Level 2 · Mathematics M2.5

Shapes and angles

Draw shapes exactly, and measure the angles between their sides.

Draw straight and parallel lines, name polygons, test right angles with folded paper, measure angles in degrees, sort shapes, find lines of symmetry, fold nets and find your way on a grid.

You are ready for this book if

You can name a triangle, a square and a rectangle and count their corners.

Written and drawn by attnom. Free to read, print and share.

M2.5.1 Chapter 1

Straight lines and parallel lines

How do I draw a line that is exactly 6 cm long?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A straight line does not bend. You draw one along the edge of a ruler.

The length of a line is how long it is. A ruler shows length in centimetres, written cm.

Draw it first

Draw a line 6 cm long

  1. A ruler marked from 0 to 8 with the word cm. Step 1 puts a cross above the 0 mark, labelled start. Step 2 puts a second cross above the 6 mark, labelled 6 cm. Step 3 draws a straight line from one cross to the other.012345678cmstart

    1 Make a small cross where the line starts. Put the 0 mark of the ruler under it.

  2. A ruler marked from 0 to 8 with the word cm. Step 1 puts a cross above the 0 mark, labelled start. Step 2 puts a second cross above the 6 mark, labelled 6 cm. Step 3 draws a straight line from one cross to the other.012345678cmstart6 cm

    2 Find 6 on the ruler. Make a second cross above it.

  3. A ruler marked from 0 to 8 with the word cm. Step 1 puts a cross above the 0 mark, labelled start. Step 2 puts a second cross above the 6 mark, labelled 6 cm. Step 3 draws a straight line from one cross to the other.012345678cmstart6 cm

    3 Hold the ruler still. Draw from cross to cross.

Do not start at the end of the ruler. Many rulers have a small gap before the 0 mark.

A ruler marked from 0 to 8 with the word cm. A cross labelled edge sits at the left end of the ruler, before the 0 mark. A line runs from that cross to the 6 mark.012345678cmedge
This line starts at the edge, not at 0. It is longer than 6 cm.

Look at a door. Its two long sides are parallel. Parallel lines always stay the same distance apart. They never meet.

Two pairs of lines. On the left, two lines slope the same way and are the same distance apart. Two short dashed lines of equal length join them. It is labelled parallel. On the right, two lines get closer. Dashed lines carry them on until they cross at a cross, labelled they meet. It is labelled not parallel.parallelthey meetnot parallel
The left pair stay the same distance apart. The right pair get closer, so they will meet.

Parallel lines can be different lengths. They can point across, up or slanted.

See it three ways

With things

Lay two sticks on the ground. Make the gap the same at both ends.

As a picture

Two straight lines, one above the other, with two dashed lines of equal length between them. Both gaps are labelled same gap.same gapsame gap

In symbols

The gap at one end = the gap at the other end. So the lines are parallel.

Try it

Draw two parallel lines

You need: a sheet of paper, a pencil, a ruler, or a book with straight sides.

  1. Draw a line 6 cm long. Put the 0 mark at the start.
  2. Measure 2 cm straight down from each end. Make a dot each time.
  3. Join the two dots with the ruler.
  4. Measure the gap in the middle. It is 2 cm too.
  5. No ruler? Draw round a book. Its two long sides are parallel.
  6. Now look around. Find parallel lines in the room.

What you will see: The gap is 2 cm at both ends and in the middle. So the lines are parallel.

If you cannot try it, this is what happens.
Two straight lines, one above the other, each 6 cm long. Dashed lines join their ends. Each dashed line is labelled 2 cm. The top line is labelled 6 cm.6 cm2 cm2 cm

A wrong idea many people have

Two lines that do not touch on the page are parallel.

Why it feels right: On a small drawing, two lines can seem to never meet.

The test that breaks it: Draw both lines longer with a ruler. If they get closer, they will meet. Then they are not parallel.

What is true: Parallel lines stay the same distance apart, however long you draw them.

Key words

straight line
a line that does not bend.
length
how long a line is.
parallel
lines that stay the same distance apart and never meet.

Back to your guess

The answer is B: Put the 0 mark on the start of the line. Stop at 6.

Length starts at the 0 mark. Many rulers have a small gap before 0. The end of the ruler is not the start.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. You draw a line 8 cm long. Where do you put the 0 mark of the ruler?

    Tap your answer.

    1. Yes, that is right. Length starts at the 0 mark. Put it on the start of the line and stop at 8.

    2. Not this one. Many rulers have a gap before 0. The end of the ruler is not the start of length. Try another.

    3. Not this one. Then you would count from 1. A line from 1 to 8 is only 7 cm long. Try another.

    See the answer

    On the start of the line

    Length starts at the 0 mark. Put it on the start of the line and stop at 8.

    • If you chose B: Many rulers have a gap before 0. The end of the ruler is not the start of length.
    • If you chose C: Then you would count from 1. A line from 1 to 8 is only 7 cm long.

  2. Mei wants a line 6 cm long. She starts at the 1 mark and stops at the 6 mark. How long is her line, in cm?

    See the answer

    5 cm

    From 1 to 6 is 6 − 1 = 5 cm. She should have started at 0.

  3. Which pair of lines is parallel?

    Three pairs of lines. Pair A has two lines that slope the same way with a gap that stays the same. Pair B has two lines that do not touch but get closer. Pair C has two lines that cross.ABC
    Tap your answer.

    1. Yes, that is right. Both lines in A slope the same way. The gap stays the same.

    2. Not this one. The lines do not touch here, but they get closer. Draw them longer and they meet. Try another.

    3. Not this one. These lines cross, so they meet. Parallel lines never meet. Try another.

    See the answer

    Pair A

    Both lines in A slope the same way. The gap stays the same.

    • If you chose B: The lines do not touch here, but they get closer. Draw them longer and they meet.
    • If you chose C: These lines cross, so they meet. Parallel lines never meet.

  4. How long is the line on the ruler, in cm?

    A ruler marked from 0 to 8 with the word cm. A line runs from a cross above the 2 mark to a cross above the 8 mark.012345678cm
    See the answer

    6 cm

    The line starts at 2, not at 0. From 2 to 8 is 8 − 2 = 6 cm.

  5. Name two pairs of parallel lines you can see around you.

    See the answer

    The long sides of a door. The lines on a page.

    Each pair stays the same distance apart and never meets.

  6. A line starts at the 3 mark on a ruler. It stops at the 9 mark. How long is it, in cm?

    A ruler marked from 0 to 9 with the word cm. A line runs from a cross above the 3 mark to a cross above the 9 mark.0123456789cm
    Tap your answer.

    1. Not this one. You read the end mark only. The line starts at 3. The first 3 cm of the ruler are not in it. Try another.

    2. Not this one. You counted the marks 3, 4, 5, 6, 7, 8, 9. Count the gaps between marks instead. Each gap is 1 cm. Try another.

    3. Yes, that is right. From 3 to 9 is 9 − 3 = 6 gaps of 1 cm. The line is 6 cm long.

    See the answer

    6 cm

    From 3 to 9 is 9 − 3 = 6 gaps of 1 cm. The line is 6 cm long.

    • If you chose A: You read the end mark only. The line starts at 3. The first 3 cm of the ruler are not in it.
    • If you chose B: You counted the marks 3, 4, 5, 6, 7, 8, 9. Count the gaps between marks instead. Each gap is 1 cm.

  7. Look at the picture. Are line A and line B parallel?

    Two straight lines, line A above line B. Two dashed lines join them, one at each end. The dashed line at the left end is labelled 4 cm. The dashed line at the right end is labelled 2 cm. The lines are labelled line A and line B.line Aline B4 cm2 cm
    Tap your answer.

    1. Not this one. Almost is not enough. The gap is 4 cm at one end. It is 2 cm at the other. Try another.

    2. Yes, that is right. The gap is 4 cm at the left. It is 2 cm at the right. The lines get closer, so they will meet.

    3. Not this one. Parallel lines can have different lengths. What counts is the gap. Here the gap changes. Try another.

    See the answer

    No. The gap is not the same at both ends.

    The gap is 4 cm at the left. It is 2 cm at the right. The lines get closer, so they will meet.

    • If you chose A: Almost is not enough. The gap is 4 cm at one end. It is 2 cm at the other.
    • If you chose C: Parallel lines can have different lengths. What counts is the gap. Here the gap changes.

  8. Amina draws two lines that do not touch. She makes both longer and they cross. What does that show?

    Tap your answer.

    1. Yes, that is right. Parallel lines stay the same distance apart and never meet. Lines that cross were never parallel.

    2. Not this one. On a small page, lines can seem never to meet. Parallel lines never cross, however long you draw them. Try another.

    3. Not this one. Parallel lines never cross, however long they run. Two lines that cross were not parallel. Try another.

    See the answer

    The lines were not parallel.

    Parallel lines stay the same distance apart and never meet. Lines that cross were never parallel.

    • If you chose B: On a small page, lines can seem never to meet. Parallel lines never cross, however long you draw them.
    • If you chose C: Parallel lines never cross, however long they run. Two lines that cross were not parallel.

Keep it

Copy this into your exercise book.

Put the 0 mark on the start of the line. Parallel lines stay the same distance apart and never meet.

Draw: A line 6 cm long. Draw a parallel line 2 cm below it. Mark the gap.

M2.5.2 Chapter 2

Polygons

Is a circle a polygon?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A polygon is a flat, closed shape with straight sides. Closed means there is no gap.

