Practice · Slide edges to a rectangle, then add what juts in · Sensim Math

Variant 1 easy answer: 44 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $9\ \text{cm}$ wide (the bottom edge) and $7\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $6\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 9 cm by 7 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 9 cm wide and 7 cm tall.
The notch adds two horizontal segments of 6 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 9 cm by 7 cm.

\text{width} = 9\text{ cm},\quad \text{height} = 7\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(9 + 7) \times 2 = 16 \times 2 = 32 \text{ cm}

32 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 6 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

6 \times 2 = 12 \text{ cm}

12 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

32 + 12 = 44 \text{ cm}

The way around is 44 cm.

Answer: 44 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 7 = 14 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 9 + 2 x 6 = 30 cm. Together that is 44 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (6 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 2 easy answer: 56 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $11\ \text{cm}$ wide (the bottom edge) and $9\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $8\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 11 cm by 9 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 11 cm wide and 9 cm tall.
The notch adds two horizontal segments of 8 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 11 cm by 9 cm.

\text{width} = 11\text{ cm},\quad \text{height} = 9\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(11 + 9) \times 2 = 20 \times 2 = 40 \text{ cm}

40 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 8 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

8 \times 2 = 16 \text{ cm}

16 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

40 + 16 = 56 \text{ cm}

The way around is 56 cm.

Answer: 56 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 9 = 18 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 11 + 2 x 8 = 38 cm. Together that is 56 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (8 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 3 easy answer: 58 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $12\ \text{cm}$ wide (the bottom edge) and $8\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $9\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 12 cm by 8 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 12 cm wide and 8 cm tall.
The notch adds two horizontal segments of 9 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 12 cm by 8 cm.

\text{width} = 12\text{ cm},\quad \text{height} = 8\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(12 + 8) \times 2 = 20 \times 2 = 40 \text{ cm}

40 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 9 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

9 \times 2 = 18 \text{ cm}

18 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

40 + 18 = 58 \text{ cm}

The way around is 58 cm.

Answer: 58 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 8 = 16 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 12 + 2 x 9 = 42 cm. Together that is 58 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (9 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 4 easy answer: 64 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $13\ \text{cm}$ wide (the bottom edge) and $10\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $9\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 13 cm by 10 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 13 cm wide and 10 cm tall.
The notch adds two horizontal segments of 9 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 13 cm by 10 cm.

\text{width} = 13\text{ cm},\quad \text{height} = 10\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(13 + 10) \times 2 = 23 \times 2 = 46 \text{ cm}

46 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 9 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

9 \times 2 = 18 \text{ cm}

18 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

46 + 18 = 64 \text{ cm}

The way around is 64 cm.

Answer: 64 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 10 = 20 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 13 + 2 x 9 = 44 cm. Together that is 64 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (9 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 5 medium answer: 72 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $14\ \text{cm}$ wide (the bottom edge) and $12\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $10\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 14 cm by 12 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 14 cm wide and 12 cm tall.
The notch adds two horizontal segments of 10 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 14 cm by 12 cm.

\text{width} = 14\text{ cm},\quad \text{height} = 12\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(14 + 12) \times 2 = 26 \times 2 = 52 \text{ cm}

52 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 10 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

10 \times 2 = 20 \text{ cm}

20 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

52 + 20 = 72 \text{ cm}

The way around is 72 cm.

Answer: 72 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 12 = 24 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 14 + 2 x 10 = 48 cm. Together that is 72 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (10 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 6 medium answer: 72 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $15\ \text{cm}$ wide (the bottom edge) and $10\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $11\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 15 cm by 10 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 15 cm wide and 10 cm tall.
The notch adds two horizontal segments of 11 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 15 cm by 10 cm.

\text{width} = 15\text{ cm},\quad \text{height} = 10\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(15 + 10) \times 2 = 25 \times 2 = 50 \text{ cm}

50 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 11 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

11 \times 2 = 22 \text{ cm}

22 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

50 + 22 = 72 \text{ cm}

The way around is 72 cm.

Answer: 72 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 10 = 20 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 15 + 2 x 11 = 52 cm. Together that is 72 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (11 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 7 medium answer: 78 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $16\ \text{cm}$ wide (the bottom edge) and $11\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $12\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 16 cm by 11 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 16 cm wide and 11 cm tall.
The notch adds two horizontal segments of 12 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 16 cm by 11 cm.

\text{width} = 16\text{ cm},\quad \text{height} = 11\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(16 + 11) \times 2 = 27 \times 2 = 54 \text{ cm}

54 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 12 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

12 \times 2 = 24 \text{ cm}

24 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

54 + 24 = 78 \text{ cm}

The way around is 78 cm.

Answer: 78 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 11 = 22 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 16 + 2 x 12 = 56 cm. Together that is 78 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (12 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 8 medium answer: 90 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $18\ \text{cm}$ wide (the bottom edge) and $14\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $13\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 18 cm by 14 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 18 cm wide and 14 cm tall.
The notch adds two horizontal segments of 13 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 18 cm by 14 cm.

\text{width} = 18\text{ cm},\quad \text{height} = 14\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(18 + 14) \times 2 = 32 \times 2 = 64 \text{ cm}

64 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 13 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

13 \times 2 = 26 \text{ cm}

26 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

64 + 26 = 90 \text{ cm}

The way around is 90 cm.

