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2D shapes Polygons Quadrilateral Scale factor Coordinate planeVectors

DecimalsHere you will learn about translation math, including how to translate 2D shapes and how to describe translations.

Students will first learn about translation math as part of geometry in 8 th grade.

A **translation** in math is a type of transformation that refers to moving a shape or object from one location to another without changing its size, shape, or orientation. A translation moves a shape in a horizontal direction (left and right) and in a vertical direction (up and down).

A translation is typically described using a **column vector** to help record the movement.

For example,

Shape A has been translated to shape B by the column vector \begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix} .

The column vector gives instructions on how to move each point of the original shape (also known as the pre-image).

\begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix} \text{ is } \; \begin{matrix} 3 \ \text{right}\\ 2 \ \text{up}\\ \end{matrix}The inverse transformation would translate shape B back to shape A using the column vector:

\begin{pmatrix} \; -3 \;\\ \; -2 \; \end{pmatrix} \text{ is } \; \begin{matrix} 3 \ \text{left}\\ 2 \ \text{down}\\ \end{matrix}The **object** or **pre-image** is the name of the original shape (or original figure). The **image** is the name of the shape after it had been translated.

When an object is translated, the object and the image have the **same orientation**. There is no rotation or reflection. The object and the image are **congruent** because they are the same shape and the same size.

Using graph paper or tracing paper can be useful when translating shapes.

How does this relate to 8 th grade math and high school math?

**Grade 8 – Geometry (8.G.A.3)**Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

**High School – Geometry – Congruence (HS.G.CO.A.5)**

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

**High School – Geometry – Congruence (HS.G.CO.B.6)**

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Prepare for math tests in your state with these Grade 3 to Grade 8 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

DOWNLOAD FREEPrepare for math tests in your state with these Grade 3 to Grade 8 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

DOWNLOAD FREEIn order to translate a 2D shape:

**Choose a vertex of the shape.****Move it left or right using the top value in the column vector.****Move it up or down using the bottom value in the column vector.****Repeat for each vertex.**

Translate shape A by the column vector \begin{pmatrix} \; 4 \;\\ \; 1 \; \end{pmatrix} and label the image B.

**Choose a vertex of the shape.**

Choose one of the vertices of the shape (a coordinate) as the point you are going to move. Letβs use the bottom right hand point of the shape.

2**Move it left or right using the top value in the column vector.**

The top number in the column vector is 4 , so move the point \bf{4} **squares to the right**.

3**Move it up or down using the bottom value in the column vector.**

The bottom number in the column vector is 1 , so move the point \bf{1} **square up**.

4**Repeat for each vertex.**

You can move each of the other points in the same way and draw in the rest of the 2D shape.

Translate shape A by the column vector \begin{pmatrix} \; -1 \;\\ \; 3 \; \end{pmatrix} and label the image B.

**Choose a vertex of the shape.**

Choose one of the vertices of the shape as the point you are going to move.

Letβs use the bottom left hand corner of the shape.

**Move it left or right using the top value in the column vector.**

The top number in the column vector is - \, 1 , so move the point \bf{1} **square to the left**.

**Move it up or down using the bottom value in the column vector.**

The bottom number in the column vector is 3 , so move the point \bf{3} **squares up**.

**Repeat for each vertex.**

You can move each of the other points in the same way and draw in the rest of the 2D shape.

Translate shape A by the column vector \begin{pmatrix} \; 0 \;\\ \; -4 \; \end{pmatrix} and label the image B.

**Choose a vertex of the shape.**

Choose one of the vertices of the shape as the point you are going to move. Letβs use the bottom left hand corner of the shape.

**Move it left or right using the top value in the column vector.**

The top number in the column vector is 0 , so there is **no horizontal movement** of the point.

**Move it up or down using the bottom value in the column vector.**

The bottom number in the column vector is - \, 4 , so move the point \bf{4} **squares down**.

**Repeat for each vertex.**

You can move each of the other points in the same way and draw in the rest of the 2D shape.

In order to describe a translation of a shape:

**Pair up two identical vertices.****Work out the horizontal movement.****Work out the vertical movement.****State the column vector.**

Describe the following translation of shape P to shape Q.

**Pair up two identical vertices.**

Pair up a point from the object and the corresponding point on the image. Here one pair of points is chosen. It should not matter which pair of points are chosen.

From the object P the point is (1, \, 1).

For the image Q the point is (3, \, 2).

