MAT-144 · Mathematical Reasoning Topic 01 · Linear Functions
Lesson 06. The Finale

Make Excel do all of this for you.

You 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.

01Cells & references 02=SLOPE() & =INTERCEPT() 03Scatter plots & trendlines
▸ THE HOOK

Last lesson, you predicted you'd hit your study-abroad goal by week 22, using y = 75x + 200. But what if you saved $80/week instead of $75? Or started with $150 instead of $200? Recalculating by hand for every "what if" is painful.

That's exactly the problem spreadsheets were invented to solve. Excel doesn't replace the math you just learned: it scales it. You write the formula once, change one number, and watch every prediction update automatically. By the end of this lesson, you'll have rebuilt your savings model in Excel and seen the future at the click of a cell.

Every cell has an address. Every formula starts with =.

A spreadsheet is a grid. Columns get letters (A, B, C…), rows get numbers (1, 2, 3…), so each cell has an address like B2 or D7, same idea as (x, y) from Lesson 4. To compute something in a cell, start the cell with an equals sign. Excel sees that and says: "this isn't text, it's a formula, calculate it."
savings_model.xlsx
B2 fx =75*A2+200
A
B
C
D
1
Week (x)
Savings (y)
2
0
200
3
1
275
4
2
350

The formula bar shows what's inside the highlighted cell B2. The grid shows what it displays.

▸ DEFINITIONS

A cell reference is a cell's address, A2, B7, etc. Use them in formulas instead of typing raw numbers, so changing one input updates everything that depends on it.

A formula in Excel always begins with =. Anything after the equals sign is computed using arithmetic operators (+ − * /) and other cell references.

Five operators you'll use constantly

Inside a formula, you combine numbers and cell references with these five symbols. The most common gotcha for beginners: multiplication is * (asterisk), not ×. There's no times-symbol on a normal keyboard, so Excel uses the asterisk you see above the 8 key.

Symbol Operation Formula Result
+Addition =3+4 7
Subtraction =9-3 6
*Multiplication =3*4 12
/Division =10/2 5
^Exponent =2^3 8

Functions that take a range, return one number

Most of the time, you don't want to do math on one cell, you want to do it on a whole column. A range is a group of cells written with a colon: A2:A6 means "all cells from A2 through A6." Excel has built-in functions that take a range and squeeze it down to a single useful number.

C2 fx =SUM(A2:A6)
A
B
C
D
1
Score
Formula
Result
2
85
=SUM(A2:A6)
434
3
92
=AVERAGE(A2:A6)
86.8
4
78
=MIN(A2:A6)
78
5
91
=MAX(A2:A6)
92
6
88
=COUNT(A2:A6)
5

Same pattern every time: feed in a range, get one number out. =SUM and =AVERAGE are the two you'll use most. =PRODUCT multiplies everything together. =MIN, =MAX, and =COUNT answer "smallest? largest? how many?" Later in the worked example, we'll meet two more, =SLOPE and =INTERCEPT, that take two ranges instead of one.

The vocabulary you actually need

  • 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.
  • Range A group of cells, written with a colon: A1:A10 means "cells A1 through A10." Use ranges inside functions.
  • 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.

Build a stopping-distance model, from real data.

A driver-ed instructor records a car's stopping distance at six different speeds. We'll use Excel to discover the linear model behind the data. using =SLOPE() and =INTERCEPT(), then chart it, predict at new speeds, and decide whether the model holds up.

"Given six (speed, distance) data points, find the linear model d = ms + b. Predict the stopping distance at a new speed, plot the data with a trendline, and interpret what the slope and intercept mean in real life."

1

Enter the data.

Set up two columns: speed in column B (the input, what you control) and distance in column C (the output, what you measure). The headers live in row 16, the six data points in rows 17 through 22.

A
B
C
D
16
speed (m/s)
distance (m)
17
10
8
18
15
22
19
20
30
20
25
46
21
30
55
22
35
70
→ s is the input, d is the output
2

Use =SLOPE() to find m.

Excel has a built-in function that computes the slope of the best-fit line straight from your data, no formula derivation needed. In an empty cell, type:

=SLOPE(C17:C22, B17:B22)
→ Excel returns m ≈ 2.43
3

Use =INTERCEPT() to find b.

Same idea, same argument order: y-values first, x-values second.

=INTERCEPT(C17:C22, B17:B22)

Excel returns b ≈ -16.14. Format both cells to two decimal places, the rubric will check.

D26 fx =SLOPE(C17:C22, B17:B22)
B
C
D
E
25
Intercept (b)
-16.14
26
Slope (m)
2.43
4

Write the equation. Predict at new speeds.

