Transformations and Tessellations: a week of ready-to-teach maths

Five days of lessons for Year 6 Space. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus regular polygons to cut out and test, squares for designing a tile, a mini-check and every answer.

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 polygon cut-outs for Day 3 and the tile squares for Day 4, one set for each pair, on card if you can.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. For Day 5, open any free dynamic geometry program if you have one; the plan works on paper too.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: a translation slides, a reflection flips and a rotation turns, and none of them changes a shape’s size or angles. Moves can be combined: two reflections make a translation or a turn. A tessellation covers a surface with copies of a shape, no gaps and no overlaps, and the copies are placed by these same moves; regular polygons tessellate on their own only when their corners fit exactly round a point.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Words to watch

The image is where a shape lands after a move. A mirror line is the line of a reflection; a centre of rotation is the point a turn goes round. A quarter turn is 90°, a half turn 180°. Regular means all sides and all angles equal.

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 names the three moves; Day 2 combines them; Day 3 finds the shapes that tessellate; Day 4 designs a new tile that tessellates; and Day 5 builds strips and rosettes from repeated moves. It builds on the Year 5 work of performing translations, reflections and rotations (AC9M5SP03) and on this year’s angles at a point (AC9M6M04). It leads to the Year 7 work on the properties of polygons (AC9M7SP02).

Materials for the week (one trip)

• All week: grid paper, tracing paper, rulers and coloured pencils.
• For Day 3: the polygon cut-outs on card, one set for each pair, and scissors.
• For Day 4: the tile squares, sticky tape and scissors.
• For Day 5: a dynamic geometry program on the board if you have one.

Words for the word wall

transformation, translation, reflection, rotation, image, mirror line, centre of rotation, quarter turn, half turn, tessellation, regular polygon, vertex, pattern, rosette

Dear families

This week in maths, Year 6 combines slides, flips and turns to make tessellations (tilings with no gaps or overlaps) and other patterns.

Try this at home

• Look for tessellations together: floor tiles, brick walls, paving, fabric and wrapping paper.
• For each one, ask: which shape repeats, and is each copy slid, flipped or turned?
• Trace a shape on paper and turn the tracing a quarter turn. Does it still fit the shape?
• Draw a strip pattern for the edge of a card using one move over and over.

Patterns we found

Fill one row a day. Write where you saw a pattern, the shape that repeats, and the moves used.

Printed from the free seegongsik Transformations and Tessellations teaching pack · seegongsik.com/au/y6/space/AC9M6SP03/pack

Slide, flip, turn

A translation slides a shape a set distance in a set direction, a reflection flips it over a mirror line, and a rotation turns it about a centre. None of them changes the shape’s size, side lengths or angles, only where it is and which way it faces. Children name the move between two positions on a grid, describe it exactly, and draw images themselves.

We are learning to

• name the transformation that takes a shape to its image,
• describe a move exactly: how far, which line, which centre and how much turn,
• explain what stays the same under every transformation.

Success criteria

• I can tell a reflection from a half turn on a grid.
• I can describe a translation as so many right or left and so many up or down.

You need

The worksheet, one per child, tracing paper, and a ruler.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with Parts B and C.

Watch for these ideas

• Calling any shape that looks upside down a reflection.
• Counting the gap between the two shapes, not the distance one corner moved, for a translation.
• Thinking a move can make a shape bigger or smaller.

Answers

• Part A: (1) a translation 4 right and 1 up; (2) a reflection in the vertical line halfway between the shapes; (3) a half turn about the dot; (4) a quarter turn anticlockwise about the dot; (5) a reflection in the horizontal line halfway between the shapes; (6) a translation 3 left and 2 down.
• Part B: the side lengths, the angles, the size and the shape stay the same; the position, and for flips and turns the way it faces, change.
• Part C: check with tracing paper; (1) reflection, (2) rotation, (3) translation.

Slide, flip, turn

A. Name the move

The shaded shape moved to the blue shape. Name the move and describe it exactly. A dot marks a centre of turn.

