Chapter 3 of 8

All 8 chapters
  1. 1Coordinates, readunderstood
  2. 2All four quarters of the grid, readunderstood
  3. 3Shapes on a grid, readunderstood
  4. 4From a rule to a line, readunderstood
  5. 5Lines like x = 3 and y = −2, readunderstood
  6. 6Steepness: the gradient, readunderstood
  7. 7The equation y = mx + c, readunderstood
  8. 8Graphs from real life, readunderstood

M3.9.3 Chapter 3

Shapes on a grid

If I slide a shape across a page, or flip it over, where do its corners go?

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

  • Find the midpoint of a line between two points
  • Move a shape on a grid by a translation
  • Reflect a shape in the x axis or the y axis

A shape on a grid is described by its corners. A triangle ABC has three corners, and each one has two coordinates. If you know the corners, you know the shape.

Three things can happen to a shape. We can find the middle of one of its lines. We can slide it. We can flip it. Each one changes the coordinates in a clear pattern.

The midpoint of a line is the point exactly halfway along it. Add the two across numbers and halve the answer. Do the same with the two up numbers.

Draw it first

Find the midpoint of a line

  1. A grid with the x axis across and the y axis up. Step 1: the points A (1, 2) and B (7, 6) joined by a line. Step 2: a dashed line up from the x axis at 4, halfway between 1 and 7. Step 3: a dashed line across from the y axis at 4, halfway between 2 and 6. Step 4: the midpoint M (4, 4) where the dashed lines cross the line.A (1, 2)B (7, 6)12345678123456780xy

    1 Join A (1, 2) and B (7, 6) with a straight line.

  2. A grid with the x axis across and the y axis up. Step 1: the points A (1, 2) and B (7, 6) joined by a line. Step 2: a dashed line up from the x axis at 4, halfway between 1 and 7. Step 3: a dashed line across from the y axis at 4, halfway between 2 and 6. Step 4: the midpoint M (4, 4) where the dashed lines cross the line.A (1, 2)B (7, 6)12345678123456780xy

    2 The across numbers are 1 and 7. Halfway is (1 + 7) ÷ 2 = 4.

  3. A grid with the x axis across and the y axis up. Step 1: the points A (1, 2) and B (7, 6) joined by a line. Step 2: a dashed line up from the x axis at 4, halfway between 1 and 7. Step 3: a dashed line across from the y axis at 4, halfway between 2 and 6. Step 4: the midpoint M (4, 4) where the dashed lines cross the line.A (1, 2)B (7, 6)12345678123456780xy

    3 The up numbers are 2 and 6. Halfway is (2 + 6) ÷ 2 = 4.

  4. A grid with the x axis across and the y axis up. Step 1: the points A (1, 2) and B (7, 6) joined by a line. Step 2: a dashed line up from the x axis at 4, halfway between 1 and 7. Step 3: a dashed line across from the y axis at 4, halfway between 2 and 6. Step 4: the midpoint M (4, 4) where the dashed lines cross the line.A (1, 2)B (7, 6)M (4, 4)12345678123456780xy

    4 The midpoint M is (4, 4). It sits on the line, halfway from each end.

Worked example

Find the midpoint of P (−5, −1) and Q (−1, 3).

  1. Add the across numbers: −5 + (−1) = −6. Half of −6 is −3.
  2. Add the up numbers: −1 + 3 = 2. Half of 2 is 1.
  3. The midpoint is (−3, 1).

Answer: (−3, 1)

A grid with the x axis across and the y axis up. The points P (−5, −1) and Q (−1, 3) joined by a line. The dot M (−3, 1) is the midpoint, halfway along the line.PQM−5−4−3−2−112345−5−4−3−2−1123450xy
M is the same distance from P as from Q.

A translation slides every point of a shape the same distance in the same direction. We say it with two numbers: how far across and how far up. A negative number means left or down.

