When building arguments, mathematically fluent students can understand and use definitions, assumptions, and facts. They can justify their statements and conclusion by relying on logic. This is why it’s important to teach logic in math. This of course includes teaching logic in geometry as well.
But if you’re a math teacher, you are probably aware that this is no easy feat! To help out, we’ve compiled several tips on teaching logic in geometry that will get you through! Read on to learn more.
What Is Logic in Geometry?
For starters, explain that in geometry (and overall in math), the term logic refers to a set of rules whereby we can draw conclusions that are valid. When discussing logic in math, we also talk about statements. These are declarative sentences that are either true or false.
Provide examples. For instance, if we consider the statement “5 is greater than 2”, we can easily come to the conclusion that it’s a true statement. Whereas the statement “2 is greater than 5” is a false statement.
Point out that statements comprise two parts – subject and predicate. A subject is often a noun or pronoun and it shows what the statement is about, whereas a predicate is the part of the sentence that tells us something about the subject. For example, in the following statement:
“Helen eats cookies every day”.
The subject is “Helen” and the predicate is “eats cookies every day”.
Make sure to also highlight that commands or questions can’t be statements. This is because we can’t determine if they’re true or not. For instance, in the command “Come here!”, we have no way of determining the truthfulness or untruthfulness of the sentence.
How to Teach Logic in Geometry
Types of Logical Connectives
After explaining what logical statements are, you can point out to students that we can create statements from simpler statements with the help of logical connectives. These include:
- Negation
- Conjunction
- Disjunction
- Conditional
- Bi-conditional
Negation
Explain that when we’re negating a given statement in math, we say that we’re determining what the opposite of a given statement is. Tell students to keep in mind that if a statement is true, its negation is false and if the statement is false, its negation is true.
Provide an example of a negation of a statement, such as:
“An octagon is a polygon that has eight sides”.
The negation of this statement is simply:
“An octagon is not a polygon that has eight sides”.
Conjunction
Explain that a conjunction is a compound statement comprising two propositions (or two simple statements) that are joined by the logical connective “and”. So if we have two simple statements B and C forming a conjunction, this would be represented as “B and C”.
The symbol we use to represent conjunctions looks like an inverted V letter. Illustrate this on the whiteboard:
B ∧ C
For a conjunction of two simple statements B and C, the statement is only true if both B and C are true. If only one of them is true, then B and C would be false.
Provide an example, such as:
B: The opposite sides of a parallelogram are parallel.
C: The opposite sides of a parallelogram are equal.
B ∧ C: The opposite sides of a parallelogram are parallel and the opposite sides of a parallelogram are equal.
Disjunction
Explain that a disjunction is a compound statement comprising two propositions (or two simple statements) that are joined by the logical connective “or”. So if we have two simple statements B and C forming a disjunction, this would be represented as “B or C”.
The symbol we use to represent disjunctions looks like a V letter. Illustrate this on the whiteboard:
B ∨ C
Point out to students that the compound statement B V C is true if the truth value of either of these simple statements is true, or when both B and C are true. A disjunction of two statements is false only when both of these simple statements are false.
Provide an example of a disjunction, such as:
“An obtuse angle is more than 90 degrees or an obtuse angle is less than 180 degrees”.
Conditional Statement
Explain that a conditional statement is a statement also known as an implication. A conditional statement is a compound statement that comprises two simple statements joined by the logical connective “if, then”. So if we have two simple statements B and C forming a conditional, this would be represented as “if B then C”.
The symbol we use to represent conditional statements resembles a rightward arrow. Illustrate this on the whiteboard:
B → C
Provide an example of a conditional statement, such as:
“If this is an obtuse angle, then it is between a right angle and a straight angle”.
Highlight that the only time a conditional statement such as B → C is false, is when the “if statement” (B) is true and the “then statement”(C) is false.
Converse Statement
Highlight that when we have a conditional statement B → C, we can create related statements, such as a converse statement or an inverse statement (see below). The converse of a conditional statement B → C is created when we reverse the hypothesis (B) and the conclusion (C).
So the converse of B → C is C → B. In other words, if the conclusion comes first, then it would cause the hypothesis.
You can provide an example, such as:
Conditional statement: “If this is an acute triangle, then the measure of all three interior angles is less than 90 degrees”.
Converse statement: “If the measure of all three interior angles is less than 90 degrees, then this is an acute triangle”.
Inverse Statement
Another way to form a related statement of a given condition statement is by creating an inverse of a statement. An inverse of a statement is when we negate both the hypothesis and the conclusion of a given conditions statement. Provide an example of this, such as:
Conditional statement: If a polygon has five sides, then it is a pentagon.
Inverse of a statement: If a polygon doesn’t have five sides, then it is not a pentagon.
Bi-conditional Statement
Explain that a biconditional statement is also known as a double implication. You can define it as a compound statement that comprises two simple statements joined by the logical connective “if and only if”.
Mathematically, a bi-conditional statement formed of two simple statements B and C can be presented as “B if and only if C”. The symbol we use for bi-conditional statements resembles a double-headed arrow. Illustrate this on the whiteboard:
B ↔ C
A bi-conditional B ↔ C is true only if both of the simple statements B and C are true, or if both of the simple statements are false. In all other cases, B ↔ C is false.
Additional Resources:
You can also decide to structure your lesson with the help of teaching videos. For instance, you can use this video that contains a detailed lecture on conditional and bi-conditional statements, as well as this one that contains a lesson on converse and inverse statements.
Finally, you may also want to link the lesson to another type of logical reasoning called inductive reasoning. Feel free to also check out our article with free resources on inductive reasoning.
Activities to Practice Logic in Geometry
Truth Tables Game
This online game by IXL will help students hone their knowledge of logical connectives and statements. It contains truth tables that students should complete. Usually one part is missing, where the student should add a T(true) or F(false).
Students can play the game individually, which makes it ideal for homeschooling parents as well. The only thing you’ll need for the game is a suitable device for each student, as well as a decent internet connection.
Logic Bags Game
This game will help children practice their recognition and improve their knowledge of conditional statements. The materials you’ll need are a number of bags, construction paper, glue, markers, and plenty of colorful cards!
To implement this activity in your classroom, you’ll need to create two “logic bags” per group. These are simply bags that contain cards with statements on them. You can write them manually, which may be a bit time-consuming, or simply print them out.
Each card contains a simple statement, that is, only one part of a conditional statement. So, if one card contains the “if” statement (hypothesis), then there must be another card that contains the “the” statement (conclusion).
For example, let’s say there’s a card in one bag with the following simple statement: “A rectangle doesn’t have four sides”. In the other bag, there can be a card with the following statement: “A rectangle is not a quadrilateral”.
Students are supposed to merge these cards by using the logical connective for conditional statements – “if, then”. In the above case, they can create a conditional statement such as: “If a rectangle doesn’t have four sides, then a rectangle is not a quadrilateral”.
Launch the game!
Divide students into groups of 3, 4. Hand out two bags to each group, in addition to markers, glue, one large construction paper per group. Explain the rules of the game to the students and point out that they’re supposed to select a card from one bag and then search for a card from the other bag so that they can construct a conditional statement whose truth value is true.
Explain that once they have managed to create a conditional statement out of the two cards, they should glue it on the construction paper. They can use the markers to add the logical connectives “if, then” on the construction paper.
In the end, you can count the number of true conditional statements that each group has created. The group with the biggest number of true statements wins the game.
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Unit 2 – Reasoning and Proof