B. What stays the same?

After any slide, flip or turn, what stays the same about the shape? What changes? __________________________

C. Draw the image

Two moves in a row

Transformations can be done one after another, and two moves often equal one. Two slides make one slide; reflecting twice in the same line puts a shape back; reflecting in two parallel lines is a slide of twice the gap between them; and reflecting in two lines at right angles is a half turn about the point where they cross. Children do two moves on a grid and find the single move that does the same.

We are learning to

• carry out two transformations one after the other,
• find a single transformation that does the same as two,
• explain why two reflections can make a translation or a rotation.

Success criteria

• I can draw a shape after two reflections in parallel lines.
• I can name the one move that does the same as two reflections.

You need

The worksheet, one per child, tracing paper, a ruler and two coloured pencils.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the first two tasks in Part A; start Session B with tasks 3 and 4.

Watch for these ideas

• Doing the second move to the start shape instead of to the first image.
• Thinking two reflections always make a rotation.
• Measuring the slide as the gap between the mirror lines instead of twice the gap.

Answers

• Part A and B: (1) a translation 3 right and 2 up; (2) no move: the shape is back where it started; (3) a translation 6 right; (4) a half turn about the point where the lines cross.
• Part C: no. Two reflections in parallel lines make a translation, twice the gap between the lines; in lines that cross they make a rotation about the crossing point, a half turn when the lines are at right angles.

Two moves in a row

A. Do both moves

Draw the first image in one colour and the second image in another.

B. One move instead of two

C. Always true?

Kai says two reflections always make a rotation. Is Kai right? Use tasks 2, 3 and 4 to explain. ________________________

Which shapes tessellate?

A shape tessellates when copies cover a surface with no gaps and no overlaps, and at every corner the angles meeting there add to 360°. A regular polygon tessellates on its own only when its corner angle divides 360° exactly: the triangle (60°), the square (90°) and the hexagon (120°). Any triangle and any quadrilateral tessellate too, using half turns, and two shapes together can fill what one cannot.

We are learning to

• test whether copies of a shape fit round a point,
• explain with angles why only some regular polygons tessellate,
• tessellate any triangle using half turns.

Success criteria

• I can explain why regular hexagons tessellate but regular pentagons do not.
• I can tessellate a triangle by turning copies about the middle of a side.

You need

The worksheet, one per child, the polygon cut-outs on card, scissors, and tracing paper.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the triangle in Part B.

Watch for these ideas

• Thinking any regular polygon will tessellate if you try hard enough.
• Squeezing shapes so they almost fit: a tessellation has no gaps and no overlaps at all.
• Thinking only regular shapes can tessellate.

Answers

• Part A: triangle 60°, 6, yes; square 90°, 4, yes; pentagon 108°, 3.33, not a whole number, no; hexagon 120°, 3, yes; octagon 135°, 2.67, not a whole number, no.
• Part B: the triangle and its half turn make a parallelogram, which slides to fill the page; the quadrilateral tiles the same way, turned about the middle of each side.
• Part C: 135° + 135° + 90° = 360°, so two octagons and a square fill every corner.

Which shapes tessellate?

A. Round a point

B. Any triangle tessellates

C. Two shapes together

Make a tile

A square tessellates, so any change that keeps its edges matching keeps it tessellating. Cut a piece from one side and slide it to the opposite side, and the bump fits the notch of the next tile along. Children make a new tile this way, check that it has the same area as the square, and tessellate it by translation, the way artists design interlocking tiles.

We are learning to

• change a square into a new tile by cutting and sliding pieces,
• explain why the new tile still tessellates,
• tessellate a tile by translation, and describe the moves.

Success criteria

• I can say which transformation moved the piece of my tile.
• I can trace my tile into a tessellation with no gaps or overlaps.

You need

The worksheet, one per child, the tile squares (cut-outs) on card, scissors, sticky tape and coloured pencils.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after making the tile; start Session B with the tracing in Part B.

