18 minutes on squared paper: use the labelled triangle ABC in the guided grid, without coordinate notation. Predict the image after moving every vertex right 2, down 3 (3 min), then construct and match all three vertices and edges (6 min). Keep A and B moves unchanged but move C right 3 instead of 2; compare whether one translation still describes the result (4 min). Restore the common move and compare lengths, angles and a marked midpoint (3 min). Explain why same direction alone is insufficient and why the whole figure must share one direction-and-distance move (2 min).
Organise the key idea
Move a whole figure in one direction by one distance; distinguish translation from rotation or reflection.
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 every labelled point on the squared grid 3 squares right, then 2 squares up. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Count grid squares; no ordered-pair notation is needed.
Apply 3 squares right, then 2 squares up at every vertex.
Check one original-to-image path at a time, then connect the matching image vertices.
Use one horizontal move of 3 squares and one vertical move of 2 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Reference answer
Every image is 3 squares right, then 2 squares up from its original point; the teacher image shows the labelled positions.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Guided attempt2
Translate every labelled point on the squared grid 2 squares right, then 3 squares down. Mark the corresponding image points and join matching edges.
For A′: right 2 squares, down ____ squares.
For B′: right ____ squares, down 3 squares.
For C′: complete the same two moves; mark A′,B′,C′ and join them in matching order.
Squared grid; each square is one unit. Use the stated directions and counts.
Count grid squares; no ordered-pair notation is needed.
Apply 2 squares right, then 3 squares down at every vertex.
Check one original-to-image path at a time, then connect the matching image vertices.
Use one horizontal move of 2 squares and one vertical move of 3 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Reference answer
Every image is 2 squares right, then 3 squares down from its original point; the teacher image shows the labelled positions.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Prerequisite3
Which idealised motion is a translation of the whole object?
A clock hand turns about its centre.
A page folds over a crease.
A drawer slides straight out without turning.
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 translation has a common direction and distance for every point.
Sliding without turning meets this condition.
Reference answer
A drawer slides straight out without turning.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Prerequisite4
A figure translates 4 grid units right. How many grid units right does each vertex move?
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.
One translation applies to the whole figure.
Every vertex moves 4 units right.
Reference answer
4.
4
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Independent practice5
Translate every labelled point on the squared grid 4 squares right, then 3 squares up. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Count grid squares; no ordered-pair notation is needed.
Apply 4 squares right, then 3 squares up at every vertex.
Check one original-to-image path at a time, then connect the matching image vertices.
Use one horizontal move of 4 squares and one vertical move of 3 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Reference answer
Every image is 4 squares right, then 3 squares up from its original point; the teacher image shows the labelled positions.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Independent practice6
Translate every labelled point on the squared grid 3 squares left, then 2 squares down. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Count grid squares; no ordered-pair notation is needed.
Apply 3 squares left, then 2 squares down at every vertex.
Check one original-to-image path at a time, then connect the matching image vertices.
Use one horizontal move of 3 squares and one vertical move of 2 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Reference answer
Every image is 3 squares left, then 2 squares down from its original point; the teacher image shows the labelled positions.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Independent practice7
The given A→A′ move is 3 squares right and 2 up. Under the same translation, mark B′ on the squared grid.
Squared grid; each square is one unit. Use the stated directions and counts.
Match A with A′ before moving B.
Count 3 horizontal and 2 vertical squares in the stated directions.
B must use the same move, not the same final location as A.
Read the given A→A′ move as right 3, up 2.
Repeat exactly that move from B; the teacher diagram locates B′.
Reference answer
B′ is 3 squares right and 2 up from B.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Independent practice8
AB=6 cm and angle ABC=42°. A translation sends A,B,C to A′,B′,C′. State A′B′ and angle A′B′C′, with a reason.
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.
Corresponding segments and angles have the same size.
Therefore the image length is 6 cm and image angle is 42°.
Reference answer
6 cm and 42°; translation preserves lengths and angles.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Exit question9
One vertex moves right 2 units, another right 3 units. Can these be corresponding moves of one translation?
