PRE-ALGEBRA

Lesson 1-7 Warm-Up

PRE-ALGEBRA Inductive Reasoning (1-7) inductive reasoning: making judgements or drawing conclusions based on patterns you observe. conjecture: The conclusion you reach by inductive reasoning. Example: Observation: The shaded triangle is rotating clockwise around the square. Conjecture: The next figure will have a shaded triangle in the bottom right corner. To write a conjecture: 1. find the pattern by determining what was done to a number to get the next number, and 2. write the rule of the pattern. Note: Begin the rule with the number you start with. Example: Write the rule for 640, 320, 160, 80, … and use your rule to find the next two numbers in the pattern. The rule is, “Start with 640 and divide by 2”. The next two number, then, are 40 and 20. What is “inductive reasoning”? What is a “conjecture”? How do you write a conjecture and use it to continue a pattern.

PRE-ALGEBRA 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 LESSON 1-7 Additional Examples

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

PRE-ALGEBRA Write a rule for the number pattern 110, 100, 90, 80,.. Find the next two numbers in the pattern. 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. The next numbers are found by subtracting 10. – 10 The first number is ,100,90,80 Inductive Reasoning LESSON 1-7 Additional Examples

PRE-ALGEBRA Inductive Reasoning (1-7) You can tell whether a conjecture is reasonable by making a prediction. Example: Is the conjecture that average hourly earnings in 2013 will be about $18.25 reasonable? The conjecture is reasonable, since average hourly earning seem to be increasing about $2.00 every five years. counterexample: a statement that proves a conjecture false (Note: you only need one counterexample to prove that a conjecture is wrong.) Example: Is the conjecture that every four-sided figure is a rectangle reasonable. Counterexample: The conjecture is incorrect. The figure above is a four-sided figure but not a rectangle. How can you tell whether a conjecture is reasonable? What is a “counterexample”? How can you use a counterexample to prove a conjecture is wrong?

PRE-ALGEBRA 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

PRE-ALGEBRA Is each conjecture correct or incorrect? If it is incorrect, give a counterexample. a. Every triangle has three sides of equal length. b. The opposite of a number is negative. The conjecture is incorrect. The figure below is a triangle, but it does not have three equal sides. The conjecture is incorrect. The opposite of –2 is 2. Inductive Reasoning LESSON 1-7 Additional Examples

PRE-ALGEBRA (continued) c. The next figure in the pattern below has 16 dots. The conjecture is correct. The diagram below shows the next figure in the pattern. Inductive Reasoning LESSON 1-7 Additional Examples

PRE-ALGEBRA Find the next three numbers in each pattern. 1.1, –1, 2, –2, 3, , 3, 7, 15, 31,... 3.–11, –8, –5, –2,... 1, 4, 7 –3, 4, –463, 127, 255 Lesson Quiz Inductive Reasoning LESSON 1-7