PAPER · REASONING · PRACTICE

Coordinates: location and translation

Use ordered pairs, quadrants, map scales and one common coordinate change.

40 minutes · Year 7, lower volume · text pp. 39–50 · textbook lessons 20–25

Textbook reference: Year 7 lower · Chapter 7 Cartesian coordinates · text lessons 20–25

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

18 minutes in the routes lab: set start (−1,−2), target (2,2). Predict endpoint and route length for RRRUUUU and UUUURRR (3 min). Run both, then insert the return pair LR before RRRUUUU, changing only the route (7 min). Record net coordinate change (3,4), common endpoint, and lengths 7,7,9; explain why a return pair changes length but not endpoint (5 min). Give a lower-bound explanation of 7 steps when movement is restricted to horizontal/vertical grid edges (3 min); this is not unrestricted straight-line distance.

Organise the key idea

Use ordered pairs, quadrants, map scales and one common coordinate change.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Translate $A(-3,-1)$ , $B(0,-1)$ , $C(-1,2)$ 4 units right, then 2 units down. Give every image coordinate and plot it.

Use the labelled axes and unit grid.

A horizontal move changes only the first coordinate.
A vertical move changes only the second coordinate.
Apply the same two changes to every vertex; do not reflect or rotate.

Every point uses the same change $(x,y)\mapsto(x+(4),y+(-2))$ .
The image coordinates are $A'(1,-3)$ , $B'(4,-3)$ , $C'(3,0)$ .
Check that corresponding coordinate differences all equal $(4,-2)$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Translate $P(-2,3)$ 5 units right, then 4 units down. Give every image coordinate and plot it.

Horizontal coordinate: $x^{\prime}=-2+5=\square$ .
Vertical coordinate: $y^{\prime}=3-4=\square$ .
Write $P'=(\square,\square)$ and plot it in the correct quadrant.

Use the labelled axes and unit grid.

The starting ordered pair is (−2,3): horizontal first.
Right 5 adds 5 to x; down 4 subtracts 4 from y.
The new x should be positive and the new y negative; use this to check the quadrant.

Every point uses the same change $(x,y)\mapsto(x+(5),y+(-4))$ .
The image coordinates are $P'(3,-1)$ .
Check that corresponding coordinate differences all equal $(5,-4)$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

Locate $P(-3,2)$ : which quadrant?

Quadrant III.
Quadrant IV.
Quadrant II.
Quadrant I.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (−,+); match it to the quadrant definition.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Locate $P(4,0)$ : on which axis or at the origin?

On the x-axis.
In a quadrant.
No location is determined.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read the horizontal coordinate first, then the vertical coordinate.
A zero second coordinate puts a point on the x-axis; a zero first coordinate puts it on the y-axis. Both zero gives the origin.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Locate $P(-4,-2)$ : which quadrant?

Quadrant III.
Quadrant I.
Quadrant II.
Quadrant IV.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (−,−); match it to the quadrant definition.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

On a map, O is the school; east is positive x, north positive y, and one unit is 100 m. A library is 300 m west and 200 m north of the school. Give its coordinates.

Use the labelled axes and unit grid.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

West means a negative horizontal coordinate: −300/100=−3.
North gives positive vertical coordinate 200/100=2.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Translate $A(-4,1)$ , $B(-2,1)$ , $C(-3,3)$ 5 units right, then 3 units down. Give every image coordinate and plot it.

Use the labelled axes and unit grid.

A horizontal move changes only the first coordinate.
A vertical move changes only the second coordinate.
Apply the same two changes to every vertex; do not reflect or rotate.

Every point uses the same change $(x,y)\mapsto(x+(5),y+(-3))$ .
The image coordinates are $A'(1,-2)$ , $B'(3,-2)$ , $C'(2,0)$ .
Check that corresponding coordinate differences all equal $(5,-3)$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

A(−1,2) becomes A′(3,−1). Under the same translation, find the image of B(2,4).