Yes; both move right.
Yes; lengths always change.
No; the displacement is not common.
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.
Equal direction alone is insufficient.
A translation also requires equal distance, but 2 differs from 3.
Reference answer
No; the displacement is not common.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Exit question10
The whole figure moves 3 squares right, then 1 squares down. State how its point P must move.
Count grid squares; no ordered-pair notation is needed.
Apply 3 squares right, then 1 squares down at every vertex.
Check one original-to-image path at a time, then connect the matching image vertices.
Use one horizontal move of 3 squares and one vertical move of 1 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Reference answer
Every image is 3 squares right, then 1 squares down from its original point; the teacher image shows the labelled positions.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Further practice11
Translate every labelled point on the squared grid 4 squares right. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Count grid squares; no ordered-pair notation is needed.
Apply 4 squares right at every vertex.
Check one original-to-image path at a time, then connect the matching image vertices.
Use one horizontal move of 4 squares and one vertical move of 0 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Reference answer
Every image is 4 squares right from its original point; the teacher image shows the labelled positions.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Further practice12
Translate every labelled point on the squared grid 4 squares down. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Count grid squares; no ordered-pair notation is needed.
Apply 4 squares down at every vertex.
Check one original-to-image path at a time, then connect the matching image vertices.
Use one horizontal move of 0 squares and one vertical move of 4 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Reference answer
Every image is 4 squares down from its original point; the teacher image shows the labelled positions.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Further practice13
P′ is marked on the squared grid after a translation left 3 and up 2. Mark the original point P.
Squared grid; each square is one unit. Use the stated directions and counts.
You are given the image, not the starting point.
Undo up 2 with down 2, and left 3 with right 3.
Check by moving your proposed P left 3 and up 2.
Reverse each direction: right 3 and down 2.
Start at P′ and make that reverse move; moving forward again must return to P′.
Reference answer
P lies 3 squares right and 2 down from P′.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Further practice14
M is the midpoint of AB. Under a translation, where is its image M′ relative to A′B′?
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.
Collinearity and the two equal half-lengths are preserved.
Thus A′M′=M′B′ on A′B′.
Reference answer
M′ is the midpoint of A′B′.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Extension15
A and B lie two squares apart on the same horizontal grid row. Both move right 3 squares to A′,B′. A learner says AA′ and BB′ must lie on two distinct parallel lines. Correct the claim.
Squared grid; each square is one unit. Use the stated directions and counts.
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.
All four marked points lie on the same horizontal row.
Corresponding move segments are equal and same-direction, but here their supporting lines coincide.
Reference answer
The displacement segments are equal and same-direction, but their supporting lines coincide here.
Squared grid; each square is one unit. Use the stated directions and counts.
Check your work
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Extension16
Translate a figure right 4 and up 1, then left 2 and down 3. Give the single equivalent translation and explain why the figure does not rotate.
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.
Horizontally, right 4 then left 2 gives right 2.
Vertically, up 1 then down 3 gives down 2. The same changes apply to every vertex.
Reference answer
Right 2 squares and down 2; every vertex receives the same net move.
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
Count equal grid squares, recognise direction, match labelled vertices and identify a midpoint; coordinates are not required.
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.
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.
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. 25–28 · textbook lessons 13–14
Original questions and explanations; the source textbook and answer PDFs are not reproduced here.
Translation: matching points and invariant lengths
Year 7, lower volume · text pp. 25–28 · textbook lessons 13–14 · 1/2
Name: __________ Class: ______ Date: __________
Worked example1
Translate every labelled point on the squared grid 3 squares right, then 2 squares up. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Guided attempt2
Translate every labelled point on the squared grid 2 squares right, then 3 squares down. Mark the corresponding image points and join matching edges.
For A′: right 2 squares, down ____ squares.
For B′: right ____ squares, down 3 squares.
For C′: complete the same two moves; mark A′,B′,C′ and join them in matching order.
