Transformations: a week of ready-to-teach maths

Five lessons for Year 7 Space. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a grid mat with two shapes to move, twelve mapping cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets at 100%: the grid mat with its two shapes (Days 1 to 4, one per pair) and the mapping cards (Day 5, one set per pair).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A transformation moves every point of a shape by the same rule, and coordinates let us write the rule down. A translation slides: (x, y) becomes (x + a, y + b). A reflection in the y-axis changes the sign of x, and a reflection in the x-axis changes the sign of y. A rotation turns every point through the same angle about a centre; about the origin, a quarter turn anticlockwise sends (x, y) to (−y, x) and a half turn sends it to (−x, −y). None of them changes a shape’s side lengths, angles or area: only where it sits and which way it faces.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write transformations

A translation is two signed numbers, across then up: (+3, −4) means 3 right and 4 down. Turns are anticlockwise unless a sheet says clockwise, and a quarter turn clockwise lands where three quarter turns anticlockwise do. The image of a point P is written P′ (“P dash”). The shapes on cut-out sheet 1 have squares the same size as the mat’s, so students can check an answer by sliding, flipping or turning the shape itself.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 slides shapes and writes the rule; Day 2 reflects them in the axes; Days 3 and 4 turn them, first about the origin and then about any point; and Day 5 chains transformations and names one from a shape and its image. It builds on plotting points in all four quadrants and describing how their coordinates change in Year 6 (AC9M6SP02), on the transformations in tessellations (AC9M6SP03) and on plotting points in AC9M7A05, and it leads into congruent shapes formed by transformations in Year 8 (AC9M8SP01).

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, a sharp pencil to hold a shape while it turns, this pack printed at 100%.
• Cut-out sheet 1, one per pair: the grid mat, a triangle and an L-shape (Days 1 to 4); keep the shapes in an envelope between lessons.
• Cut-out sheet 2, one set per pair: 12 mapping cards (Day 5).
• Tracing or baking paper if you have it, for checking turns (Days 3 and 4).

Words for the word wall

transformation, translation, reflection, rotation, image, mirror line, axis, origin, centre of rotation, quarter turn, half turn, clockwise, anticlockwise, coordinates, quadrant, orientation

Dear families

This week in maths, Year 7 is moving shapes on the Cartesian plane: sliding them (translations), flipping them (reflections) and turning them (rotations), and describing each move exactly with coordinates.

Try this at home

• Look for slides, flips and turns around you: a sliding door, a reflection in a window, a turning tap or wheel.
• On a chessboard or a tiled floor, pick a square and give a move such as “3 right and 2 back”; the other person says where it lands.
• Hold up your right hand to a mirror. Which hand does the image look like? That is what a reflection does to a shape.
• Fold a sheet of paper, cut a shape through both layers at the fold, and open it: the fold is a mirror line.

Moves this week

Fill one row a day with something you saw move, and whether it slid, flipped or turned.

Printed from the free seegongsik Transformations teaching pack · seegongsik.com/au/y7/space/AC9M7SP03/pack

Translations

A translation slides every point of a shape the same distance in the same direction, so two numbers describe it: how far across and how far up or down. Students slide shapes, write the image coordinates, and work back from a shape and its image to the translation.

We are learning to

• translate a shape and write the coordinates of its image,
• describe a translation as across, then up or down, written (+a, +b),
• find the translation that maps a shape onto its image.

Success criteria

• I can translate every corner of a shape by the same amount and draw the image.
• I can find a translation by taking a point’s coordinates away from its image’s.

You need

The worksheet, rulers, and the grid mat and triangle from cut-out sheet 1 (one per pair).

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Adding the first number to y, or reading “down 4” as +4.
• Moving some corners more than others, so the image changes shape. Every corner moves by the same amount.
• Taking the image away from the point: the translation is image minus point, not the other way round.

Answers

• Part A: P′ (1, −3), Q′ (4, −3), R′ (1, −1).
• Part B: (1) (−1, 4); (2) (−5, −3); (3) (−2, −2); (4) (−4, −1); (5) (+4, −6); (6) (−3, 4).
• Part C: (a) (+7, +3); (b) (4, 1) and (2, 4); (c) (−7, −3), since (−7, −3) + (+7, +3) = (0, 0).
• Part D: (a) (−3, +3), the two slides added; (b) no: (0, 0) would move (+3, +4) but (1, 0) would move (+4, +4), and a translation moves every point the same amount.

