Motions in Life

Mathematics is the extension of common sense by other means.

—Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

8

Algebra Uncloaked

A cocktail can be made by the bartender. But the cocktail also can be made by the chef. [And by the teacher, too, when computing course grades.]

—José Andrés, “The Culinary Miracles of Chef José Andrés,” 60 Minutes (addition by the author)

Mathematical analysis is as extensive as nature itself.… It brings together phenomena the most diverse, and discovers the hidden analogies which unite them.

—Joseph Fourier (1768–1830), The Analytical Theory of Heat

We begin by revisiting the number of days and age problem. If you recall from Chapter 2, the simplified formula that encoded the process for someone who has had a birthday in the current year is 100x + (2013 + z – y), where x is the number of days the person likes to eat out in a week, y corresponds to their year of birth, and z represents the number of years we are past 2013.

For simplicity’s sake, we will assume that we are dealing with those who have had a birthday in the current year, and that the date is June 30 of that year. If z equals 8, this corresponds to a current year of 2021, so let’s start there.

Substituting 8 for z simplifies the variable expression to 100x + (2021 – y). Here, z acts as a parameter that is constant for a given year, but changes from year to year.

EXPRESSING VARIATION WITH TABLES

We will now construct a table containing the values for the variable x, the variable y, and the variable expression 100x + (2021 – y). For the values (x = 1, y = 1930), (x = 2, y = 1952), and (x = 6, y = 1965), we have the following:

Value of x	Value of y	Value of 100x + (2021 – y)
1	1930	100(1) + (2021 – 1930) = 191
2	1952	100(2) + (2021 – 1952) = 269
6	1965	100(6) + (2021 – 1965) = 656

In the second example, for the personal three-digit number 269, 2 represents the number of days of the week the person likes to eat out and 69 represents the age of the person (born in 1952) whose birthday in 2021 is on or before June 30.

Recall that this particular problem has certain conditions that the variables must fulfill in order for the interpretation to work. For x, the value must be one of the whole numbers 1 through 7, and for y, the whole number values range from the year 1922 to the current year 2021—corresponding to individuals under the age of 100.¹ This means that the fully filled-in table will have 700 rows of data, each corresponding to the result of one of the possible number of days of the week and year of birth combinations.

For those who have yet to have a birthday by June 30, 2021, there will be a slightly different table in which the y values will range from the years 1921 to 2020.

Placing these values in a table demonstrates that we have another powerful way of expressing an ensemble of numbers. Not only are expressions able to capture and represent variable behavior, but tables can, too; and just as a specific word may be more appropriate than its synonym in a given context or sentence, so it is with these two different representations of numerically variable phenomena.

When we are recording quantitative data in an experiment, for instance, it can be helpful to first capture it in a table and see if it is possible to reverse-engineer an algebraic expression. In cases where that is hard to do exactly, a statistical approximation method known as regression may be employed to yield a formula that can estimate the data. In other situations, like the number of days and age problem, the procedure may be spelled out in such clarity that first capturing it as an algebraic formula may be the more straightforward way to proceed.

Greater algebraic awareness means becoming mindful of how algebra might structure or inform sets of data that we may encounter on the job or use in our everyday lives. Next time you encounter an IRS table, a retirement benefits table, or a motor vehicle depreciation table, consider the possibility that an algebraic expression may be responsible for generating those values behind the scenes.

SPREADSHEET AND VARIATION: FAST ALGEBRA

Let’s look at the tabular analysis in the previous section as Microsoft Excel might view it. Excel—like most spreadsheet software—is an extremely efficient tool for synthesizing numerical information in tables and maneuvering it. The next table simply lists the raw data for someone who likes to eat out 1, 2, or 6 times a week in column A, and who was born in the years 1930, 1952, or 1965, respectively, in column B. Column C is blank for the moment.

A	B	C
1	1930
2	1952
6	1965

Excel reads this table by cell addresses—for example, A3 means the cell location that is in column A and in the third row, which here contains the number 6. Similarly, B2 refers to column B and row 2, the cell that contains the value 1952. What we want to do is instruct Excel to populate column C with the values of the variable expression 100x + (2021 – y), where the entries in column A represent x values and the entries in column B represent y values.

Remember, computers cannot read as we do, meaning they can’t just visually scan a row of values and place them into the appropriate part of the expression in column C. For them to be able to successfully locate and collate data, we have to input explicit instructions formatted in specific notation.

This can be done in Excel by first entering “=100*A1 + (2021 – B1)” in C1. [Note that the asterisk (*) represents multiplication in Excel.] This can be read as “100 times (entry in A1) + 2021 – (entry in B1)” or for our example, “100(1) + (2021 – 1930),” simplifying to 100 + 91 = 191, which corresponds to the value in our earlier table for x = 1 and y = 1930. Next, we apply this same formula for the entries in (A2, B2) and (A3, B3), placing the results respectively in C2 and C3—and so on up to the formula for (A700, B700) in C700.

Once the data for columns A and B have been entered, Excel enables us to apply the speed and vigor of its computational power to all 700 situations—almost instantaneously—to calculate the values in column C:

A	B	C
1	1930	100*A1 + (2021 – B1) = 191
2	1952	100*A2 + (2021 – B2) = 269
6	1965	100*A3 + (2021 – B3) = 656
:	:	:
:	:	:

The expression “100*A1 + (2021 – B1) =” does not usually show up in the final result in C1 in Excel. Only the number 191 appears in C1, and likewise for C2, C3, and the other cells.

Thus, Excel creates a visual expression of the algebraic calculations that we discussed earlier in the number of days and age problem and the numerical ensemble generated by it. In other words, at its core, some of Excel’s processes are really a fast, pictorial type of algebra!

COMPUTING GRADES

Let’s look at another numerically variable situation with algebraic properties masked by familiarity.

Consider the case of an instructor calculating individual grades for a math class of 30 students. Throughout the course, students typically are expected to complete homework assignments, several tests during the term, and a final exam. Each of these three components contributes to the overall grade for the course, but given that an hour-long, proctored test usually has more stringent demands and expectations on students than homework assignments do, many math instructors place a higher value on test scores.

In other words, to figure out a student’s final grade, instructors typically use an uneven mixture of these three categories of assessment and assign each one a different contribution strength—or weight—based on its relative importance to the overall grade. The particular weight of each category will, of course, depend on the preferences of each individual instructor. Let’s assume that this professor decides that the average score on all of the homework assignments will contribute 25% to a student’s overall final grade, while the average score on all of the tests will contribute 55% and the score on the final exam will be the last 20%.

This situation has a lot in common with the problems we have already encountered. First of all, it is variable, because each student will have average scores for each of these categories that change depending on the student. So, instead of students having varying selections for the number of days of the week they like to eat out, their year of birth, and the years past 2013 in the problem, each student now is effectively making selections for three new variables: their homework average score, their average test score, and their final exam score.

Now the three scores don’t seem like selections in the same sense as selecting the number of days of the week you like to eat out does, because the student has long-range interaction with each of these scores and doesn’t simply “pick” them at will—rather the student “earns” these numerical values through their very effort and comprehension of the material. Nevertheless, these scores are dependent upon variable student performance—and look as such to the instructor—meaning that now the students, themselves, have become part of the variation in the problem. These possible scores range from 0 to 100, rounded to the nearest whole number.

Placing all of the average scores for six such students in a table yields the following:

Name	Homework Average	Test Average	Final Exam Score	Numerical Course Grade
Hypatia	96	95	99
Al-Samawal	90	92	94
Dardi	88	90	93
Ramus	96	88	54
Franklin	30	50	95
Darwin	78	68	72

This table represents part of the numerical symphony of scores for the entire class. As the scores vary from student to student, we can represent each category by tagging them with a different letter. Let’s say that x = homework average, y = test average, and z = final exam score.

The full numerical course grade for each student in the class then depends on how these average scores are weighted. We can use the algebraic expression 0.25x + 0.55y + 0.20z to analyze how mixing together these three types of assessments at the prescribed strengths produces this grade. From now on, we will call the numerical course grade for a student their course average. Note that in this formula we have substituted the decimal equivalents for the percentages: 0.25 for 25% (homework), 0.55 for 55% (tests), and 0.20 for 20% (final). We will switch back and forth between these equivalents depending on the situation.

