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Mathematics 2

Mathematics Department

Phillips Exeter Academy

Exeter, NH

August 2020

Phillips Exeter Academy

Introductory Math Guide for New Students

(For students, by students!)

Introduction

Annually, approximately 300 new students enroll in a Mathematics course at PEA, and

students arrive here from all over the world. As a new student, you will quickly come to

realize the distinct methods and philosophies of teaching at Exeter. One aspect of Exeter

that often catches students unaware is the math curriculum. I encourage all new students

to come to the math table with a clear mind. You may not grasp, understand, or even like

math at first, but you will have to be prepared for anything that comes before you.

During the fall of 2000, the new students avidly voiced a concern about the math cur-

riculum. Our concern ranged from grading, to math policies, and even to the very different

teaching styles utilized in the mathematics department. The guide that you have begun

reading was written solely by students, with the intent of preparing you for the task that

you have embarked upon. This guide includes tips for survival, testimonials of how we felt

when entering the math classroom, and aspects of math that we would have liked to have

known, before we felt overwhelmed. Hopefully, this guide will ease your transition into math

at Exeter. Remember, “Anything worth doing, is hard to do.” Mr. Higgins ’36.

— Anthony L. Riley ’

“I learned a lot more by teaching myself than by being taught by someone else.”

“One learns many ways to do different problems. Since each problem is different,

you are forced to use all aspects of math.”

“It takes longer for new concepts to sink in... you understand,

but because it didn’t sink in, it’s very hard to expand with that concept.”

“It makes me think more. The way the math books are setup

(i.e. simple problems progressing to harder ones on a concept)

really helps me understand the mathematical concepts.”

“When you discover or formulate a concept yourself, you remember it better

and understand the concept better than if we memorized it

or the teacher just told us that the formula was ‘xyz’.”

Homework

Math homework = no explanations and eight problems a night. For the most part, it

has become standard among most math teachers to give about eight problems a night; but

I have even had a teacher who gave ten — though two problems may not seem like a big

deal, it can be. Since all the problems are scenarios, and often have topics that vary, they

also range in complexity, from a simple, one-sentence question, to a full-fledged paragraph

with an eight-part answer! Don’t fret though, transition to homework will come with time,

similar to how you gain wisdom, as you get older. Homework can vary greatly from night to

night, so be flexible with your time — this leads to another part of doing your homework.

IN ALL CLASSES THAT MEET FIVE TIMES A WEEK, INCLUDING MATHEMATICS,

YOU SHOULD SPEND 50 MINUTES AT THE MAXIMUM, DOING HOMEWORK! No

teacher should ever expect you to spend more time, with the large workload Exonians carry.

Try your hardest to concentrate, and utilize those 50 minutes as much as possible.

i

Without any explanations showing you exactly how to do your homework, how are you

supposed to do a problem that you have absolutely no clue about? (This WILL happen!)

Ask somebody in your dorm. Another person in your dorm might be in the same class, or

the same level, and it is always helpful to seek the assistance of someone in a higher level

of math. Also remember, there is a difference between homework and studying; after you’re

through with the eight problems assigned to you, go back over your work from the last few

days.

“... with homework, you wouldn’t get marked down if you didn’t do a problem.”

Going to the Board

It is very important to go to the board to put up homework problems. Usually, every

homework problem is put up on the board at the beginning of class, and then they are

discussed in class. If you regularly put problems up on the board, your teacher will have a

good feel of where you stand in the class; a confident student will most likely be more active

in participating in the class.

Plagiarism

One thing to keep in mind is plagiarism. You can get help from almost anywhere, but

make sure that you cite your help, and that all work shown or turned in is your own, even if

someone else showed you how to do it. Teachers do occasionally give problems/quizzes/tests

to be completed at home. You may not receive help on these assessments, unless instructed

to by your teacher; it is imperative that all the work is yours.

Math Extra-Help

Getting help is an integral part of staying on top of the math program here at Exeter.

It can be rather frustrating to be lost and feel you have nowhere to turn. There are a few

tricks of the trade however, which ensure your “safety,” with this possibly overwhelming

word problem extravaganza.

Teachers and Meetings

The very first place to turn for help should be your teacher. Since teachers at Exeter have

many fewer students than teachers at other schools, they are never less than eager to help

you succeed in any way they can. You can often meet your teacher after Assembly or during

the Lunch period. You can always call or ask a teacher for help. If there is no time during

the day, it is always possible to check out of the dorm after your check-in time, to meet with

your teacher at their apartment, or house. It is easiest to do this on the nights that your

teacher is on duty in his/her dorm. Getting help from your teacher is the first and most

reliable source to turn to, for extra help.

