Week 1: Course introduction; Break-even & Math/Stats Review I

This week’s big question: what is a “model,” and why does turning a real decision into a little bit of algebra help you make a better one? Monday is entirely about how the course runs and your questions. Wednesday we meet the course’s first real tool, break-even analysis, and refresh the exact math the rest of the term relies on.

This is a course in management science: the family of tools, also called operations research or business analytics, that public, nonprofit, and business managers use to turn a messy real decision into a model they can reason about. Modern analytics pairs these methods with large amounts of data, but the mindset is the same one we start with this week. The value isn’t only the number a model spits out; it’s the why: which lever to pull, which assumption could break, which alternative wins. We will care about interpreting a result at least as much as computing it.

Before the week

Nothing is assigned for Monday. Read this before Wednesday, which carries the whole week’s content. The chapter PDF is in this week’s Canvas module.

Taylor, Introduction to Management Science (13e), Chapter 1, pp. 22–33. That is twelve pages, and it is three sections:

Pages	Section	What to take from it
22–27	The Management Science Approach to Problem Solving	The five steps, from observation to implementation
27–28	Management Science and Business Analytics	Where these methods sit next to “analytics”
28–33	Model Building: Break-Even Analysis	Fixed vs. variable cost, and the break-even quantity

Stop at the “Computer Solution” heading partway down p. 33. What follows is an Excel and QM for Windows walk-through (we do our own, in Week 8) and then a survey of techniques we meet in later weeks.

Two things to know before you open it, so the reading does not fight the lecture:

Taylor’s symbols are not ours, and one letter is actively confusing. The book writes volume as $v$, variable cost per unit as $c_v$, and fixed cost as $c_f$, so on p. 30 its profit equation is $Z = vp - c_f - vc_v$ and it finds break-even by setting that to zero and solving for $v$. This course calls the quantity $q$ and uses $v$ for the variable cost per unit, so the same model reads $\text{profit} = pq - F - vq$ and $q^* = F/(p-v)$. Taylor’s $v$ is our $q$; Taylor’s $c_v$ is our $v$. Translate as you read rather than memorizing the book’s letters.
The chapter has no summation notation in it. The $\sum$ sign in Wednesday’s Math/Stats Review is ours, not Taylor’s, so do not go hunting for it.

Session 1: Mon, Aug 24 · How this course works (TBD)

No content, no reading, no worksheet. The whole session is the course itself, and most of it is your questions.

A live tour of this site and of Canvas. Everything you need to learn lives here: the schedule, the week page behind each topic link, the self-checks, Key Concepts, and Using AI. Canvas is where you upload each week’s workbook, take the in-class checkpoints in LockDown Browser, and check your attendance record.
Where the 100 points come from. Fourteen weekly take-homes at 5 points, best 11 counted (55) · the course synthesis, submission plus presentation sessions (16) · six in-class checkpoints at 2.5 (15) · in-class participation, the 14 group practices (14). We open a real workbook on the projector so you can see what a submission actually looks like before you owe one.
Policies, in plain terms. The 3-3-3 rule (3 absences · 3 late submissions · each within 3 days), what presence means here (a sign-in signature and that session’s in-class work, where a session has one), and the full rules.
The AI tiers. Tier 0 is unaided and in class, Tier 1 is AI-as-a-tool with disclosure on the take-homes, Tier 2 is where your critique of the AI is the graded work. Using AI has the detail, and every workbook carries its tier on the Instructions tab.
How the group practice works. Most weeks close with a short worksheet you solve in a small group. Your name on that sheet is half your attendance record for the day. Groups come together in the room, and we settle them Wednesday before the first practice.
Q&A, for as long as it takes. Bring anything: the workload, Excel, how much math you need, what happens if you miss a Wednesday.

Session 2: Wed, Aug 26 · Break-even analysis, and the algebra it runs on (TBD)

Warm-up: the algebra this session runs on

We open with this. It is ungraded and you can retry every question as often as you like; the point is to find out, in five minutes, which piece of algebra you want me to slow down on. Nothing here is new: every item is prerequisite arithmetic that shows up somewhere in the next hour.

🧩 Puzzle

The club T-shirts. Your student club wants to print T-shirts. The print shop charges a flat \$300 setup (fixed, paid once no matter how many you order) plus \$6 to make each shirt (variable). You plan to sell them at \$15 each. How many shirts must you sell before the club stops losing money?

Build the skill 1: model a decision (the five-step process)

A model is a simplified description of a real system written so you can reason about it. Most of this course follows the same repeatable arc, the management science process, which we will run on the T-shirt puzzle once so you see all five steps named:

Observation: the club keeps losing money on merch.
Problem definition: how many shirts must we sell before we stop losing money?
Model construction: write the cost and revenue rules (below).
Model solution: solve for the break-even quantity.
Implementation: set the order size (or the price) and act.

When we build the model in step 3, it has named parts worth learning now, because every later week reuses them. The thing you get to choose is the decision variable, here the quantity $q$. The fixed numbers handed to you (\$300, \$6, \$15) are parameters. The quantity you want to push up or down (profit) is the objective. And a limit you cannot exceed (say, a \$1,000 print budget) would be a constraint. Break-even has no binding constraint yet, but holding these words now seeds the linear programming weeks later, where the whole game is choosing decision variables to optimize an objective subject to constraints.