The following table includes the computed course averages for the six students:

Name	x	y	z	Course Average = 0.25x + 0.55y + 0.20z
Hypatia	96	95	99	96
Al-Samawal	90	92	94	92
Dardi	88	90	93	90
Ramus	96	88	54	83
Franklin	30	50	95	54
Darwin	78	68	72	71

Using these course averages and the 10-point scale for letter grades will yield grades of A for Hypatia, Al-Samawal, and Dardi, a B for Ramus, and a C for Darwin, while Franklin would receive an F.

Range	Letter Grade
90–100	A
80–89	B
70–79	C
60–69	D
0–59	F

10-point scale

Of course, the standard way this data is represented rarely refers back to the x, y, and z variables of schoolroom elementary algebra. In Excel, this table plays out as follows:

A	B	C	D	E
Name	Homework Average	Test Average	Final Exam Score	Course Average
Hypatia	96	95	99	0.25*B2 + 0.55*C2 + 0.20*D2
Al-Samawal	90	92	94	0.25*B3 + 0.55*C3 + 0.20*D3
Dardi	88	90	93	0.25*B4 + 0.55*C4 + 0.20*D4
Ramus	96	88	54	0.25*B5 + 0.55*C5 + 0.20*D5
Franklin	30	50	95	0.25*B6 + 0.55*C6 + 0.20*D6
Darwin	78	68	72	0.25*B7 + 0.55*C7 + 0.20*D7
:	:	:	:	:
:	:	:	:	:

Pre-calculation

A	B	C	D	E
Name	Homework Average	Test Average	Final Exam Score	Course Average
Hypatia	96	95	99	96
Al-Samawal	90	92	94	92
Dardi	88	90	93	90
Ramus	96	88	54	83
Franklin	30	50	95	54
Darwin	78	68	72	71
:	:	:	:	:
:	:	:	:	:

Post-calculation

This setup is general and can be applied to class after class over the course of an instructor’s career, ultimately leading to a numerical ensemble of individual quantitative variation numbering in the thousands of students.

BIG ALGEBRA RETURNS

This is just one scenario for one instructor. Imagine a different instructor who—perhaps believing it unfair to students with test anxiety to favor tests over homework, where they can demonstrate their comprehension of the material under less pressure—decides to assign different weights to each category. What if this instructor’s grading policies are that a student’s average score on homework will count for 60% of their final grade, their average score on tests will be 30%, and the final exam will count for 10%? Assuming that x, y, and z remain the same, the formula to find a student’s final course average would change to 0.60x + 0.30y + 0.10z.

The table for final course averages now reads as follows:

Name	x	y	z	Course Average = 0.60x + 0.30y + 0.10z
Hypatia	96	95	99	96
Al-Samawal	90	92	94	91
Dardi	88	90	93	89
Ramus	96	88	54	89
Franklin	30	50	95	43
Darwin	78	68	72	74
:	:	:	:	:
:	:	:	:	:

The Excel analysis and formatting from the first case would apply here as well, with the only difference being the formula in Column E, which would be 0.60*B2 + 0.30*C2 + 0.10*D2 for the entry in the second row and so on:

A	B	C	D	E
Name	Homework Average	Test Average	Final Exam Score	Course Average
Hypatia	96	95	99	96
Al-Samawal	90	92	94	91
Dardi	88	90	93	89
Ramus	96	88	54	89
Franklin	30	50	95	43
Darwin	78	68	72	74
:	:	:	:	:
:	:	:	:	:

These results will now yield grades of A for Hypatia and Al-Samawal, B for Ramus and Dardi, and C for Darwin, while Franklin would still receive a grade of F.

Yet a third instructor who believes that the final exam is the best assessment of a student’s mastery of all of the material in the course might decide to give it much more weight. Let’s say that this instructor decides that a student’s average homework score counts for 15%, their average test score counts for 25%, and their final exam counts for 60%. The table for final course averages now reads as follows:

Name	x	y	z	Course Average = 0.15x + 0.25y + 0.60z
Hypatia	96	95	99	98
Al-Samawal	90	92	94	93
Dardi	88	90	93	92
Ramus	96	88	54	69
Franklin	30	50	95	74
Darwin	78	68	72	72
:	:	:	:	:
:	:	:	:	:

This results in A’s for Hypatia, Al-Samawal, and Dardi, C’s for Darwin and Franklin, and a D for Ramus—certainly a vastly better outcome for Franklin because he did so well on the final.

These different scenarios are reminiscent of the different scenarios for a business in Chapter 4, where instead of the selling price per unit, cost per unit, and overhead fixing a given scenario, we have the percentage contributions assigned to homework, tests, and the final now fixing a specific scenario.

This suggests that parameters may again be useful to describe not just one specific scenario but an entire class of them. We can use letters earlier in the alphabet to represent each of the percentage contributions as follows: a ≡ percentage of contribution from homework to course average, b ≡ percentage of contribution from tests to course average, and c ≡ percentage of contribution from the final exam to course average. If we continue to use x, y, and z to stand for a student’s scores in each respective category, the general expression for their course average for all possible scenarios is ax + by + cz.

The following table summarizes all three instructors’ grading policies:

Scenario	Weights/Contributions	Parameter Values	Course Average = ax + by + cz
1	25% from homework 55% from tests 20% from final exam	a = 25% b = 55% c = 20%	0.25x + 0.55y + 0.20z
2	60% from homework 30% from tests 10% from final exam	a = 60% b = 30% c = 10%	0.60x + 0.30y + 0.10z
3	15% from homework 25% from tests 60% from final exam	a = 15% b = 25% c = 60%	0.15x + 0.25y + 0.60z

Scenario table

One might now naturally ask, why go through the mathematical gymnastics to create a formula like this? Don’t most teachers know how to calculate their students’ final grades for their class without considering parameters at all?

Establishing parameters enables any instructor—who uses these three assessment categories and wants to calculate course averages—to tune the parameter values to match their grading criteria, which will give them the correct algebraic expression for the specific circumstances of their class. All that’s left is to input each student’s averages for x, y, and z. This aligns well with Excel and other software where a macro or app can be created that will empower an instructor to swiftly calculate all of the course averages for their class, seemingly all at once.

Using parameters also expands our conceptual understanding of the algebraic processes at play in computing grades. In the next sections, we will see that these processes are really representatives of an entire genre of algebraic songs with useful interpretations and renditions in many other areas.

QUANTITATIVE COCKTAILS

Our three instructors, in calculating course student averages by weighting different categories of assessment, have actually been doing something procedurally similar to a bartender mixing cocktails by combining various ingredients in different strengths. In other words, we can think of the procedures for calculating final course averages as the making of various types of “quantitative cocktails” out of different mixtures of homework averages, test averages, and final exam scores. The next table compares the recipe for our final course average “cocktail” to a piña colada cocktail, highlighting the three most important ingredients in each.

Thus, for Scenario 1 we would have the following mixture:

	Piña Colada Cocktail	Final Course Average Cocktail
Ingredient 1 (25%)	White rum	Homework average
Ingredient 2 (55%)	Pineapple juice	Test average
Ingredient 3 (20%)	Coconut cream	Final exam score

Using parameters, we could summarize all of the scenarios as follows:

	Piña Colada Cocktail	Final Course Average Cocktail
Ingredient 1 (a%)	White rum	Homework average
Ingredient 2 (b%)	Pineapple juice	Test average
Ingredient 3 (c%)	Coconut cream	Final exam score

Quantitative cocktails are, however, a bit different in the details from their alcoholic counterparts. Let’s first look at Scenario 1 where the overlap is more pronounced. Consider a quart container in which we wish to make a piña colada with the stated percentage mixture of ingredients, and suppose that we have three smaller containers each with the capacity of 25%, 55%, and 20% of a quart. We fill each of these to the brim with white rum, pineapple juice, and coconut cream, respectively. If we then pour the contents of each of these three smaller containers into the larger container and mix them, we create a full quart-size piña colada. That full piña colada correlates to a student with a perfect score in all three assignment categories—meaning that they receive the full 100 percentage points for the course average.

