# Khan's Sequences

#### Process

Phase 1: In this portion you and a partner will be observing a video which is titled Bacteria Growth. The video is scaled from 20 minutes to one second so each second in the video is roughly 20 seconds in real life. We will be creating a t chart to organize the growth of the bacteria with one side of the graph being time in seconds and the other side  the number of bacteria. After the table has been created you will be going to Desmos.com to create a table by clicking on the "add items" button in the top left corner and clicking on the table.  You and your partner will then copy your data on to your chart with x being time and y the bacteria growth as you do this desmos will be graphing your data on an x,y plane.

After watching a video on "Writing Exponential Functions from a Table" next you will write the formula generated from your data into desmos and match the results from your table to you exponential equation.

You will then predict the growth of the bacteria after 100 seconds both by looking at the graph and algebraically by using the equation you and your partner have found.

Phase 2: Many games depend on how a ball bounces. For example, if different basketballs rebounded differently, one basketball would bounce differently off of a backboard than another would, and this could cause basketball players to miss their shots. For this reason, manufacturers have to make balls’ bounciness conform to specific standards. In this lesson, you will investigate the relationship between the height from which you drop a ball and the height to which it rebounds.

Today you and your partner will be joining another group of two to make your group  of four people. You will have a recorder, a ball dropper and two spotters to help track the bounce height.

First you must find the bounce ratio of the ball you have chosen. After dropping the ball from a height of one meter determine the height after the first bounce. Divide the first bounce height with the initial release height of one meter, this will give a bounce ratio.

Imagine that you drop the ball you used  from a height of 200 cm, but this time you let it bounce repeatedly.Two students doing exercise.

As a team, discuss this situation. Then sketch a picture showing what this situation would look like. Your sketch should show a minimum of 6 bounces after you release the ball.
Predict your ball’s rebound height after each successive bounce if its starting height is 200 cm. Create a table on Desmos with these predicted heights.

Next watch  "Exponential Decay Intro" from Khan Academy and find an equation representing the bounce height.

With your equation find the bounce height of the ball on its hundredth bounce if it was released from 500 meters and kept on bouncing regardless of gravity and the release of energy.

Phase 3: After watching the "intro to sequences" from Khan Academy you will work with your partner from phase 1 to come up with the sequence for (1,1,2,3,5,8,13,21,34,...)

This is the Fibonacci Sequence.

Using the Fibonacci Sequence your next task will be to draw a Fibonacci Spiral. After watching the tutorial on how to draw it you must map out the spiral in the natural world. Images will be provided so that you can find the the sequence shoring up in place that you might not have thought about before.

#### Attachments

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• Desmos
Description: Graphing Website for all phases