Geometry Lesson

If–Then Logic and Two-Column Proofs: Conditionals, Converses, and Introductory Geometric Reasoning

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High SchooludlCCSS.MATH.CONTENT.HSG-CO.C.9,CCSS.MATH.CONTENT.HSG-CO.C.1050 min

Learning Objective

Students will be able to identify and write the converse, inverse, and contrapositive of a conditional statement; determine truth values; form biconditional statements; and construct a basic two-column algebraic or geometric proof — demonstrating understanding through at least one self-selected modality.

Lesson Overview

Students explore conditional statements and their related forms, then apply logical structure to construct introductory two-column proofs. Multiple entry points and expression modes support diverse learners throughout.

Materials

Printed graphic organizer (conditional statement chart)
Colored markers or highlighters (4 colors)
Sticky notes (for kinesthetic sorting activity)
Whiteboard or shared digital workspace (Desmos, Jamboard, or Google Slides)
Sentence-starter reference cards (laminated for reuse)
Optional: logic card sort set (printed and cut)
Optional: digital version of all tasks via Google Forms or Desmos Activity Builder

Scaffolded Task Progression

1Task 1

Identifying the Parts of a Conditional Statement

A conditional statement has the form

p \to q

('If

p

, then

q

').
-

p

is the hypothesis (the 'if' part).
-

q

is the conclusion (the 'then' part).

For each statement below, underline the hypothesis and circle the conclusion. Then rewrite it in symbolic form using

p

and

q

.

1. 'If two lines are parallel, then they never intersect.'
2. 'If

x = 5

, then

2 x = 10

.'
3. 'All right angles measure

90°

.'

Use the Venn diagram below to visualize statement 1: the small circle

p

(parallel lines) sits inside the larger circle

q

(lines that never intersect), so every

p

is also a

q

— exactly what '

p \to q

' claims.

[ Interactive graph / diagram — available in the full lesson ]

> Sentence Starter: 'The hypothesis is ___ because it follows the word IF. The conclusion is ___ because it follows the word THEN.'

> Accessibility Option: Say your answers aloud to a partner, who records them. Or draw arrows from the word 'If' to the hypothesis and from 'then' to the conclusion directly on the printed page.

2Task 2

Card Sort: Four Forms of a Conditional (Real-World Context)

Consider the conditional: 'If it is a square, then it has four equal sides.'

Match each form to its symbolic structure and write it out:

Form	Symbolic	Written Statement
Conditional	$p \to q$	If it is a square, then it has four equal sides.
Converse	$q \to p$	_____________________________
Inverse	$\neg p \to \neg q$	_____________________________
Contrapositive	$\neg q \to \neg p$	_____________________________

Step 2: Decide — is each statement True or False? Write T or F in the last column.

Step 3 (Discussion): Which two forms are always logically equivalent to each other? Which other two are logically equivalent?

> Kinesthetic Option: Write each statement on a separate sticky note. Sort them into two groups: 'Logically Equivalent Pair A' and 'Logically Equivalent Pair B.'

> Sentence Starter: 'The converse switches ___ and ___, so it says…'

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