GRN - Distances Between Points Lesson

Distances Between Points 

Remember, x comes before y in the alphabet, so you will always move right or left along the x-axis first.

Right or left first and then up or down! 

If we were to plot 7 and −7, what would the distance of each point be from zero?

We would know that from 7 to 0, it's 7 steps. From -7 to 0, it's also 7 steps. 

So what is the distance between 7 and −7? We can just think about combining the 7 steps plus 7 steps right?

We used what we know about the absolute value to combine 7 plus 7 to get that they are 14 steps away from each other. We could write that as a number sentence like this- 

|7| + |-7| = 14

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The same is true when thinking about points on a vertical number line. If we were to plot 8  and −8, what would the distance of each point be from zero on the vertical number line?

We would know that from 8 to 0, it's 8 steps. From -8 to 0, it's also 8 steps. 

So what is the distance between 8 and −8? We can just think about combining the 8 steps plus 8 steps right?

We used what we know about the absolute value to combine 8 plus 8 to get that they are 16 steps away from each other. We could write that as a number sentence like this- 

|8| + |-8| = 16

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Now you try! 

How many steps apart would 4 and -4 be on the horizontal number line? 

 

How many steps apart would 4 and -4 be on the vertical number line? 

In both examples, they are 8 steps apart. |4| + |-4| = 8 

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So when we put this together into the coordinate plane, we can think about reflecting across the x-axis and across the y-axis. 

In the example below, if we reflected the shapes across the x-axis, where would their new location be? Think about the x-axis as being horizontal. If the shape was below the x-axis, it will now be above it. If the shape was above the x-axis, it will now be below it. Where will their new locations be once they're reflected across the x-axis? How many steps apart are the new locations from the original locations?

The lightning moves from (4,-4) to (4,4). They are 8 steps apart. 

The smiley face moves from (-3,3) to (-3,-3). They are 6 steps apart. 

The triangle moves from (-1,5) to (-1,-5). They are 10 steps apart. 

The star moves from (1,3) to (1,-3). They are 6 steps apart. 

You should notice that reflecting over the x-axis means that the x-value stays the same, but that the sign changes for the y-value. 

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In the example below, it's the same picture but we are reflecting the shapes across the y-axis this time. So let's find out where their new location would be. Think about the y-axis as being vertical. If the shape was to the left of the y-axis, it will now be to the right of it. If the shape was to the right of the y-axis, it will now be to the left of it. Where will their new locations be once they're reflected across the y-axis? How many steps apart are the new locations from the original locations?

The lightning moves from (4,-4) to (-4,-4).  They are 8 steps apart. 

The smiley face moves from (-3,3) to (3,3).  They are 6 steps apart. 

The triangle moves from (-1,5) to (1,5).  They are 2 steps apart. 

The star moves from (1,3) to (-1,3).  They are 2 steps apart. 

You should notice that reflecting over the y-axis means that the sign changes for the x-value, but that the y-value stays the same.  

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