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llere per­;tric­ning urve :rom .1ine )ints con-

light gent par-

lther" :Jf the tathe-even

to be :1tions

An Example of Problem Analysis: Matching an Average 5

Questions in the reading, like the one identified above, are meant for you to consider before reading on. If the question has a correct answer (i.e., it is not asking for an opinion) and no answer is given in the reading, then an answer is supplied after the Problems for that section.

Your analysis of Question 1 should reveal that the answer to the question of whether the objects in the pair are parallel depends on the definition of parallel that you choose to adopt.

History

The concept of parallelism has played an important role in the history of geometry due to the influence of the Elements 01 Euclid. In this work, Euclid took as a postu­late (in modern language) "If two lines are cut by a transversal and the sum of the measures of the interior angles on the same side of the transversal is less than 1800

,

then the lines intersect on that side of the transversal." In Chapter 7 of this book, we discuss Euclid's Elements, and in Chapter 11, we examine this postulate in some detail. Here, it suffices to say that this postulate, which is called Euclid's parallel postulate, clearly represents the notion of parallel as "do not intersect". Euclid's parallel pos­tulate motivated a number of mathematicians to attempt to prove it from the other postulates. As a direct outgrowth of the futility of attempts over two thousand years, non-Euclidean geometries were discovered. Ibese geometries changed the ways in which mathematicians viewed geometry in particular and mathematics in general, and had a profound impact on how we think of axiom systems today.

Concept analysis includes the type of examination of alternate definitions, instances, and generalizations of a mathematical idea that we have outlincd hcrc for the "parallel" concept. rnle results of concept analysis often make us realize that mathematics is not as rigid as it is sometimes made out to be, and it may suggest changes in our formalization and interpretation of the idea. Many of the chapters of this book might be thought of as extended concept analyses of the important ideas of high school mathematics: function, equation, congruence, distance, etc. But within these concept analyses are other kinds of analyses, to which we now turn.

Problem analysis involves more than finding different ways of solving a problem. It includes looking at a problem after it has been solved and examining what has been done. Will the method of solution work for other problems? Can we extend the prob­lem? And so forth.

The problem: The average test-grade problem

To illustrate, we begin with a typical problem found in algebra books.

Jane has an average of 87 after 4 tests. What score does she need on the fifth test to average 90 for all five tests?

Solving the problem

To answer this question, the algebra student is expected to let a variable such as x stand

for Jane's score on the 5th test and to solve = 90. But many students will use arithmetic, working somewhat as follows. To average 90 points on 5 tests means to have 450 points. Jane has 348 points, so that she needs 102 points. The algebra not only verifies this answer but mimics the arithmetic. To solve the equation, we multiply both

6 Chapter 1 What is Meant by "An Advanced Perspective"?

sides by 5 and then subtract 87 . 4. Students who do the problem arithmetically will [ urally wonder why algebra is needed when they see that the algebra merely repe the arithmetic, and they are right because algebra is not needed for this problem.

Generalizing the problem

This problem means little if it ends here. First we interpret the answer. If the te contain at most 100 points, then Jane cannot average 90 for all five tests. We mi! ask: Given her current situation, what is the highest average score she can attai What is the lowest average score she can attain? 111ese questions lead us to del into the relationship between Jane's score on the 5th test and her average. Then' cannot avoid algebra.

Let A equal Jane's average for a115 tests. A = . A is a function of x,

we can write A = f(x) = k x + 69.6, so f is a linear function with slo

*. We can use the equation A to determine what Jane needs to obtain a given average, not merely the single average of 90 presented in the original proble by solving for x: x 5A 87·4.

Representing the problem

A geometric picture can add new insight into an algebraic problem. A graph A = 348 5+ x over the interval 0 ::;; x :s; 100 shows all the possible solutions (Figure ~

A

100

80

60

40 -

20

(0,69.6)

A = I(x) = 348 + x 5

(100,89.6)

I I I I x 20 40 60 80 ]00

Jane's lowest possible average is 69.6, while her highest possible average is 89.6. No, we see the power of algebra to solve an entire set of problems at once.

The linear rela tionship between x and A shows that this is a constant-increasl situation. Each point Jane earns on the 5th test adds the same amount to her aver age for the five tests. In fact, each point Jane scores on any of the tests contributes point to her average. Thus the problem analysis has shown us a different way to 1001 at the problem.