Draw it first

Slide a triangle 4 right and 2 up

  1. A grid with the x axis across and the y axis up. Step 1: the triangle with corners A (1, 1), B (3, 1) and C (1, 3). Step 2: an arrow from each corner, 4 right and 2 up. The top arrow is labelled 4 right, 2 up. Step 3: the image triangle with corners A', B' and C', drawn with hatching.ABC12345678123456780xy

    1 Draw triangle ABC with corners (1, 1), (3, 1) and (1, 3).

  2. A grid with the x axis across and the y axis up. Step 1: the triangle with corners A (1, 1), B (3, 1) and C (1, 3). Step 2: an arrow from each corner, 4 right and 2 up. The top arrow is labelled 4 right, 2 up. Step 3: the image triangle with corners A', B' and C', drawn with hatching.ABC4 right, 2 up12345678123456780xy

    2 Move every corner 4 right and 2 up. Draw an arrow from each corner. The arrows are all the same.

  3. A grid with the x axis across and the y axis up. Step 1: the triangle with corners A (1, 1), B (3, 1) and C (1, 3). Step 2: an arrow from each corner, 4 right and 2 up. The top arrow is labelled 4 right, 2 up. Step 3: the image triangle with corners A', B' and C', drawn with hatching.ABCA'B'C'4 right, 2 up12345678123456780xy

    3 Join the new corners. The new triangle is the image. It is the same size and the same way up.

Worked example

Slide the triangle with corners (−5, 2), (−3, 2), (−5, 4) by 6 right and 6 down.

  1. Add 6 to every across number. Subtract 6 from every up number.
  2. (−5, 2) goes to (1, −4). (−3, 2) goes to (3, −4).
  3. (−5, 4) goes to (1, −2).
  4. The image has corners (1, −4), (3, −4) and (1, −2).

Answer: (1, −4), (3, −4), (1, −2)

A grid with the x axis across and the y axis up. A triangle with corners A (−5, 2), B (−3, 2) and C (−5, 4), and its image A', B' and C' 6 right and 6 down with corners at (1, −4), (3, −4) and (1, −2), drawn with hatching.ABCA'B'C'−5−4−3−2−112345−5−4−3−2−1123450xy
Every corner moves 6 right and 6 down.

A reflection flips a shape over a mirror line. Each point goes to the other side of the mirror, the same distance away. The shape keeps its size but turns into its mirror image.

Draw it first

Reflect a triangle in the y axis

  1. A grid with the x axis across and the y axis up. Step 1: the triangle with corners A (2, 1), B (5, 1) and C (2, 4), and the y axis as the mirror. Step 2: a dashed line from each corner across the y axis. Step 3: the mirror image with corners A' (−2, 1), B' (−5, 1) and C' (−2, 4), drawn with hatching.ABC−5−4−3−2−112345−5−4−3−2−1123450xy

    1 Draw triangle ABC on the right of the y axis. The y axis is the mirror.

  2. A grid with the x axis across and the y axis up. Step 1: the triangle with corners A (2, 1), B (5, 1) and C (2, 4), and the y axis as the mirror. Step 2: a dashed line from each corner across the y axis. Step 3: the mirror image with corners A' (−2, 1), B' (−5, 1) and C' (−2, 4), drawn with hatching.ABC−5−4−3−2−112345−5−4−3−2−1123450xy

    2 From each corner draw a line straight across to the other side. The mirror cuts each line in half.

  3. A grid with the x axis across and the y axis up. Step 1: the triangle with corners A (2, 1), B (5, 1) and C (2, 4), and the y axis as the mirror. Step 2: a dashed line from each corner across the y axis. Step 3: the mirror image with corners A' (−2, 1), B' (−5, 1) and C' (−2, 4), drawn with hatching.ABCA'B'C'−5−4−3−2−112345−5−4−3−2−1123450xy

    3 Mark each new corner as far from the mirror as the old one. Join them.

Worked example

Reflect the triangle with corners (1, 2), (4, 2), (1, 4) in the x axis.

  1. The x axis is the mirror, so each up number changes sign. The across numbers stay.
  2. (1, 2) goes to (1, −2). (4, 2) goes to (4, −2).
  3. (1, 4) goes to (1, −4).
  4. Check one corner. (4, 2) is 2 above the x axis, and its image (4, −2) is 2 below it.
  5. The image has corners (1, −2), (4, −2) and (1, −4).

Answer: (1, −2), (4, −2), (1, −4)

A grid with the x axis across and the y axis up. A triangle with corners A (1, 2), B (4, 2) and C (1, 4) above the x axis, and its mirror image A', B' and C' below the x axis with corners at (1, −2), (4, −2) and (1, −4), drawn with hatching.ABCA'B'C'−5−4−3−2−112345−5−4−3−2−1123450xy
The x axis is the mirror. The across numbers stay and the up numbers change sign.

