Download Chapter 1.1 Patterns and Inductive Reasoning.notebook

Chapter 1.1 Patterns and Inductive Reasoning.notebook
January 03, 2013
1.1 Patterns and Inductive Reasoning
Objective: To use inductive reasoning to make conjectures
A.1.B.
Performance Standard 1.6, 3.4, 3.5 DOK­1, 2
Knowledge MA 3
Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence.
384, 192, 96, 48, …
Key vocabulary: inductive reasoning, conjectures, counterexamples
Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . .
Some are even and some are odd.
1. Make a list of the positive even numbers. 2. Make a list of the positive odd numbers. 3. Copy and extend this list to show the first 10 perfect squares.
12 = 1, 22 = 4, 32 = 9, 42 = 16, . . .
4. Which do you think describes the square of any odd number?
It is odd. It is even.
Jul 9­7:42 PM
Jul 16­8:42 PM
Examples:
Examples:
Find the next two shapes in each sequence and explain the pattern in words.
Find the next two numbers in each sequence and explain in words the pattern.
8) 1) 3, 6, 12, 24, ...____, ____,...
___________________
_______________, _______________
2) 100, 50, 25, 12.5, _____, _____,.... _______________
3) 1, 2, 4, 7, 11, _____, _____, ... ___________________
Pattern: __________________________________________
4) 7, 77, 777, 7777, _______, ______, ... _____________
5) 1, 1/3, 1/9, 1/27, _____, _____, ... _________________
9) _______________, _______________
6) 1, 4, 9, 16, _____, _____, ... _________________
Pattern: __________________________________________
7) 1, 1/9, 1/25, 1/49, ______, _____,... _______________
Dec 15­2:11 PM
Inductive reasoning: Making a prediction based on patterns 10) A plumber charges $60 for a house call and $35 per hour that she is on the job. Can you determine how much the total cost (C) will be for n(number of hours) that she will work?
Jul 22­2:41 PM
Make a conjecture about the sum of the cubes of the first 25 counting numbers.
Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern.
13
13 + 23
13 + 23 + 33
13 + 23 + 33 + 43
13 + 23 + 33 + 43 + 53
= 1
= 9
= 36
= 100
= 225
= 12
= 32
= 62
= 102
= 152
= 12
= (1 + 2)2
= (1 + 2 + 3)2
= (1 + 2 + 3 + 4)2
= (1 + 2 + 3 + 4 + 5)2
The sum of the first two cubes equals the square of the sum of the first two counting numbers.
Jul 22­2:51 PM
Jul 16­8:45 PM
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Chapter 1.1 Patterns and Inductive Reasoning.notebook
January 03, 2013
Determine if the conjecture is true/false. If false, give a counter example.
Example 12.) All animals are elephants.
Conjecture: A statement made based on inductive reasoning.
Counterexamples: An example that shows that a conjecture is false.
Example 13.) The product of 5 and any odd number is even.
Example 14.) The quotient of two proper fractions is an improper fraction.
Conjecture: If a pet has 4 legs then it is a cat.
11) Is this statement True or False? If false, give a counterexample.
Example 15.) The sum of two even numbers is even.
Jul 22­2:57 PM
Jul 22­3:00 PM
See Chapter 1 pacing guide for assignment
Jan 3­12:31 PM
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