Extending the problem: The Scoring Title Problem

In Jane's case, we sought a specific average. But now suppose that the avera( be attained is not known. We illustrate this situation with actual data, not inveIrtC:C for instructional purposes. In April, 1998, the basketball players Michael Jordar: and Shaquille (Shaq) O'Neal were vying for the season individual scoring title until the last game of the season. 'The scoring title is won by the player with the highest

An Example of Problem Analysis: Matching an Average 7

average number of points per game, calculated by dividing the total number of points by the number of games the player has played. Before the last game, Jordan

had scored 2313 points in 81 games, for an average of ::::::; game (customarily, averages in newspapers are rounded to the nearest tenth). Shaq had scored 1666 .. . 1666 points

pomts 111 59 games, for an average of 59 ::::::; 28.2 game' No one else had achance to

win the title. The Scoring Title Problem can be stated as follows:

Given the above information, with what numbers of points in their final games does Shaquille O'Neal win the scoring title over Michael Jordan?

It is easy to tryout various scenarios. You could estimate how many points Jor­dan and Shaq will score in their last game, and then calculate, on the basis of those guesses, who would win the scoring title. These calculations require only arithmetic. If there were only a couple of possibilities for the numbers of points scored, this arithmetic would give you rather quickly all of the possible outcomes. But there are many possi­bilities: Each player might reasonably score any number of points from 10 to 60.

For this reason it is useful to consider all the scenarios at one time, and algebra is needed. How can we use algebra to analyze what will happen in the final game? We show two ways, one using equations and inequalities, the other using functions. You will see that each gives a different and useful perspective on the situation. See­ing both together helps us understand the situation better, and also gives good insight into the relative role of graphs of functions and equations.

If Jordan scores more points in the last game than Shaq, will he nec­essarily win?

If Shaq scores more points in the last game than Jordan, will he nec­essaril y win?

An equations/inequalities approach

If Jordan scores j points in his last game, then he has a total of 2313 + j points in 82 . f' 2313 + points ". '1 I " . h 1 f . SI games, for an average 0 game' SImI ar y, 11 S IS t e num Jer o· pomts , 1aq

scores in his last game, he will then have 1666 + s points in 60 games, for an average . 1666 + \' points CIT d . h . . 1 of --60-' game' onseq uent y,. or an wms t e sconng tItle w 1enever

2313 + 82

1666 + s >----

60

On the other hand, Shaq wins the scoring title whenever

2313 + 82

1666 + S <----

60

Representing the extended problem

Where do we go now? First, we must ask what it means to solve inequalities like these. We cannot simply say j equals this and s equals that, because there are infinitely many solutions even if one of the variables is fixed. So we graph the boundary to the inequalities. That is the line with equation '

2313 + j 82

1666 + s

60 .. 30j + 1084

Solvmg for S III terms of j, s = --4-1-' This line is graphed in Figure 8.

8 Chapter 1 What is Meant by "An Advanced Perspective"?

s

65 . " ......... _ ....... ; ......... ". :: ..... 7""':::··· .."

/.,/';: = 30) + 1084 AI

~ 110. + IOgLl ... [,,/,/' I <l) 55 Z s>

4 [Shaq wins) '///

0 ... f·····

..... / ... / .. / . ...' .. ' (30,48

45 ..• / .. :, •. : .. /:,:;-4-.

0

t- ........

~//' s<

30j + 1084 Jorcl~n in

4-. ,,/,,/ L l] 0 35 I.' d i ....... ,/·········"· Z

(0 '"'c ,'"

I/ ... /·(~· i27.17

~ I I 15 20 25 30 35 40 45 50 55 60

j

No. of points of Michael Jordan

For j 2:: 0, each point on the line with at least one integer coordinate has me ing. If Jordan scores 0 points, Shaq still needs to score over 26.44 points to win title. That is, he needs 27 point.s or more. If Jordan scores 1 point, Shaq needs c 27.17 points-28 points or more-to win. If Jordan scores 30 points, Shaq need! points or more. We see that the lattice points (the points with integer cOOl'dina above the line in the first quadrant or on the s-axis offer all the possible ways in wl Shaq wins. TIle lattice points below the line in the first quadrant or on the j-axis sI­all the possible ways in which Jordan wins.

Redraw the graph of Figure 8 with the axes intersecting at (0,0) ( induding the line s = j. On your graph, show how Shaq could in theory win title while scoring fewer points than Jordan in the final game.

So, in theory, Shaq could have a lower average than Jordan before the fi game, and also score fewer points than Jordan in the final game, yet still have a hig average overall. (TIlis phenomenon, in which an overall average is in a differ, ordei' than the two partial averages that comprise it, is known as Simpson's parade

It happened that Jordan scored 44 points in his last game of the regular seas His game was over before Shaq's game started. When j = 44, s = 58.63 ... , wh meant that Shaq had to score 59 points in his last regular season game to win title. Shaq would have needed a personal record for him to win the scoring ti Shaq scored 39 points, which is terrific scoring, but it was not enough to win the sc ing title.