A circle is not a polygon. Its edge is curved.

Draw it first

Join straight sides to close a shape

  1. A row of polygons. Step 1 shows a triangle labelled 3 sides. Step 2 adds a shape labelled 4 sides. Step 3 adds a shape labelled 5 sides. Step 4 adds a shape labelled 6 sides.3 sides

    1 Join 3 straight lines so they close. You make a shape with 3 sides.

  2. A row of polygons. Step 1 shows a triangle labelled 3 sides. Step 2 adds a shape labelled 4 sides. Step 3 adds a shape labelled 5 sides. Step 4 adds a shape labelled 6 sides.3 sides4 sides

    2 Use 4 lines. Now the shape has 4 sides.

  3. A row of polygons. Step 1 shows a triangle labelled 3 sides. Step 2 adds a shape labelled 4 sides. Step 3 adds a shape labelled 5 sides. Step 4 adds a shape labelled 6 sides.3 sides4 sides5 sides

    3 Use 5 lines. The shape has 5 sides.

  4. A row of polygons. Step 1 shows a triangle labelled 3 sides. Step 2 adds a shape labelled 4 sides. Step 3 adds a shape labelled 5 sides. Step 4 adds a shape labelled 6 sides.3 sides4 sides5 sides6 sides

    4 Use 6 lines. The shape has 6 sides.

A shape with 4 sides is a quadrilateral. Every polygon is named by its sides.

Six polygons in two rows. The first row shows a triangle with 3 sides, a quadrilateral with 4 sides and a pentagon with 5 sides. The second row shows a hexagon with 6 sides, a heptagon with 7 sides and an octagon with 8 sides.triangle3 sidesquadrilateral4 sidespentagon5 sideshexagon6 sidesheptagon7 sidesoctagon8 sides
The names of polygons with 3 to 8 sides.

Count the corners too. A polygon has as many corners as sides.

See it three ways

With things

Lay 5 sticks end to end to make a closed shape. Count the sticks. Count the corners.

As a picture

A pentagon with a dot on each of its 5 corners. The label says 5 sides, 5 corners.5 sides, 5 corners

In symbols

Sides = corners. 5 sides, 5 corners.

Some polygons are regular. All their sides are equal. All their corners are the same size.

Two hexagons with a dot on each corner. The left one has six equal sides and six corners of the same size. It is labelled regular. The right one has sides of different lengths. It is labelled not regular.regularnot regular
Both shapes are hexagons. Only one is regular.

Equal sides are not enough. The corners must match too. Small ticks mark sides of equal length.

Three four sided shapes with small ticks on equal sides. A square has four equal sides. A rhombus has four equal sides but leans, so its corners are not all the same size. A rectangle has corners that are all the same size, but its sides are not all equal.squarerhombusrectangle
Only the square is regular. A square turned on its corner is still a square.

Try it

Make shapes from sticks

You need: 4 sticks of the same length, 2 more sticks, or strips of paper.

  1. Lay 4 sticks of the same length to make a square.
  2. Count the sides. Count the corners.
  3. Push one corner sideways so the shape leans.
  4. Look at the sides. They are all still equal.
  5. Look at the corners. They are no longer the same.
  6. Now use 6 sticks. Make a closed shape and name it.

What you will see: The sides stay equal, but the corners change. Equal sides alone do not make a regular shape.

If you cannot try it, this is what happens.
Two shapes made of four equal sticks. The left one is a square, labelled before. The right one has been pushed so it leans, labelled after. Ticks show that all four sides are equal in both.beforeafter

A wrong idea many people have

If all the sides are equal, the shape is regular.

Why it feels right: Equal sides look fair and even, so the shape seems regular.

The test that breaks it: Make a square with 4 equal sticks. Push it so it leans. The sides stay equal. The corners do not.

What is true: Regular needs equal sides and equal corners. Both must be true.

Key words

polygon
a flat, closed shape with straight sides.
quadrilateral
a polygon with 4 sides.
regular
all sides equal and all corners the same size.

Back to your guess

The answer is B: No. A polygon has only straight sides.

A polygon has straight sides only. A circle has a curved edge, so it is not a polygon.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A shape has 6 straight sides. What is it called?

    Tap your answer.

    1. Not this one. A pentagon has 5 sides. You may have missed one side when you counted. Try another.

    2. Yes, that is right. A hexagon has 6 sides. A pentagon has 5. An octagon has 8.

    3. Not this one. An octagon has 8 sides. You may have mixed up two names. Try another.

    See the answer

    Hexagon

    A hexagon has 6 sides. A pentagon has 5. An octagon has 8.

    • If you chose A: A pentagon has 5 sides. You may have missed one side when you counted.
    • If you chose C: An octagon has 8 sides. You may have mixed up two names.

  2. How many sides does this polygon have? What is it called?

    A polygon with a dot on each of its 7 corners.
    See the answer

    7 sides, a heptagon

    Count the dots. There are 7 corners, so there are 7 sides. A polygon with 7 sides is a heptagon.

  3. Which of these is not a polygon?

    Tap your answer.

    1. Not this one. A triangle is closed with 3 straight sides. It is a polygon. Try another.

    2. Yes, that is right. A circle has a curved edge. A polygon has only straight sides.

    3. Not this one. A pentagon is closed with 5 straight sides. It is a polygon. Try another.

    See the answer

    A circle

    A circle has a curved edge. A polygon has only straight sides.

    • If you chose A: A triangle is closed with 3 straight sides. It is a polygon.
    • If you chose C: A pentagon is closed with 5 straight sides. It is a polygon.

  4. Which shape is a regular polygon?

    Three four sided shapes with ticks on equal sides. Shape A is a square. Shape B is a rhombus that leans. Shape C is a rectangle that is longer than it is tall.ABC
    Tap your answer.

    1. Yes, that is right. A has equal sides and equal corners. B has only equal sides. C has only equal corners.

    2. Not this one. Its sides are equal. Its corners are not all the same size. Try another.

    3. Not this one. Its corners match. Its sides are not all equal. Try another.

    See the answer

    Shape A

    A has equal sides and equal corners. B has only equal sides. C has only equal corners.

    • If you chose B: Its sides are equal. Its corners are not all the same size.
    • If you chose C: Its corners match. Its sides are not all equal.

  5. A regular octagon has 8 equal sides. Each side is 3 cm long. How far is it all the way round, in cm?

    See the answer

    24 cm

    8 sides of 3 cm each. 8 × 3 = 24 cm.

  6. From an earlier chapter

    A line runs from the 2 mark to the 9 mark. How long is it, in cm?

    A ruler marked from 0 to 9 with the word cm. A line runs from a cross above the 2 mark to a cross above the 9 mark.0123456789cm
    See the answer

    7 cm

    The line starts at 2, not at 0. From 2 to 9 is 9 − 2 = 7 cm.

  7. Which shape is a polygon?

    Three shapes. Shape A has straight sides, but there is a small gap between two of its ends. Shape B is a circle. Shape C is a pentagon with 5 straight sides.ABC
    Tap your answer.

    1. Not this one. Its sides are straight, but the shape has a gap. A polygon must be closed. Try another.

    2. Not this one. A circle is closed, but its edge is curved. A polygon has straight sides only. Try another.

    3. Yes, that is right. C is closed and has only straight sides. A has a gap, and B has a curved edge.

    See the answer

    Shape C

    C is closed and has only straight sides. A has a gap, and B has a curved edge.

    • If you chose A: Its sides are straight, but the shape has a gap. A polygon must be closed.
    • If you chose B: A circle is closed, but its edge is curved. A polygon has straight sides only.

  8. Ravi turns a square so that it stands on a corner. Is it still a square?

    A square turned so that it stands on one corner, labelled Ravi's shape. Small ticks mark that all four sides are equal. Small squares mark four right angles.Ravi's shape
    Tap your answer.

    1. Not this one. Turning moves the whole shape. Each side keeps its length. Each corner keeps its size. Try another.

    2. Yes, that is right. Turning does not change the sides or the corners. A square on its corner is still a square.

    3. Not this one. A square is named by 4 equal sides and 4 equal corners. How it sits does not matter. Try another.

    See the answer

    Yes. Turning a shape does not change its sides or corners.

    Turning does not change the sides or the corners. A square on its corner is still a square.

    • If you chose A: Turning moves the whole shape. Each side keeps its length. Each corner keeps its size.
    • If you chose C: A square is named by 4 equal sides and 4 equal corners. How it sits does not matter.

Keep it

Copy this into your exercise book.

A polygon is a flat, closed shape with straight sides. It has as many corners as sides. Regular means equal sides and equal corners.

Draw: A hexagon. Mark each corner with a dot and count them.

M2.5.3 Chapter 3

Right angles and turns

How can I tell if a corner is a right angle?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

An angle is how much one line turns away from another. Think of a door. The wider it opens, the bigger the angle.

A right angle is a square corner, like a book corner.

Draw it first

Fold a right angle tester

  1. A piece of paper. Step 1 shows the paper with a dashed line across it, labelled first fold. Step 2 shows the folded paper with a dashed line going up from the folded edge, labelled second fold. Step 3 shows the corner you made, marked with a small square, labelled right angle.first fold

    1 Fold any piece of paper once. The fold is a straight edge.

  2. A piece of paper. Step 1 shows the paper with a dashed line across it, labelled first fold. Step 2 shows the folded paper with a dashed line going up from the folded edge, labelled second fold. Step 3 shows the corner you made, marked with a small square, labelled right angle.second fold

    2 Fold it again. Make the folded edge lie exactly along itself.

  3. A piece of paper. Step 1 shows the paper with a dashed line across it, labelled first fold. Step 2 shows the folded paper with a dashed line going up from the folded edge, labelled second fold. Step 3 shows the corner you made, marked with a small square, labelled right angle.right angle

    3 Press it flat. The corner you made is a right angle.

This is your right angle tester. It works on any piece of paper.