See it three ways

With things

Fold squared paper along the y axis. Press a pin through the point (3, 2). Open the paper. There is a second hole.

As a picture

A grid with the x axis across and the y axis up. A dot P (3, 2) and its mirror dot P' (−3, 2) on the other side of the y axis. A dashed line joins them across the mirror.P (3, 2)P' (−3, 2)−5−4−3−2−112345−5−4−3−2−1123450xy

In symbols

In the y axis, the point (3, 2) goes to (−3, 2). The across number changes sign and the up number stays.

Try it

Fold and prick

You need: squared paper, a sharp pencil, a ruler.

  1. Draw the two axes on squared paper. Plot a triangle with its corners on the right of the y axis.
  2. Fold the paper along the y axis. Press the pencil point through each corner.
  3. Open the paper. There are three new holes on the left.
  4. Join the holes with a ruler. Write the coordinates of each corner.

What you will see: Each new corner is as far from the fold as the old one, on the other side. The up numbers stay and the across numbers change sign.

If you cannot try it, this is what happens.
A grid with the x axis across and the y axis up. A triangle with corners A (2, 1), B (5, 1) and C (2, 4) on the right of the y axis. Its image with corners A', B' and C' is on the left, drawn with hatching.ABCA'B'C'−5−4−3−2−112345−5−4−3−2−1123450xy

A wrong idea many people have

A reflection in the y axis changes the sign of both numbers.

Why it feels right: The point lands on the other side of the grid, so it seems that both numbers should change.

The test that breaks it: Reflect (3, 2) in the y axis. It stays 2 up. If both numbers changed, it would be (−3, −2), which is below the x axis. A mirror in the y axis does not turn a point upside down.

What is true: A reflection in the y axis changes only the sign of the across number. A reflection in the x axis changes only the sign of the up number.

Key words

midpoint
the point exactly halfway along a line.
translation
a slide. Every point moves the same distance in the same direction.
reflection
a flip over a mirror line. The image is the mirror image.
image
the shape after it has been moved.

Back to your guess

The answer is B: The point (3, 2) is reflected in the y axis. It goes to (−3, 2).

The y axis is the mirror. A point stays the same height and lands on the other side of the mirror. So only the across number changes sign: (3, 2) goes to (−3, 2).

Check yourself

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

  1. What is the midpoint of (2, 6) and (8, 2)?

    Tap your answer.

    1. Not this one. You added the numbers but did not halve them. The midpoint is halfway. Go to M3.9.3. Try another.

    2. Yes, that is right. (2 + 8) ÷ 2 = 5 and (6 + 2) ÷ 2 = 4. The midpoint is (5, 4).

    3. Not this one. You took away instead of adding. The middle comes from adding the two numbers and halving. Go to M3.9.3. Try another.

    4. Not this one. You found half the gap, 3 across and 2 up. You did not add it to the start point. Add the two numbers, then halve. Go to M3.9.3. Try another.

    See the answer

    (5, 4)

    (2 + 8) ÷ 2 = 5 and (6 + 2) ÷ 2 = 4. The midpoint is (5, 4).

    • If you chose A: You added the numbers but did not halve them. The midpoint is halfway. Go to M3.9.3.
    • If you chose C: You took away instead of adding. The middle comes from adding the two numbers and halving. Go to M3.9.3.
    • If you chose D: You found half the gap, 3 across and 2 up. You did not add it to the start point. Add the two numbers, then halve. Go to M3.9.3.

  2. The point (3, 4) is moved 2 left and 5 down. Where does it go?

    Tap your answer.

    1. Not this one. You moved it 2 right and 5 up. Left means take away from the across number and down means take away from the up number. Go to M3.9.3. Try another.

    2. Not this one. That is only the move. Add the move to the point: 3 − 2 and 4 − 5. Go to M3.9.3. Try another.

    3. Not this one. The first number is right but down means take away. 4 − 5 is −1. Go to M3.9.2. Try another.

    4. Yes, that is right. 3 − 2 = 1 and 4 − 5 = −1. The point goes to (1, −1).

    See the answer

    (1, −1)

    3 − 2 = 1 and 4 − 5 = −1. The point goes to (1, −1).