Suppose Shaq had scored 39 points in his last gamc, and it played before Jordan's last game. Indicate (a) on the graph of Figure 8, and (b) solving an equation or inequality, how many points Jordan would have needed score to win the title.

An Example of Problem Analysis: Matching an Average 9

A functions approach

We begin this approach by representing each player's average points per game for the whole season as a function of the points he scores in the final game.

Jordan:

Shaq:

2313 + A = tU) = -----82

A = g( s) = 1666 + s 60

Ibe functions t and g are graphed in Figures 9a and 9b. The answer to Question 5 is visualized in a different way in Figure 9 than it is

in Figure 8. For any k, the x values where the graphs of t and g intersect the hori­zontalline A = k are the numbers of points Jordan and Shaq need to score in the last game to obtain a season average of k. By drawing three sides of a rectangle, we see on the horizontal axis of Figure 9, the coordinates of the points that are on the line graphed in Figure 8. For example, Figure 9 pictures 18 as the answer to Question 5 and shows that after Jordan scored 44 points, Shaq would have needed to score 59 points in his last game to match Jordan's average.

Ibere is another relationship between Figures 8 and 9. In Figure 9 we have .) 2313 + () 1666 + \' r· • • • graphed t(j and g s = -6-0-" 111e equatlOn of the hne graphed 1il

Figure 8 signifies when the two averages are the same, that is, it is tU) g(s). Since g is a linear function, g has an inverse, and g-l 0 tU) = g-I 0 g(s) = s. 111at

A

30.0

29.8

29.6

29.4

29.2

29,0 g

A 28.8

28.6

28.4

j, S

44 S9 100

(b)

out.) j, s

20 30 40 50 60 70 80 90 100 110 120

(a)

10 Chapter 1 What is Meant by "An Advanced Perspective"?

is, S = g-I 0 f(j). Thus the equation that is graphed in Figure 8, s = 30} -:1

the equation of g-I 0 f. We show this schematically.

j fU) g(s) s

(Points scored by JOrdan) __ -? (JOrdan'S season)

in last game of season average (Shaq'S season) +--__ (POints scored by S

average in last game of sea

Hf(j) = g(s), theng-I 0 fU) s.

Notice the meaning of g-l 0 f. The function f maps Jordan's last game r onto Jordan's season average. The function g maps the number of points Sha( in the last game onto his average for the season. If the season averages are equal, then g-l is mapping Jordan's average onto the number of points Shaq r to get that average. Inus the function g-l 0 f maps the number of points Jc scores in the last game onto the number of points Shaq'needs to have the same age as Jordan. In this way, function composition and function inverses both g( alize and provide an explanation for the graphs in Figures 8 and 9.

Throughout this book, you will encounter extended analyses of problem~ this one, in which we begin with a simple problem and examine its solutions and ants to deepen both the understanding of the problem and the understanding 0

underlying mathematics.

Some mathematical connections relate different areas of mathematics, such as :.; bra to geometry. Others connect the mathematics studied before college to the m ematics studied in college-level courses, such as relating school algebra to coilege-l abstract algebra. Still others show analogies between concepts, such as between ( and volume.

For Ollr example in this chapter, we exhibit a set of corresponding propertie addition and multiplication. Let p, q, and r denote real numbers and a, b, and c der corresponding positive real numbers. rnlroughout this example, the left column c tains properties arising from addition while the right column contains corresponc properties arising from multiplication. We begin by thinking of p + q as COl

sponding to abo

Commutative group properties

Addition in tbe Set of Real Numbers

p + qisreaL p+q=q+p

Multiplication in tbe Set of Positive Reals

ab is reaL ab ba a(bc) (ab)c

Closure Commutativity Associativity Identity

p + (q + r) (p + q) + r There exists a number 0 such that for all p, p + 0 = 0 + p = p.

There exists a number 1 such th(J for all a, a' 1 = 1 . a = a.

Inverse For all p, there exists a number - p

such that p + - p = O.

For all a, there exists a

1 such that a . - = 1.

a

a

An Example of Mathematical Connections: + and, 11

For those of you who are familiar with abstract algebra, these correspondences are often described with the following statements:

1. Ibe set n of real numbers with the operation + of addition is a commutative group (R, +).

2. 'nle set R+ of positive real numbers with the operation' of multiplication is a commutative group (R+, .).

