11.17: 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
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
We can use integers to show the path of the translation.
How many units did
If you count, you can see that it moved +6 units.
Here, we started with point
We can use integers to show the path of the translation.
How many units did
If you count, you can see that it moved +6 units.
How many units did
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
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
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:
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
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
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
The point is plotted at
Then we can write the movement of point
First, we count the units horizontally from one point to the next.
Then we move up vertically three units.
The integer translation is
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
x andy axis’ meet at (0, 0)
 Ordered Pair

the
x andy values used to locate points on a coordinate grid(x,y)

x axis  the horizontal axis on the coordinate grid

y axis  the vertical axis on the coordinate grid
 Coordinates

the
x andy 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
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
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.
2.
3.
4.
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*}
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Term  Definition 

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 northsouth location of a point on the Earth's surface. 
Longitude  Longitude is a coordinate that specifies the eastwest 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 onefourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the topright, and increasing counterclockwise. 
Image Attributions
Here you'll learn to describe paths between points as integer translations.