The title of this POW is “Verdania: A Math Odyssey Chapter 4: Exploring”. This problem is about one of the groups on their exploration of the island. In this problem the group comes to a huge water fall. Miguel and Jesse are so amazed by the water fall they both estimate how high the water fall is. Miguel says that the top of the water fall is about 20 feet more then 3 times the height of a young pine tree. Jesse thinks that is wrong and that the top of the falls is about 50 feet less then 4 times the height of the same young pine. Jeanie says that they are probable both correct. But assuming both are correct how tall the tree would be and how tall is the water fall? Try to think of several ways you could solve this problem. Is there one best way? The important information in this problem is that Miguel thinks that the top of the water fall is about 20 feet more then 3 times the height of a young pine and Jesse thinks that the top of the falls is about 50 feet less then 4 times the height of the same pine. Assuming both are correct how tall would the young tree have to be and how tall is the water fall? Try to think of several ways you could solve this problem. Is there one best way? The main math topics imbedded in this problem are multiplication, division, subtraction, and algebraic equations.

The steps I used to solve this problem are, first I made equations for the rules that Jesse and Miguel made up to determine the height of the falls. (M=Miguel, J=Jesse, T=Top of the water fall, and P=Height of tree.) M=T=3p+20 and J=T= 4p-50. Then I used guess and check to solve the equations. At first I estimated that the tree was 10 ft so I did M= 10*3=30+20=50ft and J=10*4=40-50=ˉ10. That did not work since the ending numbers did not turn out the same. After that I tried going up by multiples of 10 but after I reached 30 I was getting board so I thought that I should change the equation. I know that the top of the water fall had to be the same number in each equation so instead of doing guess and check I decided to solve for x or rather p(height of tree) M: T=3p+20 and J: T=4p-50. I then set about solving for p. First I set the equation up 3p+20=4p-50. Then I took away 20 from both sides 3p=4p-70. After that I took away 4p from both sides. 3p-4p=ˉp so ˉp=ˉ70. Since the answer cannot be ˉp I divided both sides by ˉ1. ˉ1/ˉ1=ˉ70ˉ1= p=70. To make sure that 70 was correct I added the number 70 into the equation where p was M: T=3*70+20= 230 and J: T=4*70-50=230. This works because both equations say that the falls where 230ft high. I new which strategies to used because I thought and talked it over with my tooter. One problem I dealt with on the way is if I had to check every single number. I was not sure about this because how could you tell if all the rest of the numbers were correct or not.

My solution is that the pine tree is 70 feet tall and the waterfall is 230 feet tall. You can solve this problem by doing guess and check, Algebraic equations, or Tile Mats. I think that the best way to solve this problem would be to do algebraic equations because tile mats would take to long to draw out and you would have a big chance of messing up. Also if you used guess and check it would work but it would take forever to find the solution. I know my solution is correct because I double checked my solution and on Friday I went over my work with one of my peers. No, there are no other possible answers to this problem.

One of the things that intrigued me was that there were many different ways to solve this problem and some of them would be good to use if the numbers were better while others would be better if the numbers were bigger. I think this problem challenged me well because it really made me think what ways would work best to solve this problem and why. I think I did not manage my time well because in the beginning I hurried through solving the problem but waited till the week end to write it up. What I would do differently next time is that I would not only try to even out the work better but also try to do different methods in solving the equation because in this POW I did not draw out the tile mats I just thought about if it would be difficult or not.

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