Investigation #6 - Post by April 19th

Here is a potpourri of problems for your last investigation write up. Many of these problems can be solved without the use of technology and with some, the use of technology can inhibit student learning.

Remember to try all of the problems before you select two for your write up.

1) Let ABC be an isosceles triangle with AB = AC, angle BAC = 20^o, measure of angle DBC = 60^o (where D is on side AC), and the measure of angle ECB = 50^o (where E is on side AB). Find the measurement of angle EDB.

2) Roses in Math?

Investigate

Pascal's triangle has many interesting properties that many people are not aware of. Your task for this investigation is to make conjectures and justifications on the patterns you observe while working with Pascal's triangle.

• Investigate the connection between pascal's triangle and computing combinations.

• Find the sum of each row of Pascal's triangle, discuss the pattern (possibly come up with a formula), and investigate how this pattern is applicable to the summation of combinations.

• If the first number in the row is a prime number (excluding the 1), then all of the other numbers in that row are divisible by that prime number. Explore the validity of this statement and come up with justification for or against the statement.

• "The hockey stick" Explore the pattern shown below. Why does it work?

• 1+6+21+56 = 84

  • 1+7+28+84+210+462+924 = 1716

  • 1+12 = 13

• Discuss another pattern not explored so far.

4) Change problems

• What is the minimum number of coins needed to guarantee that you will always have change for a purchase (using pennies, nickles, dimes, and quarters? What are the coins?

  • How does this problem change if you are allowed to use half-dollars?

  • How does this problem change if you only use pennies, nickles, and quarters? What about pennies, nickles, and dimes?

• In how many ways can 19 coins equal exactly one dollar?

• Is it possible to use 100 coins to total $5.00 if you are not allowed to use nickles?

5) Ladder Problem

Part 1.

A man is on a ladder cleaning his gutters when the ladder begins to slide down the side of his house. Assuming that the rate in which the bottom of the ladder is sliding away from the house is a constant speed, when is the best time for the man to try and jump off the ladder? In other words, is the top of the ladder falling at the same rate as the bottom is sliding away?

Part 2.

Say the man is 3/4 of the way up the ladder and it begins to slide. How would this change your answer from part 1?

Part 3.

This time the man is 1/2 of the way up the ladder and it begins to slide. How does this change your answer from part 1?

Hint: Think back to related rates from your Calculus 1 course.

6) Billiards in a Acute Triangle

Given an acute triangle and these properties:

1) The ball can only be hit in a direction that is parallel to one of the sides of the triangle.

2) When the ball bounces off the side of the triangle it must follow the path that is parallel to a different side of the triangle.

Here is a picture of what the ball does.

Will the ball ever reach the same place it started? Investigate how the ball placement effects your answer.

Does the balls path create a pattern? If so, explain it and identify any mathematical properties.

Can the balls beginning placement or the type of acute triangle change the outcome? Why?

Examine this same situation in a right triangle, a isosceles triangle, equilateral triangle, and obtuse triangle.

7) Find the area

Write a function that will calculate the area formed by the region ACHI as I moves along segment AB.

8) Regions in a Circle

Part 1)

Given a circle, if you pick any two points on the circle and and only draw a cord between adjacent points, you make two non-intersecting regions.

Now if you pick three distinct points on the circle and only make a cord between adjacent points, you make 4 non-intersecting regions.

And four distinct points would yield 5 regions.

What are the maximum number of non-intersecting regions you can make with 5 points, 6 points.... n points?

Explain at least one interesting thing you notice about the shapes and the number of non-intersecting regions? Make sure to defend your observation with mathematical reasoning.

Part 2)

Using the same concept of non-intersecting regions from part 1, let's examine what happens when the cords do no have to connect adjacent points.

Using this condition, having three distinct points would provide you with 4 non-intersecting regions, just like in part 1.

Four points would look like this:

How many regions would be produced with 5 points, 6 points....n points?

Explain at least one interesting things you notice about this situation? Make sure to defend your observation with mathematical reasoning.