side by side in one row. If there are n squares, the rectangle is 2 cm tall and 2 times n cm long.

\text{length} = 2 \times n, \quad \text{height} = 2

Laying equal 2 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 20 cm and find n: 2 times (2n + 2) = 20, so 2n + 2 = 10, so 2n = 8, so n = 4. We get a 8 cm by 2 cm rectangle.

2\,(2n + 2) = 20 \;\Rightarrow\; 2n + 2 = 10 \;\Rightarrow\; n = 4

Try row lengths until the trip around the rectangle measures 20 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 4 squares is made from 2 triangles, so the rectangle needs 4 times 2 = 8 triangles.

4 \times 2 = 8

4 squares, 2 triangles each, is 8 triangles.

Answer: 8 triangles

4 · Reviewdoes it hold up?

The rectangle is 8 cm by 2 cm, and its perimeter is 8 + 2 + 8 + 2 = 20 cm, exactly as required. 8 right triangles with legs of 2 cm cover an area of 8 times (2 times 2 divided by 2) = 16 square cm, which equals the 8 by 2 rectangle's area of 16 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 2 cm with perimeter 20 cm must be 8 cm long, which is 4 squares, hence 8 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 20 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 4 medium answer: 6 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $3\text{ cm}$ long. To make a rectangle with a perimeter of $24\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $3\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 3 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 24 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 3 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 3 cm by 3 cm square.
The finished rectangle must have a perimeter of 24 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 3 cm because the building square is 3 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 3 cm, the two triangles snap into a square that is 3 cm wide and 3 cm tall.

2 \text{ triangles} = 1 \text{ square } (3\text{ cm} \times 3\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 3 cm squares side by side in one row. If there are n squares, the rectangle is 3 cm tall and 3 times n cm long.

\text{length} = 3 \times n, \quad \text{height} = 3

Laying equal 3 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 24 cm and find n: 2 times (3n + 3) = 24, so 3n + 3 = 12, so 3n = 9, so n = 3. We get a 9 cm by 3 cm rectangle.

2\,(3n + 3) = 24 \;\Rightarrow\; 3n + 3 = 12 \;\Rightarrow\; n = 3

Try row lengths until the trip around the rectangle measures 24 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 3 squares is made from 2 triangles, so the rectangle needs 3 times 2 = 6 triangles.

3 \times 2 = 6

3 squares, 2 triangles each, is 6 triangles.

Answer: 6 triangles

4 · Reviewdoes it hold up?

The rectangle is 9 cm by 3 cm, and its perimeter is 9 + 3 + 9 + 3 = 24 cm, exactly as required. 6 right triangles with legs of 3 cm cover an area of 6 times (3 times 3 divided by 2) = 27 square cm, which equals the 9 by 3 rectangle's area of 27 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 3 cm with perimeter 24 cm must be 9 cm long, which is 3 squares, hence 6 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 24 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 5 medium answer: 8 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $3\text{ cm}$ long. To make a rectangle with a perimeter of $30\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $3\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 3 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 30 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 3 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 3 cm by 3 cm square.
The finished rectangle must have a perimeter of 30 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 3 cm because the building square is 3 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 3 cm, the two triangles snap into a square that is 3 cm wide and 3 cm tall.

2 \text{ triangles} = 1 \text{ square } (3\text{ cm} \times 3\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 3 cm squares side by side in one row. If there are n squares, the rectangle is 3 cm tall and 3 times n cm long.

\text{length} = 3 \times n, \quad \text{height} = 3

Laying equal 3 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 30 cm and find n: 2 times (3n + 3) = 30, so 3n + 3 = 15, so 3n = 12, so n = 4. We get a 12 cm by 3 cm rectangle.

2\,(3n + 3) = 30 \;\Rightarrow\; 3n + 3 = 15 \;\Rightarrow\; n = 4

Try row lengths until the trip around the rectangle measures 30 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 4 squares is made from 2 triangles, so the rectangle needs 4 times 2 = 8 triangles.

4 \times 2 = 8

4 squares, 2 triangles each, is 8 triangles.