“You could meet with the teacher for extra help anytime.”

“Extra help sessions one-on-one with the teacher. My old math text.”

ii

New Student Testimonials

“There was not a foundation to build on. There were no ‘example’ problems.”

After eight years of math textbooks and lecture-style math classes, math at Exeter was

a lot to get used to. My entire elementary math education was based on reading how to

do problems from the textbook, then practicing monotonous problems that had no real-life

relevance, one after the other. This method is fine for some people, but it wasn’t for me. By

the time I came to Exeter, I was ready for a change of pace, and I certainly got one.

Having somewhat of a background in algebra, I thought the Transition 1 course was just

right for me. It went over basic algebra and problem-solving techniques. The math books

at Exeter are very different from traditional books. They are compiled by the teachers, and

consist of pages upon pages of word problems that lead you to find your own methods of

solving problems. The problems are not very instructional, they lay the information down

for you, most times introducing new vocabulary, (there is an index in the back of the book),

and allow you to think about the problem, and solve it any way that you can. When I first

used this booklet, I was a little thrown back; it was so different from everything I had done

before — but by the time the term was over, I had the new method down.

The actual math classes at Exeter were hard to get used to as well. Teachers usually

assign about eight problems a night, leaving you time to “explore” the problems and give

each one some thought. Then, next class, students put all the homework problems on the

board. The class goes over each problem; everyone shares their method and even difficulties

that they ran into while solving it. I think the hardest thing to get used to, is being able to

openly ask questions. No one wants to be wrong, I guess it is human nature, but in the world

of Exeter math, you can’t be afraid to ask questions. You have to seize the opportunity to

speak up and say “I don’t understand,” or “How did you get that answer?” If you don’t ask

questions, you will never get the answers you need to thrive.

Something that my current math teacher always says is to make all your mistakes on the

board, because when a test comes around, you don’t want to make mistakes on paper. This

is so true, class time is practice time, and it’s hard to get used to not feeling embarrassed

after you answer problems incorrectly. You need to go out on a limb and try your best. If

you get a problem wrong on the board, it’s one new thing learned in class, not to mention,

one less thing to worry about messing up on, on the next test.

Math at Exeter is really based on cooperation, you, your classmates, and your teacher. It

takes a while to get used to, but in the end, it is worth the effort.

— Hazel Cipolle ’

iv

“At first, I was very shy and had a hard time asking questions.

“Sometimes other students didn’t explain problems clearly.”

“Solutions to certain problems by other students are sometimes not the fastest or easiest.

Some students might know tricks and special techniques that aren’t covered.”

I entered my second math class of Fall Term as a ninth grader, with a feeling of dread.

Though I had understood the homework the night before, I looked down at my paper with

a blank mind, unsure how I had done any of the problems. The class sat nervously around

the table until we were prompted by the teacher to put the homework on the board. One

boy stood up and picked up some chalk. Soon others followed suit. I stayed glued to my

seat with the same question running through my mind, what if I get it wrong?

I was convinced that everyone would make fun of me, that they would tear my work apart,

that each person around that table was smarter than I was. I soon found that I was the only

one still seated and hurried to the board. The only available problem was one I was slightly

unsure of. I wrote my work quickly and reclaimed my seat.

We reviewed the different problems, and everyone was successful. I explained my work

and awaited the class’ response. My classmates agreed with the bulk of my work, though

there was a question on one part. They suggested different ways to find the answer and we

were able to work through the problem, together.

I returned to my seat feeling much more confident. Not only were my questions cleared

up, but my classmates’ questions were answered as well. Everyone benefited.

I learned one of the more important lessons about math at Exeter that day; it doesn’t

matter if you are right or wrong. Your classmates will be supportive of you, and tolerant of

your questions. Chances are, if you had trouble with a problem, someone else in the class

did too. Another thing to keep in mind is that the teacher expects nothing more than that

you try to do a problem to the best of your ability. If you explain a problem that turns

out to be incorrect, the teacher will not judge you harshly. They understand that no one is

always correct, and will not be angry or upset with you.

— Elisabeth Ramsey ’

v

Mathematics 2

1. A 5 × 5 square and a 3 × 3 square can be cut into pieces

that will fit together to form a third square.

A B

(a) Find the length of a side of the third square.

F G

(b) In the diagram at right, mark P on segment DC so that

5

P D = 3, then draw segments P A and P F. Calculate the

3

lengths of these segments.

(c) Segments P A and P F divide the squares into pieces. Ar-

D 5 C 3 E

range the pieces to form the third square.