Lay the tester in a corner. Is the corner smaller than the tester, equal to it, or bigger?

A corner smaller than a right angle is acute. A corner bigger than a right angle is obtuse.

Three angles. The first is smaller than a right angle, labelled acute. The second has a small square in it, labelled right angle. The third is bigger than a right angle, labelled obtuse.acuteright angleobtuse
Lay your tester on each corner. Acute is smaller. Obtuse is bigger.

Turns are angles too. A quarter turn is one right angle. A half turn is two. A full turn is four.

Two straight lines cross. They make four right angles, numbered 1, 2, 3 and 4. Angle 1 is marked with a small square.1234
All four angles are right angles. Four fill a full turn. Two fill a half turn.

See it three ways

With things

Face a wall. Turn a quarter turn to the right. You have turned one right angle.

As a picture

A right angle marked with a small square. The label says quarter turn.quarter turn

In symbols

2 quarter turns = 1 half turn. 4 quarter turns = 1 full turn.

Two lines that meet at a right angle are perpendicular.

Two pairs of crossing lines. The first pair meets at a right angle, marked with a small square, and is labelled perpendicular. The second pair crosses at a slant, with no square, and is labelled not perpendicular.perpendicularnot perpendicular
Perpendicular lines make a right angle. Check with your tester.

Try it

Make a tester and use it

You need: any piece of paper, things with corners, such as a book, a door and a window.

  1. Fold the paper once.
  2. Fold it again. Make the first fold lie along itself.
  3. Press the folds flat. The corner is your tester.
  4. Lay the tester in the corner of a book. Does it fit?
  5. Try a door, a window and a table. Say smaller, equal or bigger.

What you will see: The tester fits the corners of a book and a door. It does not fit a corner that is wider or narrower.

If you cannot try it, this is what happens.
The corner of a book, drawn as two lines that meet at a right angle. A folded paper tester, labelled tester, sits in the corner and fits exactly.tester
Two right angles. The left one has short lines, labelled short arms. The right one has long lines, labelled long arms. Both have a small square.short armslong arms
Both angles are right angles. Your tester fits both.

A wrong idea many people have

An angle with longer lines is a bigger angle.

Why it feels right: A picture with long lines looks bigger on the page.

The test that breaks it: Draw a right angle with short lines. Draw another with long lines. Lay your tester on both. It fits both.

What is true: The size of an angle is how much it turns. The length of the lines does not matter.

Key words

right angle
a square corner. It is a quarter turn.
acute
smaller than a right angle.
obtuse
bigger than a right angle, but smaller than a straight line.
perpendicular
lines that meet at a right angle.

Back to your guess

The answer is B: Fold paper twice to make a tester. Lay it in the corner.

A tester checks every corner the same way. Your eyes can be fooled. A ruler measures length, not the corner.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. How do you make a right angle tester?

    Tap your answer.

    1. Yes, that is right. The second fold makes the folded edge lie along itself. That makes a square corner.

    2. Not this one. One fold makes a straight edge. It does not make a corner. Try another.

    3. Not this one. A torn corner can be any size. It may not be a right angle. Try another.

    See the answer

    Fold paper twice. Make the first fold lie along itself.

    The second fold makes the folded edge lie along itself. That makes a square corner.

    • If you chose B: One fold makes a straight edge. It does not make a corner.
    • If you chose C: A torn corner can be any size. It may not be a right angle.

  2. Compare this angle with a right angle. Is it smaller, equal or bigger?

    An angle with the letter a inside it. It is wider than a right angle.a
    Tap your answer.

    1. Not this one. Lay a tester on it. The angle is wider than the tester, so it is bigger. Try another.

    2. Not this one. A right angle is a square corner. This angle opens wider than that. Try another.

    3. Yes, that is right. The angle opens wider than a right angle. It is an obtuse angle.

    See the answer

    Bigger

    The angle opens wider than a right angle. It is an obtuse angle.

    • If you chose A: Lay a tester on it. The angle is wider than the tester, so it is bigger.
    • If you chose B: A right angle is a square corner. This angle opens wider than that.

  3. Ravi draws a right angle with lines 5 cm long. Mei draws one with lines 10 cm long. Whose angle is bigger?

    Tap your answer.

    1. Not this one. Long lines do not make a bigger angle. A tester fits both. Try another.

    2. Not this one. Both angles turn a quarter turn. The length of the lines does not change that. Try another.

    3. Yes, that is right. Both are right angles. The size of an angle is how much it turns.

    See the answer

    They are the same size

    Both are right angles. The size of an angle is how much it turns.

    • If you chose A: Long lines do not make a bigger angle. A tester fits both.
    • If you chose B: Both angles turn a quarter turn. The length of the lines does not change that.

  4. Mei turns a half turn. Then she turns a quarter turn. How many right angles has she turned in all?

    See the answer

    3 right angles

    A half turn is 2 right angles. A quarter turn is 1. 2 + 1 = 3.

  5. From an earlier chapter

    Which of these is a polygon?

    Tap your answer.

    1. Not this one. A circle has a curved edge. A polygon has only straight sides. Go to M2.5.2. Try another.

    2. Yes, that is right. A hexagon is flat and closed, and it has 6 straight sides.

    3. Not this one. A polygon must be closed. It has no gaps. Go to M2.5.2. Try another.

    See the answer

    A hexagon

    A hexagon is flat and closed, and it has 6 straight sides.

    • If you chose A: A circle has a curved edge. A polygon has only straight sides. Go to M2.5.2.
    • If you chose C: A polygon must be closed. It has no gaps. Go to M2.5.2.

  6. Find two pairs of perpendicular lines in your room. Check them with your tester.

    See the answer

    The two sides of a book corner. The edge of a door and the floor.

    In each pair the lines meet at a right angle. Your tester fits the corner.

  7. What is true of two perpendicular lines?

    Tap your answer.

    1. Yes, that is right. Perpendicular lines meet at a right angle. It does not matter which way the page is turned.

    2. Not this one. That describes parallel lines. Perpendicular lines meet at a right angle. Go to M2.5.1. Try another.

    3. Not this one. Turn the page and both lines turn. The corner is still a right angle, so they are still perpendicular. Try another.

    See the answer

    They meet at a right angle.

    Perpendicular lines meet at a right angle. It does not matter which way the page is turned.

    • If you chose B: That describes parallel lines. Perpendicular lines meet at a right angle. Go to M2.5.1.
    • If you chose C: Turn the page and both lines turn. The corner is still a right angle, so they are still perpendicular.

  8. Hana lays her tester in a corner. It fits inside, with space left beside it. Is the corner smaller, equal or bigger?

    Two lines meet at a corner. The corner is labelled corner. A square tester, labelled tester, sits inside it with its own corner on the corner. There is a gap between the tester and the upper line.cornertester
    Tap your answer.

    1. Not this one. The tester fits inside with room to spare. So the corner opens wider than the tester. Try another.

    2. Yes, that is right. The corner opens wider than the tester. A corner wider than a right angle is obtuse.

    3. Not this one. If it were equal, the tester would touch both lines. Here it does not. Try another.

    See the answer

    Bigger than a right angle

    The corner opens wider than the tester. A corner wider than a right angle is obtuse.

    • If you chose A: The tester fits inside with room to spare. So the corner opens wider than the tester.
    • If you chose C: If it were equal, the tester would touch both lines. Here it does not.

Keep it

Copy this into your exercise book.

Fold paper twice to make a right angle tester. Acute is smaller than a right angle. Obtuse is bigger. The size of an angle is how much it turns.

Draw: A right angle with a small square. Beside it, an acute angle and an obtuse angle.

M2.5.4 Chapter 4

Measuring angles in degrees

How can I say how big an angle is, with a number?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

We measure angles in degrees. The sign for degrees is °.

A right angle is 90°. A straight line is two right angles, so it is 180°.

A full turn goes all the way round. It is four right angles, so it is 360°.

A protractor measures angles. Most are a half circle with two scales of numbers.

Draw it first

Measure an angle with a protractor

  1. An angle made of two green lines that meet at a corner. Step 1 labels the corner. Step 2 puts a protractor on it, with its centre on the corner and its 0 line along the right hand line. The outer scale reads 0, 30, 60, 90, 120, 150 and 180 from the right. The inner scale reads 180, 150, 120, 90, 60, 30 and 0. Step 3 labels the angle 60° and marks the outer scale from 0 to the second line.corner

    1 Find the corner of the angle. Its two arms are the lines.

  2. An angle made of two green lines that meet at a corner. Step 1 labels the corner. Step 2 puts a protractor on it, with its centre on the corner and its 0 line along the right hand line. The outer scale reads 0, 30, 60, 90, 120, 150 and 180 from the right. The inner scale reads 180, 150, 120, 90, 60, 30 and 0. Step 3 labels the angle 60° and marks the outer scale from 0 to the second line.corner01803015060120909012060150301800

    2 Put the centre of the protractor on the corner. Put its 0 line along one arm.

  3. An angle made of two green lines that meet at a corner. Step 1 labels the corner. Step 2 puts a protractor on it, with its centre on the corner and its 0 line along the right hand line. The outer scale reads 0, 30, 60, 90, 120, 150 and 180 from the right. The inner scale reads 180, 150, 120, 90, 60, 30 and 0. Step 3 labels the angle 60° and marks the outer scale from 0 to the second line.corner0180301506012090901206015030180060°

    3 Read the scale that starts at 0 on that arm. Here it says 60°.

There are two scales. Read the one that starts at 0 on your arm.

