Unit 1 Resources & Lessons
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Daily Lesson Slides
Day 1: Building Blocks of Logic
Day 2: Flipping & Reversing Logic
Day 3: Thinking Like a Detective
Day 4: Sorting with Venn Diagrams
Extra Practice: Venn Diagrams
Unit 1: Study Guide
Unit 1 Notes: Reasoning, Logic, & Venn Diagrams
Day 1: Building Blocks of Logic
Objective: To understand statements, truth values, negations, and conditional statements.
I Do: The Core Concepts
In logic, a Proposition is a statement with a clear Truth Value of either True (T) or False (F). The Negation $\sim p$ is its opposite. We combine statements with:
- Conjunction $p \land q$: "and" - only true if BOTH are true.
- Disjunction $p \lor q$: "or" - true if AT LEAST ONE is true.
A Conditional Statement $p \rightarrow q$ is an "if-then" statement. The "if" part is the Hypothesis (p), and the "then" part is the Conclusion (q). It's only false when a true hypothesis leads to a false conclusion.
We Do: Check Your Understanding
You Do: Practice Problems
Let $p$: "It is a weekday" and $q$: "I go to school".
Day 2: Flipping & Reversing Logic
Objective: To write and determine the truth value of the converse, inverse, and contrapositive of a conditional statement.
I Do: The Core Concepts
For any conditional $p \rightarrow q$, we can create three related statements:
- Converse (flip): $q \rightarrow p$
- Inverse (negate): $\sim p \rightarrow \sim q$
- Contrapositive (flip and negate): $\sim q \rightarrow \sim p$
A statement and its contrapositive are logically equivalent.
A biconditional statement $(p \leftrightarrow q)$ uses "if and only if" and is true only when a statement and its converse are both true.
We Do: Check Your Understanding
You Do: Practice Problems
Given the true statement: "If an angle is 90°, then it is a right angle."
Day 3: Thinking Like a Detective
Objective: To use the Laws of Syllogism and Detachment to make logical conclusions.
I Do: The Core Concepts
We use two main types of reasoning. Inductive Reasoning is making a conclusion based on a pattern of examples. Deductive Reasoning uses facts and rules to reach a certain conclusion. Two laws of deductive reasoning are:
- Law of Syllogism: If $p \rightarrow q$ and $q \rightarrow r$ are true, then $p \rightarrow r$ is true.
- Law of Detachment: If $p \rightarrow q$ is true and $p$ is true, then $q$ is true.
A Counterexample is a single case that proves a statement false.
We Do: Check Your Understanding
You Do: Practice Problems
Given: "If a student is in Geometry, then they passed Algebra 1."
And: "Maria is in Geometry."
Day 4: Sorting with Venn Diagrams
Objective: To use Venn diagrams to represent and solve problems involving sets.
I Do: The Core Concepts
Venn diagrams show relationships between sets. The key to solving Venn diagram problems is to start from the very center and work your way out.
Visual Vocabulary
Intersection ($A \cap B$)
"AND" - Only the overlap
Union ($A \cup B$)
"OR" - Everything in both sets
Complement ($\sim A$)
"NOT" - Everything outside a set
Example 1: Two-Circle Problem
Problem:
In a class of 30, 18 take Art, 15 take Music, and 7 take both. How many take neither?
- Start with Both (Intersection): Place 7 in the overlap.
- Calculate 'Only Art': 18 (total Art) - 7 (both) = 11.
- Calculate 'Only Music': 15 (total Music) - 7 (both) = 8.
- Calculate Neither: 30 (total) - (11 + 7 + 8) = 30 - 26 = 4.
Example 2: Three-Circle Problem
Problem:
A survey of 100 students' favorite streaming services found: 45 watch Netflix, 40 watch Disney+, 35 watch Hulu. 15 watch Netflix & Disney+, 12 watch Disney+ & Hulu, 10 watch Netflix & Hulu, and 5 watch all three. How many watch none?
- Start at the very center (All Three): Place 5 in the central overlap.
- Calculate Two-Service overlaps (subtract the center):
- N & D only: 15 - 5 = 10
- D & H only: 12 - 5 = 7
- N & H only: 10 - 5 = 5
- Calculate 'Only' sections (subtract all overlaps from the total):
- Only N: 45 - (10 + 5 + 5) = 25
- Only D: 40 - (10 + 5 + 7) = 18
- Only H: 35 - (5 + 5 + 7) = 18
- Calculate Neither: 100 - (25+18+18+10+7+5+5) = 100 - 88 = 12.
We Do: Check Your Understanding
You Do: Practice Problems
Problem 1: 50 people were surveyed. 30 liked summer and 25 liked winter. 12 liked both.
Problem 2: At a pet store, 73 pets are available. 30 are dogs, 25 are cats, and 28 are birds. 10 are dogs and cats, 8 are cats and birds, 5 are dogs and birds, and 2 are all three. Fill out the diagram and answer the questions.
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