Multiples and powers

Tbe correspondence between addition and multiplication by no means ends here. Let m and n be fixed nonnegative integers. We see that the common properties of integer powers (sometimes called "laws of exponents"), which emanate from multi­plication, correspond to the distributive property and other properties of addition and multiplication. For instance, the fact that the zero power equals 1 (the multi­plicative identity) corresponds to the fact that the zero multiple equals 0 (the addi­tive identity).

Doubles! squares p + P = 2p ')

a' a = a~

Triples! cubes p + p + p = 3p a'a-a aJ

Multiples!powers p + p + ... + P = mp a -a-··· -a = am ~ '-~

m terms m factors Sum of multiples! mp + np = (m + n)p am -all = am-I-II

product of powers

Zero multiple/power Op = 0 aO = 1

Multiple of sum/ m( p + q) = m p + mq (ab)'" = am. bill

power of product

Multiple of multiple! m(np) = (mn)p (a")tn = allll1

power of power

What property of multiplication corresponds to 3 (2x + 5 y) = 6x + 15 y?

Inverses and inverse operations

Allow m and n to be negative integers. The first row in the table shows another way in which opposites correspond to reciprocals, and the rest follows. The third row shows that the difference in subtraction corresponds to the quotient in division.

m = -1 and inverse

Inverse of inverse

III verse operations

One-step equations

-l'p -p

-(-p)=p

p q=p+(-q)

p+q if p

r if and only r-q

a

a a 1

b = = a-b b

. . c ab = C If and only If a = b

14 Chapter 1 What is Meant by "An Advanced Perspective"?

Four conceptions of parallelism are described in this chap­ter. Which conception seems to you to be most closely related to the use of "parallel" in each situation below, and why?

a. the parallel opposite edges of a parallelepiped

b. parallel bars in gymnastics

c. In a plane, if two lines are perpendicular to the same line, then they are parallel.

d. parallel electrical circuits

e. parallel rays from the Sun

f. parallel processing (with computers)

Suppose parallel curves are defined as Leibniz did. Show that a consequence of this definition, if applied to curves in 3-space, is that there can be two curves that are parallel where one curve is a straight line and the other is not.

a. Consider the statement," In a plane, segments on fixed rays hom a point cut olf by any parallel lines are proportional." Illustrate this statement with a diagram, and describe ils hypothesis and conclusion in terms of your diagram.

b. Consider the statement," In a plane, segments on fixed par­allel lines cut off by rays from a point are proportional." Illustrate this statement with a diagram, and describe its hypothesis and conelusion in terms of your diagram.

c. Is either (or both) of a or b true? If so, why? If not, why not?

Analyze the concept of ahsolute value in a way similar to that done in this chapter for the concept of parallel. You may find it convenient to organize your analysis in the following way.

a. Define the absolute value of a real number both alge­braically and geometrically.

b. Give properties of absolute value that are easily understood using the algebraic definition, and properties of absolute value that are easily understood using the geometric definition.

c. Discuss the graph of the absolute value function f, where

f(x) = /x/ and x is a real number.

d. A function f is additive if for all x and y in its domain,

f(x + y) = f(x) + f(y)· A function f is multiplicative if

for all x and y in its domain, f(xy) = f(x) . f(y). Is the

absolute value function additive? Why or why not? Is the

absolute value function multiplicative? Why or why not?

e. Generalize absolute value to apply to complex numbers. How does this generalization relate to the algebraic and geometric conceptions of absolute value of real numbers?

f. How is absolute value generalized to apply to vectors of any dimension? To which conceptions of absolute value is this generalization linked?

g. Give an application of absolute value to a situation origi­nating outside of mathematics.

Generalize the average test-grade problem of this section as follows.

a. Suppose Jane has an average of G after 4 tests. What scc does she need on the 5th test to average H for all five tes!

b. Suppose Jane has an average of G after 1'1 tests. What scc

does she need on the (1'1 + 1 )st test to average H for

n + 1 tests?

c. Suppose Jane has an average of G after n tests. What av(

age score does she need on the next m tests to average

for all m + n tests?

d. Comment on the relative difficulty of answering parts 11

for typical high school students. \

Consider the Shaq-Jordan problem described in n . 30j + lOX4 .

chapter. Recall that the equatIOn s = 41 gIves tl

values of sand j for which the season averages of Shaq ar

Jordan would be equal.

a. Find positive integer values of sand j that satisfy this equ

tion. (A method is given in Chapter 5, hut you are nl

expected to use that method. You may use trial and errol

b. The graph of this equation is a line. What is the meanir

of the slope and s-intercept of this line?

c. Solve this equation for j. What is the meaning of

and j-intercept of the line that is the graph of the

equation?

d. Give the smallest integer number of points for Shaq and larger integer number of points for Jordan in the last gam that would have resulted in Shaq's having the higher se, son average.