Squared grid; each square is one unit. Use the stated directions and counts.
Prerequisite3
Which idealised motion is a translation of the whole object?
A clock hand turns about its centre.
A page folds over a crease.
A drawer slides straight out without turning.
Prerequisite4
A figure translates 4 grid units right. How many grid units right does each vertex move?
Translation: matching points and invariant lengths
Year 7, lower volume · text pp. 25–28 · textbook lessons 13–14 · 2/2
Name: __________ Class: ______ Date: __________
Independent practice5
Translate every labelled point on the squared grid 4 squares right, then 3 squares up. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Independent practice6
Translate every labelled point on the squared grid 3 squares left, then 2 squares down. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Independent practice7
The given A→A′ move is 3 squares right and 2 up. Under the same translation, mark B′ on the squared grid.
Squared grid; each square is one unit. Use the stated directions and counts.
Independent practice8
AB=6 cm and angle ABC=42°. A translation sends A,B,C to A′,B′,C′. State A′B′ and angle A′B′C′, with a reason.
Exit question9
One vertex moves right 2 units, another right 3 units. Can these be corresponding moves of one translation?
Yes; both move right.
Yes; lengths always change.
No; the displacement is not common.
Exit question10
The whole figure moves 3 squares right, then 1 squares down. State how its point P must move.
Translation: matching points and invariant lengths — Further practice
Year 7, lower volume · text pp. 25–28 · textbook lessons 13–14
Further practice11
Translate every labelled point on the squared grid 4 squares right. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Further practice12
Translate every labelled point on the squared grid 4 squares down. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Further practice13
P′ is marked on the squared grid after a translation left 3 and up 2. Mark the original point P.
Squared grid; each square is one unit. Use the stated directions and counts.
Further practice14
M is the midpoint of AB. Under a translation, where is its image M′ relative to A′B′?
Extension15
A and B lie two squares apart on the same horizontal grid row. Both move right 3 squares to A′,B′. A learner says AA′ and BB′ must lie on two distinct parallel lines. Correct the claim.
Squared grid; each square is one unit. Use the stated directions and counts.
Extension16
Translate a figure right 4 and up 1, then left 2 and down 3. Give the single equivalent translation and explain why the figure does not rotate.
Translation: matching points and invariant lengths — Teacher answers
Year 7, lower volume · text pp. 25–28 · textbook lessons 13–14
Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.
Prerequisites: Count equal grid squares, recognise direction, match labelled vertices and identify a midpoint; coordinates are not required.
Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.
Inquiry and discussion: 18 minutes on squared paper: use the labelled triangle ABC in the guided grid, without coordinate notation. Predict the image after moving every vertex right 2, down 3 (3 min), then construct and match all three vertices and edges (6 min). Keep A and B moves unchanged but move C right 3 instead of 2; compare whether one translation still describes the result (4 min). Restore the common move and compare lengths, angles and a marked midpoint (3 min). Explain why same direction alone is insufficient and why the whole figure must share one direction-and-distance move (2 min).
Worked example1
Translate every labelled point on the squared grid 3 squares right, then 2 squares up. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
Every image is 3 squares right, then 2 squares up from its original point; the teacher image shows the labelled positions.
Use one horizontal move of 3 squares and one vertical move of 2 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Guided attempt2
Translate every labelled point on the squared grid 2 squares right, then 3 squares down. Mark the corresponding image points and join matching edges.
For A′: right 2 squares, down ____ squares.
For B′: right ____ squares, down 3 squares.
For C′: complete the same two moves; mark A′,B′,C′ and join them in matching order.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
Every image is 2 squares right, then 3 squares down from its original point; the teacher image shows the labelled positions.
Use one horizontal move of 2 squares and one vertical move of 3 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Prerequisite3
Which idealised motion is a translation of the whole object?
A clock hand turns about its centre.
A page folds over a crease.
A drawer slides straight out without turning.
Reference answer
A drawer slides straight out without turning.
A translation has a common direction and distance for every point.
