Practice · Turn directions and distances into signed coordinates · Sensim Math

Variant 1 easy

A playground is $50$ m from our house. A flower shop is $70$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $50$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $50$ , then find where the bus stop lands.

Givens

The playground is $50$ m from the house and sits at $50$ .
The flower shop is $70$ m past the playground, going away from the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the positive direction; moving toward the house from the playground is the negative direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is positive. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

50

is on the positive side of the house, so 'from the house toward the playground' means moving in the positive direction, and 'toward the house' from the playground means moving in the negative direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

70

m farther in the positive direction from the playground.

50 + 70 = 120

Moving right on the number line adds.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

120

is the absolute value of their difference.

|120 - 0| = 120

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

120

m in the negative direction, toward the house. The trip passes the house and ends

70

m on the other side.

50 - 120 = -70

Moving left on the number line subtracts.

Answer:

-70

4 · Reviewdoes it hold up?

The bus stop at $-70$ is $120$ m from the playground at $50$ , since $50-(-70)=120$ , which matches the house-to-flower-shop distance.

Another way: Use symmetry: the house is $50$ m from the playground toward the bus stop, and the flower shop is $70$ m from the playground the other way. Walking $120 = 50+70$ m toward the house overshoots the house by $70$ m.

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 2 easy

A playground is $60$ m from our house. A flower shop is $90$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $-60$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $-60$ , then find where the bus stop lands.

Givens

The playground is $60$ m from the house and sits at $-60$ .
The flower shop is $90$ m past the playground, going away from the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the negative direction; moving toward the house from the playground is the positive direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is negative. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

-60

is on the negative side of the house, so 'from the house toward the playground' means moving in the negative direction, and 'toward the house' from the playground means moving in the positive direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

90

m farther in the negative direction from the playground.

-60 - 90 = -150

Moving left on the number line subtracts.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

-150

is the absolute value of their difference.

|-150 - 0| = 150

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

150

m in the positive direction, toward the house. The trip passes the house and ends

90

m on the other side.

-60 + 150 = 90

Moving right on the number line adds.

Answer:

90

4 · Reviewdoes it hold up?

The bus stop at $90$ is $150$ m from the playground at $-60$ , since $90-(-60)=150$ , which matches the house-to-flower-shop distance.

Another way: Use symmetry: the house is $60$ m from the playground toward the bus stop, and the flower shop is $90$ m from the playground the other way. Walking $150 = 60+90$ m toward the house overshoots the house by $90$ m.

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 3 easy

A playground is $80$ m from our house. A flower shop is $120$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $80$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $80$ , then find where the bus stop lands.

Givens

The playground is $80$ m from the house and sits at $80$ .
The flower shop is $120$ m past the playground, going away from the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the positive direction; moving toward the house from the playground is the negative direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is positive. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

80

is on the positive side of the house, so 'from the house toward the playground' means moving in the positive direction, and 'toward the house' from the playground means moving in the negative direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

120

m farther in the positive direction from the playground.

80 + 120 = 200

Moving right on the number line adds.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

200

is the absolute value of their difference.

|200 - 0| = 200

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

200

m in the negative direction, toward the house. The trip passes the house and ends

120

m on the other side.

80 - 200 = -120

Moving left on the number line subtracts.

Answer:

-120

4 · Reviewdoes it hold up?

The bus stop at $-120$ is $200$ m from the playground at $80$ , since $80-(-120)=200$ , which matches the house-to-flower-shop distance.

Another way: Use symmetry: the house is $80$ m from the playground toward the bus stop, and the flower shop is $120$ m from the playground the other way. Walking $200 = 80+120$ m toward the house overshoots the house by $120$ m.

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 4 easy

A playground is $100$ m from our house. A flower shop is $150$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $-100$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $-100$ , then find where the bus stop lands.

Givens

The playground is $100$ m from the house and sits at $-100$ .
The flower shop is $150$ m past the playground, going away from the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the negative direction; moving toward the house from the playground is the positive direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is negative. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

-100

is on the negative side of the house, so 'from the house toward the playground' means moving in the negative direction, and 'toward the house' from the playground means moving in the positive direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

150

m farther in the negative direction from the playground.

