course Mth 151
If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
007. Triangular, Square, Pentagonal Numbers
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Self-critique (if necessary):
Self-critique Rating:
Question: **** `q001. Note that there are 7 questions in this assignment.
Sketch three points A, B and C forming an equilateral triangle on a piece of paper, with point A at the lower left-hand corner, point B at the lower right-hand corner and point C at the top. Sketch the segments AB and AC.
Now double the lengths of AB and AC, and place a point at each of the endpoints of these segments. Connect these new endpoints to form a new equilateral triangle. Two sides of this triangle will have three points marked while the new side will only have its two endpoints marked. Fix that by marking that middle point, so all three sides of your new triangle are marked the same.
How many marked points were there in the original triangle, and how many are there in the new triangle?
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Your solution:
There are three points on the original triangle and there are nine marked points in the new.
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Given Solution:
The original triangle had the three points A, B and C. When you extended the two sides you marked the new endpoints, then you marked the point in the middle of the third side. So you've got 6 points marked.
Click on 'Next Picture' to see the construction. The original points A, B and C are shown in red. The line segments from A to B and from A to C have been extended in green and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and an equally spaced point has been constructed at the midpoint of that side. Your figure should contain the three original points, plus the three points added when the new side was completed.
`routine triangle2
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Self-critique (if necessary):
I didn? understand this. Was this a triangular/figurate number problem? I went to the form to see the picture and now I have a better understanding of what was being asked.
Self-critique Rating:
Question: **** `q002. Extend the two sides that meet at A by distances equal to the distance original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure?
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Your solution:
Four points on the new triangle and ten points in the whole figure.
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Given Solution:
You added the two new endpoints when you extended the sides. You then should have marked two new points on the new third side, so that each side contains 4 points including its endpoints. Your figure will now contain 10 marked points.
`routine triangle3
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Self-critique (if necessary):
This is triangular figurate numbers. I printed from the cd? in the learning lab examples of these from the section 1.2 showing 1-15 points on triangles. I now understand what is being asked.
Self-critique Rating:
Question: **** `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle?
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Your solution:
Five new points in the new triangle and 15 points all together.
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Given Solution:
You will add an endpoint to each newly extended side, so each of the new sides will contain 5 points. You will then have to add 3 equally spaced points to the new side, giving you a total of 13 points on the new triangle. In addition there are two marked points inside the triangle, for a total of 15 points.
Click on 'Next Picture' to see the construction. The line segments along two sides of the triangle have again been extended and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and equally spaced points have been constructed along that side.
`routine triangle4
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Self-critique (if necessary):
I have it.
Self-critique Rating:
Question: **** `q004. Continue the process for one more step. How many points do you have in the new triangle?
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Your solution: six new points and 21 in the whole triangle.
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Given Solution:
You will add an endpoint to each newly extended side, so each of the new sides will contain 6 points. You will then have to add 4 equally spaced points to the new side, giving you a total of 15 points on the new triangle. There are also 5 marked points inside the triangle for a total of 21 marked points.
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Self-critique (if necessary):
One thing is I haven? told you that I had to add points in the middle of the two end points I have just automatically done that and counted the whole new side.
Self-critique Rating:
Question: **** `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence?
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Your solution:
In the sequence 3,6,10,15,21 you add 3+3 and get 6. Add 4 +6 and get 10. Add 5+10 get 15. Add 6 +15 to get 21. So you must add 7 to 21 and get 28.
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Given Solution:
The differences between these numbers are 3, 4, 5, 6. The next difference, according to this pattern, should be 7, which would make the next number 28.
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Self-critique (if necessary): I used a different pattern and I suspect I used a more complicated one than you did.
Self-critique Rating:
Question: **** `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater?
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Your solution: looking at my triangle I see that each time I extended it I extended one more each time. So it had to be 7.
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Given Solution:
When you extend the triangle again, you will add two new endpoints and each side will now have 7 points. The 7 points on the new triangle will be all of the new points.
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Self-critique (if necessary):
I am on the right track I know just worded it different.
Self-critique Rating:
Question: **** `q007. How do you know this sequence will continue in this manner?
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Your solution:
It started with one point and each time an extra point was added there is no way to change that.
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Given Solution:
Each time you extend the triangle, each side increases by 1. All the new marked points are on the new side, so the total number of marked points will increase by 1 more than with the previous extension.
Good work. You understand this pattern.