Answer: 90 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 14 = 28 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 18 + 2 x 13 = 62 cm. Together that is 90 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (13 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 9 hard answer: 92 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $20\ \text{cm}$ wide (the bottom edge) and $12\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $14\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 20 cm by 12 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 20 cm wide and 12 cm tall.
The notch adds two horizontal segments of 14 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 20 cm by 12 cm.

\text{width} = 20\text{ cm},\quad \text{height} = 12\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(20 + 12) \times 2 = 32 \times 2 = 64 \text{ cm}

64 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 14 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

14 \times 2 = 28 \text{ cm}

28 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

64 + 28 = 92 \text{ cm}

The way around is 92 cm.

Answer: 92 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 12 = 24 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 20 + 2 x 14 = 68 cm. Together that is 92 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (14 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 10 hard answer: 106 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $22\ \text{cm}$ wide (the bottom edge) and $15\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $16\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 22 cm by 15 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 22 cm wide and 15 cm tall.
The notch adds two horizontal segments of 16 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 22 cm by 15 cm.

\text{width} = 22\text{ cm},\quad \text{height} = 15\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(22 + 15) \times 2 = 37 \times 2 = 74 \text{ cm}

74 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 16 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

16 \times 2 = 32 \text{ cm}

32 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

74 + 32 = 106 \text{ cm}

The way around is 106 cm.

Answer: 106 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 15 = 30 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 22 + 2 x 16 = 76 cm. Together that is 106 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (16 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 11 hard answer: 114 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $24\ \text{cm}$ wide (the bottom edge) and $16\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $17\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 24 cm by 16 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 24 cm wide and 16 cm tall.
The notch adds two horizontal segments of 17 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 24 cm by 16 cm.

\text{width} = 24\text{ cm},\quad \text{height} = 16\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(24 + 16) \times 2 = 40 \times 2 = 80 \text{ cm}

80 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 17 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

17 \times 2 = 34 \text{ cm}

34 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

80 + 34 = 114 \text{ cm}

The way around is 114 cm.

Answer: 114 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 16 = 32 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 24 + 2 x 17 = 82 cm. Together that is 114 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (17 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Variant 12 hard answer: 124 cm

The figure on the right is made up entirely of right angles. What is its perimeter, in $\text{cm}$ ?

Figure description: A C-shaped rectilinear figure in which every corner is a right angle. The smallest rectangle that surrounds the whole figure is $25\ \text{cm}$ wide (the bottom edge) and $18\ \text{cm}$ tall (the right edge). A notch cut inward from the left side creates two extra horizontal segments, each $19\ \text{cm}$ long.

Show solution

1 · Understandwhat's really being asked

A shape with nothing but right angles fits inside a 25 cm by 18 cm rectangle, with a notch cut into its left side. We need the distance around it.

Givens

The smallest surrounding rectangle is 25 cm wide and 18 cm tall.
The notch adds two horizontal segments of 19 cm each.
Every corner is a right angle.

Unknowns

The perimeter of the figure.

Constraints

Only edges of the figure count, and each is walked exactly once.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Slide the stepped edges out to the surrounding rectangle. Sliding does not change a length, so most of the outline is just that rectangle -- then handle the few edges that point inward and cannot slide out.

3 · Execute4 carry out the plan

1Slide the edges to make a rectangle

#1 Draw a Diagram 3.MD.D.8

Imagine pushing every vertical step to the right and every horizontal step up or down. The lengths do not change when you slide them. The outline becomes the smallest surrounding rectangle, which is 25 cm by 18 cm.

\text{width} = 25\text{ cm},\quad \text{height} = 18\text{ cm}

A stepped edge is a straight edge in disguise.

2Find the rectangle's perimeter

#7 Identify Subproblems 4.MD.A.3

A rectangle's perimeter is twice the width plus twice the height. Add the width and height, then double the sum.

(25 + 18) \times 2 = 43 \times 2 = 86 \text{ cm}

86 cm of the walk is accounted for.

3Add back the notch segments that did not fold in

#7 Identify Subproblems 3.MD.D.8

The two 19 cm horizontal segments form the inward notch on the left. They point into the figure, so sliding does NOT make them part of the rectangle's edge. Their length must be added to the perimeter separately.

19 \times 2 = 38 \text{ cm}

38 cm of extra walking, in and back out.

4Total the perimeter

#1 Draw a Diagram 3.MD.D.8

Add the rectangle's perimeter and the two notch segments to get the full distance around the figure.

86 + 38 = 124 \text{ cm}

The way around is 124 cm.

Answer: 124 cm

4 · Reviewdoes it hold up?

Count the edges directly instead. The verticals are the right side plus the three on the left, and they add to 2 x 18 = 36 cm. The horizontals are the top, the bottom and the two notch edges: 2 x 25 + 2 x 19 = 88 cm. Together that is 124 cm.

Another way: Note that the notch's height does not appear anywhere in the answer: making the notch taller stretches its vertical edge but shortens the left side by the same amount, so only how far it cuts in (19 cm) changes the perimeter.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the edges of a rectilinear figure once each.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the rectangle perimeter formula on the slid-out outline.

💡Takeaway. Sliding an edge sideways never changes how long it is -- so a stepped shape costs the same walk as its box, plus whatever juts inward.

Slide edges to a rectangle, then add what juts in · 12 practice problems

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