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 Canvas Files.

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

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, where your group’s practice photo lands, and where you take the two exams 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) · two exams, a midterm on Wed Oct 21 and a cumulative final in the December exam period (30) · in-class participation, the 15 group practices (15). We open a real workbook on the projector so you can see what a submission actually looks like before you owe one: three sheets, a red cell for everything you fill in, and a case brief that carries figures the model does not need, so deciding what to ignore is part of the work.
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 the ungraded practice where critiquing the AI is the task. 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 on paper in a small group; one of you photographs it and uploads the photo to Canvas before leaving. The upload is per group, but the point is yours individually: your name on that submitted sheet is half your attendance record for the day, and it only earns the point if you were in the room. Groups come together in the room, and we settle them Wednesday before the first practice.
What to bring. A laptop with Excel (enable the Solver add-in now; you need it from Week 8) and Respondus LockDown Browser installed; the decision-tree and Bayesian-network tools arrive in Weeks 5 and 6. Install links are on the home page under Software & setup.
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

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. Three of them (evaluating an expression, reading a per-unit rate off a coefficient, solving one linear equation) show up in the next hour. The square root is there for Week 2, whose standard deviation squares every miss and then takes a square root to get back to dollars. The exponent rule is general prerequisite algebra, there to check for rust rather than for any one week.

🧩 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. Build one and it has four named parts worth learning now, because every week from here reuses them. The thing you get to choose is the decision variable, here $q$, the number of shirts you order. The numbers handed to you (\$300, \$6, \$15) are parameters. The quantity you push up or down (profit) is the objective. And a limit you cannot cross (say, a \$1,000 print budget) would be a constraint. Break-even has no binding constraint yet, but holding the word now is what makes linear programming in Week 7 feel like the same thing, bigger: there the whole game is choosing decision variables to optimize an objective subject to constraints.

Most of this course follows the same repeatable arc, the management science process, which we 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.

Step 5 is the one students skip and managers care about. A break-even quantity that changes nothing was not worth computing.

Cost almost always splits into two pieces, and the split is the whole model: a fixed cost you pay once whatever the volume (the \$300 setup), and a variable cost that grows with every unit (the \$6 per shirt). Written as rules in the quantity $q$:

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

Read them as rules, not answers: hand each one a number of shirts and it hands back dollars (multiply $6q$ first, then add the 300). Drawn on a chart they are two lines. Cost starts at \$300 and climbs \$6 a shirt; revenue starts at zero and climbs \$15. At any number of shirts the gap between them is what the club makes or loses, and where the lines cross is the whole question. Reading and manipulating expressions like these (substituting values, solving for an unknown) is the core algebra skill for the whole course.

Build the skill 2: find the break-even quantity

It helps to write profit as its own object first. Profit is the gap between the two lines, revenue minus total cost, so

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

The \$9 that fell out is the contribution margin: what each shirt adds toward the \$300 the club is behind before it has sold anything. In symbols, with $p$ the price, $F$ the fixed cost, and $v$ the variable cost per unit, the same rule is

\[\text{profit} = pq - F - vq.\]

Three pieces, and you have met all three: revenue $pq$ (here $p = 15$), the fixed cost $F$ paid once whatever $q$ turns out to be ($F = 300$), and the variable cost $vq$, the per-unit cost times the same $q$ ($v = 6$). Those three letters carry through the whole course, so attach them to the T-shirt numbers now. (Taylor’s letters differ; the note under Before the week translates them.)

The break-even quantity $q^*$ is where profit is zero. Set the profit rule to zero and solve for $q$, and the answer to “how many” comes out as a division:

\[q^* = \frac{F}{p - v} = \frac{\text{fixed cost}}{\text{contribution margin}}.\]

On top is what you owe before you have sold anything; underneath is what one more shirt chips in; the quotient is how many of those chips it takes to cover it. It is a division because the question is a division: how many of these fit into that? For the club,

\[9q - 300 = 0 \;\Rightarrow\; 9q = 300 \;\Rightarrow\; q^* = \frac{300}{15 - 6} = \frac{300}{9} \approx 33.3.\]

You cannot sell a third of a shirt, so round up: the club must sell 34 shirts to stop losing money. On the chart below, that is where the gap closes. To its left is the loss wedge, where the \$300 setup is still absorbing each shirt’s \$9; to its right every shirt adds its \$9 straight to profit.

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).

Every one of those moves works through the same quantity, the gap between $p$ and $v$, and dollar for dollar the margin levers are the strong ones: one more dollar on the price (or one less per shirt) takes break-even from 33.3 to $300 / 10 = 30$, while one dollar off the setup fee moves it by a ninth of a shirt. In words: anything that widens the margin (higher price, lower variable cost) lowers break-even; raising the fixed cost raises it. Reading a model that way is sensitivity analysis, and it is most of what you will do in Excel this term. It also tells you which lever to argue for: a club that cannot raise the price can still negotiate the setup fee.