Answer: 8 triangles

4 · Reviewdoes it hold up?

The rectangle is 12 cm by 3 cm, and its perimeter is 12 + 3 + 12 + 3 = 30 cm, exactly as required. 8 right triangles with legs of 3 cm cover an area of 8 times (3 times 3 divided by 2) = 36 square cm, which equals the 12 by 3 rectangle's area of 36 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 3 cm with perimeter 30 cm must be 12 cm long, which is 4 squares, hence 8 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 30 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 6 medium answer: 6 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $4\text{ cm}$ long. To make a rectangle with a perimeter of $32\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $4\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 4 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 32 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 4 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 4 cm by 4 cm square.
The finished rectangle must have a perimeter of 32 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 4 cm because the building square is 4 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 4 cm, the two triangles snap into a square that is 4 cm wide and 4 cm tall.

2 \text{ triangles} = 1 \text{ square } (4\text{ cm} \times 4\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 4 cm squares side by side in one row. If there are n squares, the rectangle is 4 cm tall and 4 times n cm long.

\text{length} = 4 \times n, \quad \text{height} = 4

Laying equal 4 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 32 cm and find n: 2 times (4n + 4) = 32, so 4n + 4 = 16, so 4n = 12, so n = 3. We get a 12 cm by 4 cm rectangle.

2\,(4n + 4) = 32 \;\Rightarrow\; 4n + 4 = 16 \;\Rightarrow\; n = 3

Try row lengths until the trip around the rectangle measures 32 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 3 squares is made from 2 triangles, so the rectangle needs 3 times 2 = 6 triangles.

3 \times 2 = 6

3 squares, 2 triangles each, is 6 triangles.

Answer: 6 triangles

4 · Reviewdoes it hold up?

The rectangle is 12 cm by 4 cm, and its perimeter is 12 + 4 + 12 + 4 = 32 cm, exactly as required. 6 right triangles with legs of 4 cm cover an area of 6 times (4 times 4 divided by 2) = 48 square cm, which equals the 12 by 4 rectangle's area of 48 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 4 cm with perimeter 32 cm must be 12 cm long, which is 3 squares, hence 6 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 32 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 7 hard answer: 8 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $4\text{ cm}$ long. To make a rectangle with a perimeter of $40\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $4\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 4 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 40 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 4 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 4 cm by 4 cm square.
The finished rectangle must have a perimeter of 40 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 4 cm because the building square is 4 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 4 cm, the two triangles snap into a square that is 4 cm wide and 4 cm tall.

2 \text{ triangles} = 1 \text{ square } (4\text{ cm} \times 4\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 4 cm squares side by side in one row. If there are n squares, the rectangle is 4 cm tall and 4 times n cm long.

\text{length} = 4 \times n, \quad \text{height} = 4

Laying equal 4 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 40 cm and find n: 2 times (4n + 4) = 40, so 4n + 4 = 20, so 4n = 16, so n = 4. We get a 16 cm by 4 cm rectangle.

2\,(4n + 4) = 40 \;\Rightarrow\; 4n + 4 = 20 \;\Rightarrow\; n = 4

Try row lengths until the trip around the rectangle measures 40 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 4 squares is made from 2 triangles, so the rectangle needs 4 times 2 = 8 triangles.

4 \times 2 = 8

4 squares, 2 triangles each, is 8 triangles.

Answer: 8 triangles

4 · Reviewdoes it hold up?

The rectangle is 16 cm by 4 cm, and its perimeter is 16 + 4 + 16 + 4 = 40 cm, exactly as required. 8 right triangles with legs of 4 cm cover an area of 8 times (4 times 4 divided by 2) = 64 square cm, which equals the 16 by 4 rectangle's area of 64 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 4 cm with perimeter 40 cm must be 16 cm long, which is 4 squares, hence 8 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 40 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 8 hard answer: 10 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $5\text{ cm}$ long. To make a rectangle with a perimeter of $60\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $5\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 5 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 60 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 5 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 5 cm by 5 cm square.
The finished rectangle must have a perimeter of 60 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 5 cm because the building square is 5 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 5 cm, the two triangles snap into a square that is 5 cm wide and 5 cm tall.