Check your number. Ask: is the angle smaller than a right angle? If it is, the number must be less than 90.

Worked example

Draw an angle of 60°.

  1. Draw a line with a ruler. Mark a dot on it for the corner.
  2. Put the centre of the protractor on the dot. Put its 0 line along your line.
  3. Find 60 on the scale that starts at 0 on your line. Mark a dot there.
  4. Take the protractor away. Join the corner to the new dot.
  5. Check: the angle should be smaller than a right angle.

Answer: An angle of 60°. It is smaller than a right angle.

The angle you drew. Two green lines meet at a corner, labelled corner. An arc between them is labelled 60°.corner60°
The angle after step 4. It is smaller than a right angle.

See it three ways

With things

Face a wall. Turn right by a quarter turn, four times. You are back where you began.

As a picture

Three turns. A quarter turn is a right angle, marked with a small square, labelled 90°. A half turn goes from one side of a straight line to the other, labelled 180°. A full turn goes all the way round a circle, labelled 360°.90°180°360°

In symbols

90° + 90° = 180°. 4 × 90° = 360°.

Try it

Make angles with no protractor

You need: your right angle tester, 3 sticks or strips of paper of the same length, a pencil and paper.

  1. Fold your tester so one arm lies on the other.
  2. The new corner is 45°. It is half of 90°.
  3. Lay it twice in a right angle. Two fit exactly.
  4. Lay 3 equal sticks end to end to make a triangle.
  5. Each corner of this triangle is 60°. Draw round one corner.
  6. Is 60° bigger or smaller than 45°? Is it smaller than 90°?

What you will see: Two 45° corners fill a right angle. The 60° corner is bigger than 45° and smaller than 90°.

If you cannot try it, this is what happens.
Two angles. The first is 45°, labelled folded tester. The second is 60°, labelled equal sticks.45°folded tester60°equal sticks

A wrong idea many people have

You can read an angle from either scale on a protractor.

Why it feels right: Both scales have the same numbers, so either one looks right.

The test that breaks it: Look at the 60° angle in the picture. It is smaller than a right angle. So its number must be less than 90.

What is true: Read the scale that starts at 0 on your arm. Then check it against a right angle.

Key words

degrees
the unit we use to measure an angle. It is written °.
full turn
all the way round. It is 360°.
protractor
a tool with a scale for measuring angles.

Back to your guess

The answer is B: Count how many degrees it turns, with a protractor.

We measure angles in degrees. A protractor shows how many degrees an angle turns.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. How many degrees are in 2 right angles?

    See the answer

    180°

    A right angle is 90°. 2 × 90° = 180°. That is a straight line.

  2. The 0 line of the protractor lies along the right hand arm. What is the angle?

    A protractor with its centre on the corner of an angle and its 0 line along the right hand green line. The other green line crosses the outer scale one small mark after 30.01803015060120909012060150301800
    Tap your answer.

    1. Yes, that is right. The scale starts at 0 on the right hand arm. The other arm is one small mark after 30. So it is 40°.

    2. Not this one. You read the other scale. This angle is smaller than a right angle. It must be less than 90°. Try another.

    3. Not this one. You counted the 30 mark as a step. Start at 30. The arm is one small mark on, so add 10 once: 40°. Try another.

    See the answer

    40°

    The scale starts at 0 on the right hand arm. The other arm is one small mark after 30. So it is 40°.

    • If you chose B: You read the other scale. This angle is smaller than a right angle. It must be less than 90°.
    • If you chose C: You counted the 30 mark as a step. Start at 30. The arm is one small mark on, so add 10 once: 40°.

  3. Joseph turns a quarter turn. Then he turns a half turn. How many degrees has he turned in all?

    See the answer

    270°

    A quarter turn is 90°. A half turn is 180°. 90° + 180° = 270°.

  4. You fold a right angle in half to make a 45° corner. How many of these corners fill one right angle?

    See the answer

    2

    90° ÷ 45° = 2. Two 45° corners fill a right angle.

  5. Draw an angle of 60° with a protractor. Is it acute or obtuse?

    See the answer

    It is acute. 60° is smaller than a right angle.

    An acute angle is smaller than 90°. 60° is less than 90°.

  6. From an earlier chapter

    An angle is 120°. Is it acute, a right angle or obtuse?

    Tap your answer.

    1. Not this one. Acute angles are smaller than 90°. 120° is more than 90°. Go to M2.5.3. Try another.

    2. Not this one. A right angle is exactly 90°. 120° is bigger. Go to M2.5.3. Try another.

    3. Yes, that is right. An obtuse angle is bigger than 90° and smaller than 180°. 120° is in between.

    See the answer

    Obtuse

    An obtuse angle is bigger than 90° and smaller than 180°. 120° is in between.

    • If you chose A: Acute angles are smaller than 90°. 120° is more than 90°. Go to M2.5.3.
    • If you chose B: A right angle is exactly 90°. 120° is bigger. Go to M2.5.3.

  7. Ana turns all the way round, to face the way she began. How many degrees has she turned?

    Tap your answer.

    1. Not this one. That is a half turn. After 180° she would face the opposite way. Try another.

    2. Not this one. That is a quarter turn, one right angle. All the way round is four of them. Try another.

    3. Yes, that is right. All the way round is four right angles. 4 × 90° = 360°.

    See the answer

    360°

    All the way round is four right angles. 4 × 90° = 360°.

    • If you chose A: That is a half turn. After 180° she would face the opposite way.
    • If you chose B: That is a quarter turn, one right angle. All the way round is four of them.

  8. The 0 line of the protractor lies along the right hand arm. What is the angle?

    A protractor with its centre on the corner of an angle and its 0 line along the right hand green line. The other green line crosses the outer scale one small mark after 120.01803015060120909012060150301800
    Tap your answer.

    1. Not this one. You read the other scale. This angle is bigger than a right angle. It must be more than 90°. Try another.

    2. Yes, that is right. The scale starts at 0 on the right hand arm. The other arm is one small mark after 120. So it is 130°.

    3. Not this one. You stopped at the last number you could see. The arm is one small mark after it. Try another.

    See the answer

    130°

    The scale starts at 0 on the right hand arm. The other arm is one small mark after 120. So it is 130°.

    • If you chose A: You read the other scale. This angle is bigger than a right angle. It must be more than 90°.
    • If you chose C: You stopped at the last number you could see. The arm is one small mark after it.

  9. Ravi draws a line and a corner dot. The protractor centre is on the dot. Its 0 is on his line. Which mark does he join to the dot to make 60°?

    A protractor with its centre on a corner. A green line, labelled his line, runs from the corner along the 0 line to the right. The corner is labelled corner. The outer scale reads 0, 30, 60, 90, 120, 150 and 180 from the right. The inner scale reads 180, 150, 120, 90, 60, 30 and 0.01803015060120909012060150301800cornerhis line
    Tap your answer.

    1. Yes, that is right. His line lies along 0 on the outer scale. It counts up from 0 on his line. So he joins its 60.

    2. Not this one. That scale starts at 0 on the left. From his line it is 120°, not 60°. Try another.

    3. Not this one. You counted 60° down from the 90° mark. Count 60° up from 0 on his line. Try another.

    See the answer

    The 60 on the outer scale

    His line lies along 0 on the outer scale. It counts up from 0 on his line. So he joins its 60.

    • If you chose B: That scale starts at 0 on the left. From his line it is 120°, not 60°.
    • If you chose C: You counted 60° down from the 90° mark. Count 60° up from 0 on his line.

  10. Look at the two angles. Which one is bigger?

    Two angles side by side. Angle P has short arms and opens wide. Angle Q has long arms and opens narrow. They are labelled angle P and angle Q.angle Pangle Q
    Tap your answer.

    1. Not this one. Long arms do not make a bigger angle. What counts is how far the angle opens. Try another.

    2. Yes, that is right. Angle P opens wider than angle Q. Angle size is how far an angle turns. Arm length does not matter.

    3. Not this one. Arm length does not change an angle, but these two open differently. One opens wider. Try another.

    See the answer

    Angle P

    Angle P opens wider than angle Q. Angle size is how far an angle turns. Arm length does not matter.

    • If you chose A: Long arms do not make a bigger angle. What counts is how far the angle opens.
    • If you chose C: Arm length does not change an angle, but these two open differently. One opens wider.

Keep it

Copy this into your exercise book.

A right angle is 90°. A straight line is 180°. A full turn is 360°. Put the protractor centre on the corner. Put its 0 line on one arm. Read the scale from 0.

Draw: A protractor with a 60° angle. Mark where you read the number.

M2.5.5 Chapter 5

Triangles and four sided shapes

Is a square also a rectangle?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

You know shapes with 4 straight sides. Some of them have special names.

The name comes from the sides and the corners. Look for three things: equal sides, right angles and parallel sides.

