Applications of sequences and series, review video on YouTube, introduction to fractals.

1. A good resource for learning your way around the
calculator or to review what we've learned in class ...
Working with Sequences on the TI-83+ or 84+

2. A small forest of 4000 trees is under a new forestry plan. Each year 20% of the
trees will be harvested and 1000 new trees are planted.
(a) Will the forest ever disappear?
(b) Will the forest size ever stabilize? If so, how many years and with
how many trees?

3. Introduction to today's class by Mr. Green on YouTube ...
a summary of almost everything in this unit ...
Sequences and Series on YouTube

4. The Bouncing Ball
A ball is dropped from one metre, and the height is recorded after each
bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90
percent of its drop height if it is dropped onto concrete from a height of
less than two metres.
1. How high does the ball bounce on its eighth bounce?
2. How many times does the ball bounce before it rises to less than half
of its original drop height?
3. How many times does the ball bounce before it stops bouncing?
4. How far has the ball travelled as it reaches the top of its 10th bounce.
5. Construct a graph that shows the bounce height versus bounce number.

5. The Bouncing Ball
A ball is dropped from one metre, and the height is recorded after each
bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90
percent of its drop height if it is dropped onto concrete from a height of
less than two metres.
1. How high does the ball bounce on its eighth bounce?

6. The Bouncing Ball
A ball is dropped from one metre, and the height is recorded after each
bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90
percent of its drop height if it is dropped onto concrete from a height of
less than two metres.
2. How many times does the ball bounce before it rises to less than half
of its original drop height?

7. The Bouncing Ball
A ball is dropped from one metre, and the height is recorded after each
bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90
percent of its drop height if it is dropped onto concrete from a height of
less than two metres.
3. How many times does the ball bounce before it stops bouncing?

8. The Bouncing Ball
A ball is dropped from one metre, and the height is recorded after each
bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90
percent of its drop height if it is dropped onto concrete from a height of
less than two metres.
4. How far has the ball travelled as it reaches the top of its 10th bounce.

9. The Bouncing Ball
A ball is dropped from one metre, and the height is recorded after each
bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90
percent of its drop height if it is dropped onto concrete from a height of
less than two metres.
5. Construct a graph that shows the bounce height versus bounce number.

10. What is a Fractal?
Fractal Zoom on YouTube

11. The Koch Snowflake
This fractal -- the Koch Snowflake -- was developed by in 1904 by Helge von
Koch, a Swedish mathematician. The fractal is started by drawing an
equilateral triangle.
Each side of the triangle is trisected, and the middle section forms the base of
a new equilateral triangle outside the original one.
The process is then continued. The diagram below shows three generations of
the Koch Snowflake.

12. The Sierpinski Triangle
Waclaw Sierpinski, a Polish mathematician, developed another fractal known
as the Sierpinski Triangle. This fractal also starts with an equilateral triangle.
To draw the fractal, you find the midpoint of each side of the original triangle,
and then draw three segments joining the midpoints. There are now four
triangles inside the original triangle. The middle triangle is not shaded, and the
process is continued with the other three shaded triangles, as shown in the
diagram below.

13. Draw a Fractal
Use pencil and paper (metric graph paper if possible) to draw the fractal
described below.
• Draw a square with 8-cm sides in the middle of the paper.
• Position the paper horizontally (in landscape format). Extend the
diagram to the left and right by drawing a square on each side of the
original square -- touching the original square. The sides of the new
squares should be half as long as the side lengths of the original square.
• Repeat the previous step three times. Your fractal should now have
five generations, including the original square.

14. Question: Will the fractal ever be too large for this page? Explain.

15. The Rectangle ...
Draw a rectangle that measures 12 cm by 8 cm, and shade the inside of the
rectangle. Construct the midpoints of each side of the rectangle, and then draw
a quadrilateral by joining these points. Shade the quadrilateral white. Now
continue the process by finding the midpoints of the quadrilateral, drawing the
rectangle, and shading it the same colour as the first rectangle. Draw six
generations. (The initial rectangle is the first generation.)

16. The Square ...
Create a fractal that begins with a large square 20 cm on each side. Each
pattern requires that the square be divided into four equally sized squares, that
the bottom-left square be shaded, and the process continues in the upper-right
square. Repeat the process four times.

17. The Circle-Square ...
Create a fractal where a square is inscribed in a circle. The diameter of the
original circle is 16 cm. Shade the area between the circle and the square.
Inscribe a circle inside the resulting square, and then inscribe a new square
inside that circle, and shade the area between the new circle and square. Repeat
this process three times.

18. HOMEWORK
Page 282 question number 3 only.
DO NOT do question 4. Let's see how
many of your classmates read these
slides; we'll find out on Monday. ;-)