    • If you chose A: You moved it 2 right and 5 up. Left means take away from the across number and down means take away from the up number. Go to M3.9.3.
    • If you chose B: That is only the move. Add the move to the point: 3 − 2 and 4 − 5. Go to M3.9.3.
    • If you chose C: The first number is right but down means take away. 4 − 5 is −1. Go to M3.9.2.

  3. The point (−3, 5) is reflected in the x axis. Where does it go?

    Tap your answer.

    1. Not this one. That is a reflection in the y axis. The x axis is the mirror here, so the up number changes sign. Go to M3.9.3. Try another.

    2. Not this one. You changed both signs. A mirror in the x axis changes only the up number. Go to M3.9.3. Try another.

    3. Yes, that is right. The x axis is the mirror. The across number stays −3. The up number 5 changes sign to −5.

    See the answer

    (−3, −5)

    The x axis is the mirror. The across number stays −3. The up number 5 changes sign to −5.

    • If you chose A: That is a reflection in the y axis. The x axis is the mirror here, so the up number changes sign. Go to M3.9.3.
    • If you chose B: You changed both signs. A mirror in the x axis changes only the up number. Go to M3.9.3.

  4. A triangle and its image are mirror images in the y axis. The corner (3, 1) of the triangle matches which corner of the image?

    Tap your answer.

    1. Yes, that is right. In the y axis the up number stays 1. The across number 3 changes sign to −3.

    2. Not this one. That would be a reflection in the x axis. Here the y axis is the mirror. Go to M3.9.3. Try another.

    3. Not this one. You changed both signs. Only the across number changes sign for the y axis. Go to M3.9.3. Try another.

    See the answer

    (−3, 1)

    In the y axis the up number stays 1. The across number 3 changes sign to −3.

    • If you chose B: That would be a reflection in the x axis. Here the y axis is the mirror. Go to M3.9.3.
    • If you chose C: You changed both signs. Only the across number changes sign for the y axis. Go to M3.9.3.

  5. Which sentence is true about a translation?

    Tap your answer.

    1. Yes, that is right. A translation slides the whole shape. The image has the same size and the same way up.

    2. Not this one. That is a reflection. A translation only slides the shape. Go to M3.9.3. Try another.

    3. Not this one. In a translation every corner moves by the same amount, so the shape does not change. Go to M3.9.3. Try another.

    See the answer

    Every point of the shape moves the same distance in the same direction.

    A translation slides the whole shape. The image has the same size and the same way up.

    • If you chose B: That is a reflection. A translation only slides the shape. Go to M3.9.3.
    • If you chose C: In a translation every corner moves by the same amount, so the shape does not change. Go to M3.9.3.

  6. Find the midpoint of (2, 3) and (6, 9).

    See the answer

    (4, 6)

    (2 + 6) ÷ 2 = 4 and (3 + 9) ÷ 2 = 6. The midpoint is (4, 6).

  7. Move the point (2, 3) by 4 right and 5 down. What are its new coordinates?

    See the answer

    (6, −2)

    2 + 4 = 6 and 3 − 5 = −2. The point goes to (6, −2).

  8. From an earlier chapter

    In which quarter of the grid is the point (−6, −1)?

    Tap your answer.

    1. Not this one. The point has a negative across number, so it is left of the y axis. Go to M3.9.2. Try another.

    2. Not this one. The up number is −1, so the point is below the x axis. Go to M3.9.2. Try another.

    3. Yes, that is right. Both numbers are negative. The point is left of the y axis and below the x axis.

    4. Not this one. The across number is −6, so the point is left, not right. Go to M3.9.2. Try another.

    See the answer

    Bottom left

    Both numbers are negative. The point is left of the y axis and below the x axis.

    • If you chose A: The point has a negative across number, so it is left of the y axis. Go to M3.9.2.
    • If you chose B: The up number is −1, so the point is below the x axis. Go to M3.9.2.
    • If you chose D: The across number is −6, so the point is left, not right. Go to M3.9.2.

Keep it

Copy this into your exercise book.

Midpoint: add the two numbers and halve, for across and for up. Translation: add the same numbers to every corner. Reflection in the x axis: up changes sign. In the y axis: across changes sign.

Draw: Draw a triangle on a grid. Slide it 3 right and 2 up. Then reflect it in the y axis. Label the corners.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (761 KB). It opens on any phone.