2. (Continuation) Change the sizes of the squares to AD = 8 and EF = 4, and redraw

the diagram. Where should point P be marked this time? Form the third square again.

3. (Continuation) Will the preceding method always produce pieces that form a new

square? If your answer is yes, prepare a written explanation. If your answer is no, pro-

vide a counterexample — two specific squares that can not be converted to a single square.

4. Instead of walking along two sides of a rectangular field, Fran took a shortcut along the

diagonal, thus saving distance equal to half the length of the longer side. Find the length of

the long side of the field, given that the the length of the short side is 156 meters.

5. Let A = (0, 0), B = (7, 1), C = (12, 6), and D = (5, 5). Plot these points and connect

the dots to form the quadrilateral ABCD. Verify that all four sides have the same length.

Such a figure is called equilateral.

6. The main use of the Pythagorean Theorem is to find distances. Originally (

th

century

BC), however, it was regarded as a statement about areas. Explain this interpretation.

7. Two iron rails, each 50 feet long, are laid end to end with no space between them. During

the summer, the heat causes each rail to increase in length by 0.04 percent. Although this is

a small increase, the lack of space at the joint makes the joint buckle upward. What distance

upward will the joint be forced to rise? [Assume that each rail remains straight, and that

the other ends of the rails are anchored.]

D

.

8. In the diagram, AEB is straight and angles A and B are

C

right. Calculate the total distance DE + EC.

15

10

9. (Continuation) If AE = 20 and EB = 10 instead, would

10 B

DE + EC be the same?

A E 20

10. (Continuation) You have seen that the value chosen for AE determines the value of

DE + EC. One also says that DE + EC is a function of AE. Letting x stand for AE (and

30 − x for EB), write a formula for this function. Graph this formula using a graphing tool.

Locate the point on the graph that represents the shortest path from D to C through E.

Draw an accurate picture of this path, and make a conjecture about angles AED and BEC.

Consider the slopes or use a protractor to test your conjecture.

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Mathematics 2

11. Two different points on the line y = 2 are each exactly 13 units from the point (7, 14).

Draw a picture of this situation, and then find the coordinates of these points.

12. Give an example of a point that is the same distance from (3, 0) as it is from (7, 0).

Find lots of examples. Describe the configuration of all such points. In particular, how does

this configuration relate to the two given points?

13. Verify that the hexagon formed by A = (0, 0), B = (2, 1), C = (3, 3), D = (2, 5),

E = (0, 4), and F = (− 1 , 2) is equilateral. Is it also equiangular?

14. Draw a 20-by-20 square ABCD. Mark P on AB so that AP = 8, Q on BC so that

BQ = 5, R on CD so that CR = 8, and S on DA so that DS = 5. Find the lengths of the

sides of quadrilateral P QRS. Is there anything special about this quadrilateral? Explain.

15. Verify that P = (1, −1) is the same distance from A = (5, 1) as it is from B = (− 1 , 3).

It is customary to say that P is equidistant from A and B. Find three more points that are

equidistant from A and B. By the way, to “find” a point means to find its coordinates. Can

points equidistant from A and B be found in every quadrant?

16. The two-part diagram below, which shows two different dissections of the same square,

was designed to help prove the Pythagorean Theorem. Provide the missing details.

17. Inside a 5-by-5 square, it is possible to place four 3-4-5 triangles so that they do not

overlap. Show how. Then explain why you can be sure that it is impossible to squeeze in a

fifth triangle of the same size.

18. If you were writing a geometry book, and you had to define a mathematical figure called

a kite, how would you word your definition?

19. Find both points on the line y = 3 that are 10 units from (2, −3).

20. On a number line, where is

(p + q) in relation to p and q?

21. Some terminology: Figures that have exactly the same shape and

size are called congruent. Dissect the region shown at right into two

congruent parts. How many different ways of doing this can you find?

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Mathematics 2

31. I have been observing the motion of a bug that is crawling on my graph paper. When

I started watching, it was at the point (1, 2). Ten seconds later it was at (3, 5). Another ten

seconds later it was at (5, 8). After another ten seconds it was at (7, 11).

(a) Draw a picture that illustrates what is happening. What did you assume?

(b) Where was the bug 25 seconds after I started watching it? What did you assume?

(c) Where was the bug 26 seconds after I started watching it? What did you assume?

