E. Spatial Sense · Angles, Rectangles, Grids and Measurement

Reflections: Flipping Shapes on a Grid

E1.3 A1About 40 minutes

Learning goals

I can reflect a point or a shape over a mirror line on a grid.
I can explain that each point of the image is the same distance from the mirror line as the original.
I can tell the difference between a translation and a reflection.
I can predict what a shape will look like after a reflection.

Words to know

reflection
A flip. A shape is flipped over a line to make a mirror image.

line of reflection
The line a shape is flipped over. It is also called the mirror line.

mirror image
A shape that looks like the original seen in a mirror. It faces the opposite way.

image
The new shape after a move, like a translation or a reflection.

Remember from Grade 2 and the last lesson

In Grade 2 you used a mirror to check lines of symmetry. In the last lesson you translated (slid) shapes. Today you will learn a new move: the reflection, or flip.

Let's start at a lake

On a calm day, a loon floating on a lake makes a perfect upside-down picture of itself in the water. The water is like a mirror. That picture is a reflection.

In math, a reflection flips a shape over a line, called the line of reflection or mirror line. The new shape is a mirror image: the same size and shape, but facing the opposite way.

How to reflect a shape on a grid

Pick one vertex. Count how many units it is from the mirror line.
Count the same number of units on the other side of the line, going straight across. Mark the image point there.
Do this for every vertex.
Join the image points in the same order.

A point that is right on the mirror line stays where it is.

A flag on the left is reflected over the dashed mirror line (the vertical line through 5 on the x-axis). The image on the right faces the opposite way.

Worked example

Example 1: Reflect the flag

Reflect the flag over the vertical mirror line through 5 on the x-axis. The flag's vertices are (2, 1), (2, 5), (4, 4) and (2, 3).

(2, 5) is 3 units left of the line (5 − 2 = 3). Count 3 units right of the line: 5 + 3 = 8. Image: (8, 5).
(4, 4) is 1 unit left of the line. Count 1 unit right: 5 + 1 = 6. Image: (6, 4).
(2, 3) and (2, 1) are also 3 units left of the line, so their images are (8, 3) and (8, 1).
Notice: the y-numbers stay the same. The flag pointed right; the image points left.

The image has vertices (8, 1), (8, 5), (6, 4) and (8, 3). It is a mirror image of the flag.

Mirror lines that go across

A mirror line can also go across (horizontal). Then you count units up or down to the line instead. The shape flips upside down.

A triangle above the horizontal mirror line (through 3 on the y-axis) and its reflection below the line.

Worked example

Example 2: Reflect over a horizontal line

The triangle above has vertices (1, 4), (4, 4) and (1, 6). Reflect it over the horizontal line through 3 on the y-axis.

(1, 4) is 1 unit above the line. The image is 1 unit below: (1, 2).
(4, 4) is also 1 unit above. Image: (4, 2).
(1, 6) is 3 units above. The image is 3 units below the line: 3 − 3 = 0. Image: (1, 0).
This time the x-numbers stay the same, and the triangle is now upside down.

The image has vertices (1, 2), (4, 2) and (1, 0).

Translation or reflection?

After a translation (slide), the image faces the same way as the original.
After a reflection (flip), the image faces the opposite way, like a mirror image.

Three pairs of L-shapes. Pair 1 is a slide. Pair 2 is flipped over a vertical line. Pair 3 is flipped over a horizontal line.

Common mistake

Don't just copy the shape on the other side of the line. That is a slide, not a flip! Count from the line for each vertex. Parts that were close to the line stay close, and parts that were far stay far.

Math talk: test with a fold

If you are not sure, fold your paper along the mirror line and hold it up to a window. The shape and its image should land exactly on top of each other. If they do, your reflection is correct.

Just for fun

The famous artist Leonardo da Vinci wrote many of his notebook pages in mirror writing, from right to left. Every letter was a reflection! Try writing your name that way and check it in a mirror.

Hands-on activity

Paint blot and grid reflections

You need: grid paper, a pencil, a ruler, a small mirror (if you have one) and, if you like, a little paint or a washable marker.

Draw a vertical mirror line down the middle of your grid paper.
On the left side, draw a simple shape with its corners on grid points, like a flag, an arrow or the first letter of your name.
Predict what the reflection will look like. Then count units from the line for each corner and draw the image on the right side.
Stand a mirror on the line to check. Does the mirror show the same picture you drew?
Fold the paper on the line and hold it up to a window. Do the two shapes match?
Bonus: fold a blank sheet in half, open it, put a few drops of paint on one side near the fold, then press the halves together. Open it to see a reflection!

Grown-up note: cover the table for the paint, and take care with a glass mirror.

Math talk: "Which corners were the hardest to reflect? How did you check them?"

Points P, Q and R and a dashed mirror line through 4 on the x-axis, for the questions below.

Your turn

Point P is at (1, 3). How many units is it from the mirror line?

Reflect P over the mirror line. What are the coordinates of the image P′?

Reflect Q (3, 5) over the mirror line. Where is Q′?

Point R (4, 2) is on the mirror line. Where is its image?

Look at the three pairs of L-shapes. Which pair shows a translation?

Which pairs show a reflection? How can you tell?

More practice: warm-up

What is another word for a reflection?

What is the line a shape is flipped over called?

Does a reflection change the size of a shape?

After a reflection, does the image face the same way or the opposite way?

A point is 2 units to the left of a mirror line. Where is its image?

More practice: on your own

Reflect the point (2, 6) over the vertical line through 5 on the x-axis.

Reflect the point (3, 5) over the horizontal line through 3 on the y-axis.

A triangle has vertices (1, 1), (3, 1) and (1, 4). Reflect it over the vertical line through 4 on the x-axis.

Tom reflects the letter R over a vertical mirror line. Will the image look like a normal R? Explain.

Ana copies a shape on the other side of a mirror line without flipping it. What move did she really do?

Challenge

A shape is reflected over a vertical mirror line, and then its image is reflected over the same line again. Where does it end up?

A point and its image after a reflection are at (2, 4) and (8, 4). Where is the mirror line?

Draw a square with vertices (1, 1), (3, 1), (3, 3) and (1, 3). Reflect it over the vertical line through 3. Then slide the original square 2 units right. What do you notice? Why?

Check your understanding

Quiz

What is a reflection?

A point is 4 units to the right of a vertical mirror line. Where is its image?

Reflect (1, 2) over the vertical line through 3 on the x-axis.

How can you tell a reflection from a translation?

Where is the image of a point that is on the mirror line?

Reflect (4, 5) over the horizontal line through 3 on the y-axis.

For parents and teachers

Big idea: The Ontario curriculum (E1.3) asks Grade 4 students to describe and perform translations and reflections on a grid, and predict the results. A reflection makes a mirror image: every point of the image is the same distance from the mirror line as the original point, on the opposite side.

Common difficulties:

Copying instead of flipping. Children often draw the shape the same way on the other side of the line (a translation). Count from the line for each vertex.
Counting from the edge of the grid instead of from the mirror line.
Horizontal mirror lines. These feel harder. Turn the page sideways if it helps.

How to help: A small mirror is a great tool. Paint-blot butterflies, reflections in a lake and backward writing in a mirror are all fun examples. Ask your child to predict before drawing: "Which way will it face?"