Draw it first

Five shapes with four sides

  1. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.square

    1 A square has 4 equal sides and 4 right angles. Ticks and small squares show this.

  2. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.squarerectangle

    2 A rectangle has 4 right angles. Its opposite sides are equal.

  3. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.squarerectanglerhombus

    3 Push the square sideways. You get a rhombus. It has 4 equal sides.

  4. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.squarerectanglerhombusparallelogramtrapezium

    4 A parallelogram has two pairs of parallel sides. A trapezium has one pair.

A key to the marks. Equal sides: two lines, each with one tick. Right angle: two lines meeting at a small square. Parallel sides: two lines, each with an arrow head.equal sidesright angleparallel sides
Use these marks to read any shape in this chapter.

In a rhombus, the corners opposite each other are equal. In a parallelogram, the opposite sides are equal.

In some countries a trapezium is called a trapezoid. There, the word trapezium can mean a shape with no parallel sides. Ask which one your book uses.

Now test a square. It has 4 equal sides, so it is a rhombus. It has 4 right angles, so it is a rectangle.

Triangles have 3 sides. Count the sides that are equal.

Three triangles. The first is equilateral, with 3 equal sides. The second is isosceles, with 2 equal sides. The third is scalene, with no equal sides. Ticks mark the sides. Sides with the same number of ticks are equal, and the scalene sides have 1, 2 and 3 ticks.equilateral3 equal sidesisosceles2 equal sidesscaleneno equal sides
Matching ticks mark equal sides. Each corner of the equilateral triangle is 60°.

Look at the ticks. Equal sides have the same number of ticks.

Equilateral means 3 equal sides. Isosceles means 2 equal sides. Scalene means no equal sides.

See it three ways

With things

Lay 4 equal sticks end to end in a loop. Make a square. Then push it sideways.

As a picture

A square and a rhombus side by side, each with four ticks showing four equal sides. The square has small squares at its corners. The labels are square and rhombus.squarerhombus

In symbols

Square: 4 equal sides and 4 right angles. This rhombus: 4 equal sides and no right angles.

Try it

Push a square into a rhombus

You need: 4 sticks of the same length, such as pencils or straws, your right angle tester from chapter 3.

  1. Lay the sticks in a loop so the ends touch. Make a square.
  2. Check each corner with your tester. All four fit.
  3. Push the loop sideways. Keep every end touching.
  4. Check the corners again. Two are smaller than your tester. Two are bigger.

What you will see: The sticks did not change, so the 4 sides stay equal. Only the corners changed. The square became a rhombus.

If you cannot try it, this is what happens.
A square and a rhombus. The tester fits the corner of the square. At the corner of the rhombus a dashed line shows a gap. The labels are tester fits and gap shows.tester fitsgap shows

A wrong idea many people have

A square is not a rectangle.

Why it feels right: Each shape has one name. So people think it is only one thing.

The test that breaks it: Check the rules of a rectangle: 4 right angles, equal opposite sides. Test a square with your tester. All four corners fit. Its opposite sides match.

What is true: A square follows every rule of a rectangle. It is a special rectangle. It is also a rhombus.

Key words

rhombus
a shape with 4 equal sides.
parallelogram
a shape with two pairs of parallel sides.
trapezium
a shape with one pair of parallel sides.

Back to your guess

The answer is B: Yes. A square has everything a rectangle needs.

A rectangle has 4 right angles and equal opposite sides. A square has both, so a square is a special rectangle. Size does not matter.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which shape has 4 equal sides and no right angles?

    Three shapes. Shape A is a square with 4 ticks and 4 small squares. Shape B is a rectangle with 1 tick on each long side and 2 ticks on each short side. Shape C is a rhombus with 4 ticks and no small squares. The labels are A, B and C.ABC
    Tap your answer.

    1. Not this one. A has 4 equal sides, but it also has 4 right angles. Look for the shape with no small squares. Try another.

    2. Not this one. B has right angles, and its sides are not all equal. Check the ticks. Try another.

    3. Yes, that is right. C is a rhombus. Its 4 sides are equal and it has no right angles.

    See the answer

    Shape C

    C is a rhombus. Its 4 sides are equal and it has no right angles.

    • If you chose A: A has 4 equal sides, but it also has 4 right angles. Look for the shape with no small squares.
    • If you chose B: B has right angles, and its sides are not all equal. Check the ticks.

  2. A triangle has sides of 5 cm, 5 cm and 8 cm. What is it called?

    A triangle with its sides labelled 5 cm, 5 cm and 8 cm. The two shorter sides look equal.5 cm5 cm8 cm
    Tap your answer.

    1. Not this one. Equilateral means all 3 sides are equal. The 8 cm side is longer. Try another.

    2. Yes, that is right. Two sides are equal, 5 cm and 5 cm. The third side is different. That makes it isosceles.

    3. Not this one. Scalene means no sides are equal. Two sides here are both 5 cm. Try another.

    See the answer

    isosceles

    Two sides are equal, 5 cm and 5 cm. The third side is different. That makes it isosceles.

    • If you chose A: Equilateral means all 3 sides are equal. The 8 cm side is longer.
    • If you chose C: Scalene means no sides are equal. Two sides here are both 5 cm.

  3. Each corner of a triangle with 3 equal sides is 60°. What do the three corners add up to?

    See the answer

    180°

    60 + 60 + 60 = 180. That is the same turn as a straight line.

  4. Which sentence is true?

    Tap your answer.

    1. Yes, that is right. A square has 4 right angles and equal opposite sides. That is all a rectangle needs.

    2. Not this one. A rectangle can have long sides and short sides. Only some rectangles have 4 equal sides. Try another.

    3. Not this one. A rhombus can lean over, and then it has no right angles. A square is the rhombus with 4 right angles. Try another.

    See the answer

    A square is a rectangle.

    A square has 4 right angles and equal opposite sides. That is all a rectangle needs.

    • If you chose B: A rectangle can have long sides and short sides. Only some rectangles have 4 equal sides.
    • If you chose C: A rhombus can lean over, and then it has no right angles. A square is the rhombus with 4 right angles.

  5. A shape has 4 sides. Only one pair of sides is parallel. What is it?

    Tap your answer.

    1. Not this one. A parallelogram has two pairs of parallel sides, not one. Try another.

    2. Yes, that is right. A trapezium has exactly one pair of parallel sides.

    3. Not this one. A rectangle has two pairs of parallel sides, not one. Try another.

    See the answer

    trapezium

    A trapezium has exactly one pair of parallel sides.

    • If you chose A: A parallelogram has two pairs of parallel sides, not one.
    • If you chose C: A rectangle has two pairs of parallel sides, not one.

  6. From an earlier chapter

    A corner is bigger than a right angle. It is smaller than a straight line. What is it called?

    Tap your answer.

    1. Not this one. Acute is smaller than a right angle. Test the corner with your tester. Go to M2.5.3. Try another.

    2. Not this one. A right angle matches your tester exactly. This corner is bigger. Go to M2.5.3. Try another.

    3. Yes, that is right. Obtuse is bigger than a right angle. It is smaller than a straight line.

    See the answer

    obtuse

    Obtuse is bigger than a right angle. It is smaller than a straight line.

    • If you chose A: Acute is smaller than a right angle. Test the corner with your tester. Go to M2.5.3.
    • If you chose B: A right angle matches your tester exactly. This corner is bigger. Go to M2.5.3.

  7. Which shape is both a rhombus and a rectangle?

    Four shapes. Shape A leans, and has two pairs of equal sides of different lengths, marked with one and two ticks. Shape B leans and has one tick on every side. Shape C has matching ticks on opposite sides and a small square at each corner. Shape D has one tick on every side and a small square at each corner. The labels are A, B, C and D.ABCD
    Tap your answer.

    1. Not this one. A has equal opposite sides, but not all four are equal. It has no small squares, so no right angles. Try another.

    2. Not this one. B has 4 equal sides, so it is a rhombus. It has no right angles, so it is not a rectangle. Try another.

    3. Not this one. C has 4 right angles, so it is a rectangle. Its sides are not all equal, so it is not a rhombus. Try another.

    4. Yes, that is right. D has 4 equal sides, so it is a rhombus. It has 4 right angles, so it is a rectangle. It is a square.

    See the answer

    Shape D

    D has 4 equal sides, so it is a rhombus. It has 4 right angles, so it is a rectangle. It is a square.

    • If you chose A: A has equal opposite sides, but not all four are equal. It has no small squares, so no right angles.
    • If you chose B: B has 4 equal sides, so it is a rhombus. It has no right angles, so it is not a rectangle.
    • If you chose C: C has 4 right angles, so it is a rectangle. Its sides are not all equal, so it is not a rhombus.

Keep it

Copy this into your exercise book.

Square: 4 equal sides, 4 right angles. Rectangle: 4 right angles. Rhombus: 4 equal sides. Parallelogram: two pairs of parallel sides. Trapezium: one pair.

Draw: A square, a rectangle and a rhombus. Mark equal sides with ticks and right angles with small squares.

M2.5.6 Chapter 6

Lines of symmetry

When do the two halves of a folded shape match?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Fold a leaf down its middle. Often the two halves almost match. A shape is symmetrical when its two folded halves match exactly.

Fold a paper rectangle in different ways. Some folds work. Some do not.

Draw it first

Fold a rectangle three ways

  1. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.