32. The point on segment AB that is equidistant from A and B is called the midpoint of

AB. For each of the following, find coordinates for the midpoint of AB:

(a) A = (− 1 , 5) and B = (3, −7) (b) A = (m, n) and B = (k, l)

33. Write a formula for the distance from A = (− 1 , 5) to P = (x, y), and another formula

for the distance from P = (x, y) to B = (5, 2). Then write an equation that says that P

is equidistant from A and B. Simplify your equation to linear form. This line is called the

perpendicular bisector of AB. Verify this by calculating two slopes and one midpoint.

34. Find the slope of the line through

(a) (3, 1) and (3 + 4t, 1 + 3t) (b) (m − 5 , n) and (5 + m, n

35. Is it possible for a line ax + by = c to lack a y-intercept?

To lack an x-intercept? Explain.

36. The sides of the triangle at right are formed by the graphs

of 3 x + 2y = 1, y = x − 2, and − 4 x + 9y = 22. Is the triangle

isosceles? How do you know?

37. Pat races at 10 miles per hour, while Kim races at 9 miles

per hour. When they both ran in the same long-distance race

y

x

last week, Pat finished 8 minutes ahead of Kim. What was the length of the race, in miles?

Briefly describe your reasoning.

38. (Continuation) Assume that Pat and Kim run at p and k miles per hour, respectively,

and that Pat finishes m minutes before Kim. Find the length of the race, in miles.

39. A bug moves linearly with constant speed across my graph paper. I first notice the bug

when it is at (3, 4). It reaches (9, 8) after two seconds and (15, 12) after four seconds.

(a) Predict the position of the bug after six seconds; after nine seconds; after t seconds.

(b) Is there a time when the bug is equidistant from the x- and y-axes? If so, where is it?

40. What is the relation between the lines described by the equations − 20 x + 12 y = 36 and

− 35 x + 21 y = 63? Find a third equation in the form ax + by = 90 that fits this pattern.

41. Rewrite the equation 3 x − 5 y = 30 in the form ax + by = 1. Are there lines whose

equations cannot be rewritten in this form?

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Mathematics 2

42. Consider the linear equation y = 3.62(x − 1 .35) + 2 .74.

(a) What is the slope of this line?

(b) What is the value of y when x = 1.35?

(c) This equation is written in point-slope form. Explain the terminology.

(d) Use a graphing tool to graph this line.

(e) Find an equation for the line through (4. 23 , − 2 .58) that is parallel to this line.

(f) Use point-slope form to write the equation of a line that has slope − 1. 25 and that goes

through the point (− 3. 75 , 8 .64).

D

43. The dimensions of rectangular piece of paper ABCD are

A

AB = 10 and BC = 9. It is folded so that corner D is matched

G

with a point F on edge BC. Given that length DE = 6, find

EF , EC, F C, and the area of EF C.

44. (Continuation) The lengths EF , EC, and F C are all func-

tions of the length DE. The area of triangle EF C is also a

function of DE. Using x to stand for DE, write formulas for

H

E

these four functions.

B F C

45. (Continuation) Find the value of x that maximizes the area of triangle EF C.

46. The x- and y-coordinates of a point are given by the equations shown below. The

position of the point depends on the value assigned to t. Use your graph

x = 2 + 2t

paper to plot points corresponding to the values t = −4, −3, −2, −1, 0, 1,

y = 5 − t

2, 3, and 4. Do you recognize any patterns? Describe them.

47. Plot the following points on the coordinate plane: (1, 2), (2, 5), (3, 8). Write equa-

tions, similar to those in the preceding exercise, that produce these points when t-values are

assigned. There is more than one correct answer.

48. Given that 2 x − 3 y = 17 and 4 x + 3y = 7, using mental math (that is, without using

paper, pencil, or calculator), find the value of x.

49. A slope can be considered to be a rate. Explain this interpretation.

50. Find a and b so that ax + by = 1 has x-intercept 5 and y-intercept 8.

51. Given points A = (− 2 , 7) and B = (3, 3), find two points P that are on the perpendicular

bisector of AB. In each case, what can you say about segments P A and P B?

52. Explain the difference between a line that has no slope and a line whose slope is zero.

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Mathematics 2

66. Is there anything wrong with the figure shown at right?

67. Show that a 9-by-16 rectangle can be transformed into a

square by dissection. In other words, the rectangle can be cut

into pieces that can be reassembled to form the square. Do it

with as few pieces as possible.

68. At noon one day, Corey decided to follow a straight course

in a motor boat. After one hour of making no turns and traveling

at a steady rate, the boat was 6 miles east and 8 miles north of its

point of departure. What was Corey’s position at two o’clock?

How far had Corey traveled? What was Corey’s speed?