Examine the general case of the Jordan-Shaq problem wit variables defined as in the following table.

Jordall

Shalf

J S

Points in Last Game

j s

a. Copy the table and fill in the right column.

Average Points per Game for

b. The functions f and g in this chapter are linear functions

Interpret the meaning of the slope and intercept of each 0

these functions.

E . I t' . I S+f J-t-j I c. quatlllg t 1e unction va ues ~ = -;;-;-, we can so v(

for s in terms of j or solve for j in terms of s. Do both ane

interpret the meaning of the slope and intercept of these

functions.

At the beginning of the 2002 Major League Basebal} son, Sammy Sosa had hit 450 horne runs in his careert,. major league career record for home runs is Hank Aa;()o;s 755. Suppose Sosa plays m more seasons.

a. How many home runs must he average per season to sur­pass Hank Aaron's career total?

b. Generalize part 3 in some way and discuss your gene­

ralization. Consider the following classic problem.

A substance is 99% water. Some water evaporates, leaving a substance that is 98% water. How much of the water has evaporated?

a. Getting the initial answer. Solve the given problem.

b. The classic status of the problem. Based on your answer to part 3, indicate why you think this is called a classic problem.

c. A numerical approach. In answering part a you may have used algebra, setting up unknowns and solving equations.

Answer part a again, this time using no algebra, but only

concrete, numerical reasoning. (For example, if the original

substance had 1 unit of solid stuff and 99 units water, the

evaporated substance still has 1 unit of solid stuff: so .... )

d. A diagrammatic approach. Solve the problem yet again,

this time using a diagram as your basic reasoning tool.

(For example, a simple rectangle can be divided into two

regions representing the water and the solid in the orig­

inal substance. A different rectangle can represent the

evaporated substance .... ) e. GeneraLizing the problem. Solve the problem again, but

this time replace the numerical values 99% and 98% in the statement of the problem with general parameters. cnlis is a first step to generalizing the problem.)

f. A functions approach. The algebraic solution from part e is fully general, yet it is not fully revealing about why evap­orating half the water has lowered the proportion of water only about 1 %. In a sense, this solution is too genera] to focus on the essentials of this problem. Solve the problem again, but this time keep the specific numerical value 1 % of the drop in water content part of the solution, and express the proportion of water evaporated as a function of the original proportion of water in the substance.

g. Another functions approach. The approach outlined in the discussion of part f is not the only functions approach pos­sible. You can also express the proportion of water as a function of the absolute amount of water in the substance, letting the fixed amount of solute S = 1. Try this.

ANSWERS TO QUEST!ONS THE CHAPTER TEXT}:

This will happen if the point (j, s) is above the line 30j + 10K4 . ,. 8 I h I' . s = 41 graphed m FIgure yet be ow t e me s = J.

These lines intersect at about (98.5,98.5). 'Thus it is possible

that Jordan could score more points than Shaq and still lose

the title, but unlikely, since Shaq would have to score more

than 98 points.

The lattice points (j, s) in the shaded region of Figure 10 indicate situations in which Jordan scores more points than

Shaq in the final game yet loses the scoring title.

Chapter 1 Problems 15

h. Summarize what you have learned from working on this

problem analysis. Refer to the correspondence between addition and mul­

tiplication described in this chapter. Six properties are given below. Into which column would each fall, addition or multi­plication? Write the corresponding property that would go in the other column. We occasionally use different letters for the variables so that you cannot rely on the letters as guides.

~ a. XU

b. a(b + c) = ab + ac

c. Vp."V(j = '1jiq d. 111e geometric mean of p and q is Vji(j.

e. (p - q) - (r - s) = (p q) + (s - r)

f.

From the distributive property mx + /'IX = (m + n)x,

using group properties, we can deduce Ox = 0 for all x. Here

is a proof:

By the distributive property, for all real x, m, and /'I,

mx + nx (m + n)x.

So mx + Ox = (m + O)x.

Since ° is the additive identity,

mx + Ox = mx.

So Ox is the additive identity. So Ox = O.

Refer to the correspondence between addition and multipli­cation described in this chapter. Give the corresponding prop­erties and proof for the other column.

Five properties of inequality are given below. Follow the directions of Problem 10. Again, different letters may be used for the variables so that you cannot rely on the letters as guides.

a. a < b <==> a + c < b + c

b. If a > ° and b > 0, then a + b > O.

c. If a < 0 and b < 0, then a + b < O.

d. If x > ° and m < 0, then mx < O.

e. If x < y and n > 0, then =;; < *'

s

s = ----''-\----

s j