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

A changes by (4,−3).
Apply the same change to B: (2+4,4−3)=(6,1).

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Locate $P(0,-5)$ : on which axis or at the origin?

In a quadrant.
No location is determined.
On the y-axis.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read the horizontal coordinate first, then the vertical coordinate.
A zero second coordinate puts a point on the x-axis; a zero first coordinate puts it on the y-axis. Both zero gives the origin.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Translate $P(2,-3)$ 4 units left, then 2 units up. Give every image coordinate and plot it.

A horizontal move changes only the first coordinate.
A vertical move changes only the second coordinate.
Apply the same two changes to every vertex; do not reflect or rotate.

Every point uses the same change $(x,y)\mapsto(x+(-4),y+(2))$ .
The image coordinates are $P'(-2,-1)$ .
Check that corresponding coordinate differences all equal $(-4,2)$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Locate $P(2,5)$ : which quadrant?

Quadrant III.
Quadrant IV.
Quadrant I.
Quadrant II.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (+,+); match it to the quadrant definition.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Locate $P(3,-4)$ : which quadrant?

Quadrant I.
Quadrant II.
Quadrant III.
Quadrant IV.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (+,−); match it to the quadrant definition.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Locate $P(0,0)$ : on which axis or at the origin?

At the origin.
In a quadrant.
No location is determined.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read the horizontal coordinate first, then the vertical coordinate.
A zero second coordinate puts a point on the x-axis; a zero first coordinate puts it on the y-axis. Both zero gives the origin.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Plot P(−2,3) and Q(3,−2). Explain why the points differ.

Use the labelled axes and unit grid.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Move horizontally first, then vertically.
P uses negative x, positive y; Q uses positive x, negative y.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

From (−1,−2), walk right 4, left 1, up 3 and down 2 grid steps. Find the endpoint and total grid-route length.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Net horizontal and vertical changes are (3,1).
Endpoint is (−1+3,−2+1)=(2,−1); all steps count toward 4+1+3+2=10.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

In a map with the same east/north directions and unit, a park is (2,3) relative to the old origin O. The new origin is the old point (1,−1). Find the park coordinates relative to the new origin.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Subtract the new origin coordinates from the old park coordinates.
(2−1,3−(−1))=(1,4). The park itself has not moved.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Use signed numbers, horizontal/vertical direction, ordered pairs and a stated unit length.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 39–50 · textbook lessons 20–25

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Coordinates: location and translation

Year 7, lower volume · text pp. 39–50 · textbook lessons 20–25 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Translate $A(-3,-1)$ , $B(0,-1)$ , $C(-1,2)$ 4 units right, then 2 units down. Give every image coordinate and plot it.

Use the labelled axes and unit grid.

Guided attempt2

Translate $P(-2,3)$ 5 units right, then 4 units down. Give every image coordinate and plot it.

Horizontal coordinate: $x^{\prime}=-2+5=\square$ .
Vertical coordinate: $y^{\prime}=3-4=\square$ .
Write $P'=(\square,\square)$ and plot it in the correct quadrant.

Use the labelled axes and unit grid.

Prerequisite3

Locate $P(-3,2)$ : which quadrant?

Quadrant III.
Quadrant IV.
Quadrant II.
Quadrant I.

Prerequisite4

Locate $P(4,0)$ : on which axis or at the origin?

On the x-axis.
In a quadrant.
No location is determined.

Coordinates: location and translation

Year 7, lower volume · text pp. 39–50 · textbook lessons 20–25 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Locate $P(-4,-2)$ : which quadrant?

Quadrant III.
Quadrant I.
Quadrant II.
Quadrant IV.

Independent practice6

On a map, O is the school; east is positive x, north positive y, and one unit is 100 m. A library is 300 m west and 200 m north of the school. Give its coordinates.

Use the labelled axes and unit grid.

Independent practice7

Translate $A(-4,1)$ , $B(-2,1)$ , $C(-3,3)$ 5 units right, then 3 units down. Give every image coordinate and plot it.

