Chapter 1
Linear Functions
Functions, slope, and linear modeling — plus the Excel moves (formulas, cell references, basic graphs) you'll lean on the rest of the term.
This week is about three ideas that show up everywhere once you notice them: functions, slope, and linear modeling. By Sunday you'll write functions, read slope from a table, and use a linear equation to predict numbers your gut already half-knows.
You'll also start using Excel. The Excel moves are small but they make every other week of this course possible. If Excel feels rusty, lean on the Scribe walkthroughs.
By the end of this chapter, you'll be able to…
- 1.1Explore the basics of functions and their uses.
- 1.2Use spreadsheet applications: functions, cell references, and basic graphing.
- 1.3Use rounding and estimation to solve application problems.
- 1.4Apply deductive and inductive reasoning to solve problems.
- 1.5Define the slope as a rate of change.
- 1.6Use linear modeling to predict outcomes from data.
Sections
- 1.1What is a function?A function is one of the simplest, most useful ideas in math, and you've been using them your whole life without knowing the name.Read →
- 1.2Two ways to reasonEvery problem you solve uses one of two reasoning moves. Knowing which is which makes you a sharper thinker, in math, and in everything else.Read →
- 1.3Close enoughRounding makes hard numbers easy. Estimation strings them together to predict, and protect you from, every total you'll ever calculate.Read →
- 1.4Slope as a rate of changeEvery "per" you've ever said (dollars per hour, miles per gallon, feet per second) is a slope. It's how mathematicians measure change.Read →
- 1.5Linear modeling and predictionA linear model is just slope plus a starting point. It's everything you've learned in this topic, packed into a single line of math, and it's how a number becomes a forecast.Read →
- 1.6Spreadsheets, the finaleYou learned the math. Now learn to automate it. A spreadsheet is just a giant grid of functions, and you already know how to write functions.Read →
Chapter summary
Formulas, key concepts, and the kind of one-page reference you'd want during a problem set.
This chapter is the math toolkit you will use for everything else: functions (rules that take an input and return an output), slope as the rate of change, linear modeling for predictions, and the Excel moves that make the rest of the course possible. The math is intentionally accessible — what you are really building is the language of every later chapter.
Chapter glossary
All key terms introduced across this chapter, in the order they appear in the reading.
- Function
- The rule itself. We name it with a letter, usually f, but g and h show up too.
- Input
- The value you feed in. Almost always called x.
- Output
- What comes out. Often called y, or written as f(x).
- f(x)
- Read out loud as "f of x." It means: "the output of the function f when the input is x." It does not mean f times x.
- Evaluate
- To compute a function's output for a specific input. "Evaluate f at x = 7" means: substitute 7 wherever you see x in the rule, then simplify. Same idea as plugging a value in.
- Domain
- All inputs that are allowed. (Working 0 hours makes sense. Working −5 hours doesn't.)
- Range
- All possible outputs you can actually get out.
- Deductive
- Top-down. Apply a general rule to a specific case. "All large drinks are $5.50. This is large. So it costs $5.50."
- Inductive
- Bottom-up. Generalize a pattern from specific examples. "My large drink has been $5.50 every time. Probably it'll be $5.50 today."
- Conjecture
- A pattern-based guess from inductive reasoning. Not yet proven, but reasonable.
- Counterexample
- A single case that breaks a conjecture. One counterexample is enough, it only takes one black swan to disprove "all swans are white."
- Premise
- A starting statement assumed true. Deductive reasoning chains premises to a conclusion.
- Conclusion
- What you end up with. In deduction, it's certain. In induction, it's probable.
- Place value
- A digit's position. In 4,587: 4 is thousands, 5 is hundreds, 8 is tens, 7 is ones. To the right of the decimal point, the places keep going: in 7.4839, the 4 is tenths, 8 is hundredths, 3 is thousandths, 9 is ten-thousandths.
- Round up / down
- Replace a number with the nearest higher or lower "nice" value. Look at the digit just to the right of your target place, 5 or more, round up; less than 5, round down. The rule is the same whether you're rounding 4,587 to the nearest hundred or 7.4839 to the nearest hundredth.
- Estimate
- An approximate answer. Faster than the exact calculation, and good enough most of the time.
- Compatible numbers
- Numbers chosen because they're easy to combine in your head. 27 + 38 becomes 30 + 40 = 70.
- Order of magnitude
- The rough size of a number. Is the answer in the tens? Hundreds? Thousands? This is what your gut catches when something feels "way off."
- Sanity check
- A fast estimate you do after a calculation to confirm the answer isn't absurd. If your calculator says $4,892 for groceries, your sanity check should scream.
- Slope (m)
- A number that describes how steep a line is and which direction it tilts. The letter m is the standard symbol, there's no deep reason for it, it's just tradition.
- Rise
- The vertical change between two points. Up is positive, down is negative.
- Run
- The horizontal change between two points. Right is positive, left is negative.
- Rate of change
- The plain-English version of slope. "60 mph," "$15 per hour," "3 inches per year", all rates of change.
- Δ (delta)
- A Greek letter meaning "change in." So Δy / Δx reads as "change in y over change in x", slope, in two characters.
- Ordered pair
- A point on a graph, written (x, y). Slope needs two of them, that's why the formula has subscripts 1 and 2.
- Linear model
- An equation that describes how one quantity changes at a constant rate as another quantity changes. Always takes the form y = mx + b.
- Slope-intercept form
- The format y = mx + b. It's called this because m is the slope and b is the y-intercept, both ingredients are right there in the equation, no rearranging needed.
- y-intercept (b)
- Where the line crosses the y-axis, the value of y when x = 0. In word problems, it's the starting amount, base fee, or fixed cost.
- Independent variable
- The input. The thing you control or that "drives" the change. Goes on the x-axis.
- Dependent variable
- The output. The thing that depends on the input. Goes on the y-axis.
- Extrapolation
- Using your model to predict beyond your data. Powerful, but also where models go wrong if the underlying pattern changes.
- Cell
- A single box in the grid. Can hold text, a number, or a formula.
- Cell reference
- The address of a cell, like B2. Column letter first, row number second.
- Formula
- Anything that starts with =. Excel calculates the result. Examples: =A1+5, =B2*3, =A2*0.10+25.
- Function (Excel)
- A built-in shortcut, like =SUM(A1:A10), =AVERAGE(B2:B20), =SLOPE(y, x), or =INTERCEPT(y, x). Same framing as the math f(x) from Lesson 1 (give it inputs, get one output), but more specific: each Excel function is a named tool with a fixed behavior. Use them when you'd rather not write the math by hand.
- Fill / drag-down
- Click a cell with a formula, grab the small square at its corner, drag it down. Excel copies the formula to every cell you drag over. adjusting cell references automatically.
- Trendline
- A best-fit line drawn through scattered data points on a chart. Right-click a data series → Add Trendline → choose Linear. Check Display Equation on chart to see y = mx + b printed alongside it. The fastest way to read slope and intercept off real data.
- Chart
- A graph generated from your cells. Select the data, click Insert Chart, choose Scatter or Line. Excel draws it for you.