Teacher Guide: If-Then Statements

Learn to write conditional statements in if-then form, find the hypothesis and conclusion, and decide when a conditional is false.

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Class quiz

10 questions on Logical Reasoning. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Identify the hypothesis and the conclusion of a conditional statement
Rewrite everyday and mathematical sentences in if-then form and in the notation p → q
Determine the truth value of a conditional from the truth values of its parts
Find a counterexample to show that a conditional statement is false
Explain why examples alone cannot prove a conditional statement

Prerequisites

• Introduction to Logical Reasoning: statements and their truth values
• Deductive Reasoning: using a rule (premise) to reach a conclusion
• Comparing numbers with <, > and ≥ and basic arithmetic

Discussion Starters

1. Your parent says: "If you clean your room, then you can go out." You did not clean your room, but you were allowed out anyway. Did your parent lie?
2. Find a rule at school or in a game. Can you write it as an if-then statement?
3. Why is one counterexample enough to disprove a claim, but a hundred good examples are not enough to prove it?
4. "All cats are animals" and "If something is a cat, then it is an animal": are these the same statement?
5. Can you think of a statement that is true, but becomes false when you swap the "if" part and the "then" part?

Common Misconceptions

A conditional with a false hypothesis is false (or "cannot be decided")

Remediation: Use the promise model: "If you score at least 80, you get a sticker." Ask whether the teacher broke the promise when a student with 70 points gets a sticker anyway. Then let students check the F rows in the interactive truth table.

"If $p$ , then $q$ " also means "if $q$ , then $p$ "

Remediation: Give the pair "if a shape is a square, then it is a rectangle" and the swapped version, and ask students to draw a counterexample for the swapped one. Tell them this idea is studied in detail in the next lesson, Converse and Inverse.

A few examples that work prove a statement

Remediation: Test $2^n + 1$ for $n = 1, 2, 3$ together, or $n^2 + n + 1$ for $n = 2, 3, 4$ . The pattern breaks at $n = 3$ and $n = 4$ , which shows why mathematicians ask for proof.

Differentiation Ideas

For Struggling Students:

• Give sentence frames: "If ___, then ___." and let students underline the hypothesis in one color and the conclusion in another
• Start with everyday rules (games, school) before mathematical statements
• Use the four-student sticker table so the T → F case is concrete

For On-Level Students:

• Rewrite "every", "all" and "when" sentences in if-then form and in the notation p → q
• Decide whether number claims are true and give counterexamples for the false ones
• Complete the truth table for p → q in quiz mode without help

For Advanced Students:

• Find the smallest counterexample to claims such as " $n^2 + n + 1$ is always prime"
• Explore the card puzzle: which cards must you turn over to test an if-then rule?
• Write a true conditional whose swapped version is false, as a preview of the converse

Standards Alignment

MP3 (CCSS.MATH.PRACTICE.MP3)
Construct viable arguments and critique the reasoning of others
MP6 (CCSS.MATH.PRACTICE.MP6)
Attend to precision

Lesson Resources

visualInteractive Truth Table
Students switch to the → operator, see the only false row light up and then fill in the table themselves in quiz mode
activityBroken Promises
In pairs, students write class rules as if-then statements and act out the four cases to find which one breaks the rule
worksheetCounterexample Hunt
A list of number claims; students mark each as true or give the smallest counterexample

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A conditional statement, also called an if-then statement, joins two statements with the words if and then:

\text{If } p\text{, then } q.

The part after "if" is the hypothesis ( $p$ ), the condition.
The part after "then" is the conclusion ( $q$ ), what must follow.

In symbols we write

p \to q

and read it as "if

p

, then

q

" or "

p

implies

q

Example: If a number ends in 0, then it is divisible by 10.

Hypothesis $p$ : a number ends in 0
Conclusion $q$ : the number is divisible by 10

Think of a conditional as a promise. The promise is broken only when the hypothesis happens and the conclusion does not. So

p \to q

is false in exactly one case:

p

is true and

q

is false. In all other cases it is true.

$p$	$q$	$p \to q$
T	T	T
T	F	F
F	T	T
F	F	T

The interactive truth table in this lesson lists the same four rows in the opposite order, starting with F, F. Both orders are fine.

An example where the hypothesis is true and the conclusion is false is called a counterexample. One counterexample is enough to show that a conditional is false.

Worked Examples

Identify the hypothesis and the conclusion: "If a triangle has a right angle, then it is a right triangle."

Find the part after "if"

"a triangle has a right angle" → Hypothesis $p$ : a triangle has a right angle

Find the part after "then"

"it is a right triangle" → Conclusion $q$ : the triangle is a right triangle

Write it in symbols

The condition comes first, the result second → $p \to q$

Answer: Hypothesis: a triangle has a right angle. Conclusion: it is a right triangle. In symbols: $p \to q$ .

Common Mistakes

Taking the first part of the sentence as the hypothesis, as in "I take an umbrella if it rains" → hypothesis "I take an umbrella"

Why it's wrong: The hypothesis is the part that follows the word "if" (or "when", "whenever"), wherever it stands in the sentence.

Correct: Find the condition first: "it rains". Then write: If it rains, then I take an umbrella.

Thinking that $p \to q$ is false whenever the hypothesis is false

Why it's wrong: A conditional only promises something for the case where $p$ happens. If $p$ does not happen, the promise cannot be broken.

Correct: $p \to q$ is false only in the case T → F. When $p$ is false, the conditional is true, whatever $q$ is.