But what happens if a student doesn’t have a perfect average on each category of assignment, such as with Ramus? To see how the metaphor looks for someone with less than perfect scores, we continue with the perfect scores situation.

Let’s think of this situation as one where we have an initially empty 1-quart container representing the final course average and three smaller containers—the 25%, 55%, and 20% quart containers—which correspond, respectively, to homework average, test average, and the final. Now imagine each of these smaller containers as being filled with a fluid of the same type, in contrast with the piña colada case, where the constituents are different in each container. Let’s take this fluid here to be water.

So, for the student with perfect scores, the three smaller containers are filled to the brim with water, meaning that when they are each emptied into the quart container, this larger container is filled 100% to the brim.

[Artwork provided courtesy of William Hatch]

For the student with less than perfect scores, each of the smaller containers are not completely filled with water, and thus when they are combined in the larger container, the amount of liquid isn’t a full quart. In Ramus’ case, the homework container is 96% full, the test container is 88% full, and the container for the final exam score is filled up 54% of the way. Pouring these quantities of liquid together into the larger quart container to mix Ramus’ quantitative cocktail, we find that it’s filled up to 83% of its total capacity. This, of course, corresponds to Ramus’ course average, or a B grade.

[Artwork provided courtesy of William Hatch]

If it were possible to have digital readouts on each of the containers that showed the percentage of liquid in each container—rounded off to whole number values—then we could represent the score for each category of assessment by the corresponding percentage level of fluid in each of the smaller containers. We could then find out the course average by combining the water from the three containers into the quart container and reading its display in lieu of calculating on paper or in Excel. In short, mixing the fluids this way could serve as a type of special-purpose calculator.² The table for Scenario 1 is given here for a quart container:

A	B	C	D	F	E
Name	Homework Average Container	Test Average Container	Final Exam Score Container	Quart Container	Course Average
Hypatia	96% full	95% full	99% full	96% full	96
Al-Samawal	90% full	92% full	94% full	92% full	92
Dardi	88% full	90% full	93% full	90% full	90
Ramus	96% full	88% full	54% full	83% full	83
Franklin	30% full	50% full	95% full	54% full	54
Darwin	78% full	68% full	72% full	71% full	71
:	:	:	:	:	:
:	:	:	:	:	:

INTERPRETATIONS

Before moving on to other examples, let’s revisit our quantitative piña colada from another point of view. If the white rum, pineapple juice, and coconut cream are mixed well in the quart container, then we end up with a different situation than we started with. If we now were to take the quart piña colada mixture and pour it back into the smaller 25%, 55%, and 20% containers, each of the smaller portions would no longer consist of the original single ingredient, but instead will now be piña coladas all by themselves.

If we were to once again recombine them into the quart container, this would still be an equivalent process to the one when the ingredients were separate. This illustrates the essence of what the initial blending of the ingredients tells us. That is, we can think of the blending of the three separate ingredients as an operation on these ingredients that produces a new combined object—the cocktail drink itself.

Applying this perspective to Ramus’ fluid levels, we know that mixing together the 96%, 88%, and 54% full smaller containers yields a quart container that is 83% full. We could now reapportion this fluid into each of the smaller containers in such a way that each is an individual copy of the larger container in that it is 83% full, too. This is one of the things that this blended average tells us—83% perfectly blends all of Ramus’ scores over the semester. That is, if he received a score of 83 on every single assignment that he did that semester—all of the homework assignments, all of the tests, and the final—then his average would match the average he actually received with all of the varying individual scores and differently weighted assessments.

Though receiving an identical score on every assignment didn’t, of course, happen for Ramus—unlike a well-mixed cocktail where perfect blending can occur—it still can be very useful to think of the collective average in this way. We will leverage the ideas in the last few sections where appropriate.

CLASS SIZES AS PARAMETERS

Three professors give the same common final exam to their respective Math 116 classes: one with 25 students (Section 001), the second with 55 students (Section 002), and the third with 20 students (Section 003). Before leaving for summer break, the professors each record their class averages on the exam as follows:

Section	Final Exam Average
001	96
002	88
003	54

Later during the summer, the department chair wants to obtain an overall average on the final exam for the students in all three sections to report to college administration. Unfortunately, she doesn’t have the individual test scores for each student, meaning that she can’t find the average by simply adding up all 100 individual scores and then dividing by 100. Now that the three professors are on vacation, she’d rather not bother them to get the scores, so she must figure out another way.

One option would be to simply calculate the average of the three class averages. Doing this yields

which rounds off to a 79 average.

The problem with this approach is that there were more than twice as many students—and therefore twice as many test scores—in Section 002 than in either of the other two sections, so its average should carry more impact or weight, and the current calculation doesn’t give it that. How can we factor in this greater impact?

One thing the department chair does have is easy access to the student enrollment numbers for each class. She adds this information to the table:

Section	Exam Average	Number of Students
001	96	25
002	88	55
003	54	20
Total Number of Students		100

What the department chair wants to do is to assign a value to each class based on the relative size of its student population, where a class with more students is given greater weight. This turns out to be another type of quantitative cocktail where, instead of the ingredients being three types of assessments with different contribution values, they are three final exam averages for three classes with different enrollment values.

	Piña Colada Cocktail	Final Course Average Cocktail	Final Exam Average Cocktail
Ingredient 1	White rum	Homework average	Section 001 average
Ingredient 2	Pineapple juice	Test average	Section 002 average
Ingredient 3	Coconut cream	Final exam score	Section 003 average

This time, however, instead of the department chair having the power to determine the concentration of each ingredient, there is a straightforward way to decide what the contribution for each class should be. She can simply calculate the proportion of students in a given class out of the total number of students who took the course:

Computing this for each class gives their weight:

	Final Exam Average Cocktail	Number of Students	Section Contribution to Overall Exam Average	Final Exam Average
Ingredient 1	Section 001	25	0.25 or 25%	96
Ingredient 2	Section 002	55	0.55 or 55%	88
Ingredient 3	Section 003	20	0.20 or 20%	54

The percentage concentration of each ingredient corresponds to the parameter values, meaning that a = 0.25, b = 0.55, and c = 0.20 as in Scenario 1 from the previous sections of this chapter. The regular variables are the averages for the final exam in each class, corresponding to x = final exam average in Section 001, y = final exam average in Section 002, and z = final exam average in Section 003.

Thus, the expression ax + by + cz becomes 0.25x + 0.55y + 0.20z, and after substituting the final exam averages with x = 96, y = 88, and z = 54, the department chair can calculate that the final exam average for all 100 students is 0.25(96) + 0.55(88) + 0.20(54) = 83 (rounded off).

Note that this value differs from the final exam average of 79 that we obtained when each class contributed equally to the average. It’s also worth noting that if the department chair had been able to add all one hundred individual scores and then divide by 100, she would have arrived at the same value of 83, or close to it.³ We can think of this 83 as being the score that all of the different individual scores of the 100 students blend to when we mix them together—as if each and every one of them got the identical score of 83 on the final.

The department chair has performed a powerful mathematical maneuver in calculating the overall final exam average without actually seeing a single individual student exam score. It doesn’t mean, however, that the individual student exams and scores were never seen in the process—obviously, the three professors saw the individual exams in each of their respective classes and graded them accordingly. What the technique employed here allowed the department chair to do, was to piggyback off of the efforts of her colleagues in a nontrivial way to gain new and useful information. That is, she acquired it by using an algebraic formula to process the three class averages differently, completely bypassing the need to do a direct computation involving 100 numbers.

This is conceptually similar to how the formula for the number of days and age problem enabled us to generate those special three-digit numbers in place of following each individual step in the original procedure presented in Chapter 1.