-100 - 150 = -250

Moving left on the number line subtracts.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

-250

is the absolute value of their difference.

|-250 - 0| = 250

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

250

m in the positive direction, toward the house. The trip passes the house and ends

150

m on the other side.

-100 + 250 = 150

Moving right on the number line adds.

Answer:

150

4 · Reviewdoes it hold up?

The bus stop at $150$ is $250$ m from the playground at $-100$ , since $150-(-100)=250$ , which matches the house-to-flower-shop distance.

Another way: Use symmetry: the house is $100$ m from the playground toward the bus stop, and the flower shop is $150$ m from the playground the other way. Walking $250 = 100+150$ m toward the house overshoots the house by $150$ m.

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 5 medium

A playground is $150$ m from our house. A flower shop is $100$ m from the playground, in the direction that runs from the playground toward our house. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $-150$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $-150$ , then find where the bus stop lands.

Givens

The playground is $150$ m from the house and sits at $-150$ .
The flower shop is $100$ m from the playground, going toward the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the negative direction; moving toward the house from the playground is the positive direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is negative. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

-150

is on the negative side of the house, so 'from the house toward the playground' means moving in the negative direction, and 'toward the house' from the playground means moving in the positive direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

100

m from the playground in the positive direction, toward the house.

-150 + 100 = -50

Moving right on the number line adds.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

-50

is the absolute value of their difference.

|-50 - 0| = 50

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

50

m in the positive direction, toward the house. The trip stops

100

m short of the house.

-150 + 50 = -100

Moving right on the number line adds.

Answer:

-100

4 · Reviewdoes it hold up?

The bus stop at $-100$ is $50$ m from the playground at $-150$ , since $-100-(-150)=50$ , which matches the house-to-flower-shop distance.

Another way: Track distances first: the flower shop is $100$ m from the playground toward the house, which is $150$ m away, so the shop is $50$ m from the house. Walking $50$ m from the playground toward the house stops $100$ m before the house. Since the house side of the playground is the positive side, the bus stop is at $-100$ .

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 6 medium

A playground is $120$ m from our house. A flower shop is $200$ m from the playground, in the direction that runs from the playground toward our house. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $120$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $120$ , then find where the bus stop lands.

Givens

The playground is $120$ m from the house and sits at $120$ .
The flower shop is $200$ m from the playground, going toward the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the positive direction; moving toward the house from the playground is the negative direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is positive. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

120

is on the positive side of the house, so 'from the house toward the playground' means moving in the positive direction, and 'toward the house' from the playground means moving in the negative direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

200

m from the playground in the negative direction, toward the house.

120 - 200 = -80

Moving left on the number line subtracts.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

-80

is the absolute value of their difference.

|-80 - 0| = 80

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

80

m in the negative direction, toward the house. The trip stops

40

m short of the house.

120 - 80 = 40

Moving left on the number line subtracts.

Answer:

40

4 · Reviewdoes it hold up?

The bus stop at $40$ is $80$ m from the playground at $120$ , since $120-40=80$ , which matches the house-to-flower-shop distance.

Another way: Track distances first: the flower shop is $200$ m from the playground toward the house, which is $120$ m away, so the shop is $80$ m from the house. Walking $80$ m from the playground toward the house stops $40$ m before the house. Since the house side of the playground is the negative side, the bus stop is at $40$ .

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 7 medium

A playground is $200$ m from our house. A flower shop is $50$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $-200$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $-200$ , then find where the bus stop lands.

Givens

The playground is $200$ m from the house and sits at $-200$ .
The flower shop is $50$ m past the playground, going away from the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the negative direction; moving toward the house from the playground is the positive direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is negative. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

-200

is on the negative side of the house, so 'from the house toward the playground' means moving in the negative direction, and 'toward the house' from the playground means moving in the positive direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

50

m farther in the negative direction from the playground.

-200 - 50 = -250

Moving left on the number line subtracts.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

-250

is the absolute value of their difference.

|-250 - 0| = 250

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

250

m in the positive direction, toward the house. The trip passes the house and ends

50

m on the other side.

-200 + 250 = 50

Moving right on the number line adds.

Answer:

50

4 · Reviewdoes it hold up?