Using a number that does not satisfy the hypothesis as a counterexample, for example $x = 0$ against "if $x > 3$ , then $x > 1$ "

Why it's wrong: For $x = 0$ the conclusion $x > 1$ is false, but the hypothesis $x > 3$ is false too, so this case cannot break the statement.

Correct: A counterexample must make the hypothesis true and the conclusion false. For "if $x > 3$ , then $x > 1$ " no such number exists, so the statement is true.

Believing that one example that works proves the statement

Why it's wrong: "If $n$ is a whole number, then $2^n + 1$ is prime" works for $n = 1$ ( $3$ ) and $n = 2$ ( $5$ ), but $n = 3$ gives $9$ , which is not prime.

Correct: Examples can only support a statement. To prove it you need a general argument, but a single counterexample is enough to disprove it.

Why It Matters

If-then statements are everywhere, often hidden in ordinary sentences:

Shopping: "If your order is at least 40 euros, then delivery is free."
Rules and games: "If you roll a 6, then you roll again."
Science: "If water is heated to 100 °C at sea level, then it boils."
Programming: every app is full of commands like "if the password is correct, then open the account".
Mathematics: almost every rule and theorem is a conditional, for example "if a triangle has a right angle, then $a^2 + b^2 = c^2$ " (where $c$ is the longest side).

In the lesson Deductive Reasoning you used if-then rules as premises. Now you will learn how they are built, how to write them yourself and how to test them. This is the starting point for the converse, the inverse and the contrapositive, which come next in the lessons Converse and Inverse and Contrapositive.

Real World Applications

Online Shopping

Shops use if-then rules for free delivery, discounts and gifts. To use the offer, you must make the hypothesis true.

Example:

"If your order is at least 40 dollars, then delivery is free." An order of 52 dollars makes the hypothesis true, so delivery must be free.

1Try It Yourself

A shop promises: if your order is at least 40 dollars, then delivery is free. Your basket has a T-shirt for 18 dollars and a cap for 15 dollars.

How many more dollars do you need to spend so that the hypothesis becomes true?

Step 1: Write the mathematical expression

Subtract your total from 40:

Testing Claims in Mathematics

Mathematicians test a conditional by searching for a counterexample. If they find one, the claim is false.

Example:

"If a number is prime, then it is odd" is false, because 2 is prime and even.

2Try It Yourself

A classmate claims: if a whole number is a multiple of 3, then it is odd.

What is the smallest positive counterexample?

Step 1: Write the mathematical expression

The multiples of 3 are 3, 3 + 3, 3 + 3 + 3, ... Write the second one:

Smart Homes and Programming

A thermostat runs on a conditional: if the room is colder than a set value, then the heating turns on.

Example:

"If the temperature is below 18 °C, then the heating turns on." At 17 °C the hypothesis is true, so the heating must turn on. At 20 °C nothing is promised.

3Try It Yourself

The rule is: if the temperature is below 18 °C, then the heating turns on. At 7:00 the room is 22 °C and it cools by 3 °C every hour.

What is the temperature after 2 hours, and is the heating on?

Step 1: Write the mathematical expression

Start at 22 and subtract 3 twice:

Key Takeaways

1A conditional statement has the form "if $p$ , then $q$ ", written $p \to q$
2The hypothesis $p$ is the condition after "if"; the conclusion $q$ is the result after "then"
3Many sentences hide a conditional: "every A is B" means "if something is A, then it is B"
4 $p \to q$ is false only when $p$ is true and $q$ is false (T → F); in the other three cases it is true
5A counterexample makes the hypothesis true and the conclusion false, and one is enough to disprove a statement
6Examples that work support a statement, but they do not prove it

Frequently Asked Questions

Why is "if $p$ , then $q$ " true when $p$ is false?

Because the statement only makes a promise about the case where

p

happens. If you say "if it rains, I take an umbrella" and it does not rain, you have not broken your word, whatever you do.

Does the word "then" always have to be there?

No. "If it rains, I stay home", "I stay home when it rains" and "Rain means I stay home" are all the same conditional. Writing it with "if" and "then" just makes the hypothesis and the conclusion easy to see.

How many counterexamples do I need to show a statement is false?

Just one. A conditional claims that the conclusion follows every time the hypothesis is true, so a single case with T → F is enough to break it.

What happens if I swap the hypothesis and the conclusion?

You get a new statement that can have a different truth value. "If a shape is a square, then it is a rectangle" is true, but "if a shape is a rectangle, then it is a square" is false. Swapping and negating parts is the topic of the next lessons, Converse and Inverse and Contrapositive.

Glossary

Conditional statement: A statement of the form "if $p$ , then $q$ ", also called an if-then statement
Hypothesis: The condition of a conditional, the part after "if" ( $p$ )
Conclusion: The result of a conditional, the part after "then" ( $q$ )
Implication: Another name for a conditional; $p \to q$ is read " $p$ implies $q$ "
Counterexample: A case where the hypothesis is true and the conclusion is false; it shows that a conditional is false
Truth value: Whether a statement is true (T) or false (F)

Formula Card

Conditional

p \to q

Read "if $p$, then $q$": $p$ is the hypothesis, $q$ is the conclusion

Only false case

T \to F = F

A conditional is false only when the hypothesis is true and the conclusion is false

Counterexample

p \text{ true and } q \text{ false}

One such case is enough to disprove $p \to q$

Teacher Guide: If-Then Statements

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Online Shopping

Testing Claims in Mathematics

Smart Homes and Programming

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Conditional

Only false case

Counterexample

More in This Topic

Introduction to Logical Reasoning

Deductive Reasoning

Converse and Inverse

Contrapositive