Cost almost always splits into two pieces: a fixed cost you pay regardless of volume (the \$300 setup), and a variable cost that grows with each unit (the \$6 per shirt). Written as functions of the quantity $q$:

\[\text{Cost}(q) = 300 + 6q, \qquad \text{Revenue}(q) = 15q.\]

Reading and manipulating expressions like these (substituting values, solving for an unknown) is the core algebra skill for the whole course. The club loses money whenever cost is above revenue, and profits once revenue pulls ahead.

Build the skill 2: find the break-even quantity

It helps to write profit as its own object first. Profit is revenue minus total cost, so

\[\text{Profit}(q) = \underbrace{15q}_{\text{revenue}} - \underbrace{(300 + 6q)}_{\text{cost}} = (15 - 6)q - 300 = 9q - 300.\]

The break-even quantity $q^*$ is just where profit is zero. Setting $9q - 300 = 0$ gives

\[9q = 300 \;\Rightarrow\; q^* = \frac{300}{15 - 6} = \frac{300}{9} \approx 33.3.\]

The number $15 - 6 = 9$ that fell out of the algebra is the contribution margin. The \$9 each shirt adds toward covering the fixed cost. So the general formula is

\[q^* = \frac{\text{fixed cost}}{\text{price} - \text{variable cost}} = \frac{\text{fixed cost}}{\text{contribution margin}}.\]

Written with symbols that is $q^* = F/(p - v)$, where $F$ is the fixed cost, $p$ the price per unit, and $v$ the variable cost per unit. Those three letters carry through the whole course, so attach them to the T-shirt numbers now: $F = 300$, $p = 15$, $v = 6$.

Drag the price, variable-cost, and fixed-cost sliders below and watch the cost and revenue lines, and the shaded loss (red, below break-even) and profit (green, above it) zones, move with them. Each line carries its own equation, so you can watch $C = 300 + 6q$ and $R = 15q$ rewrite themselves as you drag, and check the picture against the algebra above. At \$15 the lines cross right around 33–34 shirts:

The three levers (what-if). A model earns its keep when you ask “what changes the answer?” There are exactly three levers here, and each moves the break-even quantity in a predictable direction:

Price up → break-even down. Raise the shirt price from \$15 to \$18 and the margin jumps to \$12, so $q^{\ast} = 300 / (18 - 6) = 25$ shirts. A higher price makes break-even easier.
Variable cost up → break-even up. If the supplier raises the per-shirt cost from \$6 to \$9, the margin shrinks to \$6 and $q^{\ast} = 300 / (15 - 9) = 50$ shirts. A higher variable cost makes break-even harder.
Fixed cost down → break-even down. Negotiate the setup fee from \$300 to \$180 and $q^{\ast} = 180 / 9 = 20$ shirts. Lower fixed cost makes break-even easier (and proportionally so).

In words: anything that widens the margin (higher price, lower variable cost) lowers break-even; raising the fixed cost raises it. That is the heart of sensitivity analysis, and it is exactly the kind of “what-if” you will do in Excel all term.

Build the skill 3: functions, slope, and intercept

Math review. You already have what you need. The rest of this session leans on four pieces of prerequisite algebra, and every one showed up in the break-even model above. Order of operations (PEMDAS): to evaluate $\text{Cost}(q) = 300 + 6q$ you multiply $6q$ before adding 300. Solving a single linear equation for the unknown: $9q = 300$ gives $q = 300/9$, exactly how we found break-even. Reading the slope and intercept of a line: the \$6 rate and the \$300 setup are nothing but the slope and intercept of $C = 6q + 300$. The summation sign: a tidy way to add a long list. If any of these feel rusty, that is normal. Einstein reportedly told a student: “do not worry about your difficulties in mathematics; I can assure you mine are still greater.” We will keep the algebra concrete and tied to decisions.

A function like $f(q) = 300 + 6q$ is a rule turning an input into an output; its graph is a line with a slope (the \$6-per-shirt rate) and an intercept (the \$300 fixed cost). The break-even chart above was just two such lines (cost and revenue) and the answer was where they crossed.
Slope is “rise over run”: how much the output changes per one-unit change in the input. Intercept is the output when the input is zero. Spotting these in a cost or revenue formula is how you read a model at a glance.

Build the skill 4: summation notation

A second puzzle: the weighted total. Five neighborhoods have populations $p_1, \dots, p_5$ and different vaccination rates $r_1, \dots, r_5$. You need the total number vaccinated across all five. How can you write “multiply each neighborhood’s population by its rate and add them all up” without writing five separate terms?

Summation notation compresses long sums: $\sum_{i=1}^{5} p_i r_i$ means “add $p_i r_i$ for $i = 1, 2, \dots, 5$.” The symbol below the $\sum$ is where you start, the one above is where you stop, and the expression to the right is the term you repeat. So the total vaccinated is exactly $\sum_{i=1}^{5} p_i r_i$.

