# linear interpolation the power of prediction!. scenario 1: birthday cakes you had a birthday cake...

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We ASSUME (predict): That your cousin ate the birthday cake! But do we know for 100% certainty? No! But, its a pretty good guess.

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Linear Interpolation

The Power of Prediction!

Scenario 1: Birthday Cakes

You had a birthday cake sitting on your kitchen table. Your cousin came over. When you came back into the kitchen, part of the birthday cake was gone and your cousin had frosting on his face.What happened?

We ASSUME (predict):

That your cousin ate the birthday cake!But do we know for 100% certainty?No! But, its a pretty good guess.

We ASSUME (predict):

Hypothesize, guess, predict.These are all synonyms.

How does this relate to our grass?

There are 2 math words for prediction:

Interpolate and Extrapolate.

Interpolate:

Predicting within the data (like with your cousin and the birthday cake).Predicting what has already happened.GRASS: Predicting how tall your grass was on Sunday when we were not here.

Extrapolate:

Predicting what will happen.Predicting outside the data.Predicting how tall your grass could grow given NO LIMITING FACTORS.

How do we do this?

We use our scatter plots (graphs)!

How tall was my grass on day 7?

1: Find points on either side.

2: find the slope.

2: find the slope.

2: find the slope.

2: find the slope.

We found the slope!

The slope between the two known days is . So what is the height (y value) at day 7?Plug it back in!

Take slope and one (x, y)

Take slope and one (x, y)

And plug into y = mx + b

And plug into y = mx + b

And plug into y = mx + b

And plug into y = mx + b

And plug into y = mx + b

And plug into y = mx + b

The y-intercept b = -2 ??

Yes! But our grass was never -2 inches tall!We are basing our calculation on just when the grass was growing. Based on that information, math will give us a negative y-intercept.This is where life and math clash (not the only time!)

Our equation then is.

Our equation then is.

Our equation then is.

And day 7s height is...

And day 7s height is...

We just interpolated!

We figured out how tall our grass may have been on day 7!We can also use our y = mx + b formula to predict into the future!

IMPORTANT!

As grass growth slows, your line of fit (your equation) will change. Why?How?

Classwork:

1: Find the slope of the line formed between your day 5 and day 8.

2: plug your slope and one of the two points back into y = mx + b to find b.

3: Construct YOUR graphs y = mx + b formula to predict day 7.