Sliding without turning meets this condition.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Prerequisite4
A figure translates 4 grid units right. How many grid units right does each vertex move?
Reference answer
4.
4
One translation applies to the whole figure.
Every vertex moves 4 units right.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Independent practice5
Translate every labelled point on the squared grid 4 squares right, then 3 squares up. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
Every image is 4 squares right, then 3 squares up from its original point; the teacher image shows the labelled positions.
Use one horizontal move of 4 squares and one vertical move of 3 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Independent practice6
Translate every labelled point on the squared grid 3 squares left, then 2 squares down. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
Every image is 3 squares left, then 2 squares down from its original point; the teacher image shows the labelled positions.
Use one horizontal move of 3 squares and one vertical move of 2 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Independent practice7
The given A→A′ move is 3 squares right and 2 up. Under the same translation, mark B′ on the squared grid.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
B′ is 3 squares right and 2 up from B.
Read the given A→A′ move as right 3, up 2.
Repeat exactly that move from B; the teacher diagram locates B′.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Independent practice8
AB=6 cm and angle ABC=42°. A translation sends A,B,C to A′,B′,C′. State A′B′ and angle A′B′C′, with a reason.
Reference answer
6 cm and 42°; translation preserves lengths and angles.
Corresponding segments and angles have the same size.
Therefore the image length is 6 cm and image angle is 42°.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Exit question9
One vertex moves right 2 units, another right 3 units. Can these be corresponding moves of one translation?
Yes; both move right.
Yes; lengths always change.
No; the displacement is not common.
Reference answer
No; the displacement is not common.
Equal direction alone is insufficient.
A translation also requires equal distance, but 2 differs from 3.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Exit question10
The whole figure moves 3 squares right, then 1 squares down. State how its point P must move.
Reference answer
Every image is 3 squares right, then 1 squares down from its original point; the teacher image shows the labelled positions.
Use one horizontal move of 3 squares and one vertical move of 1 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Further practice11
Translate every labelled point on the squared grid 4 squares right. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
Every image is 4 squares right from its original point; the teacher image shows the labelled positions.
Use one horizontal move of 4 squares and one vertical move of 0 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Further practice12
Translate every labelled point on the squared grid 4 squares down. Mark the corresponding image points and join matching edges.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
Every image is 4 squares down from its original point; the teacher image shows the labelled positions.
Use one horizontal move of 0 squares and one vertical move of 4 squares, in the stated directions.
Repeat this move at every labelled point, then match original and image labels.
Corresponding moves must have the same direction and number of squares; matching lengths and angles remain unchanged.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Further practice13
P′ is marked on the squared grid after a translation left 3 and up 2. Mark the original point P.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
P lies 3 squares right and 2 down from P′.
Reverse each direction: right 3 and down 2.
Start at P′ and make that reverse move; moving forward again must return to P′.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Further practice14
M is the midpoint of AB. Under a translation, where is its image M′ relative to A′B′?
Reference answer
M′ is the midpoint of A′B′.
Collinearity and the two equal half-lengths are preserved.
Thus A′M′=M′B′ on A′B′.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Extension15
A and B lie two squares apart on the same horizontal grid row. Both move right 3 squares to A′,B′. A learner says AA′ and BB′ must lie on two distinct parallel lines. Correct the claim.
Squared grid; each square is one unit. Use the stated directions and counts.
Reference answer
The displacement segments are equal and same-direction, but their supporting lines coincide here.
All four marked points lie on the same horizontal row.
Corresponding move segments are equal and same-direction, but here their supporting lines coincide.
Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.
Squared grid; each square is one unit. Use the stated directions and counts.
Extension16
Translate a figure right 4 and up 1, then left 2 and down 3. Give the single equivalent translation and explain why the figure does not rotate.
Reference answer
Right 2 squares and down 2; every vertex receives the same net move.
Horizontally, right 4 then left 2 gives right 2.
Vertically, up 1 then down 3 gives down 2. The same changes apply to every vertex.
Use the given conditions and show a justified intermediate step.
