Minds On

Notice and wonder

Explore the two pairs of shapes provided.

2 triangles. They are different sizes, and oriented differently. 2 rectangles that are the same size, but oriented differently.

How are the triangles the same? How are they different?

How are the rectangles the same? How are they different?

Record your ideas in a notebook or a method of your choice.

Action

Task 1: Transformations of a shape

What is a transformation?

A transformation on a shape results in a change to its position or its size. As a shape transforms, its vertices (points on the grid) move. The transformation describes the results of the movement.

Transformations include:

  • translations – slides
  • rotations – turns
  • reflections – flips

A triangle with vertices at points ( 1 , 2 ) , ( 3 , 2 ) , and ( 2 , 5 ) on a Cartesian plane undergoes translation, rotation, and reflection. The first triangle is translated 4 units to the right and has vertices at points ( 5 , 2 ) , ( 7 , 2 ) , and ( 6 , 5 ) . The second triangle is rotated to the left and has vertices at points ( 1 , 2 ) , ( 1 , 4 ) , and ( - 2 , 3 ) . The third triangle has been reflected along the y-axis and has vertices at points ( 1 , - 2 ) , ( 3 , - 2 ) , and ( 2 , - 5 ) .

Task 2: Translations

What is a translation?

A translation is a type of transformation that moves all points of a shape the same distance and direction across a grid. When the translation is completed, there will be a new shape, congruent to the one that was translated. Congruent means the figures are the exact same size and shape.

The image on the left contains two squares of the same size with a check mark underneath them and the word “Congruent.” The image on the right contains two squares of different sizes with an x underneath them and the words “Not congruent.

To complete a proper translation, you must make sure all points on the shape have moved:

  • the same distance
  • the same direction

As the shape is translated, the original shape slides to create the translated shape. The direction and distance the shape moves is called the translation vector. The new shape is called the translation image. A translation vector is a value that describes the direction and distance moved.

When expressing a translation vector, you can do so using arrows. For example: (5→ 3↓) would mean the shape moves 5 to the right and 3 down.

Examine the following translation vectors. Explain in your own words what it is instructing a shape to do. Record your answers using a method of your choice, and then check your answer.

Question 1. (8→, 5↑)

Move 8 to the right and 5 up.

Question 2. (9←, 10↓)

Move 9 to the left and 10 down.

Question 3. (7→, 0)

Move 7 to the right.

Examine the following statements. Record your answers using a method of your choice, and then check your answer.

Question 1. 4 to the left and 3 up

(4←, 3↑)

Question 2. 0 and 6 up

(0, 6↑)

Question 3. 1 to the right, then 10 down

(1→, 10↓)

Labelling translations

Translation vectors can be used to identify how to translate a shape on a Cartesian plane.

The following is a Cartesian plane with a square that has been translated into a second square. Examine the two squares in the grid.

  • What do they have in common?
  • What are the differences between them?

Record your thoughts using a method of your choice.

A Cartesian plane with a square in Quadrant 2 that has been translated into a second square in Quadrant 3.

One square is labelled A, B, C, D. The other square is labelled A’, B’, C’, D’.

When points are transformed (reflected, translated, or rotated) the image is often depicted using a tick mark or apostrophe known as a prime symbol. This ( ’ ) is how you can distinguish between the original shape and the translated shape.

Completing translations

Complete the following translation:

Question 1. For the coordinate (-3, 4), translate 2 units right and 4 units down, (2→, 4↓).

Record the coordinates of the translated point, and then check your answer.

The following grid shows the translated point at (-1, 0)

Three points are plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by ones. The point of origin is (0, 0).

The letter A is at the coordinate: (-3, 4). One arrow points to the right two units to (-1,4). A second arrow points down four units to the final coordinate (-1, 0).

Question 2. Translate the points A (-2, 3), B (-4, -1) and C (-2, -5) : 4 units right and 3 units up (4→, 3↑).

Record the original and translated points using a method of your choice.

Six points are plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by 1s. The point of origin is (0, 0). The letter A is at the coordinates (-2, 3), the letter B is at (-4, -1), and the letter C is at (-2, -5). The letter A’ is at the coordinates (2, 6), the letter B’ is at (0, 2), and the letter C’ is at (2, -2).

