# 11.17: Integer Translations, Paths Between Points

**At Grade**Created by: CK-12

**Practice**Integer Translations, Paths Between Points

Have you ever tried to track someone's movements by using a map? Take a look at this situation.

Caro is having a party. She has decided to use a coordinate grid as a map to show a treasure hunt. Caro loves math and thinks that this could be a way to incorporate two of her favorite things together - scavenger hunts and mathematics. She draws a coordinate grid and plots the first two points on the map.

The first point is \begin{align*}B\end{align*} and the second point is \begin{align*}B'\end{align*}.

Here is what the grid looked like.

Caro wants to draw more points on her map, but first she must record the movement of the path as an integer translation.

Do you know how to do this?

**This Concept will teach you how to describe paths of points as integer translations. Then we'll return to this problem at the end of the Concept.**

### Guidance

A ** translation** is when a figure or a point is moved on a coordinate grid. It is when you slide a figure or a point on a grid. We can use integers to assist us in indentifying different translations.

**Here, we started with point \begin{align*}A\end{align*}. Then point \begin{align*}A\end{align*} was moved on the coordinate grid. It has been translated to a new location. When we slide or translate a point or figure, the new point has a little symbol next to it. Here is how we write a translation.**

\begin{align*}A\end{align*} **to** \begin{align*}A'\end{align*}

**We can use integers to show the path of the translation.**

**How many units did \begin{align*}A\end{align*} move on the \begin{align*}x\end{align*} axis?**

If you count, you can see that it moved +6 units.

\begin{align*}A\end{align*} **to** \begin{align*}A'\end{align*}

**We can use integers to show the path of the translation.**

**How many units did \begin{align*}A\end{align*} move on the \begin{align*}x\end{align*} axis?**

If you count, you can see that it moved +6 units.

**How many units did \begin{align*}A\end{align*} move on the \begin{align*}y\end{align*} axis?**

If you count, you can see that it moved -2 units (remember that negative 'y' means down).

**We write the translation as (6, -2).**

**Try one of these on your own. Write the path of \begin{align*}B\end{align*} as an integer translation.** **How many units did \begin{align*}A\end{align*} move on the \begin{align*}y\end{align*} axis?**

If you count, you can see that it moved -2 units (remember that negative 'y' means down).

**We write the translation as (6, -2).**

Try some of these on your own. Write the path of \begin{align*}B\end{align*} as an integer translation.

#### Example A

If you start at (3,5) and move three units across and four units up, where are you? Write this as an integer translation.

**Solution: \begin{align*}(6,9)\end{align*}**

#### Example B

True or false. You move horizontally before vertically when figuring out the path of a translation.

**Solution: True**

#### Example C

True or false. A translation with a path of \begin{align*}(-2,3)\end{align*} means that starting at the original point, you move two places to the left and three places up.

**Solution: True**

Here is the original problem once again.

Caro is having a party. She has decided to use a coordinate grid as a map to show a treasure hunt. Caro loves math and thinks that this could be a way to incorporate two of her favorite things together - scavenger hunts and mathematics. She draws a coordinate grid and plots the first two points on the map.

The first point is \begin{align*}B\end{align*} and the second point is \begin{align*}B'\end{align*}.

Here is what the grid looked like.

Caro wants to draw more points on her map, but first she must record the movement of the path as an integer translation.

Do you know how to do this?

To write the movement of point \begin{align*}B\end{align*} as an integer translation, we must first write where it started.

The point is plotted at \begin{align*}(-2,-1)\end{align*}.

Then we can write the movement of point \begin{align*}B\end{align*} to \begin{align*}B'\end{align*} as an integer translation.

First, we count the units horizontally from one point to the next.

\begin{align*}5\end{align*}

Then we move up vertically three units.

\begin{align*}3\end{align*}

The integer translation is \begin{align*}(5,3)\end{align*}.

**This is our answer.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Quadrants
- the four sections of a coordinate grid

- Origin
- the place where the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axis’ meet at (0, 0)

- Ordered Pair
- the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values used to locate points on a coordinate grid \begin{align*}(x,y)\end{align*}

- \begin{align*}x\end{align*} axis
- the horizontal axis on the coordinate grid

- \begin{align*}y\end{align*} axis
- the vertical axis on the coordinate grid

- Coordinates
- the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values of an ordered pair

- Longitude
- vertical measure of degrees on a map

- Latitude
- horizontal measure of degrees on a map

### Guided Practice

Here is one for you to try on your own.

A point is plotted at \begin{align*}(-2,1)\end{align*} if the integer translation is \begin{align*}(3,-4)\end{align*}, what are the coordinates of the next point?

**Answer**

To figure this out, we must first plot the first point. Then we move horizontally three units and vertically down four units.

The final point is plotted at \begin{align*}(1,-3)\end{align*}.

### Video Review

Here is a video for review.

Khan Academy: Quadrants of Coordinate Plane - This video has content which supports student success in this Concept.

### Practice

Directions: Use integers to identify each translation.

1. \begin{align*}A\end{align*} to \begin{align*}A'\end{align*}

2. \begin{align*}B\end{align*} to \begin{align*}B'\end{align*}

3. \begin{align*}C\end{align*} to \begin{align*}C'\end{align*}

4. \begin{align*}D\end{align*} to \begin{align*}D'\end{align*}

5. \begin{align*}E\end{align*} to \begin{align*}E'\end{align*}

6. \begin{align*}F\end{align*} to \begin{align*}F'\end{align*}

7. \begin{align*}G\end{align*} to \begin{align*}G'\end{align*}

8. \begin{align*}H\end{align*} to \begin{align*}H'\end{align*}

9. \begin{align*}H\end{align*} to \begin{align*}E\end{align*}

10. \begin{align*}E\end{align*} to \begin{align*}D\end{align*}

11. \begin{align*}E'\end{align*} to \begin{align*}G\end{align*}

12. \begin{align*}E'\end{align*} to \begin{align*}C\end{align*}

13. \begin{align*}B\end{align*} to \begin{align*}A'\end{align*}

14. \begin{align*}B\end{align*} to \begin{align*}G'\end{align*}

15. \begin{align*}C'\end{align*} to \begin{align*}D'\end{align*}

axis

The axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.axis

The -axis is the vertical number line of the Cartesian plane.Coordinates

The coordinates of a point represent the point's location on the Cartesian plane. Coordinates are written in ordered pairs: .Latitude

Latitude is a coordinate that specifies the north-south location of a point on the Earth's surface.Longitude

Longitude is a coordinate that specifies the east-west location of a point on the Earth's surface.Ordered Pair

An ordered pair, , describes the location of a point on a coordinate grid.Origin

The origin is the point of intersection of the and axes on the Cartesian plane. The coordinates of the origin are (0, 0).Quadrants

A quadrant is one-fourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the top-right, and increasing counter-clockwise.### Image Attributions

Here you'll learn to describe paths between points as integer translations.

## Concept Nodes:

axis

The axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.axis

The -axis is the vertical number line of the Cartesian plane.Coordinates

The coordinates of a point represent the point's location on the Cartesian plane. Coordinates are written in ordered pairs: .Latitude

Latitude is a coordinate that specifies the north-south location of a point on the Earth's surface.Longitude

Longitude is a coordinate that specifies the east-west location of a point on the Earth's surface.Ordered Pair

An ordered pair, , describes the location of a point on a coordinate grid.Origin

The origin is the point of intersection of the and axes on the Cartesian plane. The coordinates of the origin are (0, 0).Quadrants

A quadrant is one-fourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the top-right, and increasing counter-clockwise.