U6L15 Weighted Averages

U6L15 Weighted Averages

By Matthew Hamilton

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What is the midpoint of the segment connecting (1,2) and (5,2)?

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What is the midpoint of the segment connecting (5,2) and (5,10)?

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What is the midpoint of the segment connecting (1,2) and (5,10)?

Open Ended

How did you get your answers?

Open Ended

How does the word "average" relate to finding midpoints?

Open Ended

What do you think these formulas could help us calculate?

$$\frac{1}{2}A+\frac{1}{2}B\ \ \ \ \ or\ \ \ \ \ \frac{1}{2}\left(A+B\right)$$  

Multiple Select

What operations are we doing, mathematically, in these formulas?

$$\frac{1}{2}A+\frac{1}{2}B\ \ \ \ \ or\ \ \ \ \ \frac{1}{2}\left(A+B\right)$$  

• adding

• subtracting

• multiplying

• dividing

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Which point partitions AB in a 1 : 2 ratio?

In other words, if we call that point C, which point makes the ratio AC : CB = 1 : 2?

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Which point partitions AB in a 2 : 1 ratio?

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Calculate $$C=\frac{1}{3}A+\frac{2}{3}B$$  

​Distribute all the x values first, then all the y values.

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Step 4 is the hardest part

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For 2 new points K and L, write an expression for the point that partitions segment KL in a 3:1 ratio.

​The first number is the second numerator.

The second number is the first numerator.

The sum is the ​denominator.

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What's the formula?

The midpoint is the average of two coordinates because it is the location that is exactly in the middle.

Average

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But sometimes we don't want the point that is exactly in the middle. Sometimes we want a point that is closer to one point than the other. That is called a weighted average.

Weighted Average

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I wouldn't recommend this formula usually, but the structure is similar to how we calculate weighted averages.

The Midpoint

It's a point that is a little closer to B than A.

Partioning a line segment

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Why that formula?

Let's do some math!

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Consider just the horizontal part (the blue line) from A to B. How long is it?

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So the total horizontal distance is 6 units.

If we go only 2/3 of that distance, how many units do we go?

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The distance from 2 to 8 was 6 because 6 = 8-2.

If we didn't know where A and B are...

The distance from A to B is C because C = ___

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We wanted to go 2/3 of 6 units, so we got 4 by calculating 2/3(6)=4.

If the distance from A to B is "B-A", write an expression that says we only want to go 2/3 of that distance.

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So now we know that we can add 2/3(B-A) to our starting point to get our ending point.

If we want to travel from A to B, where is our starting point?

Open Ended

So our new formula becomes $$A+\frac{2}{3}\left(B-A\right)$$  . What does this say, in your own words?

In your own words

Open Ended

What steps can we take to rewrite $$A+\frac{2}{3}\left(B-A\right)$$   so that it becomes $$\frac{1}{3}A+\frac{2}{3}B$$  ?

2 steps

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Write an expression for the point that partitions A and B in a 1 : 4 ratio.

The first number is the second numerator.

The second number is the first numerator.

The sum is the denominator.​

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Ratio to formula

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Find the point that partitions AB in a 1:4 ratio. Label it B'.

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Find the point that partitions AD in a 1:4 ratio. Label it D'.

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Find the point that partitions AC in a 1:4 ratio. Label it C'.

Multiple Choice

Is AB'C'D' a dilation of ABCD?

• Yes

• No

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AB'C'D' is a dilation of ABCD from center A by scale factor

Partitioning line segments creates dilated versions of the original.

​If we partition multiple line segments in the same ratio from the same center, we get similar figures.

​Cool, right?!

Similar Figures

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