The bus stop at $50$ is $250$ m from the playground at $-200$ , since $50-(-200)=250$ , which matches the house-to-flower-shop distance.

Another way: Use symmetry: the house is $200$ m from the playground toward the bus stop, and the flower shop is $50$ m from the playground the other way. Walking $250 = 200+50$ m toward the house overshoots the house by $50$ m.

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 8 medium

A playground is $240$ m from our house. A flower shop is $90$ m from the playground, in the direction that runs from the playground toward our house. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $240$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $240$ , then find where the bus stop lands.

Givens

The playground is $240$ m from the house and sits at $240$ .
The flower shop is $90$ m from the playground, going toward the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the positive direction; moving toward the house from the playground is the negative direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is positive. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

240

is on the positive side of the house, so 'from the house toward the playground' means moving in the positive direction, and 'toward the house' from the playground means moving in the negative direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

90

m from the playground in the negative direction, toward the house.

240 - 90 = 150

Moving left on the number line subtracts.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

150

is the absolute value of their difference.

|150 - 0| = 150

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

150

m in the negative direction, toward the house. The trip stops

90

m short of the house.

240 - 150 = 90

Moving left on the number line subtracts.

Answer:

90

4 · Reviewdoes it hold up?

The bus stop at $90$ is $150$ m from the playground at $240$ , since $240-90=150$ , which matches the house-to-flower-shop distance.

Another way: Track distances first: the flower shop is $90$ m from the playground toward the house, which is $240$ m away, so the shop is $150$ m from the house. Walking $150$ m from the playground toward the house stops $90$ m before the house. Since the house side of the playground is the negative side, the bus stop is at $90$ .

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 9 hard

A playground is $100$ m from our house. A flower shop is $250$ m from the playground, in the direction that runs from the playground toward our house. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $-100$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $-100$ , then find where the bus stop lands.

Givens

The playground is $100$ m from the house and sits at $-100$ .
The flower shop is $250$ m from the playground, going toward the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the negative direction; moving toward the house from the playground is the positive direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is negative. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

-100

is on the negative side of the house, so 'from the house toward the playground' means moving in the negative direction, and 'toward the house' from the playground means moving in the positive direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

250

m from the playground in the positive direction, toward the house.

-100 + 250 = 150

Moving right on the number line adds.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

150

is the absolute value of their difference.

|150 - 0| = 150

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

150

m in the positive direction, toward the house. The trip passes the house and ends

50

m on the other side.

-100 + 150 = 50

Moving right on the number line adds.

Answer:

50

4 · Reviewdoes it hold up?

The bus stop at $50$ is $150$ m from the playground at $-100$ , since $50-(-100)=150$ , which matches the house-to-flower-shop distance.

Another way: Track distances first: the flower shop is $250$ m from the playground toward the house, which is $100$ m away, so the shop is $150$ m from the house. Walking $150$ m from the playground toward the house goes $100$ m to the house and $50$ m beyond it. Since the house side of the playground is the positive side, the bus stop is at $50$ .

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 10 hard

A playground is $300$ m from our house. A flower shop is $250$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $300$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $300$ , then find where the bus stop lands.

Givens

The playground is $300$ m from the house and sits at $300$ .
The flower shop is $250$ m past the playground, going away from the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the positive direction; moving toward the house from the playground is the negative direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is positive. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

300

is on the positive side of the house, so 'from the house toward the playground' means moving in the positive direction, and 'toward the house' from the playground means moving in the negative direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

250

m farther in the positive direction from the playground.

300 + 250 = 550

Moving right on the number line adds.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

550

is the absolute value of their difference.

|550 - 0| = 550

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

550

m in the negative direction, toward the house. The trip passes the house and ends

250

m on the other side.

300 - 550 = -250

Moving left on the number line subtracts.

Answer:

-250

4 · Reviewdoes it hold up?

The bus stop at $-250$ is $550$ m from the playground at $300$ , since $300-(-250)=550$ , which matches the house-to-flower-shop distance.

Another way: Use symmetry: the house is $300$ m from the playground toward the bus stop, and the flower shop is $250$ m from the playground the other way. Walking $550 = 300+250$ m toward the house overshoots the house by $250$ m.