Now drag the levers yourself. The price, variable-cost, and fixed-cost sliders below move 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 at $300/9 \approx 33.3$ shirts, and the readout under the chart reports the same break-even:

Build the skill 3: functions, slope, and intercept

Math review. You already have what you need. This session leans on four pieces of prerequisite algebra. Three of them showed up in the break-even model above, and the fourth is next. 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. The warm-up also checked two rules nothing today needs. Week 2 needs the first: a square root undoes a square ($\sqrt{225} = 15$ because $15^2 = 225$), which is how its standard deviation gets back to dollars after squaring every miss. The second is general algebra rather than a Week 2 tool: dividing powers of the same base subtracts the exponents ($x^7 / x^3 = x^4$). 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.

$C(q) = 300 + 6q$ is a linear function: a rule that turns an input (shirts) into an output (dollars) at a constant rate, so its graph is a line, the cost line you have been reading since Build the skill 1. Its slope is the \$6-per-shirt rate: between 20 and 60 shirts the line climbs $6 \times 40 = \$240$, forty more shirts and \$240 more cost. Its intercept is the output at zero shirts, the \$300 you owe having sold nothing. The break-even chart above was just two such lines, 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 inside a cost or revenue formula is how you read a model at a glance, before solving anything.

Build the skill 4: summation notation

A second question on the same order. The plan said \$15 a shirt. Suppose the order sells out in five batches, each at its own price: the activity fair, a member discount, the home game, a dorm’s bulk order, and a few online. Batch $i$ sold $q_i$ shirts at $p_i$ dollars each, and you need the club’s total revenue. How do you write “multiply each batch’s count by its own price, then add them all up” without writing five separate terms?

Summation notation is the whole answer, and it is nothing but three parts: $\sum_{i=1}^{5} p_i q_i$. The symbol below the $\sum$ is where you start (the first value of the index, $i = 1$), the one above is where you stop (the last value, 5), and the expression to the right is the term you repeat, written once ($p_i q_i$). So the total revenue is exactly $\sum_{i=1}^{5} p_i q_i$.

Now with real numbers, because the notation only clicks after you have computed one. Put the five batches in a table, one row per value of the index $i$. The last column is the term $p_i q_i$, computed row by row:

$i$	Batch	Price $p_i$	Shirts $q_i$	Term $p_i q_i$
1	Activity fair	\$15	10	150
2	Member discount	\$10	40	400
3	Home game	\$15	20	300
4	Dorm order	\$12	25	300
5	Online	\$18	5	90
	Total		100	1,240

Read the last column downward and add: $150 + 400 + 300 + 300 + 90 = 1{,}240$ dollars of revenue on 100 shirts. That single number is what $\sum_{i=1}^{5} p_i q_i$ means, and setting $i$ to a row number is all the notation ever asks you to do. The whole table collapses to

\[\sum_{i=1}^{5} p_i q_i = 1{,}240.\]

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 the home game, so $p_3 = 15$ and $q_3 = 20$, and the term is their product, $300$. When the term is a formula in $i$ rather than a table lookup, you substitute each value in turn instead: $\sum_{i=1}^{3} (i + 1) = 2 + 3 + 4 = 9$. Either way, run $i$ through its range and add.
A sum of products is not a product of sums. The price the club really averaged is $1{,}240 / 100 = \$12.40$, not the plain average of the five prices (\$14), and certainly not $100 \times 14 = \$1{,}400$ of revenue. A sum of products keeps each batch’s own count attached to its own price. 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 an assumption: it is the price for this order size. Order fewer and the print shop may charge more per shirt, and the fixed/variable split shifts under you. It also has to arrive on the basis $q$ is counted in: a shop that quotes \$72 a dozen still gives you $v = 6$, but only once you divide.
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, in Week 13, is for.)
Index bookkeeping is silent when it fails. Start the sum at $i = 0$ instead of $i = 1$, or line up prices against the wrong batches, and $\sum p_i q_i$ is a different number. Spreadsheets hide the $\sum$ inside SUMPRODUCT, where the same misalignment returns a confident, wrong total.

Back to the puzzle

34 shirts clears the break-even point of $q^* \approx 33.3$, and 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 $300/9 \approx 33.3$ shirts’ worth of margin before you profit). That “why” is what management science adds over guessing.

The five-batch sale also shows a lever moving under you. The club planned on \$15 a shirt and averaged \$12.40, so the margin it really earned was $12.40 - 6 = \$6.40$ and the break-even it really faced was $300 / 6.40 \approx 46.9$, so 47 shirts, not 34. Break-even at the plan and break-even at what actually happened are two different numbers, and the gap between them is the price lever. That comparison is exactly what this week’s take-home asks you to run.

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. Next week the future stops being certain: weight each outcome by its probability instead of by a count of shirts, add the products exactly as you just did, and the total is the expected value.

In-class group practice (wrap-up)

To close the session, work a fresh break-even by hand in your group.

Write every member’s name on the sheet. You have 15 minutes for the practice, then two minutes to photograph your group’s own attempt before the solutions are shown. One member uploads the photo to Canvas by the end of class; participation credit is for attempting the work.

The instructor uses the final eight minutes to demonstrate the handout’s method, check the main results with units, and explain a common mistake. This demonstration closes the session.

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.