69. (Continuation) Assume that the fuel tank initially held 12 gallons, and that the boat

gets 4 miles to the gallon. How far did Corey get before running out of fuel? When did this

happen? When radioing the Coast Guard for help, how should Corey describe the boat’s

position?

70. Suppose that numbers a, b, and c fit the equation a

+ b

= c

, with a = b. Express c

in terms of a. Draw a good picture of such a triangle. What can be said about its angles?

71. The Krakow airport is 3 km west and 5 km north of the city center. At 1 pm, Zuza

took off in a Cessna 730. Every six minutes, the plane’s position changed by 9 km east and

7 km north. At 2:30 pm, Zuza was flying over the town of Jozefow. In relation to the center

of Krakow, (a) where is Jozefow? (b) where was Zuza after t hours of flying?

72. Golf balls cost $0.90 each at a local golf club, which has an annual $ 25 membership

fee. At the sporting-goods store down the road, the price is $1.35 per ball for the same

brand. Where you buy your golf balls depends on how many you wish to buy. Explain, and

illustrate your reasoning by drawing a graph.

73. Draw the following segments. What do they have in common?

from (3, −1) to (10, 3); from (1. 3 , 0 .8) to (8. 3 , 4 .8); from π, 2 to 7 + π, 4 + 2.

74. (Continuation) The directed segments have the same length and the same direction.

Each represents the vector [7, 4]. The components of the vector are the numbers 7 and 4.

(a) Find another example of a directed segment that represents this vector. The initial point

of your segment is called the tail of the vector, and the final point is called the head.

(b) Which of the following directed segments represents [7, 4]? from (− 2 , −3) to (5, −1);

from (− 3 , −2) to (11, 6); from (10, 5) to (3, 1); from (− 7 , −4) to (0, 0).

75. Is it possible for a positive number to exceed its reciprocal by exactly 1? One number

that comes close is

, because

is

. Is there a fraction that comes closer?

76. Points (x, y) described by the equations x = 1 + 2t and y = 3 + t form a line. Is the

point (7, 6) on this line? How about (− 3 , 1)? How about (6, 5 .5)? How about (11, 7)?

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Mathematics 2

77. The perimeter of an isosceles right triangle is 24 cm. How long are its sides?

78. The x- and y-coordinates of a point are given by the equations shown below. Use your

graph paper to plot points corresponding to t = −1, 0, and 2. These points should appear

to be collinear. Convince yourself that this is the case, and calculate the

x = −4 + 3t

slope of this line. The displayed equations are called parametric, and t is

y = 1 + 2t

called a parameter. How is the slope of a line determined from its parametric

equations?

79. Find parametric equations to describe the line that goes through the points A = (5, −3)

and B = (7, 1). There is more than one correct answer to this question.

80. Show that the triangle formed by the lines y = 2x − 7, x + 2y = 16, and 3 x + y = 13

is isosceles. Show also that the lengths of the sides of this triangle fit the Pythagorean

equation. Can you identify the right angle just by looking at the equations?

81. Leaving home on a recent business trip, Kyle drove 10 miles south to reach the airport,

then boarded a plane that flew a straight course — 6 miles east and 3 miles north each

minute. What was the airspeed of the plane? After two minutes of flight, Kyle was directly

above the town of Greenup. How far is Greenup from Kyle’s home? A little later, the plane

flew over Kyle’s birthplace, which is 50 miles from home. When did this occur?

82. A triangle has vertices A = (1, 2), B = (3, −5), and C = (6, 1). Image triangle A

B

C

is obtained by sliding triangle ABC 5 units to the right (in the positive x-direction, in other

words) and 3 units up (in the positive y-direction). It is also customary to say that vector

[5, 3] has been used to translate triangle ABC. What are the coordinates of image points A

B

, and C

? By the way, “C prime” is the usual way of reading C

83. (Continuation) When vector [h, k] is used to translate trian-

gle ABC, it is found that the image of vertex A is (− 3 , 7). What

are the images of vertices B and C?

84. It is a simple matter to divide a square into four smaller

squares, and — as the figure at right shows — it is also possible

to divide a square into seventeen smaller squares. In addition to

four and seventeen, what numbers of smaller squares are possible?

The smaller squares can be of any size whatsoever, as long as they

fit neatly together to form one large square.

85. Caught in another nightmare, Blair is moving along the line y = 3x + 2. At midnight,

Blair’s position is (1, 5), the x-coordinate increasing by 4 units every hour. Write parametric

equations that describe Blair’s position t hours after midnight. What was Blair’s position

at 10:15 pm when the nightmare started? Find Blair’s speed, in units per hour.

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