2 \text{ triangles} = 1 \text{ square } (5\text{ cm} \times 5\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 5 cm squares side by side in one row. If there are n squares, the rectangle is 5 cm tall and 5 times n cm long.

\text{length} = 5 \times n, \quad \text{height} = 5

Laying equal 5 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 60 cm and find n: 2 times (5n + 5) = 60, so 5n + 5 = 30, so 5n = 25, so n = 5. We get a 25 cm by 5 cm rectangle.

2\,(5n + 5) = 60 \;\Rightarrow\; 5n + 5 = 30 \;\Rightarrow\; n = 5

Try row lengths until the trip around the rectangle measures 60 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 5 squares is made from 2 triangles, so the rectangle needs 5 times 2 = 10 triangles.

5 \times 2 = 10

5 squares, 2 triangles each, is 10 triangles.

Answer: 10 triangles

4 · Reviewdoes it hold up?

The rectangle is 25 cm by 5 cm, and its perimeter is 25 + 5 + 25 + 5 = 60 cm, exactly as required. 10 right triangles with legs of 5 cm cover an area of 10 times (5 times 5 divided by 2) = 125 square cm, which equals the 25 by 5 rectangle's area of 125 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 5 cm with perimeter 60 cm must be 25 cm long, which is 5 squares, hence 10 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 60 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Variant 9 hard answer: 12 triangles

There is a right triangle whose two legs (the two sides that form the right angle) are each $6\text{ cm}$ long. To make a rectangle with a perimeter of $84\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $6\text{ cm}$ .

Show solution

1 · Understandwhat's really being asked

We have a right triangle whose two perpendicular legs are each 6 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 84 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 6 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 6 cm by 6 cm square.
The finished rectangle must have a perimeter of 84 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 6 cm because the building square is 6 cm on a side.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives the target perimeter.

3 · Execute4 carry out the plan

1Combine two triangles into a square

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 6 cm, the two triangles snap into a square that is 6 cm wide and 6 cm tall.

2 \text{ triangles} = 1 \text{ square } (6\text{ cm} \times 6\text{ cm})

Two matching right triangles fit together along the long side to make a square.

2Line the squares up in a row

#7 Identify Subproblems 3.OA.A.3

Join the 6 cm squares side by side in one row. If there are n squares, the rectangle is 6 cm tall and 6 times n cm long.

\text{length} = 6 \times n, \quad \text{height} = 6

Laying equal 6 cm squares end to end multiplies the length by the number of squares.

3Match the perimeter

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 84 cm and find n: 2 times (6n + 6) = 84, so 6n + 6 = 42, so 6n = 36, so n = 6. We get a 36 cm by 6 cm rectangle.

2\,(6n + 6) = 84 \;\Rightarrow\; 6n + 6 = 42 \;\Rightarrow\; n = 6

Try row lengths until the trip around the rectangle measures 84 cm.

4Count the triangles

#7 Identify Subproblems 3.OA.A.3

Each of the 6 squares is made from 2 triangles, so the rectangle needs 6 times 2 = 12 triangles.

6 \times 2 = 12

6 squares, 2 triangles each, is 12 triangles.

Answer: 12 triangles

4 · Reviewdoes it hold up?

The rectangle is 36 cm by 6 cm, and its perimeter is 36 + 6 + 36 + 6 = 84 cm, exactly as required. 12 right triangles with legs of 6 cm cover an area of 12 times (6 times 6 divided by 2) = 216 square cm, which equals the 36 by 6 rectangle's area of 216 square cm, so the triangles tile it perfectly.

Another way: Instead of guess and check, reason directly: the only single-row rectangle of height 6 cm with perimeter 84 cm must be 36 cm long, which is 6 squares, hence 12 triangles.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 84 cm to find the length.

💡Takeaway. Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!

Perimeter plus tiles needed to build it · 9 practice problems

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