**Work out the horizontal movement.**

Using the grid or the x -coordinates, work out the horizontal movement. There is a shift of \bf{2} **to the right**.

**Work out the vertical movement.**

Using the grid or the y -coordinates, work out the vertical movement. There is a shift of \bf{1} **upwards**.

**State the column vector.**

Now you need to write the horizontal shift and the vertical shift in a column vector.

\begin{matrix} 2 \ \text{right}\\ 1 \ \text{up}\\ \end{matrix} \; \text{ is } \begin{pmatrix} \; 2 \;\\ \; 1 \; \end{pmatrix}

Shape P has been translated to shape Q by the column vector \begin{pmatrix}
\; 2 \;\\
\; 1 \;
\end{pmatrix} .

Describe the translation of shape A to shape B.

**Pair up two identical vertices.**

Pair up a point from the object and the corresponding point on the image. Here one pair of points is chosen. It should not matter which pair of points are chosen.

From the object A the point is (6, \, 4).

For the image B the point is (1, \, 2).

**Work out the horizontal movement.**

Using the grid or the x -coordinates, work out the horizontal movement. There is a shift of \bf{3} **to the left**.

**Work out the vertical movement.**

Using the grid or the y -coordinates, work out the vertical movement. There is a shift of \bf{4} **downwards**.

**State the column vector.**

Now you need to write the horizontal shift and the vertical shift in a column vector.

\begin{matrix} 3 \ \text{left}\\ 4 \ \text{down}\\ \end{matrix} \; \text{ is } \begin{pmatrix} \; -3 \;\\ \; -4 \; \end{pmatrix}

Shape A has been translated to shape B by the column vector \begin{pmatrix}
\; -3 \;\\
\; -4 \;
\end{pmatrix} .

Describe the translation of shape A to shape B.

**Pair up two identical vertices.**

Pair up a point from the object and the corresponding point on the image. Here one pair of points is chosen. It should not matter which pair of points are chosen.

From the object A the point is (1, \, 1).

For the image B the point is (1, \, 6).

**Work out the horizontal movement**

Using the grid or the x -coordinates, work out the horizontal movement. There is **no horizontal shift**.

**Work out the vertical movement.**

Using the grid or the y -coordinates, work out the vertical movement. There is a shift of \bf{5} **upwards**.

**State the column vector.**

Now you need to write the horizontal shift and the vertical shift in a column vector.

\begin{matrix} 0 \ \text{left or right}\\ 5 \ \text{up}\\ \end{matrix} \; \text{ is } \begin{pmatrix} \; 0 \;\\ \; 5 \; \end{pmatrix}

Shape A has been translated to shape B by the column vector \begin{pmatrix}
\; 0 \;\\
\; 5 \;
\end{pmatrix} .

- Start by introducing translations in the coordinate plane. Explain that translating a shape means moving it horizontally (along the x -axis) or vertically (along the y -axis), while the shape remains the same in size and orientation.

- Connect translation examples to everyday actions, like sliding a book across a table or moving a piece on a board game, to make it relatable.

- Connect translation to their previous knowledge of coordinate systems from 6 th and 7 th grade. Explain that just like they used coordinates to locate points, they will now use translation vectors to move these points to new coordinates.

- Once students are comfortable with single translations, give worksheets that include multiple translations, first a horizontal translation then a vertical translation or vice versa. This reinforces the idea of movement along both the x and y axes.

- Introduce the translation rule, which is typically written as (x, \, y) \rightarrow(x+a, \, y+b), where a is the horizontal shift (left or right), and b is the vertical shift (up or down).

- Break down translations into steps, calculating each of the new coordinates based on the translation rule. This helps students understand how each axis is affected independently.

**Not checking the scale of the coordinate grid**

Be careful to consider the scaling on the axes.

For example,

What is the column vector for the translation of shape P to shape Q?

If you count the squares, it may appear that shape P has been translated by the column vector \begin{pmatrix} \; 6 \;\\ \; 2 \; \end{pmatrix}.

But if by considering the scale on the axes, the correct column vector is \begin{pmatrix} \; 3 \;\\ \; 1 \; \end{pmatrix} .

**Misinterpreting the column vector**

Remember, the top number is for horizontal movement. A positive number moves the shape to the right and a negative number moves the shape to the left.

The bottom number is for vertical movement. A positive moves the shape upwards and a negative number moves the shape downwards.

**Confusing object and image**

The original shape is the object or pre-image and the translated shape is the image. Make sure you know which shape is the original shape and start there when describing transformations such as translations.