Plugging the values into y = mx + b. but using context-appropriate variables, since this isn't a generic graph but a real situation:

d = 2.43s − 16.14

Now use the equation to predict stopping distance at speeds the data didn't include. At 40 m/s (about 90 mph):

d = 2.43(40) − 16.14 = 97.20 − 16.14 ≈ 81 m

Roughly the length of an Olympic swimming pool. In Excel, you'd put predicted speeds in column E and the prediction formula =2.43*E17-16.14 in column F, then drag down.

5

Make a scatter plot. Add a trendline.

Select your data range (B16:C22). Click Insert → Scatter, pick the X-Y points version, not the connected line. Then right-click any data point in the chart, choose Add Trendline, select Linear, and check Display Equation on chart. Excel draws the best-fit line and prints the equation right next to it.

0 10 20 30 40 50 Speed (m/s) 0 25 50 75 100 Distance (m) PREDICTED (40, 81) d = 2.43s − 16.14 TRENDLINE

The trendline runs through the cloud of data points, exactly the same equation Excel computed in steps 2 and 3. Two paths to the same answer, which is a useful sanity check.

6

Interpret the slope and intercept.

The numbers don't matter unless you can translate them back to the real world.

Slope (m = 2.43): for every 1 m/s increase in speed, stopping distance increases by 2.43 m. That makes sense, go faster, need more distance to stop. ✓

Y-intercept (b = −16.14): at 0 m/s, the model predicts −16.14 m of stopping distance. That does not make physical sense. A car that isn't moving stops in 0 meters, not negative ones.

So is the model wrong? Not quite, it's a linear approximation that fits the data well between 10 and 35 m/s but breaks down outside that range. This is the extrapolation warning from Lesson 5: trust your model inside the data, be skeptical outside it.

→ slope: makes sense. intercept: does not.

Three problems. Read, write, predict.

First identify a cell reference, then write a formula from scratch, then read what a chart is telling you.
PROBLEM 01 ☆ ☆   warm-up

Look at the spreadsheet below. What's the cell address of the cell that contains 350?

A
B
C
D
1
Week
Savings
2
0
200
3
1
275
4
2
350
cell:
PROBLEM 02 ★ ★ ☆   write the formula

A phone plan charges $0.10 per minute plus a $25 base fee. Cell A2 holds the number of minutes used. Write the formula for cell B2 that computes the total bill.

B2 =
PROBLEM 03 ★ ★ ★   write the slope formula

Cells A2:A11 hold the year (1 through 10). Cells B2:B11 hold the corresponding home values. Write the Excel formula in cell C2 that returns the slope of the trendline.

C2 =

Three fast questions before we wrap.

Tap an answer. You'll see right away whether it stuck.

Q1. Why does every Excel formula start with =?

Why B? Without the =, Excel reads 75*A2+200 as the literal text "75*A2+200" and just displays those characters. The = is the signal that says: "this is a formula, calculate it." Forgetting the equals sign is the single most common Excel mistake, and it's an easy fix once you know to look for it.

Q2. Cell A1 contains the number 50. Cell A2 contains the formula =A1*2. If you change A1 to 80, what happens to A2?

Why C? This is the whole point of cell references. Because A2 contains =A1*2 (referring to A1, not the literal value 50), Excel recalculates whenever A1 changes. New A1 (80) × 2 = 160. This is what makes "what if?" analysis fast. change one number, watch every dependent cell update at once.

Q3. You write =A2+10 in cell B2, then drag the formula down to B3, B4, and B5. What formula ends up in B5?

Why D? When you drag a formula down, Excel adjusts the row numbers in the cell references to match. B2 had =A2+10; B3 becomes =A3+10; B4 becomes =A4+10; B5 becomes =A5+10. The 10 stays put because it's a literal number, not a reference. This auto-adjustment is exactly why we use cell references instead of typing values directly, one formula, twenty-five rows.
▸ TOPIC 01 · COMPLETE

You just built a small but real mathematical toolkit.

Six lessons ago, "function" was a vague term and slope was something to memorize. Now you can read a real-world situation, build a model, predict outcomes, sanity-check the result, and automate the whole thing in a spreadsheet. That's the same workflow used in business forecasting, science labs, healthcare planning, sports analytics: anywhere people turn numbers into decisions.

You won't always be calculating slopes by hand. But the next time someone says "$15 per hour," "about $50," or "if this trend continues," you'll know exactly what they mean. and how to check it.

Functions Reasoning Estimation Slope Linear models Spreadsheets
Continue to Topic 02

Different angle? Need another rep? These are optional — tap any that look helpful.

▸ Browse all Topic 1 resources