    1 Draw a rectangle that is not a square.

  2. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.fold 1: halves match

    2 Fold it down the middle. The halves match. The fold is a line of symmetry.

  3. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.2 lines of symmetry

    3 Fold it the other way. The halves match again. So it has 2 lines.

  4. A rectangle that is not a square, folded three ways. Step 2 shows a dashed vertical fold and the label fold 1: halves match. Step 3 adds a dashed horizontal fold and the label 2 lines of symmetry. Step 4 shows a dashed fold from corner to corner and the label diagonal: no match.diagonal: no match

    4 Fold from one corner to the opposite corner. This fold is a diagonal. The corners do not meet. It is not a line of symmetry.

A line of symmetry is a fold. The two halves match exactly.

Six shapes with their lines of symmetry drawn as dashed lines. The labels are square, 4 lines; rectangle, 2 lines; rhombus, 2 lines; equilateral, 3 lines; isosceles, 1 line; and scalene, 0 lines.square4 linesrectangle2 linesrhombus2 linesequilateral3 linesisosceles1 linescalene0 lines
Dashed lines are lines of symmetry. A scalene triangle has none.

A regular hexagon has 6 lines of symmetry. A circle has more than you can count.

Look at the rhombus. Its two diagonals are lines of symmetry. A rectangle is different.

A line of symmetry also works as a mirror line. What is on one side is a mirror picture of the other.

See it three ways

With things

Fold a sheet in half and open it. Draw half a house on one side. Count the steps from each corner to the fold. Count the same steps on the other side.

As a picture

A grid of dots with a dashed mirror line down the middle. The left half of a house is drawn with solid lines. Its mirror twin on the right is drawn with dashed lines. The label is mirror line.mirror line

In symbols

A corner 2 steps left of the line has a twin. The twin is 2 steps right, at the same height.

Try it

Tear a symmetrical shape

You need: a sheet of paper, a pencil.

  1. Fold the paper in half. Press the fold flat.
  2. Draw half of a leaf or a tree. Its middle must touch the fold.
  3. Tear along your line through both layers. Tear, do not cut.
  4. Open the shape. Look at the fold.

What you will see: The two halves match. The fold is a line of symmetry. You made a symmetrical shape.

If you cannot try it, this is what happens.
A six sided shape with a dashed vertical line down its middle. The two halves match. The label is fold line.fold line

A wrong idea many people have

The diagonal of a rectangle is a line of symmetry.

Why it feels right: The diagonal cuts the rectangle into two triangles. They are the same size.

The test that breaks it: Fold a paper rectangle corner to corner. One corner sticks out past the edge. The corners do not meet.

What is true: Halves that are the same size are not enough. They must match when you fold. A rectangle that is not a square has 2 lines.

Key words

symmetrical
a shape with at least one line of symmetry.
line of symmetry
a fold line where the two halves match exactly.
mirror line
a line of symmetry, used to draw a mirror picture.

Back to your guess

The answer is B: Only a line that makes two halves that match when folded.

A line works only when the two halves match when you fold. Many lines through the middle fail the test. The fold decides.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. How many lines of symmetry does a square have?

    Tap your answer.

    1. Not this one. You found the two folds through the middle. A square also has two diagonal folds. Fold paper to check. Try another.

    2. Yes, that is right. A square folds 4 ways. Down the middle, across the middle, and along each diagonal.

    3. Not this one. You counted both ends of each line. One line is one fold. Count each fold once. Try another.

    See the answer

    4

    A square folds 4 ways. Down the middle, across the middle, and along each diagonal.

    • If you chose A: You found the two folds through the middle. A square also has two diagonal folds. Fold paper to check.
    • If you chose C: You counted both ends of each line. One line is one fold. Count each fold once.

  2. A rectangle is 8 cm long and 4 cm wide. How many lines of symmetry does it have?

    See the answer

    2

    The sides are not equal, so it is not a square. It folds down the middle and across the middle. The diagonals do not work.

  3. Which shape has no line of symmetry?

    Three shapes. A is a square. B is an isosceles triangle. C is a scalene triangle. The labels are A, B and C.ABC
    Tap your answer.

    1. Not this one. A square folds 4 ways. Try the folds on paper. Try another.

    2. Not this one. An isosceles triangle folds down its middle. The halves match. Try another.

    3. Yes, that is right. C is a scalene triangle. Its sides all differ, so no fold makes the halves match.

    See the answer

    Shape C

    C is a scalene triangle. Its sides all differ, so no fold makes the halves match.

    • If you chose A: A square folds 4 ways. Try the folds on paper.
    • If you chose B: An isosceles triangle folds down its middle. The halves match.

  4. Corner X is 3 steps left of the mirror line. How many steps right is its twin?

    A grid of dots with a dashed mirror line. The left half of a shape is drawn with solid lines. The corner X is at the top left of the shape. The labels are mirror line and X.mirror lineX
    See the answer

    3 steps

    The twin is as far from the mirror line as X. It is on the other side. X is 3 steps away, so its twin is 3 steps away.

  5. Ana folds a paper rectangle corner to corner. The corners do not meet. What does this show?

    Tap your answer.

    1. Yes, that is right. The two triangles are the same size. They do not match when folded. A line of symmetry needs a match.

    2. Not this one. The diagonal fails. Folds down the middle and across the middle work. Try another.

    3. Not this one. The fold was fine. Halves of the same size do not always match when folded. Try another.

    See the answer

    The diagonal is not a line of symmetry.

    The two triangles are the same size. They do not match when folded. A line of symmetry needs a match.

    • If you chose B: The diagonal fails. Folds down the middle and across the middle work.
    • If you chose C: The fold was fine. Halves of the same size do not always match when folded.

  6. From an earlier chapter

    How many corners does this polygon have?

    A polygon with 7 straight sides and a dot on each corner.
    See the answer

    7

    A polygon has as many corners as sides. Count the dots. There are 7. It is a heptagon.

  7. Look at the rhombus. How many lines of symmetry does it have?

    A rhombus that leans. One small tick on each side marks that all four sides are equal. The label is rhombus.rhombus
    Tap your answer.

    1. Yes, that is right. A rhombus folds along each diagonal and the halves match. Folds across the middle of its sides do not match.

    2. Not this one. It leans, but a lean does not stop a fold. Fold a paper rhombus corner to corner. The halves match. Try another.

    3. Not this one. This rhombus has 4 equal sides, but only 2 folds match. A square has 4 lines. Try another.

    See the answer

    2

    A rhombus folds along each diagonal and the halves match. Folds across the middle of its sides do not match.

    • If you chose B: It leans, but a lean does not stop a fold. Fold a paper rhombus corner to corner. The halves match.
    • If you chose C: This rhombus has 4 equal sides, but only 2 folds match. A square has 4 lines.

  8. Corner X is 3 steps left of the mirror line. Where is its twin?

    A grid of dots with a dashed mirror line. The left half of a house is drawn with solid lines. A dot marks the corner X, at the left end of the roof step. The labels are mirror line and X.mirror lineX
    Tap your answer.

    1. Not this one. You counted 1 from the left edge of the grid. Count steps from the mirror line instead. Try another.

    2. Yes, that is right. The twin is as far from the mirror line as X. It is on the other side. It keeps the same height.

    3. Not this one. This mirror line stands up. It flips left and right only. The twin stays at the same height. Try another.

    See the answer

    3 steps right, at the same height as X

    The twin is as far from the mirror line as X. It is on the other side. It keeps the same height.

    • If you chose A: You counted 1 from the left edge of the grid. Count steps from the mirror line instead.
    • If you chose C: This mirror line stands up. It flips left and right only. The twin stays at the same height.

Keep it

Copy this into your exercise book.

A line of symmetry is a fold where the halves match. A square has 4. A rectangle has 2. A scalene triangle has 0. A diagonal of a rectangle fails.

Draw: A rectangle with its 2 lines of symmetry dashed. Add its diagonal and write "no".

M2.5.7 Chapter 7

Solid shapes and nets

Will any 6 squares joined together fold into a cube?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Pick up a box. It has flat sides, straight lines and sharp points.

Draw it first

Faces, edges and corners of a cube

  1. A cube drawn with its hidden edges dashed. Step 2 outlines the front side and labels it face. Step 3 outlines the left front line and labels it edge. Step 4 puts a dot on the top right back point and labels it corner.

    1 Draw a cube. Dashed lines are edges you cannot see.

  2. A cube drawn with its hidden edges dashed. Step 2 outlines the front side and labels it face. Step 3 outlines the left front line and labels it edge. Step 4 puts a dot on the top right back point and labels it corner.face

    2 A flat side is a face. A cube has 6 faces.

  3. A cube drawn with its hidden edges dashed. Step 2 outlines the front side and labels it face. Step 3 outlines the left front line and labels it edge. Step 4 puts a dot on the top right back point and labels it corner.faceedge

    3 A line where two faces meet is an edge. A cube has 12 edges.

  4. A cube drawn with its hidden edges dashed. Step 2 outlines the front side and labels it face. Step 3 outlines the left front line and labels it edge. Step 4 puts a dot on the top right back point and labels it corner.faceedgecorner

    4 A point where edges meet is a corner. A cube has 8 corners.

Some books call a corner a vertex. Two or more are vertices.

A cube and a triangular prism with their counts. The cube: 6 faces, 12 edges, 8 corners. The triangular prism: 5 faces, 9 edges, 6 corners. The labels are cube, 6 faces, 12 edges, 8 corners, triangular prism, 5 faces, 9 edges and 6 corners.cube6 faces12 edges8 cornerstriangular prism5 faces9 edges6 corners
A box like a matchbox has the same counts as a cube.