Observe that the department chair’s situation is mathematically identical to the calculation for Ramus’ course average in Scenario 1. These two Excel tables demonstrate this:

A	B	C	D	F
Name	Homework Average	Test Average	Final Exam Score	Course Average
Ramus	96	88	54	0.25*B2 + 0.55*C2 + 0.20*D2 = 83
Course	Section 001 Average	Section 002 Average	Section 003 Average	Final Exam Average for All Three Sections
Math116	96	88	54	0.25*B5 + 0.55*C5 + 0.20*D5 = 83

We can see how these two particular versions of the same type of mathematical idea—or song—play out. The first produces a single number that represents an individual student’s performance derived from their various scores across the three categories of homework, tests, and the final exam. The other generates a single number that represents the results of many individual students on one final exam across three class sections.

Before moving on, it is worth pointing out again that these two renditions of this idea differ in the way that the contribution weights are assigned. In the case of the individual course average for a student, the instructor had quite a bit of freedom to weight the value of each assessment category as they saw fit, influenced by factors such as school policies and their own educational philosophy. In the second case of the collective final exam average for three classes, the contributions were already predetermined for the department chair on the basis of a particular section’s enrollment out of 100 students. The existence of such inherent parameter values often occurs with this type of problem.

In the end, however, mathematics doesn’t ultimately care how the parameters were obtained. Once they have been established, the math involved in the calculation takes care of itself.

A THIRD VERSION: GRADE POINT AVERAGE (GPA)

Imagine that a student has just received their report card for spring semester, earning a semester GPA of 2.75:

Course	Grade	Credits
Chemistry	A	4
Calculus	C	4
Economics	B	4
Composition	C	4
Spring Semester GPA: 2.75

How would we perform a calculation by hand to verify that the GPA is correct?

One formula for calculating GPA reads as follows:

From the student’s report card, we see that they have taken a total of 16 credits for the spring semester, 4 of which were at an A grade, 4 at a B grade, 8 at a C grade, 0 at a D grade, and 0 at an F grade. Plugging this information into the GPA formula, along with the point values for each grade, yields the following for the student’s spring semester GPA:

This 2.75 GPA matches the report card.

What this calculation does is help us find the contribution value of each grade by adding all of the credits under the umbrella of that grade and then multiplying its numerical point value and credits. The varying contribution values for each letter grade are then blended together—for this student, that means 16 grade points for the A in Chemistry, 12 grade points for the B in Economics, and 8 grade points each for the C’s in Calculus and Composition—to yield the total grade points for the semester. These points are then divided by the total credits taken to give the GPA, illustrating that a GPA is just another type of quantitative cocktail with different numerical ingredients blended together in varying concentrations.

We can interpret this GPA by observing that the 44 total grade points for the semester out of 16 semester credits blend to 2.75 grade points per credit, out of 4 possible grade points per credit. Thus, if we multiply 16 credits times the 2.75 grade points per credit, we get back the 44 total grade points.

The uneven distribution of 44 grade points (among the one A, one B, and two C grades) gets blended to this 2.75 grade points per credit and by extension to the 2.75 grade points per course. We can think of the student as having made slightly less than a B grade in each of the four classes.

In its present form, however, the formula may seem a bit mysterious and isn’t an exact corollary to the earlier examples of quantitative cocktails that we’ve worked through. In both of those examples, we had a total of something—such as 100% of a grade or 100 students—spread out over categories like “types of assessment” or “class sections.” The distribution of this spread defined the scenario, which we then captured by tuning the parameters to the appropriate values. In both cases, the values of the parameters (a, b, c) and the variables (x, y, z) were each represented as percentages.

The grade point values used to calculate a GPA are generally not used that way. If we viewed these values as percentages, we would have that an A would be 100% on a 4-point scale, a B would be 75%, a C 50%, a D 25%, and an F 0%. Thus, a 2.75 GPA on the percentage scale would be 68.75%, corresponding to 2.75 out of 4 as a percent. Note that these percentages are different from the percentages that we generally associate with each letter grade on the grading scale in an individual class. Because our goal here is to use algebra as an aid in understanding, we will stay with the standard way of representing a GPA for a 4-point scale. So how, then, do we interpret this formula as a quantitative cocktail in the sense of the previous two examples?

Let’s first see if we can maneuver the formula to calculate GPA a bit, in order to better understand how it works. We need to use a property of fractions to do this, so let’s first review how to add two or more fractions with the same denominator:

These are several instances of an ensemble of numerical expressions, so we can use algebra to represent them all in a single form as

For example 1, we have x = 3, y = 2, z = 0, u = 0, v = 0, and w = 7.

For example 2, we have x = 1, y = 3, z = 0, u = 0, v = 0, and w = 8.

For example 3, we have x = 5, y = 1, z = 7, u = 4, v = 2, and w = 24.

This expression states the well-known arithmetic principle that when you add fractions with a common denominator, you simply add the numerators and retain the common denominator.

We have chosen to give the addition of five fractions here, as the general example, because this corresponds to the five fractions that we will get from GPA Formula (1). Recall from Chapter 3 that this algebraic formula is an identity rather than an equation to be solved.

An effective way to think about the equals sign in an identity is like a simple doorway. You can come into a room one way through a door, and you may leave the room through the same door going the opposite way. Often when an identity is presented to us, the tendency is to read it from left to right, the way we read in English. However, unlike English, we can also read an identity from right to left as both sides are equivalent. To remind ourselves of this, we can simply rewrite it with the left-hand side becoming the right-hand side and vice versa:

This is the principle we need to be able to maneuver GPA Formula (1) into a more familiar form. Before applying this identity, let’s first simplify the formula by plugging in the grade point values for each letter grade:

Now, applying the fraction identity [where the common denominator w = total credits and the numerators x = (A credits) (4), y = (B credits) (3), z = (C credits) (2), u = (D credits) (1), and v = (F credits) (0)] yields the following:

GPA Formula (2)

Looking at the first term, we can rewrite as For example, can be written as where each simplifies respectively to and 2·3, which in turn both equal 6.

The fraction is simply the percentage of credits taken that were A credits out of the total number of credits in the semester. For the spring report card, this becomes which is 0.25 or 25%. We can think of the four other fractions in this same fashion as the percentage of credits that were B credits and so on. Substituting this information into GPA Formula (2) and moving the grade point numbers in front of each term gives the following:

GPA Formula (3)

This time, the fixed elements of the expression are the grade point values and the variable elements are the percentages of credits received under the umbrella of each letter grade. So, we can let these fixed values define the scenario in this case—it would be a different scenario if we gave plus and minus values for grades such as A+ or A–.

Returning to the parameter form of the previous quantitative cocktails, this time with five categories instead of three, we obtain the formula ax + by + cz + du + ev. Substituting in the parameter values for the worth of each letter grade, the expression simplifies to 4x + 3y + 2z + 1u + 0v.

It is important to note that in this case using constant grade point values as parameters means that, for this particular interpretation, they do not add up to 100%; rather, it is the variables, as percentages of credits taken for a certain grade, that add up to 100%. Thus, in this sense, the roles of parameters and variables are reversed for GPA Formula (3).

The variables are now as follows:

x = Percentage of credits taken where an A grade was received

y = Percentage of credits taken where a B grade was received

z = Percentage of credits taken where a C grade was received

u = Percentage of credits taken where a D grade was received

v = Percentage of credits taken where an F grade was received

Let’s now apply GPA Formula (3) to another report card, this time to a case where the student is taking more courses and the courses are worth different amounts of credits:

Course	Grade	Credits
Geology	A	4
Calculus III	F	4
Anatomy/Physiology	B	5
Computer Science	C	4
Political Science	A	3
Total Credits		20
Fall Semester GPA: ??

Fall semester report card

Variables	Total Credits	Percentage (Out of 20 Credits)
x = total A grade credits	7	35%
y = total B grade credits	5	25%
z = total C grade credits	4	20%
u = total D grade credits	0	0%
v = total F grade credits	4	20%

Fall semester credit totals

Placing this information into the expression 4x + 3y + 2z + 1u + 0v yields

fall semester GPA = 4(0.35) + 3(0.25) + 2(0.20) + 1(0) + 0(0.20) = 2.55.

You can check the calculation of the GPA this way by using the original formula, GPA Formula (1), to verify that they match.