Question 3. Translate the following triangle 5 units left and 4 units down (5←, 4↓).

Record the coordinates of the original shape and the translated shape.

A triangle is plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by ones. The point of origin is (0, 0). The letter A is at the coordinates (3, 9), the letter B is at (3, 3), and the letter C is at (9, 3).

Question 4. Translate the parallelogram 6 units right and 6 units up (6→, 6↑).

Label the translated shape appropriately.

Record the coordinates of the original shape and the translated shape.

A parallelogram is plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by ones. The point of origin is (0, 0). The letter A is at the coordinates (-7, -4), the letter B is at (-2, -4), the letter C is at (-9, -9), and the letter D is at (-4, -9).

Question 5. A trapezoid has vertices A (3, -5), B (6, -5), C (2, -8), and D (7, -8). Predict where the trapezoid will be when it has been translated (8←, 6↑).

Record the new coordinates of the translated shape.

Describe your translation in words in a notebook or a method of your choice

A trapezoid is plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by ones. The point of origin is (0, 0). The letter A is at the coordinates (3, -5), the letter B is at (6, -5), the letter C is at (2, -8), and D is at (7, -8).

Task 3: Reflections

What is a reflection?

A reflection is a type of transformation that flips a shape or figure over a line.

When flipped, it forms a congruent shape.

This reflected shape is referred to as the reflection image, as it is the mirror image of the original shape.

Reflections involve a line called the “reflection line.” This line acts as a mirror.

All points on the original shape are flipped over the reflection line to create the reflection image, also referred to as the mirror image.

All points on the reflected image will be the same distance from the reflection line as the original points.

Reflections are symmetrical.

An example of a reflection is in the following grid. The reflection line is labelled. You will notice that each point on the triangle is the same distance from the reflection line.

Grid with a triangle and its mirror image. There is an arrow pointing to the dotted line in the middle labelled “reflection line.”

Steps to completing a reflection:

  1. Measure all points (by counting the spaces on the grid or using a ruler) on the original shape from the reflection line
  2. Measure that same distance on the other side of the reflection line (directly across), and place a dot on each coordinate
  3. Connect the dots to form the shape

Helpful hints:

  • If the reflection is the x-axis, then change each (x, y) to be (x, -y). Here is an example 7, 4 then becomes 7, -4.
  • If the reflection is the y-axis, then change each (x, y) to be (-x, y). Here is an example: 1, 2 then becomes -1, 2.
If the reflection is… Then… Example
The x-axis Change each (x, y) to be (x, -y) (7, 4) becomes (7, -4)
The y-axis Change each (x, y) to be (-x, y) (1, 2) becomes (-1, 2)

Labelling reflections

Similar to translations, when you complete a reflection, the mirror image or shape that is the reflected shape will have a tick ( ’ ) in the top right corner of the letter. This is how you can distinguish between the original shape and the translated image.

Cartesian plane with a reflection of point A. A is at ( 5 ,   3 ) . The reflected point is labelled as A’. A’ is at ( 5 ,   - 3 ) .

Notice that A and A’ are the same distance from the x-axis (the reflection line).

Completing reflections

Examine each of the points and shapes, and complete a reflection of each one following the criteria provided. Use the first two as examples. Label the shapes appropriately. Work with a partner or independently, and record your ideas using a method of your choice.

Question 1. Reflect point A (3, 5), B (6, 5), C (3, 1), and D (6, 1) on the x-axis. Label the original and reflected points appropriately, and connect the dots of each shape.

A rectangle and its reflection are plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by ones. The point of origin is ( 0 ,   0 ) . The original rectangle has coordinates A ( 3 ,   5 ) , B ( 6 ,   5 ) , C ( 3 ,   1 ) , and D ( 6 ,   1 ) . The reflected rectangle has coordinates A’ ( 3 ,   - 5 ) , B’ ( 6 ,   - 5 ) , C’ ( 3 ,   - 1 ) , D’ ( 6 ,   - 1 ) .

