Lesson 23 — Activity 2: Using Scale to Reproduce 2-D Shapes



Digging Deeper

Before you go on to this activity, click here if you'd like to play a game of plotting coordinates.



Getting Ready


Sometimes it is necessary to make a picture larger or smaller. You could do this on a photocopier or a computer, but what if you didn't have access to either of these? An easy way to make a picture larger or smaller is to use coordinates and a scale on graph paper. Scale means to use proportions to accurately reduce or enlarge objects.


You could make this image larger or smaller.
Image courtesy of www.imagesgoogle.com

Let's say you had a square with corners at (–2,10) (2,10) (2,6), and (–2,6) and you wanted to make it half of its original size. You could show this as the formula (1/2x,1/2y). All this means is that every x and y value is going to be half of what it originally was.

Our original square looks like this:


Now see how this can be changed.

Let's figure out the coordinates of the corners of our new picture. Take each coordinate and divide each number by 2.


(–2 ÷ 2, 10 ÷ 2) = (–1,5)


(2 ÷ 2, 10 ÷ 2) = (1,5)


(2 ÷ 2, 6 ÷ 2) = (1,3)


(–2 ÷ 2, 6 ÷ 2) = (–1,3)



The new square has coordinates of (–1,5) (1,5) (1,3), and (–1,3) and looks like this:


You can see that the square has been reduced in size.


You can also enlarge a picture. Let's say you had a triangle with corners at (1,7) (4,3), and (–2,3) and you wanted to double the size of it. Your formula would be (2x,2y). This means that you will double every x and y value.

Our original picture looks like this:

Now see how this can be changed.


First we need to figure out the coordinates of the corners of our new picture. Take each coordinate and multiply each by 2.

(1 x 2, 7 x 2) = (2,14)


(4 x 2, 3 x 2) = (8,6)


(–2 x 2, 3 x 2) = (–4,6)


The new triangle has coordinates of (2,14) (8,6), and (–4,6) and looks like this:


You can see that the triangle has been increased in size.

What if you only want to move the picture a certain way? Perhaps you want to move a picture over and up a bit? You can do that by either adding or subtracting from your original values. For example, lets say you had a rectangle with corners at (–5,13) (–1,13) (–1,5), and (–5,5) and you want to move it over to the right 3 places and down 2 places. Your formula would be (x + 3,y – 2). This means that you will add 3 to every x value and subtract 2 from every y value.

Our original picture looks like this:

Now see how this can be moved.


First we need to figure out the coordinates of the corners of our new picture. Take each coordinate and add 3 to the x value and subtract 2 from each y value.


(–5 + 3, 13 – 2) = (–2,11)


(–1 + 3, 13 – 2) = (2,11)


(–1 + 3, 5 – 2) = (2,3)


(–5 + 3, 5 – 2) = (–2,3)


The new rectangle has coordinates of (–2,11) (2,11) (2,3), and (–2,3) and looks like this:

You can see that the rectangle has been moved on the grid paper.

You will look more closely at moving an image in an upcoming lesson.