Use the labelled axes and unit grid.

Independent practice8

A(−1,2) becomes A′(3,−1). Under the same translation, find the image of B(2,4).

Exit question9

Locate $P(0,-5)$ : on which axis or at the origin?

In a quadrant.
No location is determined.
On the y-axis.

Exit question10

Translate $P(2,-3)$ 4 units left, then 2 units up. Give every image coordinate and plot it.

Coordinates: location and translation — Further practice

Year 7, lower volume · text pp. 39–50 · textbook lessons 20–25

Further practice11

Locate $P(2,5)$ : which quadrant?

Quadrant III.
Quadrant IV.
Quadrant I.
Quadrant II.

Further practice12

Locate $P(3,-4)$ : which quadrant?

Quadrant I.
Quadrant II.
Quadrant III.
Quadrant IV.

Further practice13

Locate $P(0,0)$ : on which axis or at the origin?

At the origin.
In a quadrant.
No location is determined.

Further practice14

Plot P(−2,3) and Q(3,−2). Explain why the points differ.

Use the labelled axes and unit grid.

Extension15

From (−1,−2), walk right 4, left 1, up 3 and down 2 grid steps. Find the endpoint and total grid-route length.

Extension16

In a map with the same east/north directions and unit, a park is (2,3) relative to the old origin O. The new origin is the old point (1,−1). Find the park coordinates relative to the new origin.

Coordinates: location and translation — Teacher answers

Year 7, lower volume · text pp. 39–50 · textbook lessons 20–25

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Use signed numbers, horizontal/vertical direction, ordered pairs and a stated unit length.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

Inquiry and discussion: 18 minutes in the routes lab: set start (−1,−2), target (2,2). Predict endpoint and route length for RRRUUUU and UUUURRR (3 min). Run both, then insert the return pair LR before RRRUUUU, changing only the route (7 min). Record net coordinate change (3,4), common endpoint, and lengths 7,7,9; explain why a return pair changes length but not endpoint (5 min). Give a lower-bound explanation of 7 steps when movement is restricted to horizontal/vertical grid edges (3 min); this is not unrestricted straight-line distance.

Worked example1

Translate $A(-3,-1)$ , $B(0,-1)$ , $C(-1,2)$ 4 units right, then 2 units down. Give every image coordinate and plot it.

Use the labelled axes and unit grid.

Reference answer

$A'(1,-3)$ , $B'(4,-3)$ , $C'(3,0)$ .

Every point uses the same change $(x,y)\mapsto(x+(4),y+(-2))$ .
The image coordinates are $A'(1,-3)$ , $B'(4,-3)$ , $C'(3,0)$ .
Check that corresponding coordinate differences all equal $(4,-2)$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the labelled axes and unit grid.

Guided attempt2

Translate $P(-2,3)$ 5 units right, then 4 units down. Give every image coordinate and plot it.

Horizontal coordinate: $x^{\prime}=-2+5=\square$ .
Vertical coordinate: $y^{\prime}=3-4=\square$ .
Write $P'=(\square,\square)$ and plot it in the correct quadrant.

Use the labelled axes and unit grid.

Reference answer

$P'(3,-1)$ .

Every point uses the same change $(x,y)\mapsto(x+(5),y+(-4))$ .
The image coordinates are $P'(3,-1)$ .
Check that corresponding coordinate differences all equal $(5,-4)$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the labelled axes and unit grid.

Prerequisite3

Locate $P(-3,2)$ : which quadrant?

Quadrant III.
Quadrant IV.
Quadrant II.
Quadrant I.

Reference answer

Quadrant II.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (−,+); match it to the quadrant definition.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Locate $P(4,0)$ : on which axis or at the origin?

On the x-axis.
In a quadrant.
No location is determined.

Reference answer

On the x-axis.

Read the horizontal coordinate first, then the vertical coordinate.
A zero second coordinate puts a point on the x-axis; a zero first coordinate puts it on the y-axis. Both zero gives the origin.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Locate $P(-4,-2)$ : which quadrant?

