Units in Formulas and Graphs

Lesson 2 of 2: Formula Unit Contracts and Scale Choices

In this lesson:

• Check unit consistency before substituting into formulas
• Choose and interpret scale and origin in graphs

Lesson 2 Learning Objectives Today

By the end of this lesson:

1. Use units consistently in formulas — know expected units before substituting
2. Identify and resolve unit mismatches in formulas
3. Choose and interpret scale and origin in graphs
4. Explain how scale choice shapes a graph's visual story

Formulas Return Numbers, Not Warnings

Plug numbers into . You get a number.

Meters? Feet? Seconds-squared?

The formula doesn't warn you about wrong units. It just returns a number — and you might believe it.

How do you catch the error?

Units Multiply Through the Formula

; ft, ft:

— units multiply just like numbers do.

Area is in square units because units multiply — not by convention.

Every Formula Has a Unit Contract

• Each variable has expected units
• Both sides must match in units
• Wrong input units → wrong output unit, even if the number looks reasonable

Rule: Identify expected units → convert → compute → verify output unit

Convert All Inputs Before Applying Any Formula

Problem: m, cm. Find the area.

Tracing Units Through

Correct: , s → ✓

Wrong: min instead:

Minutes break the unit contract — the answer unit is nonsense.

Spot the Mismatch: Simple Interest Formula

Simple interest:

• dollars
• per year
• months

Before computing: what unit mismatch do you see?

Name the conversion needed before substituting.

Simple Interest: Wrong Setup vs. Right

Wrong: months, no conversion

Unit: ✗

Correct: Convert months yr first

dollars

Unit: ✓

The Three-Step Rule for Formulas

Before using any formula with given values:

1. Identify the expected units for each variable
2. Convert all inputs to those expected units
3. Compute and verify the output unit matches the desired answer

Never skip step 1 — it's what makes step 2 possible.

Your Turn: Volume Formula Unit Audit

; given m, cm, m

1. Flag the unit mismatch
2. Convert all inputs to meters
3. Compute the correct volume with units

Show each step before computing.

Same Data, Two Different Visual Stories

What Every Graph Axis Must Show

Every axis needs:

1. Quantity name — what is being measured
2. Unit — in what
3. Evenly spaced tick marks at consistent intervals

✓ "Time (hours)" — not just "t"
✓ "Height (meters)" — not just "h"
✗ "Test score" — missing unit and scale context

Probe: Is This Graph Lying?

A bar chart shows four companies' annual revenues.

The y-axis reads: 0 ... 5,000 ... 5,100 ... 5,200 (millions)

The bars look dramatically different in height.

Actual differences: between 2% and 4%.

Is this graph lying? What makes it misleading?

Scale Choice Depends on the Question

No single right scale — match it to the question.

Worked Example: Plant Growth Graph Scale

Heights (cm): 2, 6, 11, 17, 21, 28 over 6 weeks

• x-axis: "Time (weeks)", 0–6, intervals of 1
• y-axis: "Height (cm)", 0–30, intervals of 5

Why 0–30, not 0–100? Data reaches 28 cm — a wider range compresses the trend into a nearly flat line.

Critique This Graph: Find Two Problems

A graph with these features:

• y-axis: 3,500 to 4,200, no label
• x-axis: "Year"
• The line looks nearly vertical

1. What is missing from the y-axis?
2. What makes this scale misleading?

Choose the Right Graph for the Purpose

Revenue grew from $980k to $1,020k over three years.

Graph A: y: 0–1,500,000 → nearly flat line
Graph B: y: 970,000–1,030,000 → steep upward trend

Which graph fits each purpose?

1. Investor comparing this company to competitors
2. Internal progress review

Three Unit-Reasoning Traps in Practice

Formula trap: Substituting wrong units silently

→ looks fine, produces nonsense

Graph trap 1: Missing axis labels → "Height" tells you nothing without a unit

Graph trap 2: Steep visual line ≠ large absolute change — check the scale numbers

Key Ideas: Formulas and Graphs

✓ Check expected units before substituting into any formula

✓ Convert all inputs to matching units first

✓ Graph scale is a choice — match it to the question

✓ Label every axis: quantity name and unit

Steep visual slope ≠ large absolute change

What Comes Next: Choosing Quantities to Measure

You can now check units in computations, formulas, and graphs.

The next lesson adds a harder question: how do you choose what to measure?

Modeling from scratch requires these skills — plus judgment about what quantity matters most.