Writing a Claim, Example, and Justification in Quarto

Structure a short mathematical writeup: setup, claim, example, and justification

This lab walks through producing a structured mathematical writeup in Quarto — a setup, a clearly marked claim, an example, and a justification — using ordinary headings and bold labels. It is the practical companion to Structuring Mathematical Writing, the short conceptual reading for Week 3.

You should already be comfortable with the Week 2 workflow: writing inline and displayed math in a .qmd file and rendering it to PDF. If your hw02/ writeup rendered and the equations looked right, you are ready. Week 3 adds structure on top of that, not new syntax.

What you’ll have at the end

  • A new hw03/ subfolder in your math-software-portfolio/.
  • A .qmd source file in that folder containing a definition/setup, a clearly marked claim, an example, and a justification, with at least one displayed or aligned equation used in the reasoning.
  • A rendered PDF of that file, inspected on two levels — formatting, and structure/logic.
  • A short AI Use Note in the standard three-line Tool / Purpose / Verification format (only required if you used AI assistance).

The exact assignment prompt and submission details for Week 3 live in the Assignments/LMS space.

1. Create a Week 3 portfolio subfolder

Inside your math-software-portfolio/ folder, create a new subfolder called hw03/, next to your existing hw01/ and hw02/. The full path will look like:

  • macOS / Linux: ~/Documents/math-software-portfolio/hw03/
  • Windows: C:\Users\YourName\Documents\math-software-portfolio\hw03\

This is where your Week 3 source file and rendered PDF will live.

2. Open the hw03/ folder in VS Code

From VS Code: File → Open Folder… and pick the hw03/ folder. Opening the folder (not a single file) lets the Quarto extension treat your work as a workspace — the Quarto: Preview command, the file explorer, and the integrated terminal all behave better with a folder open.

3. Start from a simple template

In VS Code’s Explorer sidebar, create a new file inside hw03/. For this walkthrough, call it week03-structured-writing.qmd. (The exact file name for the submitted assignment is specified in the Assignments/LMS space.)

Paste this starter content. It already lays out the four parts as headings and bold labels, so you are filling in structure rather than inventing it:

---
title: "Week 3 — A structured mathematical writeup"
author: "YOUR NAME"
format:
  pdf: default
---

# Setup

**Definition.** State what your objects are and define any notation.

# Claim and example

**Claim.** State precisely what you will argue.

**Example.** Give one concrete case that illustrates the claim.

# Justification

**Justification.** Argue, in prose plus at least one displayed
equation, why the claim holds in general.

The YAML at the top is the title block; format: pdf produces a PDF. The # lines are section headings. The **Definition.**, **Claim.**, **Example.**, and **Justification.** lead-ins are the bold labels that make the structure visible — this is the required Week 3 mechanism. You do not need theorem environments or raw .tex.

4. Confirm the headings give you a visible skeleton

Before writing any mathematics, render the template as-is (see step 8 for how). You should see three sections and four bold labels in the PDF. That skeleton is the whole point: a reader can already tell what kinds of statement are coming. Now you fill it in.

5. Write the definition / setup

Replace the placeholder after **Definition.** with your real setup. Name your objects and define any notation before you use it. For the running example “the sum of two even integers is even,” the setup is:

**Definition.** An integer $n$ is *even* if $n = 2k$ for some integer
$k$. Let $m$ and $n$ be even integers.

Everything you use later — here, “even” and the variables \(m\) and \(n\) — is introduced here first. A symbol that appears in your justification but never in your setup is a red flag.

6. Write the claim

Replace the placeholder after **Claim.** with one precise sentence stating what you will argue:

**Claim.** The sum $m + n$ is even.

Keep it to a single clear statement. “Precise” means a reader knows exactly what you are asserting, with nothing left vague.

7. Include an example, then write the justification

Example. Give a concrete case. It illustrates the claim and builds intuition; it does not replace the justification.

**Example.** If $m = 4$ and $n = 6$, then $m + n = 10$, which is even.

Justification. Now argue the general case, in prose, with at least one displayed equation carrying a step of the reasoning:

**Justification.** Since $m$ and $n$ are even, we can write $m = 2a$
and $n = 2b$ for integers $a$ and $b$. Then

$$
m + n = 2a + 2b = 2(a + b).
$$

Because $a + b$ is an integer, $m + n$ has the form $2 \times (\text{an
integer})$, which is the definition of even. Therefore $m + n$ is even.

Two things to notice, because they are exactly what gets graded:

  • The displayed equation is a step of the argument, with prose before it (“we can write…”) and prose after it (“because \(a + b\) is an integer…”). It is not decoration.
  • The connective words — since, then, because, therefore — are the logic. They tell the reader how each statement follows from the last. Do not drop them.

