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Algebra2TrigH

Algebra2TrigH
Algebra2TrigH

June, 2012 Dear prospective Algebra 2 and Trigonometry Honors students,

I very much look forward to beginning our mathematical journey together in the fall and would like to take this opportunity to welcome you to this wonderful course. I’m delighted that you are on board and hope that you will look back on this course as a significant milestone in your growth as a student of mathematics. I hope that you will find this course to be rewarding, enjoyable, and appropriately challenging. There is no doubt in my mind that this course offers us the opportunity to explore some very interesting mathematical ideas which combined with your previous experiences will form the foundation of your future studies in mathematics.

What follows is some information regarding an assignment which I would like you to complete over the summer.

Algebra 2 and Trigonometry Honors Summer Assignment

As a way of connecting those topics which you have studied in Geometry this year with concepts and skills which we will be developing further in Algebra 2 and Trigonometry Honors, you are asked to complete a summer assignment in preparation for the course next year. This is an important assignment designed to get your mind engaged in studying a range of interesting problems prior to beginning our adventure together in Algebra 2 and Trigonometry Honors.

Details of the Assignment

The summer packet consists of seven questions based on ideas, concepts, and skills many of which you are most likely to have encountered in your prior Geometry and Algebra 1 courses. Nonetheless, you should not be surprised if some problems might appear unfamiliar to you and/or may require some creative problem solving. It is intended that this problem set be more than just a set of routine exercises.

A total of five hours should be more than sufficient to make a solid attempt at this assignment. It is suggested that you not spend more than 30 minutes on any given problem. You may work on the problems in any order, though clearly it would be beneficial to focus on one problem at any given time. In the event that a particular problem appears to present some challenges, you may find it helpful to make an initial attempt and then revisit the problem at a later time. Nonetheless, you are expected to devote quality time in completing this assignment. You are welcome to refer to books or notes from previous courses, but I ask that no other resources (ex. internet resources) be used in completing this assignment. Please note that you are asked to complete this assignment without assistance from other individuals.

The work should be done in pencil and neatly presented on loose-leaf paper. It is particularly important that you show all work as you will be graded on the correctness of your methods in addition to the accuracy of your final answers. A partial solution attempt may be invaluable in terms of the process and thinking behind your efforts. A problem left blank or not attempted is clearly of less value.

So that it is fresh in your mind, you are strongly advised to complete this assignment during the 2-3 week period prior to the start of school. Beyond the fact that this work is a graded assignment, it will serve as an important transition into the kind of thinking which you can expect to be engaged in next year in the Algebra 2 and Trigonometry Honors course.

The assignment will be due on the first day of class next year. Should you have any questions regarding the course or the summer assignment, you may contact me via email (acaglieris@https://www.wendangku.net/doc/9e14886475.html,). You are encouraged to seek any clarifications before you leave for summer break. I look forward to seeing you in the fall.

Thank you in advance for your efforts.

Have a wonderful summer!

Dr. Caglieris

NAME: _______________

ALGEBRA 2 and Trigonometry Honors

SUMMER ASSIGNMENT PACKET Instructions (Important, please read carefully):

A total of five hours should be more than sufficient to make a solid attempt at this assignment. It is suggested that you not spend more than 30 minutes on any given problem. Nonetheless, you are expected to devote quality time in completing this assignment. You are welcome to refer to books or notes from previous courses, but I ask that no other resources (ex. internet resources) be used in completing this assignment. Please note that you are asked to complete this assignment without assistance from other individuals.

The work should be done in pencil and neatly presented on loose-leaf paper. It is particularly important that you show all work as you will be graded on the correctness of your methods in addition to the accuracy of your final answers.

So that it is fresh in your mind, you are strongly advised to complete this assignment during the 2-3 week period prior to the start of school. Beyond the fact that this work is a graded assignment, it will serve as an important transition into the kind of thinking which you can expect to be engaged in next year in the Algebra 2 and Trigonometry Honors course.

The assignment will be due on the first day of class next year.

Please read the questions carefully. Note that the use of a calculator is only permitted on problems

where the image appears. All other questions should be attempted without the use of a calculator and answers should be given in exact form.

_____________________________________________________________________________________

1.

a. Given the points A ( 1, 4) and B ( 6, 2) in the x-y coordinate plane describe in words the set of points in the x-y plane that are equidistant from point A and point B.

b. Provide an equation to describe the set of points that are equidistant from point A and point B.

c. Find the slope of the line containing the points A, and B.

d. Find the length of the segment from point A to point B. Leave your answer in exact form.

e. Explain the relationship between the set of points equidistant from point A and point B and the segment containing the points A, and B. Justify your answer algebraically using the distance formula. _____________________________________________________________________________________ 2.

Given the circle with equation: 225x y +=

a. Determine the equation of the line tangent to the circle at the point (1, 2). Leave your answer in point slope form.

b. Determine the x-intercept and the y-intercept of the tangent line obtained in part (a).

c. Find the measure of the acute angle formed by the tangent line and the x-axis. Give your answer to the nearest tenth of a degree.

d. Find the area of the triangle formed by the tangent line, the x-axis and the y-axis.

e. Find the length of the segment containing the x-intercept and the y-intercept as its endpoints. Leave your answer in exact form. _____________________________________________________________________________________

A spider crawls so that its position in the xy-plane at time t (in seconds), where 0t 3 can be described by the following equations:

2x t =-

()2

2y t =-

When x and y are given as separate equations like the ones above, they are referred to as parametric equations and t is called the parameter. So for example at time t=0 the spider is at the point (-2, 4) a) On a neatly labeled and scaled set of axes, sketch the path of the spider on the interval from t= 0 to t = 4 seconds. You may find it helpful to set up a table of values prior to sketching the path of the spider:

Time t (seconds) x y

0 -2 4

1

2

3

4

b) Determine a single equation in terms of x and y to describe the path of the spider. _____________________________________________________________________________________

An ant crawls so that its position in the xy-plane at time t (in seconds), where 0t 3 can be described by the parametric equations: 342x t =- 322

y t =- So for example at time t=2 the spider is at the point (-1, 1)

c) On a neatly labeled and scaled set of axes, sketch the path of the ant on the interval from t= 0 to t = 4 seconds. You may find it helpful to set up a table of values prior to sketching the path of the spider:

Time t (seconds) x y

1

2 -1 1

3

4

d)Determine a single equation in terms of x and y to describe the path of the ant.

e)From your sketch in (b) and (d) determine whether the path of the spider and the path of

the ant cross, and if so determine the coordinates of those points at which the paths

cross. Justify your answer algebraically.

f)Determine whether the spider and the ant will actually collide. In other words, is/are

there any value(s) of t such that the ant and the spider are at the same point, at the same

time? Justify your answer algebraically.

_____________________________________________________________________________________

4.

a)Suppose you have 4 feet of wire to be divided into two pieces, one used to form a square and

the other to form a circle. How much of the wire should be used for the square and how much should be used for the circle in order to enclose the maximum area?

b)What is significant in terms of your answer to part (a).

c)Suppose you have 4 feet of wire to be used to form a square and a circle as in (a). How much of

the wire should be used for the square and how much should be used for the circle in order to enclose the minimum area?

d)What is significant about your answer in (c). More specifically, what is the relationship between

the circle and the square if the enclosed area is to be minimized?

More important than arriving at the exact final answer what is most important is your

thinking and approach to this interesting problem. Describe how you approached solving this problem.

___________________________________________________________________________________

5.

Suppose a checker piece is placed on a white square, marked A in the first row.

A

F

In how many ways (or how many different paths are there) for the checker piece to move from the square in the first row (marked A), to the square in the bottom row (marked F)?

Keep in mind that the checker cannot move backwards, and can only move diagonally down the board. One possible path is shown on the board to give you an example.

Here’s some advice on possible strategies to employ when confronted with an unfamiliar

problem:

·Devise and solve a smaller or simpler version of the original problem to get the idea

·Try working backwards as well as forwards

·Consider special cases

·Use symmetry

·Keep good records of your thinking, attempts, etc.

These heuristics as they are known were originally proposed by the mathematician George Polya in his well known book “How To Solve It”.

More important than arriving at the exact final answer what is most important is your

thinking and approach to this interesting problem. Describe how you approached solving this problem.

________________________________________________________________________

6.

Stage 0 Stage 1 Stage 2 Stage 3

https://www.wendangku.net/doc/9e14886475.html,/images_1/fractal_math_patterns/simple-fractal/koch-curve-

diagram.jpg

The Koch snowflake is a very beautiful example of a fractal. The word fractal was first used by the mathematician Benoit Mandelbrot. Beyond their aesthetic beauty, fractals possess several interesting properties.

One of these properties is that a fractal can be generated through a succession of

geometric iterations. Here is a visual explanation of the iteration through the four

stages of the Koch Snowflake for one side of the equilateral triangle.

Stage 0:

Stage 1: The original segment is

reduced by a factor of 3, then four

copies of the segment are joined

together to form a kind of tent.

Stage 2: The process is repeated.

www.scielo.br/img/fbpe/bjp/v28n2/28n2a7f5.gif

We are now in a position to do an interesting investigation:

Calculator may be helpful here, though it may also be helpful to work with exact values to

see the emerging pattern.

a)Suppose the length of the side of the initial equilateral triangle is 1 unit. Calculate the perimeter

of the different evolutions of the Koch Snowflake at each stage. You will want to refer to the

diagrams on the previous page:

Stage 0 1 2 3 4

Perimeter 3

b)What do you think happens to the perimeter of the snowflake as we increase the number of

stages?

c)Again, assuming that the length of the side of the original equilateral triangle is 1 unit. Calculate

the area of the snowflake at each stage. You will want to refer to the diagrams on the previous page:

So for stage 0, all you need to do is calculate the area an equilateral triangle in which each side is of length 1. Verify the answer below.

For stage 1, the area from stage 0 is increased by 3 small equilateral triangles.

For stage 2, the area from stage 1 is increased by 12 even smaller equilateral triangles.

Continue this process, calculating the area at each stage and completing the table below:

d)What do you think happens to the value of the area of the snowflake as we increase the number

of stages?

______________________________________________________________________________

7. How about some Geometric Probability

a)

Assuming you hit the region inside the circle, what is the probability that you hit the sector containing the Peddie falcon? Leave your answer in exact form. b)

Assuming you hit the region inside the circle, what is the probability that your dart lands inside the triangle IOB? Leave your answer in exact form. c)

Assuming you hit the region inside the circle, what is the probability that your dart lands inside the triangle OGH? Leave your answer in exact form. d)

Assuming you hit the region inside the circle, what is the probability that your dart lands inside the trapezoid CFED? Leave your answer in exact form. e)

Assuming you hit the region inside the circle, what is the probability that your dart lands on the segment containing the old Peddie logo? Leave your answer in exact form. f)

What is the probability that you hit the region inside the square but do not manage to get your dart to land inside the circle? Leave your answer in exact form.

_____________________________________________________________________________________

H

Thank you for all of your hard work on this

summer packet!

Enjoy the rest of your summer!

I look forward to seeing you in September!

Dr. Caglieris

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