Developing Expert Voices 2009

1. “ Our den skipping rocks” by kaostep Geometric Sequences

2. “ Our den skipping rocks” by kaostep You come across a group of people skipping rocks by a lake.

3. You tell these people that there is a mathematical way to graph the movement of the rock. “ Our den skipping rocks” by kaostep You come across a group of people skipping rocks by a lake.

4. You also tell them that you can calculate the total vertical distance of the rock, the height of the 4 th skip and when the height of the skip will reach below 1. “ Our den skipping rocks” by kaostep You come across a group of people skipping rocks by a lake. You tell these people that there is a mathematical way to graph the movement of the rock.

5. They don’t believe you, so you say you will prove it to them. “ Our den skipping rocks” by kaostep You come across a group of people skipping rocks by a lake. You tell these people that there is a mathematical way to graph the movement of the rock. You also tell them that you can calculate the total vertical distance of the rock, the height of the 4 th skip and when the height of the skip will reach below 1.

6. Lets assume that the rocks path across the water is a geometric sequence.

7. This means that the rocks “skip height” is decreasing at a constant rate. It seems reasonable to say that its height is decreasing by ¾ a skip and the first skip reached a height of 24 cm. Lets assume that the rocks path across the water is a geometric sequence.

8. This means that the rocks “skip height” is decreasing at a constant rate. It seems reasonable to say that its height is decreasing at ¾ a skip and the first skip reached a height of 24 cm. You could figure out all the values by going: 24 * (3/4) = 18 18 * (3/4)= 13.5 However, it would be a lot easier to make a T-chart. Lets assume that the rocks path across the water is a geometric sequence.

10. This T-chart displays the information in a way much easier to read. With this information, you can now make your graph.

12. By looking at the graph we can see that the total vertical distance will be around 86cm. However, the skip went up and down so every skip will have double the vertical distance. For example, the first skip reached a “skip height” of 24 cm going up but then it came down another 24 cm, that is a total of 48 cm. So, the graph shows us that it reached 86cm but then multiply by 2 to get the total vertical distance. Note: Do not include the distance from which the person skipping the rock released the rock.

13. Now if we go back to our T-chart and add everything up it should be around 86. So.... 24 + 18 + 13.5 + 10.125 + 7.5938 + 5.6953 + 4.2715 + 3.2036= 86.3892 cm * 2= 172.7784 cm By looking at the graph we can see that the total vertical distance will be around 86cm. However, the skip went up and down so every skip will have double the vertical distance. For example, the first skip reached a “skip height” of 24 cm going up but then it came down another 24 cm, that is a total of 48 cm. So, the graph shows us that it reached 86cm but then multiply by 2 to get the total vertical distance. Note: Do not include the distance from which the person skipping the rock released the rock.

14. Now if we go back to our T-chart and add everything up it should be around 86. So.... 24 + 18 + 13.5 + 10.125 + 7.5938 + 5.6953 + 4.2715 + 3.2036= 86.3892 cm * 2= 172.7784 cm By looking at the graph we can see that the total vertical distance will be around 86cm. However, the skip went up and down so every skip will have double the vertical distance. For example, the first skip reached a “skip height” of 24 cm going up but then it came down another 24 cm, that is a total of 48 cm. So, the graph shows us that it reached 86cm but then multiply by 2 to get the total vertical distance. Note: Do not include the distance from which the person skipping the rock released the rock. To find the height of the 4 th term we can go back to either our graph or our T-chart.

15. Now if we go back to our T-chart and add everything up it should be around 86. So.... 24 + 18 + 13.5 + 10.125 + 7.5938 + 5.6953 + 4.2715 + 3.2036= 86.3892 cm * 2= 172.7784 cm By looking at the graph we can see that the total vertical distance will be around 86cm. However, the skip went up and down so every skip will have double the vertical distance. For example, the first skip reached a “skip height” of 24 cm going up but then it came down another 24 cm, that is a total of 48 cm. So, the graph shows us that it reached 86cm but then multiply by 2 to get the total vertical distance. Note: Do not include the distance from which the person skipping the rock released the rock. To find the height of the 4 th term we can go back to either our graph or our T-chart. Looking at our graph, we know it should be just a bit over 10. Looking at our T-chart we know it should be 10.125 cm .

16. There are two ways you can figure out on which skip the “skip height” will be below 1.

17. The first way, which is the way most people would do it, is to continue the T-chart. It would look like this…. There are two ways you can figure out on which skip the “skip height” will be below 1.

18. The first way, which is the way most people would do it, is to continue the T-chart. It would look like this…. There are two ways you can figure out on which skip the “skip height” will be below 1.

19. However, there is an easier way to figure this out.

20. However, there is an easier way to figure this out. You may have noticed a pattern in the T-chart. This pattern is: If you take the 1 st term ( 24 ) and multiply it by ¾ to the exponent of one less then the term you’re looking for, it will give you the value of that term. For example: 24 * (3/4)^(8-1)= 3.2036 (the value of the 8 th term) What I did there was, I took the first term and multiplied it by ¾ . I then raised ¾ to the term I was looking for minus 1 . This equation looks like this: A= 1 st term R= The ratio by which each term is decreasing, in this case ¾. n= The term you are looking for. So.... AR^(n-1) is our equation.

21. However, there is an easier way to figure this out. If you take the 1 st term ( 24 ) and multiply it by ¾ to the exponent of one less then the term you’re looking for, it will give you the value of that term. For example: 24 * (3/4)^(8-1)= 3.2036 (the value of the 8 th term) What I did there was, I took the first term and multiplied it by ¾ . I then raised ¾ to the term I was looking for minus 1 . This equation looks like this: A= 1 st term R= The ratio by which each term is decreasing, in this case ¾. n= The term you are looking for. So.... AR^(n-1) is our equation. You’ll find that with just a bit of guess and check you will have the answer very easily. You may have noticed a pattern in the T-chart. This pattern is:

22. So, if the 8 th term was 3.2036 then lets find the value of the 15 th term.

23. So, if the 8 th term was 3.2036 then lets find the value of the 15 th term. Plug it into the equation: 24 * (3/4)^(15-1)= 0.4276

24. So, if the 8 th term was 3.2036 then lets find the value of the 15 th term. Plug it into the equation: 24 * (3/4)^(15-1)= 0.4276 That’s too low, so lets try 12.... 24 * (3/4)^(12-1)= 1.0136

25. So, if the 8 th term was 3.2036 then lets find the value of the 15 th term. Plug it into the equation: 24 * (3/4)^(15-1)= 0.4276 The 12 th term was very close so lets try the next one up, the 13 th term.... 24 * (3/4)^(13-1)= 0.7602 That’s too low, so lets try 12.... 24 * (3/4)^(12-1)= 1.0136

26. Tahdaa! You’ve got the answer. On the 13 th skip, the “skip height” is less than one. So, if the 8 th term was 3.2036 then lets find the value of the 15 th term. Plug it into the equation: 24 * (3/4)^(15-1)= 0.4276 The 12 th term was very close so lets try the next one up, the 13 th term.... 24 * (3/4)^(13-1)= 0.7602 That’s too low, so lets try 12.... 24 * (3/4)^(12-1)= 1.0136

27. “ Skipping Rocks” by Flickr user Clay1976 Good Job! These people are thoroughly amazed. They are definitely going to spread the word of the goodness of math.