Cycle1L4: Conjectures, Counterexamples, Conditional Statements

Geometry

August 26, 2024

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Unit 2: Reasoning in Geometry
Lesson 2: Conjectures, Counterexamples &
Conditional Statements

LO: SWBAT verify that a conjecture is false
using a counterexample, including conditional
statements.
DOL: Given 5 problems, students will correctly
verify that a conjecture is false using a
counterexample, including conditional
statements in at least 4 of 5 questions.

GEOM.4C Verify that a conjecture is false using a counterexample.
GEOM.4B Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the
connection between a biconditional statement and a true conditional statement with a true converse

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Lesson Vocabulary

Conjecture

A conclusion reached by using
inductive reasoning. It can be
true or false.

Counterexample

An example that shows that
conjecture is false. You can
prove that a conjecture is false
by finding one counterexample.

Conditional Statement

A conditional statement is a
statement that connects two
events based on a condition,
written in the form “If p, then q”.

Hypothesis

A part of a conditional
statement that follows “If”.

Example: If I go to school
everyday…

Conclusion

A part of a conditional statement
that follows “then”.

Example: then I will have a good
attendance.

Venn diagram

A diagram representing
mathematical or logical sets
pictorially as circles or closed
curves within an enclosing
rectangle, common elements are
the areas of overlap.

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Examples of False Conjectures

1. Everything that has ears can hear.

2. All multiples of 7 are odd numbers.

3. Any three points can be connected to form a triangle.

Give at least one counterexample for each.

​4 min

Multiple Select

Which Conjecture is False?

(Choose all that apply)

•  Any three points that are non-collinear are coplanar.

• All multiples of 4 are even numbers.

• Through any two points, there is exactly one line.

• All positive fractions are between 0 and 1.

• All girls like ice cream.

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Conditional Statement

Example: If you multiply any number by 2,
then the product is always greater than the
original number.

Has two parts:Hypothesis and Conclusion

Symbolic: p q which means “If p…then q”,
where p contains the hypothesis and q contains the
conclusion.

​2 min

Labelling

Label the sentence with the conditional statement parts.

Drag labels to their correct position on the image

Conclusion

Hypothesis

Labelling

Label the sentence with the conditional statement parts.

Drag labels to their correct position on the image

Hypothesis

Conclusion

Labelling

Label the sentence with the conditional statement parts.

Drag labels to their correct position on the image

Hypothesis

Conclusion

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Give One Hypothesis …Get One Conclusion

Give One, Get One
1.

On an index card, write your name and come up with
your own hypothesis.

2.

Get up, find a partner, switch index cards, then write a
conclusion to your partner’s hypothesis.

3.

Get your index card back and decide if the statement
is true or false. Be ready to share your answer when
asked.

Activity Time!

Repeat as needed!

​8 min

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Not all statements are written in
“If…then” form. How do you write
statements in “If…, then…” form?

​1 min

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Writing Conditional Statements

Step 1: Identify the hypothesis and the conclusion.

How can you write the statement in “If…then” form?

Vertical angles share a vertex.

Step 2: Rewrite in the If..then form.

What do you notice about the conditional statement?

Notice that there are added words to make the sentence make sense but not change the meaning.

If two angles are vertical, then they share a vertex.

​3 min

Open Ended

Rewrite the statement in “If…,then…” format.

Open Ended

Rewrite the statement in “If…,then…” format.

Open Ended

Rewrite the statement in “If…,then…” format.

​2 min

​1 min

Labelling

Label the Venn diagram to represent each statement.

"If an angle measures 100°, then it is obtuse."

Drag labels to their correct position on the image

Obtuse Angle

Labelling

Label the Venn diagram to represent each statement.

"If you are the captain of your team, then you are a junior or a senior."

Drag labels to their correct position on the image

Team captain

Junior or Senior

Labelling

Label the Venn diagram to represent each statement.

"If a number is divisible by 2, then it is an even number."

Drag labels to their correct position on the image

Even number

Divisible by 2

Labelling

Label the Venn diagram to represent each statement.

"If you play a sport, then you are an athlete."

Drag labels to their correct position on the image

sport

athlete

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Can you reverse the process?
Can we write a conditional
statement from a diagram?

​1 min

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quadrilateral
(q)

square

(p)

Example: Write a conditional from a given
Venn diagram.

Statement: If a polygon is a square, then
___________________.
Or If it is a square, then ________________.

it is a quadrilateral

it is a quadrilateral

​2 min

Drag and Drop

Write a conditional statement that is illustrated by each Venn diagram.


If you are a ​

Placeholder

then you are a ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

pitcher,

Baseball player

Drag and Drop

Write a conditional statement that is illustrated by each Venn diagram.



If a number is a ​

Placeholder

, then it is an ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

whole number

integer

Drag and Drop

Write a conditional statement that is illustrated by each Venn diagram.



If a shape is a ​ ​

Placeholder

, then it is a ​

Placeholder

Drag these tiles and drop them in the correct blank above

parallelogram

rectangle

Demonstration of Learning

DOLGiven 5
problems,
students will
correctly verify
that a conjecture
is false using a
counterexample,
including
conditional
statements in at
least 4 of 5
questions.

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