Now with real numbers, because the notation only clicks once you have run it once by hand. Put five neighborhoods in a table, one row per value of the index $i$. The last column is the term $p_i r_i$, computed row by row:

$i$	Neighborhood	Population $p_i$	Vaccination rate $r_i$	Term $p_i r_i$
1	Eastside	4,000	0.60	2,400
2	Riverbend	2,500	0.80	2,000
3	Hilltop	1,200	0.50	600
4	Old Mill	3,000	0.45	1,350
5	Northgate	1,800	0.75	1,350
	Total	12,500		7,700

Read the last column downward and add: $2{,}400 + 2{,}000 + 600 + 1{,}350 + 1{,}350 = 7{,}700$ people vaccinated. That single number is what $\sum_{i=1}^{5} p_i r_i$ means. The whole table collapses to

\[\sum_{i=1}^{5} p_i r_i = 7{,}700.\]

Two things worth noticing in the table, because both come back all term:

The index is a row number, not a quantity. $i = 3$ is not the number three; it points at Hilltop. So $p_3 = 1{,}200$ and $r_3 = 0.50$, and the term is their product, $600$.
This is not the same as multiplying the totals. The overall rate is $7{,}700 / 12{,}500 = 0.616$, not the plain average of the five rates ($0.62$), and certainly not $12{,}500 \times 0.62 = 7{,}750$. A sum of products keeps each neighborhood’s own size attached to its own rate. That is exactly why Excel has SUMPRODUCT and not just SUM, and it is the arithmetic behind every expected value in Week 2.

This single symbol appears all course long, in expected value, in objective functions for optimization, and in forecasting error measures. Getting comfortable now pays off every week.

In practice: the numbers come from somewhere, and they have to line up

Where do the inputs come from? The \$6 “per shirt” hides assumptions: it’s the price for this order size. Order fewer and the print shop may charge more per shirt; the fixed/variable split can shift.
Hidden fixed costs. Real setups include design time, a table at the activity fair, and leftover inventory, none of which the tidy \$300 covers. A break-even that ignores them is optimistic.
Break-even is a volume, not a guarantee. The model says you must clear 34 shirts to stop losing money, but it says nothing about whether the market, the donors, or the crowd will actually absorb that many. For a public or nonprofit manager deciding whether to run a program at all, this is the single most decision-relevant caveat: the break-even quantity is a target to compare against expected demand, not a promise that demand exists. (Estimating that demand is what forecasting, later in the course, is for.)
Index bookkeeping matters. Off-by-one in the summation limits ($i=1$ vs $i=0$) silently changes the answer, and $\sum p_i r_i$ only makes sense if neighborhood $i$’s population and rate really are the $i$-th entries of each list. Spreadsheets hide the $\sum$ inside SUMPRODUCT, where the same misalignment gives a confident, wrong total.

Back to the puzzle

Since you can’t sell a third of a shirt, round up: the club must sell 34 shirts to stop losing money ($q^* \approx 33.3$). Each shirt past that adds its \$9 contribution margin straight to profit; at 50 shirts the club nets $15(50) - (300 + 6\cdot50) = 750 - 600 = \$150$. The model didn’t just give a number; it told you why (the \$300 setup eats 33 shirts’ worth of margin before you profit). That “why” is what management science adds over guessing.

That number drives a real decision a manager makes: it sets the order size to place, helps you pick a price (run the slider until break-even is comfortably below expected demand), or tells you whether to run the program at all. Spot the same fixed-vs-variable split next time you read a headline about a stadium, a clinic, or a startup. It is everywhere. And notice the habit you just used: set up a model, then ask what-if. The rest of the term turns that exact habit into bigger tools: decision analysis, optimization, project scheduling, and forecasting. Slopes, intercepts, and $\sum$ are the alphabet those tools are written in.

In-class group practice (wrap-up)

To close the session, work a fresh break-even by hand in your group. Write your group’s names on the sheet and hand it to your TA before you leave. It counts toward participation.

Download: Week 1 group practice (Word)

After-class check

Questions spanning models, break-even analysis, functions and lines, and summation. Retry freely; nothing is submitted.

Weekly take-home (graded: submit on Canvas)

This week’s take-home is a break-even model for a student bike-repair pop-up, and the model is not built for you. The workbook’s model sheet opens with a case brief: the treasurer’s raw figures, one of them priced by the dozen, one that is money coming in rather than a cost, and one that belongs to last semester. In the Your model block you decide what q counts, sort the cost lines into fixed and variable, and write the cost and profit relationships yourself. Only then do you solve for the break-even volume and chart revenue against cost.

The last part is where it stops being arithmetic. The pop-up beat the break-even volume your model predicts and still cleared almost nothing, because one month ran a discount. You work out the fee it really averaged, run your own break-even a second time at that fee, and explain the gap between what the model promised at the real volume and what the treasurer actually banked. Because this is the first week, the work area keeps its headings and a worked first row. One short response plus the required AI-use disclosure go on the Free responses sheet, and the Instructions tab explains every step and why it matters. Upload the completed .xlsx on Canvas. That one file is your entire submission.