Lesson Focus
This issue practices Making Conjectures in Geometry.
Synopsis
A conjecture is a mathematical statement that you believe is true based on observations, examples, or patterns. Mathematicians often use inductive reasoning to form conjectures by studying several specific cases and identifying a consistent relationship. A good conjecture clearly states the general rule suggested by the evidence.
In this issue, you practice moving from observed examples to reasonable general statements. You also learn an important limitation: many supporting examples do not prove that a conjecture is always true. A single counterexample can show that a conjecture is false. This distinction between forming a conjecture and proving a statement is central to geometric reasoning and prepares you for deductive proof.
Practice Problems
Problem 1. A conjecture is best described as a ____.
A. proved theorem
B. careless mistake
C. reasonable guess based on evidence
D. calculator answer
E. definition
Problem 2. What can show a conjecture is false?
A. counterexample
B. another supporting example
C. a title
D. a longer proof
E. a variable name
Problem 3. You test several rectangles and find equal diagonals. What may you make?
A. a conjecture
B. a quotient
C. a percent
D. an equation only
E. a table of contents
Problem 4. Which is a counterexample to ‘All even numbers are multiples of 4’?
A. 4
B. 8
C. 12
D. 14
E. 20
Problem 5. Which statement is a conjecture?
A. The next term may be 21.
B. The answer key is printed.
C. A triangle has three sides by definition.
D. The page number is 2.
E. The word geometry has 8 letters.
Problem 6. Why do examples not prove a conjecture true for all cases?
A. Examples are never useful.
B. They may not cover every possible case.
C. Examples are always false.
D. They are not numbers.
E. They must be decimals.
Problem 7. What is the role of evidence in making a conjecture?
A. It supports a reasonable prediction.
B. It guarantees a theorem.
C. It removes all need for proof.
D. It changes the problem.
E. It replaces logic.
Google Form
Answer Key
- Problem 1: C
- Problem 2: A
- Problem 3: A
- Problem 4: D
- Problem 5: A
- Problem 6: B
- Problem 7: A