Formatting the GPA formula into an explicit parameter form has its advantages, as it enables us to structure this calculation in a familiar conceptual framework and leverage our understanding of the first two examples. If we have a firm grasp on how earlier quantitative cocktails were mixed, we can cross-apply that understanding to this circumstance.

This is not unlike how we use percentages to transport understanding in arithmetic. Percentages provide a common framework to understand the relationship between parts and a whole. When we are confronted with an unfamiliar numerical relationship—such as 46,659 people voting in an election out of 155,530 eligible voters—percentages set up a ratio that enables us to understand that relationship in the familiar context of parts out of 100. In this case, that ratio would be which equals 0.30. We can interpret this as 30%, which means that the ratio of the number of people who voted to the total number of people eligible to vote is the same as if 100 people were eligible and only 30 of them voted.

So, if the parameter viewpoint in GPA Formula (3) gives us a similarly standard frame of reference, why use the original formula at all?

Though the parameter viewpoint may offer some conceptual advantages, the original GPA Formula (1) has certain computational advantages that become apparent when the ratios do not yield a simple decimal. For example, if the Anatomy/Physiology course were only worth 4 credits instead of 5, the total number of credits in the student’s fall semester would be 19 instead of 20. The percentage of A credits to total credits would then be 7 out of 19, or 0.3684210526.… To use this number in the parameter formula, we would most likely want to round it off to something like 0.3684 or 36.84%, which would result in a loss of information for the calculation. Because the prime number 19 is not a very decimal-friendly denominator, the percentages for the other grades would have the same issue, amplifying the scale of information loss in the problem and ultimately leading to a less accurate result for the resulting GPA.

GPA Formula (1) avoids these issues by dealing with whole numbers throughout the majority of the calculation, saving rounding off until the final step after the division is performed. This will give it a more accurate answer in cases such as the 19 credits variant, where we could end up rounding numbers as many as five times if we were to use GPA Formula (3). So, it’s best to use the generally more straightforward GPA Formula (1) once we understand the concepts that underpin the calculation of a GPA.

It is also worth pointing out that using the fraction maneuver on the GPA calculation here would have been difficult to perform or understand for someone who only knew how to compute numerical fractions with a calculator. In order to successfully understand and work with fractions when variable quantities are involved, we need a deeper understanding of fractions and what the operations mean, which is just one illustration of how arithmetic is such an important conceptual foundation for algebra.

A FOURTH VERSION: EFFECTIVE PERCENTAGE RATE OF RETURN ON INVESTMENTS

An investor is considering whether or not to invest their money in three stocks or make a less volatile investment that guarantees them a 4% return over the next year. They know that stocks come with the potential for higher gains, but they come with more risks too, some of which could lead to a lesser return or even a loss. The investor has a computer application that allows them to forecast different scenarios to see how much their money will increase from spreading it out over the three stocks, each with a possibly different percentage return. Though pleased with the program, it seems like a bit of a black box to them, and they want to know more about how the returns on investment are calculated. Over the course of their research, they find the following formula for calculating the effective percentage rate of return:

(% of money invested in Stock A) (% return from Stock A)

+ (% of money invested in Stock B) (% return from Stock B)

+ (% of money invested in Stock C) (% return from Stock C).

Let’s say that the investor places 25% of their money in the stock for Company A, 55% in Company B’s stock, and the final 20% of their money in Company C’s stock—with expected percentage returns of 3.0%, 5.0%, and 9.5%, respectively. Then, the effective percentage increase of their investment can be calculated as

(0.25) (3.0%) + (0.55) (5.0%) + (0.20) (9.5%) = 5.4%.

This gives an effective percentage rate of return of 5.4% on all of the money in this scenario, meaning that if they invested $50,000 for the year, their money would have increased by 50000(0.054) = $2700 in the stocks versus 50000(0.04) = $2000 with the safer investment.

The formula in this format helps the investor understand the situation a bit more than simply plugging numbers into the application, but they still don’t have a firm handle on it. The calculation looks both familiar and strange to them at the same time.

Firstly, the investor is taking a certain amount of money and distributing it over three stocks. In the case of a $50,000 investment, that distribution would be $12,500 (25%) in Stock A, $27,500 (55%) in Stock B, and $10,000 (20%) in Stock C. The calculation shows them the overall projected increase in their finances, blending the results of their three individual investments expected to increase by 3.0%, 5.0%, and 9.5%, respectively. This suggests that once again we are dealing with a quantitative cocktail. The following table compares this to the other renditions of quantitative cocktails in the chapter:

	Final Course Average Cocktail	Final Exam Average Cocktail	GPA Cocktail	Effective Return Cocktail
Ingredient 1	Homework Average	Section 001 Average	% of credits receiving an A	% return of Stock A
Ingredient 2	Test Average	Section 002 Average	% of credits receiving a B	% return of Stock B
Ingredient 3	Final Exam Average	Section 003 Average	% of credits receiving a C	% return of Stock C
Ingredient 4			% of credits receiving a D
Ingredient 5			% of credits receiving an F

Thinking back to earlier cocktails, let’s take a look at how this one is mixed and establish our parameters and variables. The levers that we can adjust to fix the scenario are the percentages of money invested in Stock A, Stock B, and Stock C. The values that change, then, are the increases and decreases in each stock’s performance. Thus, we will consider the former as parameters and the latter as variables. For the expression ax + by + cz, this results in the following interpretation:

a = (percent of total amount of money invested in Stock A),

b = (percent of total amount of money invested in Stock B),

c = (percent of total amount of money invested in Stock C),

x = (percent return from Stock A),

y = (percent return from Stock B),

z = (percent return from Stock C).

For the investment scenario here, we already know that a = 0.25, b = 0.55, and c = 0.20, and so the formula becomes 0.25x + 0.55y + 0.20z—which should look familiar because this, too, is operationally identical to Scenario 1 for the course average cocktail.

Let’s now consider six investors each investing in a portfolio of three stocks, each with a different mixture of stocks than the other five. For example, Al-Samawal invests in three different stocks than Ramus, but they both allocate their money over their three unique stocks in the same way: 25% of their money in their Stock A, 55% in their Stock B, and 20% in their Stock C. The following table gives the combined increases on each total investment derived from the respective increases in specific stocks:

Name	x (Stock A)	y (Stock B)	z (Stock C)	Effective % Return = 0.25x + 0.55y + 0.20z
Hypatia	9.6%	9.5%	9.9%	9.605%
Al-Samawal	9.0%	9.2%	9.4%	9.19%
Dardi	8.8%	9.0%	9.3%	9.01%
Ramus	9.6%	8.8%	5.4%	8.32%
Franklin	3.0%	5.0%	9.5%	5.4%
Darwin	7.8%	6.8%	7.2%	7.13%

For example, if Ramus’ stocks increased in value by 9.6%, 8.8%, and 5.4%, respectively, then the money that he invested in all three has a combined return of 8.32%. This means that we could think of the three individual investments working together as one big single investment that yielded an 8.32% return. As such, if Ramus invested a total of $50,000 in his three-stock portfolio, then his money would have increased by 50000(0.0832) = $4160—more than twice as much as the increase in the safe 4% investment. Note that our initial investor’s circumstances with a 5.4% overall increase match Franklin’s in the table.

It is also possible for a stock’s value to decrease. When this happens, we can use the same formula and simply introduce negative numbers to represent the losses. For example, if Dardi kept his investment percentages the same but his portfolio performance changed such that Stock A increased in value by 6.7%, Stock B decreased by 6.2%, and Stock C decreased by 3.5%, his return could be represented as follows:

x (Stock A)	y (Stock B)	z (Stock C)	Effective % Return = 0.25x + 0.55y + 0.20z
6.7%	–6.2%	–3.5%	0.25(6.7%) + 0.55(–6.2%) + 0.20(–3.5%) = –2.435%

This means that if he invested $50,000, then his money would have changed in value by 50000(–0.02435) = –$1217.50, which represents a loss of $1217.50. This, of course, would be a worse situation than placing his money in the safe investment with a guaranteed 4% increase on his money.