Translations

A. Slide the triangle

Translate triangle PQR by (+6, −4): 6 right and 4 down. Draw the image P′Q′R′ and fill in the table. Check with your cut-out triangle on the mat.

B. Fill the gaps

C. Work backwards

A triangle has corners (−5, −2), (−3, −2), (−5, 1). A translation moves the corner (−5, −2) to (2, 1).

1. What is the translation?   ____________
2. Where do the other two corners land?   ______________________
3. Which point does this translation move to (0, 0)?   ____________

D. Stretch

1. A shape slides by (+2, −1) and then by (−5, +4). Which single translation does the same?   ____________
2. Can one translation move (0, 0) to (3, 4) and also move (1, 0) to (5, 4)? Explain.

Reflections in the axes

A reflection flips a shape over a mirror line, and every image point is the same distance from the line as its original, on the other side. With an axis as the mirror the rule is exact: in the y-axis x changes sign, and in the x-axis y changes sign.

We are learning to

• reflect a shape in the x-axis or the y-axis and write the image coordinates,
• state which coordinate changes sign for each axis, and why,
• decide from a point and its image which axis, if either, is the mirror.

Success criteria

• I can reflect each corner and explain why its image is the same distance from the axis.
• I can say which axis maps a point onto another, or show that neither does.

You need

The worksheet, rulers, and the grid mat and L-shape from cut-out sheet 1.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Changing the sign of the wrong number: in the y-axis it is x that changes, because the mirror runs up and down.
• Sliding the shape across the axis instead of flipping it, so the image faces the same way as the shape.
• Thinking a point on the mirror moves. It stays where it is.

Answers

• Part A: in the y-axis A′ (−1, 1), B′ (−3, 1), C′ (−2, 4); in the x-axis A′ (1, −1), B′ (3, −1), C′ (2, −4).
• Part B: (1) (5, 2); (2) (6, 3); (3) (0, −4), on the mirror; (4) (−2, 5); (5) (4, 0), on the mirror; (6) (1, 6).
• Part C: (1) the y-axis; (2) the x-axis; (3) neither; (4) the x-axis; (5) the y-axis; (6) neither. Pair 3 is a half turn about the origin; pair 6 swaps the numbers, a reflection in the line y = x.
• Part D: (1) (2, 5), (2, −5), (−2, 5), (−2, −5): 5 above or below the x-axis, 2 left or right of the y-axis; (2) for every corner (x, y) there is a corner (−x, y).

Reflections in the axes

A. Flip the L-shape

Reflect the L-shape in the y-axis, then reflect the original in the x-axis. Draw both images and fill in the table. Check by flipping your cut-out L-shape over each axis.

B. Reflect each point

C. Which axis?

Is the second point the reflection of the first in the x-axis, the y-axis, or neither? If neither, say what happened to the numbers.

D. Stretch

1. A point and its reflection in the x-axis are 10 units apart, and the point is 2 units from the y-axis. Find every point it could be.
2. Draw a shape that is its own image when reflected in the y-axis. What must be true of its corners?

Rotations about the origin

A rotation turns every point through the same angle about a centre, and each point keeps its distance from the centre. About the origin, quarter and half turns follow exact rules, which students find by turning a shape.

We are learning to

• turn a shape a quarter turn or a half turn about the origin,
• find and use the coordinate rule for each turn,
• name a turn about the origin from a point and its image.

Success criteria

• I can turn a shape about the origin and record where its corners land.
• I can match the rules (−y, x), (−x, −y) and (y, −x) to their turns, and use them.

You need

The worksheet, the grid mat and L-shape, and a sharp pencil to hold the corner on the origin.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Swapping the numbers but forgetting the sign: a quarter turn anticlockwise takes (x, y) to (−y, x), not (y, x).
• Turning clockwise when the sheet says anticlockwise, the way the hands of a clock do not go.
• Thinking any point at the same distance must be a quarter or half turn: (1, 3) to (3, 1) is neither.

Answers

• Part A (quarter turn anticlockwise, half turn, quarter turn clockwise): B (−1, 2), (−2, −1), (1, −2); C (−3, 1), (−1, −3), (3, −1); the corner on the origin stays put.
• Part B, in the same order: rules (−y, x), (−x, −y), (y, −x); (5, 2): (−2, 5), (−5, −2), (2, −5); (−3, 4): (−4, −3), (3, −4), (4, 3); (−6, −1): (1, −6), (6, 1), (−1, 6); (1, −4): (4, 1), (−1, 4), (−4, −1).
• Part C: (1) quarter turn anticlockwise; (2) half turn; (3) quarter turn clockwise; (4) none of these: a clockwise turn of less than a quarter; (5) quarter turn anticlockwise; (6) half turn.
• Part D: (1) (2, 4), (−4, 2), (−2, −4), (4, −2), back to the start; three quarter turns anticlockwise match one quarter turn clockwise; (2) only (0, 0).

