Using patterns to make conjectures 1-7 Inductive Reasoning Using patterns to make conjectures

Inductive reasoning Inductive reasoning is making conclusions based on observed patterns. The conclusion you reach is called a conjecture. The conjecture can be proven false with a counterexample.  Tips: all you need is one counter example to prove a conjecture is not true

Observation: The circles are rotating counterclockwise Inductive Reasoning Lesson 1-7 Additional Examples Use inductive reasoning. Make a conjecture about the next figure in the pattern. Then draw the figure. Observation: The circles are rotating counterclockwise within the square. Conjecture: The next figure will have a shaded circle at the top right.

Inductive Reasoning Write a rule for each number pattern. Lesson 1-7 Additional Examples Write a rule for each number pattern. a. 0, – 4, – 8, –12, . . .   Start with 0 and subtract 4 repeatedly. b. 4, – 4, 4, – 4, . . .   Alternate 4 and its opposite. c. 1, 2, 4, 8, 10, . . .   Start with 1. Alternate multiplying by 2 and adding 2.

Inductive Reasoning Lesson 1-7 Additional Examples Write a rule for the number pattern 110, 100, 90, 80, . . . Find the next two numbers in the pattern. The first number is 110. 110, 100, 90, 80 The next numbers are found by subtracting 10.   – 10 The rule is Start with 110 and subtract 10 repeatedly. The next two numbers in the pattern are 80 – 10 = 70 and 70 – 10 = 60.

Inductive Reasoning Lesson 1-7 Additional Examples A child grows an inch a year for three years in a row. Is it a reasonable conjecture that this child will grow an inch in the year 2015? No; children grow at an uneven rate, and eventually they stop growing.

Inductive Reasoning Lesson 1-7 Additional Examples Is each conjecture correct or incorrect? If it is incorrect, give a counterexample. a. All students who attend Rivermont live in Iowa. b. Every triangle has three sides of equal length. The conjecture is incorrect. The figure below is a triangle, but it does not have three equal sides. c. The opposite of a number is negative. The conjecture is incorrect. The opposite of –2 is 2.

Inductive Reasoning d. The next figure in the pattern below has 16 dots. The conjecture is correct. The diagram below shows the next figure in the pattern.

Homework pg. 38-39 (2-22 eve)due Monday, September 12th!

1.8 Look for a pattern

Tip There are many ways to find a solution to a problem that involves a pattern, such as tables or a tree diagram

Problem Solving Strategy: Look for a Pattern Each student on a committee of five students shakes hands with every other committee member. How many handshakes will there be in all? The pattern is to add the number of new handshakes to the number of handshakes already made. 4 the number of handshakes by 1 student 4 + 3 = 7 the number of handshakes by 2 students

One option for solving is making a Table Make a table to extend the pattern to 5 students. Student Number of original handshakes Total number of handshakes 1 4 2 3 4 + 3 7 + 2 9 + 1 5 10 + 0 = 7 = 9 = 10 There will be 10 handshakes in all.

Example 2 You have a penny, a nickel, a dime, and a quarter. You give away three coins. How many different amounts of money can you give away? Name the values.

Example 3 You can cut a pizza into two piece with one straight cut. With 2 cuts you can get 4 pieces. Three cuts gives you maximum of 7 pieces. What is the maximum number of pieces with 4 cuts? With 5 cuts?

Consider this situation: A bacterium divides into two new bacteria every 30 min. If you start out with a single bacterium, how many bacteria will there be after two hours? How about after 5 hours? How did you solve this problem? Can you write a rule for how to solve it?

Consider this situation: Each student on a committee shakes hands with every other committee member at the end of a meeting. How many total handshakes will there be if there are 3 committee members? How many handshakes will there be if there are 5 committee members? What if there are 10 committee members?

Homework pg. 42-43 (1-8 all, skip 5)due Monday September 12th!

1-10 The coordinate plane

Defintions Coordinate Plane – formed by the intersection of two number lines 2. x-axis – the horizontal number line y-axis – the vertical number line 4. Quadrants – the x and y axes divide the coordinate plane into 4 sections Origin – the point where the x and y axes intersect 6. Ordered Pair – gives the coordinates (x , y) and location of a point x-coordinate – shows the position left or right of the y-axis 8. y-coordinate – shows the position above or below the x-axis

Label Terms on the Graph

Graphing Practice Graph each point (2,-3), (-2,5), (0,4), (3, 0) (-1, -1)

Homework Pg. 54-55 (41-44, 54, 57-60, 61-63) due September 12th!