Question 2. Reflect the trapezoid with the y-axis as the reflection line. Label the new coordinates appropriately and connect the dots of the new shape. Describe your translation reflection.

Cartesian plane with a trapezoid on it. Its coordinates are: A (1, 4), B (5, 4), C (2, 1) and D (4, 1).

A trapezoid and its reflection are plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by ones. The point of origin is ( 0 ,   0 ) . The original trapezoid has coordinates A ( 1 ,   4 ) , B ( 5 ,   4 ) , C ( 2 ,   1 ) , and D ( 4 ,   1 ) . The reflected trapezoid would be A’ ( - 1 ,   4 ) , B’ ( - 5 ,   4 ) , C’ ( - 2 ,   1 ) , D’ ( - 4 ,   1 ) .

Question 3. Predict where the parallelogram would be if it was reflected on the y-axis. Then, reflect the parallelogram with the y-axis as the reflection line. Label the new coordinates appropriately and connect the dots of the new shape. Record your thoughts using a method of your choice.

Cartesian plane with a parallelogram on it. The coordinates are: A (2, 5), B (6, 8), C (6, -4) and D (2, -7).

Task 4: Translations and reflections

Now that you have learned to complete translations and reflections, we are going to learn how to combine them and complete transformations that include both translations and reflections.

Record your ideas in a notebook or a method of your choice.

Complete each of the activities to translate and reflect the shapes provided, the first has an answer if you press ‘Suggested Answer.’

Question 4. Translate the triangle: 2 units left and 3 units down. Then reflect it over the y-axis.

Cartesian plane with a triangle labelled A (-4, 1), B (-1, 1), and C (-4, 4).

A triangle with both translation and reflection are plotted on a Cartesian plane. The x-axis ranges from –10 to 10. The y-axis ranges from –10 to 10. Both axes increase by ones. The point of origin is ( 0 ,   0 ) . The original triangle has coordinates A ( - 4 ,   1 ) , B ( - 1 ,   1 ) , and C ( - 4 ,   4 ) . The translated triangle would be ( - 6 ,   - 2 ) , ( - 3 ,   - 2 ) , ( - 6 ,   - 1 ) , and the reflected and then translated triangle would be A’ ( 6 ,   - 2 ) , B’ ( 3 ,   - 2 ) , and C’ ( 6 ,   1 ) .

Question 5. Translate the rectangle: 4 units right and 1 unit up. Then reflect it over the y-axis.

Cartesian plane with a rectangle labelled A (3, 4), B (4, 4), C (4, -5), and D (3, -5).

Question 6. Translate the rhombus: 5 units right and 3 unit up. Then reflect it over the x-axis.

Cartesian plane with a rhombus labelled A (2,-3), B (4, -2), C (6, -3), and D (4, -4).

Consolidation

Designing a room plan

Translation in action

Online or on paper, create a blank Cartesian plane onto which you can design a room for students to learn in.

Your task is to decorate the classroom with various pieces of furniture in it.

Determine the location of the furniture by labeling and indicating the coordinates of each item, or by creating a physical model.

Include at least five pieces of furniture on your room plan.

You can also record a detailed description of your classroom design and the location of each item, in a notebook or another method of your choice.

Transforming a room plan

There are various pieces of furniture around the room you designed.

A sample design of a living room space with a variety of shapes including squares, rectangles, a triangle and a pentagon

Your job is to rearrange your room plan. You can move the furniture around the room however you would like. All furniture must be transformed (i.e., translated or reflected) at least once. You must perform some translations and some reflections. Record all the coordinates of the transformed objects, or describe the transformations using a method of your choice.

If you would like, you can complete the activity using TVO Mathify. You can also use your notebook, or the following fillable and printable Creating a Plan for a Classroom document.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

TVO Mathify (Opens in a new window) Activity (Open PDF in a new window)

Questions to reflect on

Record your response to the following questions in a notebook or a method of your choice:

  1. How are translating and reflecting shapes different? How are they similar?
  2. What strategies can be helpful to complete translations and reflections?

Reflection

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

I feel...

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.

Connect with a TVO Mathify tutor

Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.

TVO Mathify (Opens in a new window)