A prism has the same shape all along its length. A triangular prism has 2 triangle faces and 3 rectangle faces.

A square based pyramid has 5 faces, 8 edges and 5 corners.

Take a box apart and lay it flat. The flat shape is a net. A net folds up into the solid.

Only some shapes of 6 squares are nets of a cube. Look at these three.

Three shapes made of 6 squares, with dashed fold lines. The first is a cross shape and folds into a cube, labelled yes. The second is a row of 6 and does not, labelled no. The third is a block of 2 by 3 and does not, labelled no.yesnono
Dashed lines are folds. The row of 6 wraps round into a tube.

See it three ways

With things

Hold a box, such as a matchbox. Touch each face, edge and corner as you count.

As a picture

A cube drawn with hidden edges dashed. The labels are 6 faces, 12 edges and 8 corners.6 faces12 edges8 corners

In symbols

Faces 6. Edges 12. Corners 8. The top has 4 edges and the bottom has 4. Four edges go up.

Try it

Fold a net into an open box

You need: a square of paper that is 15 cm wide, a ruler, a pencil, scissors.

Be safe: Ask a grown up to help with the scissors.

  1. Mark every 5 cm along each side. Join the marks with a ruler. You get 9 squares.
  2. Cut out the 4 corner squares. A cross of 5 squares is left.
  3. Fold up along the 4 sides of the middle square. The 4 arms stand up.
  4. Hold the upright sides together at the corners. You have an open box.

What you will see: The net has 5 squares, and the box has 5 faces. It is open on top, so it has no sixth face.

If you cannot try it, this is what happens.
A cross of 5 squares with dashed fold lines. The middle square is the base. The four arms fold up as sides. The labels are base and side.baseside

A wrong idea many people have

Any 6 squares joined edge to edge fold into a cube.

Why it feels right: A cube has 6 faces. A net has 6 squares.

The test that breaks it: Draw 6 squares in a row. Fold them. Four squares make a tube. The last two land on top of the first two.

What is true: No two faces may land in the same place. A cube has 11 different nets. A row of 6 is not one of them.

Key words

face
a flat side of a solid shape.
edge
a line where two faces meet.
corner
a point where edges meet.
net
a flat shape that folds up into a solid.

Back to your guess

The answer is B: No. Only some joined squares fold into a cube.

Only some work. When you fold, no two squares may land in the same place. A cube has 11 different nets.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A cube has 6 faces. Each face has 4 edges. Each edge joins 2 faces. How many edges does a cube have?

    See the answer

    12

    6 faces with 4 edges each is 24. Each edge was counted twice, so 24 ÷ 2 = 12.

  2. A triangular prism has two triangle ends. A triangle has 3 corners. How many corners does the prism have?

    See the answer

    6

    Each triangle end has 3 corners. 3 + 3 = 6. There are no other corners.

  3. Which solid folds up from this net?

    A net made of three rectangles in a row, with a triangle joined above and below the middle rectangle. Dashed lines show the folds.
    Tap your answer.

    1. Not this one. A cube net has 6 squares. This net has triangles and rectangles. Try another.

    2. Yes, that is right. The net has 2 triangles and 3 rectangles. That is 5 faces, the faces of a triangular prism.

    3. Not this one. A pyramid net has a square and 4 triangles. This net has 3 rectangles. Try another.

    See the answer

    a triangular prism

    The net has 2 triangles and 3 rectangles. That is 5 faces, the faces of a triangular prism.

    • If you chose A: A cube net has 6 squares. This net has triangles and rectangles.
    • If you chose C: A pyramid net has a square and 4 triangles. This net has 3 rectangles.

  4. Which shape folds into a cube?

    Two shapes made of 6 squares. Shape A is a row of 6 squares. Shape B is a cross shape with a row of 4 and one square above and below. The labels are A and B.AB
    Tap your answer.

    1. Not this one. Six in a row wrap into a tube. The last squares land on the first ones. Two faces are missing. Try another.

    2. Yes, that is right. B folds so every square lands in a different place. A does not.

    3. Not this one. Both have 6 squares. But a row of 6 wraps into a tube. Try another.

    See the answer

    Shape B

    B folds so every square lands in a different place. A does not.

    • If you chose A: Six in a row wrap into a tube. The last squares land on the first ones. Two faces are missing.
    • If you chose C: Both have 6 squares. But a row of 6 wraps into a tube.

  5. A square based pyramid has a square base with 4 edges. 4 more edges go up to the top corner. How many edges?

    See the answer

    8

    4 edges on the base and 4 edges going up. 4 + 4 = 8.

  6. From an earlier chapter

    Mei measures a line with a ruler. Where should the start of the line be?

    Tap your answer.

    1. Yes, that is right. Put one end of the line at the 0 mark. Read the number at the other end.

    2. Not this one. Many rulers have a gap before 0. Start at the 0 mark, not at the edge. Go to M2.5.1. Try another.

    3. Not this one. The line would read 1 cm too long. Measuring starts at 0. Go to M2.5.1. Try another.

    See the answer

    At the 0 mark

    Put one end of the line at the 0 mark. Read the number at the other end.

    • If you chose B: Many rulers have a gap before 0. Start at the 0 mark, not at the edge. Go to M2.5.1.
    • If you chose C: The line would read 1 cm too long. Measuring starts at 0. Go to M2.5.1.

  7. How many faces does this triangular prism have?

    A triangular prism lying on a rectangle, with its hidden edges dashed. The label is triangular prism.triangular prism
    Tap your answer.

    1. Yes, that is right. It has 2 triangle ends and 3 rectangle sides. 2 + 3 = 5 faces.

    2. Not this one. You counted the 3 rectangle sides. The 2 triangle ends are faces too. Try another.

    3. Not this one. A cube has 6 faces. A prism with triangle ends has 2 triangles and 3 rectangles. Try another.

    See the answer

    5

    It has 2 triangle ends and 3 rectangle sides. 2 + 3 = 5 faces.

    • If you chose B: You counted the 3 rectangle sides. The 2 triangle ends are faces too.
    • If you chose C: A cube has 6 faces. A prism with triangle ends has 2 triangles and 3 rectangles.

  8. Lucía folds this net into a box with no lid. How many faces does the box have?

    A cross of squares with dashed fold lines. The label is net.net
    Tap your answer.

    1. Not this one. You counted the sides only. The middle square is the base, and it is a face too. Try another.

    2. Not this one. A box with a lid has 6 faces. This net has no square for a lid. Try another.

    3. Yes, that is right. Each square of the net becomes one face. The net has 5 squares, so the box has 5 faces.

    See the answer

    5

    Each square of the net becomes one face. The net has 5 squares, so the box has 5 faces.

    • If you chose A: You counted the sides only. The middle square is the base, and it is a face too.
    • If you chose B: A box with a lid has 6 faces. This net has no square for a lid.

  9. How many corners does this cube have?

    A cube drawn with its hidden edges dashed. The label is cube.cube
    Tap your answer.

    1. Not this one. That is the number of edges. A corner is a point where edges meet. Try another.

    2. Yes, that is right. The front face has 4 corners and the back face has 4. 4 + 4 = 8.

    3. Not this one. That is the number of faces. A corner is a point, not a flat side. Try another.

    See the answer

    8

    The front face has 4 corners and the back face has 4. 4 + 4 = 8.

    • If you chose A: That is the number of edges. A corner is a point where edges meet.
    • If you chose C: That is the number of faces. A corner is a point, not a flat side.

Keep it

Copy this into your exercise book.

A cube has 6 faces, 12 edges and 8 corners. A triangular prism has 5, 9 and 6. A net is a solid laid flat. Not every 6 squares make a cube net.

Draw: A cube with its hidden edges dashed. Label a face, an edge and a corner.

M2.5.8 Chapter 8

Grids and directions

How do I tell a friend which box to look in?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

A grid is a set of boxes in rows and columns. It names a place exactly.

Draw it first

Find the box B3

  1. A grid with columns A, B, C, D and E across the top and rows 1, 2, 3 and 4 down the side. Step 2 outlines column B and is labelled column B. Step 3 outlines row 3 and is labelled row 3. Step 4 marks the box where they meet with a dot and is labelled B3.ABCDE1234

    1 Letters name the boxes across. Numbers name the boxes down.

  2. A grid with columns A, B, C, D and E across the top and rows 1, 2, 3 and 4 down the side. Step 2 outlines column B and is labelled column B. Step 3 outlines row 3 and is labelled row 3. Step 4 marks the box where they meet with a dot and is labelled B3.ABCDE1234column B

    2 A column goes up and down. This is column B.

  3. A grid with columns A, B, C, D and E across the top and rows 1, 2, 3 and 4 down the side. Step 2 outlines column B and is labelled column B. Step 3 outlines row 3 and is labelled row 3. Step 4 marks the box where they meet with a dot and is labelled B3.ABCDE1234row 3

    3 A row goes across. This is row 3.

  4. A grid with columns A, B, C, D and E across the top and rows 1, 2, 3 and 4 down the side. Step 2 outlines column B and is labelled column B. Step 3 outlines row 3 and is labelled row 3. Step 4 marks the box where they meet with a dot and is labelled B3.ABCDE1234B3

    4 Column B and row 3 meet in one box. Its name is B3.

Say the column first, then the row. B3 means column B, row 3.

