geometry vocabulary 1a geometry, like much of mathematics and science, developed when people began...
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Geometry Vocabulary 1A
Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. In this course, you will study many amazing patterns that were discovered by people throughout history. You will also learn to recognize and describe patterns of your own.
Vocabulary
A conjecture is an unproven statement that is based on observations.
Inductive reasoning is a process that involves looking for patterns and making conjectures.
A counterexample is an example that shows a conjecture is false.
Find a Pattern
Sketch the next figure in the pattern.
Find a Pattern
Sketch the next figure in the pattern.
Find a Pattern
Describe a pattern in the sequence of numbers. Predict the next number.
a) 5, 3, 1, -1,…
b) 0, 3, 8, 15, 24 …
Find a Pattern
Describe a pattern in the sequence of numbers. Predict the next number.
c) ½, ¼, ⅛, …
d) 1, 2, 6, 24, ….
Inductive Reasoning
Much of the reasoning in geometry consists of three stages:
① Look for a Pattern Look at several examples. Use diagrams and tables to help discover a pattern.
② Make a Conjecture Use the examples to make a general conjecture.
③ Verify the Conjecture Use logical reasoning to verify that the conjecture is true in all cases.
Making a Conjecture
Complete the conjecture. Conjecture: The product of two consecutive even integers is divisible by _________.
Solution: List some examples and look for a pattern.
Use various consecutive even integers. 2 x 4 = 8, 6 x 8 = 48, 4 x 6 = 24, 10 x 12 = 120
Making a Conjecture
All the numbers from those examples (8, 48, 24, 120)are divisible by 8.
So, the conjecture can be completed with:The product of two consecutive even integers is divisible by 8.
Making a Conjecture
Complete the following conjecture.
Conjecture: For any two numbers a and b, the product of (a + b) and (a – b) is always equal to __________.
List some specific examples and look for a pattern.
Finding a Counterexample
Show the following conjecture is false by finding a counterexample.
Conjecture: All odd numbers are prime.
Solution: The conjecture is false. Here is a counterexample: The number 9 is odd and it is not a prime number.
Finding a Counterexample
Show the following conjecture is false by finding a counterexample.
Conjecture: The difference of two whole numbers is a whole number.