Rotations about the origin

A. Turn the L-shape

Lay your cut-out L-shape on the drawing, hold its corner on the origin with a pencil point, and turn it. Draw the quarter turn anticlockwise and the half turn, and fill in the table.

B. Find the rules, then use them

From your table, complete each rule. Then use the rules to fill in the rest.

C. Which turn?

Each second point is the first turned about the origin. Is it a quarter turn anticlockwise, a half turn, a quarter turn clockwise, or none of these?

D. Stretch

1. Turn (4, −2) a quarter turn anticlockwise, then turn the image again, and again, and again. Write each point. What do you notice? Which turn lands where three quarter turns anticlockwise do?
2. Which point is its own image under a half turn about the origin?   ____________

Rotations about any point

The centre of a turn need not be the origin. To turn a point about a centre, take the arrow from the centre to the point, turn the arrow, and add it back on to the centre. A half turn’s centre is halfway between each point and its image.

We are learning to

• turn a shape about a given point that is not the origin,
• use the arrow from the centre to find each image point,
• find the centre of a half turn from a point and its image.

Success criteria

• I can turn a shape a quarter or a half turn about a given point and check it with the cut-out.
• I can find a half turn’s centre as the point halfway between a point and its image.

You need

The worksheet, the grid mat and L-shape, and a sharp pencil to hold the centre.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Turning about the origin out of habit when the centre is another point.
• Turning the point’s own coordinates instead of the arrow, or not adding the centre back on.
• Expecting a half turn’s centre to lie on the shape: it is halfway between each point and its image.

Answers

• Part A: K stays at (−2, 2); quarter turn A′ (−2, −1), B′ (−4, 2); half turn A′ (1, 2), B′ (−2, 0).
• Part B (arrow, turned, image): (1) (+3, +2), (−2, +3), (0, 4); (2) (−2, +2), (−2, −2), (−1, 0); (3) (+4, −1), (+1, +4), (0, 3); (4) (−2, −3), (+3, −2), (1, −2).
• Part C: (1) (−1, −2); (2) (2, −1); (3) (1, −1); (4) (−4, 4).
• Part D: (1) (1, 3) to (3, −3) to (7, 3): a translation (+6, +0), twice the gap between the centres; (2) (1, 1): the arrow (+3, −1) turns to (+1, +3), landing on (2, 4).

Rotations about any point

A. Turn about a corner

Lay your cut-out L-shape on the drawing and hold corner K with a pencil point. Turn it a quarter turn anticlockwise, then a half turn from the start. Draw both images and fill in the table.

B. Turn the arrow

Turn each point a quarter turn anticlockwise about its centre: find the arrow from the centre, turn it, and add it back on.

The first row, in grey, is done for you.

C. Half turns

A half turn sends each point to the other side of the centre, the same distance away. Fill the gaps.

D. Stretch

1. Give (1, 3) a half turn about (2, 0), then a half turn about (5, 0). Try two points of your own. Which single transformation does the same?
2. A quarter turn anticlockwise sends (4, 0) to (2, 4). Where is the centre?   ____________

Combining transformations

Students name a transformation from a shape and its image, and test whether order matters. Then the quiz.

We are learning to

• name a transformation in full: its two numbers, its mirror axis, or its centre, turn and direction,
• apply two transformations in a row and test whether the order matters.

Success criteria

• I can name the transformation on a mapping card and check it on one corner.
• I can show with one point that the order of a translation and a reflection matters.

You need

The worksheet, the mapping cards (one set per pair), and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 1; start the second with problem 2.

Watch for these ideas

• Naming a reflection without its axis, or a turn without its direction.
• Mixing up a reflection with a half turn: both move a shape across, but only the reflection flips it.
• Assuming the order never matters. It does for a translation and a reflection, though not for two translations.