Dardi is free to forecast other values of x, y, and z while keeping his investment percentages the same, thereby leaving the parameters alone. Alternatively, he could change the scenario entirely by distributing his money differently among the three stocks—changing investment percentages a, b, and c—or by diversifying even further by distributing his money over five stocks instead of three, which would mean the addition of two more parameters and two more variables, as happened with the GPA formulas.

In the next section, we briefly discuss how mathematicians handle situations that require them to deal with larger numbers of variables.

SUBSCRIPTED VARIABLES

When we calculated GPAs, we were confronted with a situation that had more variables than our traditional variables x, y, and z could cover. This meant that we had to introduce more letters to handle them—u and v. However, there were only two additional variables to contend with in that case. What should we do if we have 10, 15, or even 30 variables? Clearly, assigning variables according to Descartes’ method of using later letters in the alphabet for variables and those early in the alphabet for parameters will founder under such an explosion.⁴

These situations can and do happen, and mathematicians throughout history have come up with creative protocols to deal with them. The medieval Indians on the subcontinent came up with a color scheme to handle situations involving multiple unknowns, although parameters as we know them today appear not to have existed for them. Their primary unknown was yāvattāvat, or “as much as so much,” which approximates the x that we use today. When they encountered situations involving several unknowns, they innovated by using colors. Describing their scheme in the 1100s, Indian mathematician Bhaskara II wrote:

[One] unknown (yāvattāvat) is the color black, another blue, yellow, and red. [Colors] beginning with these have been imagined by the best of teachers to be the designations of the measures of the unknowns in order to accomplish their calculation.⁵

In order to further simplify matters, they frequently abbreviated to only the first syllable. In the case of yāvattāvat, they would use yā.⁶

In modern times, mathematicians have developed other resourceful ways to handle multiple variables and parameters, usually injecting subscripted numerals into the notational scheme. We will discuss one such scheme that solves the problem at hand while maintaining consistency with the Descartes protocol.

Let’s consider the case where we have five variables. We decided earlier to choose the letters x, y, z, u, and v to represent them, but using this new notation, we can instead use a subscripted format of x₁, x₂, x₃, x₄, and x₅. Although we repeat the use of x for each new variable, this does not imply a derivative relationship between these variables. We can think of the subscripts as telling us that x₁ represents the first variable instead of x, x₂ represents the second variable instead of y, and so on up to x₅ representing the fifth variable instead of v. We can replace parameters a, b, c, d, and e in a similar fashion with the subscripted format a₁, a₂, a₃, a₄, and a₅.

This means that the expression for GPA translates in the following way:

ax + by + cz + du + ev → a₁x₁ + a₂x₂ + a₃x₃ + a₄x₄ + a₅x₅.

Note that this subscripted formula possesses a convenience that the un-subscripted one does not. For instance, if you were asked with which variable the parameter d was associated, you might have to go back and do a bit of comparing to see that the answer is u. However, when utilizing subscripts, if you were asked with which variable the parameter a₄ was associated, you could immediately say x₄. If we simplify the GPA expression to 4x₁ + 3x₂ + 2x₃ + 1x₄ + 0x₅—where a₁ = 4, a₂ = 3, a₃ = 2, a₄ = 1, and a₅ = 0—the fall semester GPA table would become the following:

Variables	Total Credits	Percentage (Out of 20 Credits)
x1 = total A grade credits	7	35%
x2 = total B grade credits	5	25%
x3 = total C grade credits	4	20%
x4 = total D grade credits	0	0%
x5 = total F grade credits	4	20%

Fall semester credit totals

Performing the calculations using the relabeled formula will still yield the same GPA of 2.55.

We can see how this new notational system can scale to situations involving even more variables—for instance, in a case involving 10 variables, we would simply use the 10 subscripted variables x₁, x₂, x₃,…, x₈, x₉, x₁₀.

A FEW MORE EXAMPLES

There are many more examples of quantitative cocktails beyond those we have discussed here. Like our GPA calculation, some of them can be simplified, doing one grand division at the end of the calculation, to avoid rounding decimals multiple times in the process. However, here we will discuss them from the point of view of the subscripted expression: a₁x₁ + a₂x₂ + a₃x₃ + a₄x₄ + a₅x₅.

SURVEY SCALES

If you’ve taken a survey recently, you may immediately recognize the potential for variation within them. Respondents are given a series of questions and are expected to answer them according to a rating scale. Many surveys use psychometric rating scales, often called Likert scales after their inventor, Wyoming-born Rensis Likert. Some scales use the following ratings and number assignments: 1 = very unsatisfied, 2 = unsatisfied, 3 = neutral, 4 = satisfied, 5 = very satisfied, or 1 = strongly disagree, 2 = disagree, 3 = neutral, 4 = agree, 5 = strongly agree. Then, for any given question, the individual answers from all respondents are blended into a collective result.

	Survey Question Score Cocktail 1	Survey Question Score Cocktail 2	Variables
Ingredient 1	% of answers receiving very unsatisfied	% of answers receiving strongly disagree	x1
Ingredient 2	% of answers receiving unsatisfied	% of answers receiving disagree	x2
Ingredient 3	% of answers receiving neutral	% of answers receiving neutral	x3
Ingredient 4	% of answers receiving satisfied	% of answers receiving agree	x4
Ingredient 5	% of answers receiving very satisfied	% of answers receiving strongly agree	x5

Using the parameter assignments a₁ = 1, a₂ = 2, a₃ = 3, a₄ = 4, and a₅ = 5, the formula for calculating the blended scores for each question a₁x₁ + a₂x₂ + a₃x₃ + a₄x₄ + a₅x₅ becomes

1x₁ + 2x₂ + 3x₃ + 4x₄ + 5x₅.

For example, if 5 respondents answer Question 7 on the survey with very unsatisfied, 6 answer unsatisfied, 8 answer neutral, 21 answer satisfied, and 10 answer very satisfied, we can calculate the following percentages out of 50 responses: x₁ = 10%, x₂ = 12%, x₃ = 16%, x₄ = 42%, and x₅ = 20%. This will yield an overall score for Question 7 of 1(0.10) + 2(0.12) + 3(0.16) + 4(0.42) + 5(0.20) = 3.5, meaning that the collective feeling on this question is halfway between neutral and satisfied.

Recalling course averages, we can think of all the individual survey responses mixing together to yield a result that would be equivalent to a circumstance where each respondent selected 3.5. Of course, in the context of the problem, possible responses can only be whole numbers, so we can also think about this as an even split of respondents scoring 3 and 4. The other questions would be scored in the same fashion.

This procedure can be applied to many rating systems, including those that use stars to evaluate various products and services such as restaurants, hotels, books, or films. Some such rating systems, however, may employ additional methodology besides what has been discussed here to generate their final ratings.

SURVEY CATEGORIES

Consider the results of a survey of 1000 households to determine the average number of televisions they have in the home:

Number of TVs per Household	Number of Households
0	40
1	90
2	460
3	310
4	100
Total Households:	1000

We could summarize the overall result of all of these responses using a similar formula a₀x₀ + a₁x₁ + a₂x₂ + a₃x₃ + a₄x₄ with a slight adjustment, this time with parameters a₀ = 0, a₁ = 1, a₂ = 2, a₃ = 3, and a₄ = 4 and variables x₀ = percentage of households with zero TVs, x₁ = percentage of households with one TV, and so on up to x₄ = percentage of households with four TVs. You’ll notice that in this case we’ve chosen to begin with a₀ and x₀ rather than a₁ and x₁ so that the parameter and variable subscripts match the number of TVs.

Placing the values in for the parameters yields the expression 0x₀ + 1x₁ + 2x₂ + 3x₃ + 4x₄. The percentage of households in each category of TV ownership are x₀ = 4%, x₁ = 9%, x₂ = 46%, x₃ = 31%, and x₄ = 10%. Substituting these values into our general formula yields 0(0.04) + 1(0.09) + 2(0.46) + 3(0.31) + 4(0.10) = 2.34. This means that, for the 1000 households surveyed, the average one has about 2.3 TVs in the home, or every three households have about seven TVs between them.