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 11 hard

A playground is $250$ m from our house. A flower shop is $400$ m from the playground, in the direction that runs from the playground toward our house. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $-250$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $-250$ , then find where the bus stop lands.

Givens

The playground is $250$ m from the house and sits at $-250$ .
The flower shop is $400$ m from the playground, going toward the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the negative direction; moving toward the house from the playground is the positive direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is negative. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

-250

is on the negative side of the house, so 'from the house toward the playground' means moving in the negative direction, and 'toward the house' from the playground means moving in the positive direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

400

m from the playground in the positive direction, toward the house.

-250 + 400 = 150

Moving right on the number line adds.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

150

is the absolute value of their difference.

|150 - 0| = 150

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

150

m in the positive direction, toward the house. The trip stops

100

m short of the house.

-250 + 150 = -100

Moving right on the number line adds.

Answer:

-100

4 · Reviewdoes it hold up?

The bus stop at $-100$ is $150$ m from the playground at $-250$ , since $-100-(-250)=150$ , which matches the house-to-flower-shop distance.

Another way: Track distances first: the flower shop is $400$ m from the playground toward the house, which is $250$ m away, so the shop is $150$ m from the house. Walking $150$ m from the playground toward the house stops $100$ m before the house. Since the house side of the playground is the positive side, the bus stop is at $-100$ .

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Variant 12 hard

A playground is $400$ m from our house. A flower shop is $150$ m from the playground, in the direction that runs from our house toward the playground. A bus stop is reached by starting at the playground and going toward our house a distance equal to the distance between our house and the flower shop. If our house is placed at the origin of a number line and the playground at the point $400$ , what is the coordinate of the bus stop?

Your answer

Show solution

1 · Understandwhat's really being asked

Places along one straight road are described by direction and distance. Put them on a number line with the house at $0$ and the playground at $400$ , then find where the bus stop lands.

Givens

The playground is $400$ m from the house and sits at $400$ .
The flower shop is $150$ m past the playground, going away from the house.
The bus stop is that house-to-flower-shop distance from the playground, heading toward the house.

Unknowns

The coordinate of the bus stop.

Constraints

Moving from the house toward the playground is the positive direction; moving toward the house from the playground is the negative direction.

2 · Planchoose the strategy

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

Draw the number line and fix which way is positive. Then each sentence becomes one move: find the flower shop, measure the house-to-flower-shop distance, and make that move from the playground.

3 · Execute4 carry out the plan

1Fix the directions

#1 Draw a Diagram 6.NS.C.5

The playground at

400

is on the positive side of the house, so 'from the house toward the playground' means moving in the positive direction, and 'toward the house' from the playground means moving in the negative direction.

Once one point is placed, every direction gets a sign.

2Place the flower shop

#7 Identify Subproblems 7.NS.A.1

The flower shop is

150

m farther in the positive direction from the playground.

400 + 150 = 550

Moving right on the number line adds.

3Measure house to flower shop

#7 Identify Subproblems 7.NS.A.1

The distance between

0

and

550

is the absolute value of their difference.

|550 - 0| = 550

Distance is never negative.

4Place the bus stop

#7 Identify Subproblems 7.NS.A.1

From the playground, move

550

m in the negative direction, toward the house. The trip passes the house and ends

150

m on the other side.

400 - 550 = -150

Moving left on the number line subtracts.

Answer:

-150

4 · Reviewdoes it hold up?

The bus stop at $-150$ is $550$ m from the playground at $400$ , since $400-(-150)=550$ , which matches the house-to-flower-shop distance.

Another way: Use symmetry: the house is $400$ m from the playground toward the bus stop, and the flower shop is $150$ m from the playground the other way. Walking $550 = 400+150$ m toward the house overshoots the house by $150$ m.

Standardsmin grade 7

7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Adding and subtracting signed numbers as moves on a number line and finding a distance as an absolute difference.
6.NS.C.5 Understand positive and negative numbers — Using positive and negative numbers for opposite directions.

💡Takeaway. Pick a direction to be negative, and every 'go this way' sentence becomes plain addition or subtraction.

Turn directions and distances into signed coordinates · 12 practice problems

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