Quadrant III.
Quadrant I.
Quadrant II.
Quadrant IV.

Reference answer

Quadrant III.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (−,−); match it to the quadrant definition.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

On a map, O is the school; east is positive x, north positive y, and one unit is 100 m. A library is 300 m west and 200 m north of the school. Give its coordinates.

Use the labelled axes and unit grid.

Reference answer

$(-3,2)$ .

West means a negative horizontal coordinate: −300/100=−3.
North gives positive vertical coordinate 200/100=2.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the labelled axes and unit grid.

Independent practice7

Translate $A(-4,1)$ , $B(-2,1)$ , $C(-3,3)$ 5 units right, then 3 units down. Give every image coordinate and plot it.

Use the labelled axes and unit grid.

Reference answer

$A'(1,-2)$ , $B'(3,-2)$ , $C'(2,0)$ .

Every point uses the same change $(x,y)\mapsto(x+(5),y+(-3))$ .
The image coordinates are $A'(1,-2)$ , $B'(3,-2)$ , $C'(2,0)$ .
Check that corresponding coordinate differences all equal $(5,-3)$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the labelled axes and unit grid.

Independent practice8

A(−1,2) becomes A′(3,−1). Under the same translation, find the image of B(2,4).

Reference answer

$B'(6,1)$

A changes by (4,−3).
Apply the same change to B: (2+4,4−3)=(6,1).

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Locate $P(0,-5)$ : on which axis or at the origin?

In a quadrant.
No location is determined.
On the y-axis.

Reference answer

On the y-axis.

Read the horizontal coordinate first, then the vertical coordinate.
A zero second coordinate puts a point on the x-axis; a zero first coordinate puts it on the y-axis. Both zero gives the origin.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Translate $P(2,-3)$ 4 units left, then 2 units up. Give every image coordinate and plot it.

Reference answer

$P'(-2,-1)$ .

Every point uses the same change $(x,y)\mapsto(x+(-4),y+(2))$ .
The image coordinates are $P'(-2,-1)$ .
Check that corresponding coordinate differences all equal $(-4,2)$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Locate $P(2,5)$ : which quadrant?

Quadrant III.
Quadrant IV.
Quadrant I.
Quadrant II.

Reference answer

Quadrant I.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (+,+); match it to the quadrant definition.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Locate $P(3,-4)$ : which quadrant?

Quadrant I.
Quadrant II.
Quadrant III.
Quadrant IV.

Reference answer

Quadrant IV.

Read the horizontal coordinate first, then the vertical coordinate.
The sign pair is (+,−); match it to the quadrant definition.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Locate $P(0,0)$ : on which axis or at the origin?

At the origin.
In a quadrant.
No location is determined.

Reference answer

At the origin.

Read the horizontal coordinate first, then the vertical coordinate.
A zero second coordinate puts a point on the x-axis; a zero first coordinate puts it on the y-axis. Both zero gives the origin.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Plot P(−2,3) and Q(3,−2). Explain why the points differ.

Use the labelled axes and unit grid.

Reference answer

P is in quadrant II; Q is in quadrant IV. Ordered coordinates cannot be interchanged.

Move horizontally first, then vertically.
P uses negative x, positive y; Q uses positive x, negative y.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the labelled axes and unit grid.

Extension15

From (−1,−2), walk right 4, left 1, up 3 and down 2 grid steps. Find the endpoint and total grid-route length.

Reference answer

Endpoint (2,−1); route length 10 grid units.

Net horizontal and vertical changes are (3,1).
Endpoint is (−1+3,−2+1)=(2,−1); all steps count toward 4+1+3+2=10.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

In a map with the same east/north directions and unit, a park is (2,3) relative to the old origin O. The new origin is the old point (1,−1). Find the park coordinates relative to the new origin.

Reference answer

$(1,4)$

Subtract the new origin coordinates from the old park coordinates.
(2−1,3−(−1))=(1,4). The park itself has not moved.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.