BASEBALL SLUGGING PERCENTAGE

The official website of Major League Baseball defines slugging percentage as follows:

Slugging percentage represents the total number of bases a player records per at-bat.… Slugging percentage differs from batting average in that all hits are not valued equally. While batting average is calculated by dividing the total number of hits by the total number of at-bats [note that a walk does not count as an at-bat], the formula for slugging percentage is: (1B + 2Bx2 + 3Bx3 + HRx4)/AB.⁷

A player’s slugging percentage could just as easily be called their slugging average. The idea behind a slugging percentage is that it differentiates between the type of hits a batter gets. Thus, a player who gets more extra base hits will generally fare better with slugging percentage than a player who hits only singles. The traditional batting average does not distinguish between the different types of hits, meaning that a single and a home run have the same value in the calculation.

The calculation for slugging percentage is operationally similar to a GPA; however, there are some key differences which include the following:

• The values for slugging percentage are given in thousandths, so where a GPA would be given as 0.52 the equivalent slugging percentage would be given as 0.520 (pronounced 520).

• Slugging percentages for very good baseball players are far lower than GPAs for very good students. That is, though it is not unusual for very good students to get straight A’s (or a 4.0 GPA), good players don’t get “straight home runs.” The highest slugging percentage in major league baseball history for a single season is 0.863 by Barry Bonds in 2001.⁸ A 0.863 GPA is below a straight D average.

If we were to convert slugging percentage into a quantitative cocktail, we would have the following:

	Slugging Percentage Cocktail	Variables
Ingredient 1	% of at-bats that are outs	x1
Ingredient 2	% of at-bats that are singles	x2
Ingredient 3	% of at-bats that are doubles	x3
Ingredient 4	% of at-bats that are triples	x4
Ingredient 5	% of at-bats that are home runs	x5

The parameter values would therefore be a₁ = 0, a₂ = 1, a₃ = 2, a₄ = 3, and a₅ = 4, and applying them to our general formula will yield

slugging percentage = 0x₁ + 1x₂ + 2x₃ + 3x₄ + 4x₅.

Note that the formula on the MLB website leaves out the 0 value term in the numerator for when an out occurs. This is not a problem because multiplying by 0 would make this term disappear anyway. The outs made still get recorded in the total number of at-bats in the denominator.

The possibility of relabeling this formula starting at x₀ and ending at x₄, as we did in “Survey Categories,” remains an option here as well.

Let’s see how the slugging percentage plays out in the following 25 at-bat scenario where a player recorded 16 outs, 5 singles, 2 doubles, 1 triple, and 1 home run, adding up to 16 total bases. Out of 25 at-bats, this would give our variables the following values: x₁ = 64%, x₂ = 20%, x₃ = 8%, x₄ = 4%, and x₅ = 4%. The slugging percentage formula therefore becomes

0(0.64) + 1(0.20) + 2(0.08) + 3(0.04) + 4(0.04) = 0 + 0.2 + 0.16 + 0.12 + 0.16 = 0.64.

We would interpret this as a slugging percentage of 0.640, which would be considered an extremely good slugging percentage. To put it in perspective, this means that this batter on average gets 0.64 bases for every at-bat. We can check this by multiplying the 25 at-bats by 0.64, which yields 16 total bases out of 25 at-bats, which matches the scenario. Note that the straight batting average would be given by or 0.360.

Another way to view what the slugging percentage is telling us is to divide the slugging percentage by the batting average. This will give us the number of bases per hit that the player gets on average. Here that would be or approximately 1.78 bases per hit. You can also check this directly by dividing the 16 bases by the 9 hits. This means that we can think of this batter as getting close to a double for every hit.

The ratio of number of bases per hit by itself is not as useful as the slugging percentage because it doesn’t take outs into account. For instance, a player who gets 1 hit out of every 50 at-bats, but that hit is a home run, will have a ratio of 4 bases per hit, and yet they’ll get out 98% of the time. The batter’s slugging percentage, which does account for outs, would be a very low 0.080, or 0.08 bases per at-bat.

ENHANCED REASONING

Back in Chapter 1, we discussed the hypothetical situation of a child discovering new words through rhymes rather than synonyms. The former ties words together based on having similar sounds, whereas the latter method ties together words based on common meanings.

Both turn out to be useful and systematic ways to learn and remember new words.

Similar to that child, in this chapter we have tied together diverse numerical phenomena that share common algebraic and conceptual features in order to learn more about them. We have done this before in the classroom word problems of Chapter 5, where phenomena were connected by similar types of equations, and in Chapter 7, where five classroom and real-world problems were linked together around an identical simplified equation. Linking together conceptually similar entities is something that nearly all of us do, with the process forming an important foundation to analogical and metaphorical thinking more generally. What distinguishes and intensifies what we have been doing here is the way that we have incorporated algebra into our thinking.

The Spanish philosopher José Ortega y Gasset described it thus:

The metaphor is perhaps one of man’s most fruitful potentialities. Its efficacy verges on magic, and it seems a tool for creation which God forgot inside one of His creatures when He made him.… The metaphor alone furnishes an escape; between the real things, it lets emerge imaginary reefs, a crop of floating islands.⁹

What algebra does par excellence is sharpen these tools to a much finer point. Put another way, algebra singularly weaponizes metaphorical and analogical reasoning, rendering it more precise and operational. This is true of mathematics in general, but algebra forms one of the strongest alloys used to forge this mighty mathematical sword.

We have employed this enhanced reasoning twice in this chapter, first by tying together some types of table and spreadsheet analysis with algebraic reasoning and second by examining different variations on the theme of quantitative cocktails. What useful knowledge has been gained by doing this?

In Chapter 6 we discussed, as part of the algebraic experience, how algebraic thinking can help us gain more insight into numerically varying phenomena while also giving us greater technical confidence and comfort in dealing with such situations. The goal of this chapter has been to see this premise in action, as well as highlight the emerging possibilities for deeper understanding and new connections that become possible through the strategic deployment of algebraic technique.

Let’s revisit some of the major conceptual takeaways from this chapter that emphasize the strengths of algebraic thinking.

EXPRESSING VARIATION WITH TABLES AND SPREADSHEETS

By connecting certain aspects of tables and spreadsheets to algebraic expressions, we identified a conduit between these two important ways of expressing numerical information. Moreover, the use of specific examples allowed this connection to be exploited in a bit more detail and depth. The examples we analyzed are only a few of the hundreds and thousands possible—some relevant to everyday life and some, like the number of days and age problem, more gimmicky than practical.

Although the number of days and age problem may have no direct bearing on everyday life, the relationships, properties, and strategies essential to understanding the problem—like symbolically and efficiently organizing ensembles of numbers—can and do have direct applicability to a wide range of circumstances. The analysis of data in tables and spreadsheets will inevitably come up time and again for most of us, and so the hope is that by highlighting the relationship between algebra, tables, and spreadsheets, we can strengthen our understanding of all three tools and their interaction.

QUANTITATIVE COCKTAILS

By exploring problems that blend distinct, unequal portions together into a whole, we have uncovered additional conceptual treasures.

PRECISION/INTUITIVE INTERPLAY: Algebra helps us operationalize and control our mixing of diverse quantities with mathematical precision. Although our general ability to recognize patterns makes it possible for us to intuit connections between affected phenomena, it is algebra—with its vocabulary of variables and parameters—that gives us a language in which to explicitly identify the substance of that connection. Qualitative metaphors like cocktail mixing also strengthen conceptual comprehension. The precise and qualitative features of this enhanced reasoning can work together in powerful, almost magical conjunction.

ALGEBRAIC SUPERHIGHWAY: When we use algebraic language to express connections and establish relationships, we are able to link seemingly isolated phenomena into a powerful symbolic superhighway. Much like an interstate connecting isolated towns can enhance commerce and exchange within that network, so too can our understanding of diverse phenomena be greatly strengthened by algebra.

We tried to do this explicitly in the case of calculating GPAs by illustrating its similarities to calculating course averages, in hopes that we could leverage our existing understanding to better conceptualize the new circumstances. Mathematicians and scientists do this on a regular basis, using words such as realizations, representations, isomorphisms, homeomorphisms, and canonical forms to describe the sophisticated linking together of diverse ideas, objects, and behaviors for conceptual insight and computational advantage.