Answers with working

1. A ¼ turn anticlockwise; B translation (+5, −5); C ½ turn; D reflection in the x-axis; E ¼ turn clockwise; F reflection in the y-axis; G translation (+5, +0); H ½ turn; I reflection in the y-axis; J ¼ turn clockwise; K reflection in the x-axis; L ¼ turn anticlockwise; every turn is about the origin.
2. (a) (3, 1), then (−3, 1); (b) (−1, 4), then (1, 1): the order matters.
3. (−4, 6) both ways; the one translation (−2, +3) does the same.
4. (5, 2) to (5, −2) to (−5, −2); (−3, 4) to (−3, −4) to (3, −4): a half turn about the origin.
5. a translation (+7, −3), or a quarter turn clockwise about the origin; one point cannot tell them apart.
6. lengths, angles and area stay; the position moves; a rotation turns the facing, and a reflection flips it.
7. a quarter turn anticlockwise about the origin: (1, 0) to (0, 1), (3, 0) to (0, 3), (1, 1) to (−1, 1).
8. (1, 4) and (3, −2); no: a quarter turn sends (4, 1) to (−1, 4), and the reflection flips the shape.

Combining transformations

Give a reason with every answer. Write a translation as (+a, +b), a reflection by its axis, and a turn by its size and direction.

Problems (7 and 8 are the stretch)

1. Name the transformation on each mapping card.

   A	B	C
   D	E	F
   G	H	I
   J	K	L

2. Start at (1, 4). (a) Translate by (+2, −3), then reflect in the y-axis. (b) Now reflect in the y-axis first, then translate by (+2, −3). Does the order matter?
3. Translate (−2, 3) by (+4, +1) and then by (−6, +2). Then do the two translations the other way round. What do you notice?
4. Reflect (5, 2) in the x-axis, then reflect the image in the y-axis. Do the same with (−3, 4). Which single transformation does the same as the two reflections?
5. A transformation sends (−2, 5) to (5, 2). Give two different transformations that could do this. Why is one point not enough to tell which?
6. Which of these stays the same under every translation, reflection and rotation: the side lengths, the angles, the area, the position, the way the shape faces? Explain any that can change.
7. Stretch. The triangle (1, 0), (3, 0), (1, 1) maps onto the triangle (0, 1), (0, 3), (−1, 1). Which single transformation does it? Check every corner.
8. Stretch. A reflection in the line y = x swaps the numbers: (x, y) becomes (y, x). Reflect (4, 1) and (−2, 3) in this line. Is it the same as a quarter turn about the origin? Explain.

Grid mat and two shapes to move

Days 1 to 4, one per pair. Cut out the triangle and the L-shape along their outlines and keep the grid whole. The shapes’ squares match the grid’s, so a shape can be slid, flipped over or turned on the mat to check an answer. Write any corner letters on both sides of a shape before flipping it, and hold a corner down with a pencil point to turn about that corner.

Mapping cards

Day 5, one set per pair. On each card the solid shape is the start and the dashed shape is its image. Name the one transformation that maps the start onto the image.

What we know: Transformations

Work on your own. Turns are anticlockwise unless the question says otherwise. Show your working.

1. Translate (−4, 2) by (+5, −3).   ____________
2. A translation moves (5, −1) to (−2, 3). What is the translation?   ____________
3. Reflect (−3, 5) in the x-axis.   ____________
4. Reflect (4, −2) in the y-axis.   ____________

5. Turn (3, 5) a quarter turn anticlockwise about the origin.   ____________
6. Turn (−1, 3) a half turn about the origin.   ____________
7. Turn (4, 1) a half turn about the point (2, 2).   ____________
8. The corners of triangle ABC run anticlockwise, and the corners of its image A′B′C′ run clockwise. Was the transformation a translation, a reflection or a rotation? Explain.

Answers and marking guide

Answers with working

1. (1, −1): 5 is added to x and 3 taken from y.
2. (−7, +4): the image take away the point, x and y separately.
3. (−3, −5): the x-axis is the mirror, so x stays and y changes sign.
4. (−4, −2): the y-axis is the mirror, so y stays and x changes sign.
5. (−5, 3): a quarter turn anticlockwise sends (x, y) to (−y, x).
6. (1, −3): a half turn changes both signs.
7. (0, 3): the arrow from the centre is (+2, −1); turned half way round it is (−2, +1), and the centre plus that arrow is the image.
8. A reflection: it flips the shape, so the corners run the other way round. A translation or a rotation keeps their order.

A quick three-level guide

Eight questions, four ideas. A student at standard translates, reflects in either axis and turns about the origin using coordinates, turns about another centre with the arrow, and tells the transformations apart by what they do to a shape.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: translations, reflections in the axes, turns about the origin, turns about other centres, and combining.