Maps show direction too. North is at the top of a map. South is at the bottom.

East is on the right. West is on the left. These four are the compass points.

Four arrows leave a middle point. The arrow up points to the label north. The arrow right points to east. The arrow down points to south. The arrow left points to west.northeastsouthwest
Go round the way a clock hand turns: north, east, south, west.

Face north. East is on your right. West is on your left. South is behind you.

To find north where you live, use a map or a compass. Ask a neighbour if you are not sure.

A route is the path you follow from one box to another. Say it in boxes and compass points.

A grid with columns A, B, C and D and rows 1, 2 and 3. A route starts at A1, goes east along row 1 to D1, then south down column D to D3. The labels are start A1 and end D3.ABCD123start A1end D3
East 3 boxes, then south 2 boxes.

Worked example

Start at A1. Go east 3 boxes. Then go south 2 boxes. Which box are you in?

  1. East is to the right, along the letters. From A, go 3 boxes: B, C, D.
  2. You are in column D. You are still in row 1.
  3. South is down, along the numbers. From row 1, go 2 boxes: 2, 3.
  4. Column D and row 3 make the box D3.

Answer: D3

See it three ways

With things

Draw a grid on paper. Put a bean in one box. Tell a friend its name, column first.

As a picture

A map grid with columns A, B, C, D and E and rows 1, 2, 3 and 4. A tree is in D1, a hut in B2, a well in C3 and a pond in E4. The labels are tree, hut, well and pond.ABCDE1234treehutwellpond

In symbols

The well is in column C and row 3. Its name is C3.

Try it

Walk a route on a floor grid

You need: floor tiles, or chalk or a stick for drawing boxes, a stone to mark north.

  1. Make a grid 4 boxes wide and 4 boxes long.
  2. Put the stone by one edge. That edge is north, like the top of a map.
  3. Face the stone. Write A to D along that edge, from left to right.
  4. Write 1 to 4 along a side. Start next to the stone. East is on your right.
  5. Stand in B2. Walk east 2 boxes. Say your box. You are in D2.
  6. Walk south 1 box, then west 3 boxes. Say your box each time. You end in A3.

What you will see: Each step changes one thing. East and west change the column letter. North and south change the row number.

If you cannot try it, this is what happens.
A grid with columns A, B, C and D and rows 1, 2, 3 and 4. A route starts at B2, goes east to D2, then south to D3, then west to A3. The labels are start B2 and end A3.ABCD1234start B2end A3

A wrong idea many people have

The order of a grid position does not matter.

Why it feels right: Adding can swap places. 2 + 3 and 3 + 2 are equal.

The test that breaks it: Draw a grid. Count 2 boxes across and 3 down. Mark the box. Now count 3 across and 2 down. Mark that box too. They are different boxes.

What is true: A position has a fixed order. Say the column first, then the row. 2 across and 3 down is B3. 3 across and 2 down is C2.

Key words

column
boxes in a line going up and down.
row
boxes in a line going across.
compass points
north, east, south and west.
route
the path you follow from one place to another.

Back to your guess

The answer is C: Say the column and the row, like B3.

A column alone names a whole stripe of boxes. A column and a row meet in one box. Everyone must say them in the same order. This book gives the column first.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which box has the well?

    A map grid with columns A, B, C, D and E and rows 1, 2, 3 and 4. A tree is in D1, a hut in B2, a well in C3 and a pond in E4. The labels are tree, hut, well and pond.ABCDE1234treehutwellpond
    Tap your answer.

    1. Yes, that is right. The well is in column C and row 3. Column first, then row, so C3.

    2. Not this one. This book says the column letter comes first, then the row number. Write C3. Try another.

    3. Not this one. Look at the letter above the well. It is C, not B. Box B3 is under the hut. Try another.

    See the answer

    C3

    The well is in column C and row 3. Column first, then row, so C3.

    • If you chose B: This book says the column letter comes first, then the row number. Write C3.
    • If you chose C: Look at the letter above the well. It is C, not B. Box B3 is under the hut.

  2. Start at B1. Go east 2 boxes. Then go south 3 boxes. Which box are you in?

    See the answer

    D4

    East 2 from B is C, D. South 3 from row 1 is 2, 3, 4. So you are in D4.

  3. Mei is in box E3. She walks west 2 boxes. Which box is she in?

    See the answer

    C3

    West is left, along the letters. From E, go back 2 boxes: D, C. The row stays 3.

  4. You face north. Where is east?

    Tap your answer.

    1. Yes, that is right. Face north. East is on your right. West is on your left. South is behind you.

    2. Not this one. West is on your left when you face north. East is the other side. Try another.

    3. Not this one. South is behind you when you face north. East is to the side. Try another.

    See the answer

    On your right

    Face north. East is on your right. West is on your left. South is behind you.

    • If you chose B: West is on your left when you face north. East is the other side.
    • If you chose C: South is behind you when you face north. East is to the side.

  5. From an earlier chapter

    Ravi faces north. He turns a quarter turn clockwise, so he faces east. How many degrees did he turn?

    See the answer

    90°

    A quarter turn is one right angle. A full turn is 360°, and 360 ÷ 4 = 90.

  6. Tariq is in box B2. He walks east 2 boxes. Which box is he in?

    Tap your answer.

    1. Not this one. You counted the box you stand in as 1. Step 2 boxes away from B: C, then D. Try another.

    2. Not this one. That changes the row number. East and west change the letter. North and south change the number. Try another.

    3. Yes, that is right. East is to the right, along the letters. From B, go 2 boxes: C, D. The row stays 2.

    See the answer

    D2

    East is to the right, along the letters. From B, go 2 boxes: C, D. The row stays 2.

    • If you chose A: You counted the box you stand in as 1. Step 2 boxes away from B: C, then D.
    • If you chose B: That changes the row number. East and west change the letter. North and south change the number.

  7. Ana walked from A1 to D3. She went east 3 boxes, then south 2 boxes. How does she walk back?

    A grid with columns A, B, C and D and rows 1, 2 and 3. A route starts at A1, goes east along row 1 to D1, then south down column D to D3. The labels are start A1 and end D3.ABCD123start A1end D3
    Tap your answer.

    1. Yes, that is right. Going back, the last walk comes first. South 2 becomes north 2. Then east 3 becomes west 3.

    2. Not this one. Those are the same directions as the way there. To go back, turn each direction round. Try another.

    3. Not this one. The directions are right, but the numbers are swapped. She went south 2 boxes and east 3. Try another.

    See the answer

    North 2 boxes, then west 3 boxes

    Going back, the last walk comes first. South 2 becomes north 2. Then east 3 becomes west 3.

    • If you chose B: Those are the same directions as the way there. To go back, turn each direction round.
    • If you chose C: The directions are right, but the numbers are swapped. She went south 2 boxes and east 3.

  8. Which name tells where the dot is?

    A grid with columns A, B, C, D and E and rows 1, 2, 3 and 4. A dot sits in one box, in column D and row 2.ABCDE1234
    Tap your answer.

    1. Not this one. You read the row first. This book says the column letter comes first, then the row number. Try another.

    2. Yes, that is right. The dot is in column D. It is in row 2, counting down from the top. Column first, so D2.

    3. Not this one. You counted the rows from the bottom. Rows are numbered from the top, so this is row 2. Try another.

    See the answer

    D2

    The dot is in column D. It is in row 2, counting down from the top. Column first, so D2.

    • If you chose A: You read the row first. This book says the column letter comes first, then the row number.
    • If you chose C: You counted the rows from the bottom. Rows are numbered from the top, so this is row 2.

Keep it

Copy this into your exercise book.

Name a box by column, then row: B3. North is up, east is right, south is down, west is left. East and west change the letter. North and south change the number.

Draw: A grid of 4 columns and 3 rows. Label it A to D and 1 to 3. Mark B2 with a dot.

Key words, A to Z

Every key word in this book, what it means, and the chapter that teaches it.

acute
smaller than a right angle. M2.5.3
column
boxes in a line going up and down. M2.5.8
compass points
north, east, south and west. M2.5.8
corner
a point where edges meet. M2.5.7
degrees
the unit we use to measure an angle. It is written °. M2.5.4
edge
a line where two faces meet. M2.5.7
face
a flat side of a solid shape. M2.5.7
full turn
all the way round. It is 360°. M2.5.4
length
how long a line is. M2.5.1
line of symmetry
a fold line where the two halves match exactly. M2.5.6
mirror line
a line of symmetry, used to draw a mirror picture. M2.5.6
net
a flat shape that folds up into a solid. M2.5.7
obtuse
bigger than a right angle, but smaller than a straight line. M2.5.3
parallel
lines that stay the same distance apart and never meet. M2.5.1
parallelogram
a shape with two pairs of parallel sides. M2.5.5
perpendicular
lines that meet at a right angle. M2.5.3
polygon
a flat, closed shape with straight sides. M2.5.2
protractor
a tool with a scale for measuring angles. M2.5.4
quadrilateral
a polygon with 4 sides. M2.5.2
regular
all sides equal and all corners the same size. M2.5.2
rhombus
a shape with 4 equal sides. M2.5.5
right angle
a square corner. It is a quarter turn. M2.5.3
route
the path you follow from one place to another. M2.5.8
row
boxes in a line going across. M2.5.8
straight line
a line that does not bend. M2.5.1
symmetrical
a shape with at least one line of symmetry. M2.5.6
trapezium
a shape with one pair of parallel sides. M2.5.5