- Transformations
- Rotations
- Reflection in math
- Dilations
- Center of dilation

1. Translate the shaded shape by the column vector \begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix} .

Choose a point on the vertex of the shape and move it 3 to the right and 2 up. Do the same with all the points, or carefully draw in the rest of the image.

\begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}Β \text{ is } \; \begin{matrix} 3 \ \text{right}\\ 2 \ \text{up}\\ \end{matrix}

2. Translate the shaded shape by the column vector \begin{pmatrix} \; -4 \;\\ \; 1 \; \end{pmatrix} .

Choose a point on the vertex of the shape and move it 4 to the left and 1 up. Do the same with all the points, or carefully draw in the rest of the image.

\begin{pmatrix} \; -4 \;\\ \; 1 \; \end{pmatrix} \text{ is } \; \begin{matrix} 4 \ \text{left}\\ 1 \ \text{up}\\ \end{matrix}

3. Translate the shaded shape by the column vector \begin{pmatrix} \; 3 \;\\ \; 0 \; \end{pmatrix} .

Choose a point on the vertex of the shape and move it 3 to the right and 0 up or down. Do the same with all the points, or carefully draw in the rest of the image.

\begin{pmatrix} \; 3 \;\\ \; 0 \; \end{pmatrix} \text{ is } \; \begin{matrix} 3 \ \text{right}\\ 0 \ \text{up or down}\\ \end{matrix}

4. Describe the transformation of shape A to shape B.

Translation by \begin{pmatrix} \; 2 \;\\ \; 3 \; \end{pmatrix}

Translation by \begin{pmatrix} \; -3 \;\\ \; -2 \; \end{pmatrix}

Translation by [ \begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}

Translation by \begin{pmatrix} \; -2 \;\\ \; -3 \; \end{pmatrix}

Choose a point on the object shape A and its corresponding point on the image shape B. Count the shift in the horizontal direction. Count the shift in the vertical direction.

\begin{matrix} 2 \ \text{right}\\ 3 \ \text{up}\\ \end{matrix} \; \text{ is } \begin{pmatrix} \; 2 \;\\ \; 3 \; \end{pmatrix}

5. Describe the transformation of shape A to shape B.

Translation by \begin{pmatrix} \; 4 \;\\ \; 3 \; \end{pmatrix}

Translation by \begin{pmatrix} \; -3 \;\\ \; 4 \; \end{pmatrix}

Translation by \begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix}

Translation by \begin{pmatrix} \; -3 \;\\ \; -4 \; \end{pmatrix}

Choose a point on the object shape A and its corresponding point on the image shape B. Count the shift in the horizontal direction. Count the shift in the vertical direction.

\begin{matrix} 4 \ \text{right}\\ 3 \ \text{down}\\ \end{matrix} \; \text{ is } \begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix}

6. Describe the transformation of shape A to shape B.

Translation by \begin{pmatrix} \; 4 \;\\ \; -1 \; \end{pmatrix}

Translation by \begin{pmatrix} \; 1 \;\\ \; -4 \; \end{pmatrix}

Translation by \begin{pmatrix} \; 4 \;\\ \; 1 \; \end{pmatrix}

Translation by \begin{pmatrix} \; 1 \;\\ \; 4 \; \end{pmatrix}

Choose a point on the object shape P and its corresponding point on the image shape Q. Count the shift in the horizontal direction. Count the shift in the vertical direction.

\begin{matrix} 1 \ \text{right}\\ 4 \ \text{down}\\ \end{matrix} \; \text{ is } \begin{pmatrix} \; 1 \;\\ \; -4 \; \end{pmatrix}

A translation in math is a type of transformation where a shape or object is moved from one position to another on the Cartesian plane without changing its size, shape, or orientation. Each point of the shape moves the same distance in the same direction.

A translation is described by a vector (a, \, b), where a represents the horizontal shift (positive for right, negative for left) and b represents the vertical shift (positive for up, negative for down).

If point A \, (x, \, y) is translated by a vector (a, \, b), the new coordinates of point Aβ are given by x'=x+a and y'=y+b.

Some common mistakes include mixing up the x -axis and y -axis when applying the translation, forgetting to apply the translation to every point of the shape to find the new points, misinterpreting the direction of negative or positive values in the translation vector.

Yes, real-world examples of translations in math include sliding a book across a table, shifting a piece on a chessboard, or moving an image across a computer screen.

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[FREE] Common Core Practice Tests (3rd to 8th Grade)

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