8. Render to PDF

Render the file exactly as in Week 2. Either method works:

  • Preferred — Quarto: Preview. Open week03-structured-writing.qmd in the editor, then press Ctrl/Cmd + Shift + P and run Quarto: Preview, or press Ctrl/Cmd + Shift + K. (If the Quarto extension shows a Preview button in the upper-right editor toolbar, that works too.)

  • Always works — terminal. Open VS Code’s integrated terminal (View → Terminal), make sure it is in the hw03/ folder, then run:

    quarto render week03-structured-writing.qmd

A PDF should appear next to your .qmd. If VS Code shows no Quarto commands, check the lower-right corner of the window: if it says Plain Text, click it and choose Quarto (or Markdown), and confirm the Quarto extension is installed. If the terminal says “No valid input files passed to render,” the terminal is in the wrong folder or the filename is misspelled.

9. Inspect the PDF on two levels

This is the load-bearing step. Open the PDF and check it twice.

Formatting (the Week 2 check).

  • Are the variables italic and the operators upright?
  • Did \frac, subscripts, and superscripts come out the way you intended?
  • If you used an aligned block, do the lines line up at =?

Structure and logic (new in Week 3).

  • Can you find the definition, the claim, the example, and the justification at a glance? Are the bold labels visually distinct?
  • Does your justification argue the claim you actually stated — not a weaker or different one?
  • Does every displayed equation support a step of the reasoning?
  • Is every symbol introduced in the setup before it appears?
  • Read the justification with the equations covered: do the prose steps (“since… therefore…”) still reach the claim on their own?

Fix anything that is off in the source, and re-render.

Common problems

Skim this before you start; come back when something feels wrong.

A claim with no reasoning

  • Symptom. You wrote a bold **Claim.**, an example, and then stopped. There is no justification, or the “justification” just restates the claim.
  • Fix. Add a real argument: start from the setup and reach the claim by steps. An example is not a justification — one case does not establish the general statement.

A justification that proves the wrong thing

  • Symptom. Your argument is internally reasonable, but it establishes something other than (or weaker than) the claim you stated.
  • Fix. Re-read your **Claim.** sentence, then re-read the last line of your justification. They should match. If the justification ends somewhere else, either fix the argument or restate the claim to match what you actually showed.

Equation dump — equations with no prose

  • Symptom. A column of displayed equations and almost no sentences. It compiles and looks mathematical, but a reader cannot tell what is being claimed or why each line follows.
  • Fix. Put prose before each displayed equation (what it is) and after it (what it gives you). If an equation has no sentence explaining why it is there, add one or delete it.

AI-generated reasoning you cannot explain

  • Symptom. You asked an assistant for a proof, pasted in proof-shaped text, and cannot explain one of the steps in your own words.
  • Fix. Proof-shaped is not proof-correct. Read every step, confirm it follows from your setup and reaches your claim, and rewrite anything you cannot justify yourself. Record what you checked in the Verification line of your AI Use Note. If you cannot explain a step, it does not belong in your writeup.

Labels that do not stand out

  • Symptom. Your definition, claim, example, and justification all look like ordinary paragraphs, so a reader cannot tell the parts apart.
  • Fix. Use the bold lead-ins (**Definition.**, **Claim.**, **Example.**, **Justification.**) and/or section headings so each part is visually distinct in the rendered PDF.

Vague proof prose (“clearly”, “obviously”)

  • Symptom. A step is hand-waved with “clearly” or “obviously” instead of being argued.
  • Fix. Those words usually hide the one step that needs writing. Replace them with the actual reason: why is it clear? Write that sentence.

Symbols used but never defined

  • Symptom. A variable or piece of notation appears in your claim or justification but was never introduced in the setup.
  • Fix. Introduce and define every symbol in the setup before you use it. If a symbol shows up out of nowhere, the reader cannot follow.

Renders correctly but the logic is wrong

  • Symptom. The PDF is clean and well formatted, and the argument is still invalid — a step does not follow, or the example is doing the work the justification should do.
  • Fix. There is no error message for broken logic. The only way to catch it is to read the rendered PDF as a skeptical reader and check that each step truly follows and the argument reaches the stated claim.

What this prepares you to do

When you finish this lab you should be able to:

  • organize a short mathematical writeup into a setup, a claim, an example, and a justification,
  • mark those parts clearly with headings and bold labels,
  • write a justification whose prose carries the logic and whose displayed equations carry real steps,
  • avoid equation-dump writing,
  • inspect a rendered PDF for structure and logic, not just formatting,
  • use AI assistance for outlines, prose, and examples while verifying the reasoning yourself.

The Week 3 assignment in the course LMS uses exactly this workflow. The course LMS holds the exact prompt, the file-naming convention, and the submission area.