THE TRADITIONAL ARITHMETIC AVERAGE HAS SIBLINGS: Calculating more sophisticated averages has helped establish a more contextual view of how they relate to the traditional average and what that comparison tells us.

For instance, to find the average of the three numbers 30, 50, and 95, we would calculate to arrive at rounding to 58. However, by performing a symbolic maneuver, we could rethink this division as

These three numbers correspond to Franklin’s scores in the three assessment categories of homework, tests, and the final from the section on computing grades.

This maneuver now shows us that in the formula ax + by + cz, we have the parameters a, b, and c each equal to or which yields The variables x, y, and z retain their meaning, which here makes them equal to 30, 50, and 95, respectively.

Thus, the traditional average itself is a quantitative cocktail corresponding to the situation where each category contributes the same amount to the total. In the case of three numbers, each contributes to the total. If there were five numbers—or categories—each number would contribute or 20% to the total, and so on.¹⁰

The traditional average is generally determined from a single division for the same reasons that one grand, final division is preferable when calculating GPAs—we obtain a more accurate result. Thus, rather than standing alone as an isolated entity, the traditional average turns out to be one of an entire family of different types of averages. This family of averages is more commonly called weighted averages, a conceptually important mathematical category in its own right.

GATEWAY TO NEW IDEAS: All versions of an idea are not created equal. Identifying the effective percentage rate of returns on stock investments as a quantitative cocktail opened up a new situation—the possibility that the impact of one category can take away from the impact of another category and that effects can be subtractive. In the first three examples, the impact from the various categories all reinforced one another.

A CHILD SEARCHING FOR NEW WORDS: Though we discovered that effects from a given category can be subtractive as well as additive through a specific example, we could have inferred as much directly from the formula ax + by + cz itself. That is, nothing prevented us from asking the question: Can we use this formula if one or more of the products—ax, by, or cz—are negative?

This question may have led to examples that fit the scenario in question, which may in turn have pointed us to “effective percentage rates of returns.” This type of critical inquiry and mathematical probing can and does occur in the higher levels of the subject and is a useful catalyst for generating new ideas and discoveries, just as an exploratory approach to language produces new words and opens new possibilities for a child.

COMPUTATIONAL GAIN: When we set out to purchase a new piece of technology such as a digital camera or a computer, we have predetermined criteria in mind that form the basis of our decision-making, some more important to us than others. For a computer, these various properties may be the type of microprocessor, the speed of the microprocessor, RAM, hard-drive capacity, or price. For a digital camera, they could be aspects like image resolution, battery life, ISO range, cost, or a built-in flash.

For the consumer, these categories can help them assess many individual devices, just as homework, tests, and the final exam enable teachers to assess their students. Like a teacher, the consumer can assign contribution percentage values—or weights—to each of these categories or properties based on their relative importance to them, giving each device their own personal “grades.” In the case of a digital camera, some consumers will place a higher premium on image resolution over ISO range, and so on. A built-in flash may not matter at all to certain consumers, leading them to dismiss that property from consideration just like the instructor who doesn’t factor attendance into a student’s final grade. In some cases, the scores for the camera categories will be on a scale from 0 to 5 or 0 to 10 or 0 to 100, and will be rated by an expert in a magazine or online.

Let’s say, for example, that a given consumer ranks categories of importance for their digital camera purchase as follows: 50% for image resolution, 20% for battery life, and 30% for ISO range. The formula to rate prospective cameras will then be 0.50(image resolution score) + 0.20(battery life score) + 0.30(ISO range score). In our standardized language, this would establish parameters a = 0.50, b = 0.20, and c = 0.30 with variables x = image resolution score, y = battery life score, and z = ISO range score—resulting in the expression 0.50x + 0.20y + 0.30z.

Now, let’s assume that a magazine rates a particular model of camera at a 6 for image resolution, an 8 for battery life, and a 7 for ISO range on a 10-point scale. The overall score for this camera on the consumer’s grading scale would therefore be 0.50(6) + 0.20(8) + 0.30(7) = 3 + 1.6 + 2.1 = 6.7. In a similar fashion, the customer could give personal grades to other cameras and then make their determination by comparing those values. Of course, just like instructors grading students, other customers will weigh the importance of each category differently, resulting both in different parameter values and in different scores or grades for the cameras.

Computers, cars, houses, and even colleges can be scored and compared this way. It’s worth remembering that algebra doesn’t remove subjectivity and shouldn’t remove common sense from the decision-making process—emotions, gut instincts, and other intangibles still matter—but it may help to impose a bit of consistency, organization, and clarity on what may otherwise seem like a tidal wave of options.

CONCEPTUAL GAIN: A quantitative cocktail that affects multiple tens of thousands of students every year is the well-known U.S. News & World Report College Rankings. The following table lists the 2021 categories and their weights for universities and liberal arts colleges¹¹:

	U.S. News & World Report National Universities and Liberal Arts Colleges Cocktail	Variables
Ingredient 1	Graduation and retention rates (22%)	x1
Ingredient 2	Social mobility (5%)	x2
Ingredient 3	Graduation rate performance (8%)	x3
Ingredient 4	Undergraduate academic reputation (20%)	x4
Ingredient 5	Faculty resources (20%)	x5
Ingredient 6	Student selectivity (7%)	x6
Ingredient 7	Financial resources (10%)	x7
Ingredient 8	Average alumni giving rate (3%)	x8
Ingredient 9	Graduate indebtedness (5%)	x9

The parameter assignments are respectively

a₁ = 0.22, a₂ = 0.05, a₃ = 0.08, a₄ = 0.20, a₅ = 0.20, a₆ = 0.07, a₇ = 0.10, a₈ = 0.03, a₉ = 0.05.

Plugging these values in yields a University and College Ranking Score:

0.22x₁ + 0.05x₂ + 0.08x₃ + 0.20x₄ + 0.20x₅ + 0.07x₆ + 0.10x₇ + 0.03x₈ + 0.05x₉.

Once scores for each of the variables are known for a given school, we can plug them into this formula and complete the calculation. However, the U.S. News & World Report applies a bit of additional methodology to obtain the final scores that they publish.

Now if you are so inclined, you have the tools to change the parameter assignments for each of the various categories and come up with your own personal ranking system for colleges and universities based on the relative importance of each category to you—thus no longer being a spectator in this particular area, but a player with the agency to interact with and manipulate algebra for your own needs.¹²

CONCLUSION

The narratives as discussed in this chapter would be hard pressed to enter our consciousness without some awareness of algebra. Crucial to it all is the recognition that when we encounter algebraic ideas outside of the classroom, they are usually cloaked in other garb. They rarely come at us in the x’s, y’s, and z’s of school algebra.

The critical “tell” of algebraic possibilities in the wild is the presence of numerical variation of some sort. Sometimes what we are interested in may only be a single instance of a variable situation. The objects that generate those variations can be individual people distinguished by their performances in a class, grades in several classes, income, survey responses, or batting results; or they could be variations in company stock prices, ratings of different electronic objects, and ratings of different colleges and universities, as well as a host of other possibilities.

Once it has been established that variation is present in a situation, either by personal investigation or more likely through formulas obtained elsewhere—the internet, by word of mouth, or a book—it sometimes becomes possible to replace explanations—often given in words, via strange symbols, via visual diagrams, or as tables of values—by the familiar x’s, y’s, and z’s, or subscripted symbols and parameters, too, if needed.

This standard notation offers a valuable orienting principle, as a landmark does, to understand and interact with numerical phenomena. Sometimes the abbreviations in their original form are powerful enough symbols in their own right, or are so common that we tend not to retag the variable, as with E = mc². Either way, algebra is there to play some role in helping us to better understand and interpret relevant quantitative problems—but only if we want it to. It is reminiscent of the possibilities afforded us by the panorama of history to better understand and interpret our own historical times, where we have powerful tools at our disposal that offer us insight to make better, more informed decisions for the future.

But for individuals to take advantage of the warmth and brightness from either of these conceptual lights, we must first know where to look, appreciate their power to enlighten, learn how they work, and, ultimately, have the courage to flip the switch and turn them on.