Watch for these ideas

• Sliding the piece to the next side round instead of the opposite side, so the edges no longer match.
• Turning or flipping the piece as it moves, which needs a different tessellation.
• Leaving gaps between traced tiles, or overlapping them.

Answers

• Part A: (1) a translation 4 right; (2) a translation 4 up; (3) the same, 16 squares, because every piece cut off is added back.
• Part B: tessellations vary; each tile is the last one slid 4 squares across or 4 squares up.
• Part C: (1) a rotation (a quarter turn about the corner); (2) the copies must be turned too, four quarter turns round each corner where they meet.

Make a tile

A. Cut and slide

1. Which move took the first piece across? __________ 2. Which move took the second piece up? __________

3. Is the new tile’s area bigger, smaller or the same as the square’s? Why? ______________________________

B. Tessellate it

C. A turning tile

Ari cut a piece from the bottom of a square and turned it a quarter turn about the bottom-left corner onto the left side.

1. What kind of move is that? ______________ 2. How must the copies be placed to tessellate? __________________

Strips and rosettes

Repeating one move, or a combination of moves, builds a pattern. Along a strip, a motif can be slid, flipped in upright lines, turned half turns, or reflected in the middle line and slid (a glide). Round a point, a motif turned again and again by 360° divided by the number of copies makes a rosette. Children name the moves in strips, continue a strip, and build a rosette on paper or in dynamic geometry software.

We are learning to

• name the transformations used to make a strip pattern,
• continue a pattern by repeating a transformation,
• build a rosette by repeating a turn about a point.

Success criteria

• I can tell a strip made by flips from one made by half turns.
• I can work out the turn needed for a rosette of 6 copies.

You need

The worksheet, one per child, tracing paper, a protractor, and dynamic geometry software on the board if you have it.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the strip in Part B.

Watch for these ideas

• Calling every upside-down copy a reflection, so half turns and glides are mixed up.
• Turning by the number of copies instead of 360° divided by it.
• Turning each copy about its own middle instead of about one shared centre.

Answers

• Part A: A reflections in upright lines; B translations only; C a reflection in the middle line, then a slide (a glide reflection); D half turns about points on the middle line.
• Part B: each copy is the mirror image of the one before, and every second copy is the first slid 8 squares along.
• Part C: 4 copies, 90°; 6 copies, 60°; 8 copies, 45°.

Strips and rosettes

A. Name the moves

The shaded shape is the first copy. Which move, or moves, made each strip?

B. Continue the strip

The shape was reflected in the first dashed line. Reflect each new copy in the next line along.

C. Rosettes

Draw a small shape near the top of a circle on the back, then trace and turn it to make a rosette of 6.

Regular polygons

Print on card, one sheet for each pair. Every shape has the same side length, so any two fit edge to edge.

Teacher note: 7 triangles, 5 squares, 4 pentagons, 4 hexagons, 3 octagons, one more of each than fit round a point, so pairs can see whether the last one fits, leaves a gap or overlaps.

Tile squares

Print on card and cut round the dark edges. Each child takes one square to make a tile; the light lines help cut neat pieces.

How to make a sliding tile

1. Cut a piece out of the left side. Slide it straight across to the right side and tape it on.
2. Cut a piece out of the bottom. Slide it straight up to the top and tape it on.
3. Trace round the tile, slide it so the bump fits the notch, and trace again.

What we know: Transformations and Tessellations

Work on your own. Show your thinking if you can.

Answers and marking guide

Answers

1. A rotation: a half turn about the dot.
2. A reflection in a horizontal line halfway between the shapes.
3. Back where it started.
4. A translation of 8 squares, at right angles to the lines.
5. triangle, square, hexagon.
6. No: 360° ÷ 144° = 2.5, not a whole number, so copies leave a gap or overlap.
7. Half turns about points on the middle line.
8. 72°.

A quick three-level guide

Eight questions, four ideas. A child at standard names and combines the three moves and uses angles to explain which shapes tessellate.

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: naming moves, combining moves, which shapes tessellate, designing a tile, and strips and rosettes.