2.2 Reasoning and Proof
Inductive Reasoning
The nth Term –Students enjoy using inductive reasoning to find missing terms in a pattern. They are good at
finding the next term, or the tenth term, but have trouble finding a generic term or rule for the number sequence. If
the sequence is linear (the difference between terms is constant), they can use methods they learned in Algebra for
writing the equation of a line.
Key Exercise: Find a rule for the nth term in the following sequence.
13 , 9 , 5 , 1 ,....Answer: The sequence is linear, each term decreases by 4. The first term is 13, so the point( 1 , 13 )can be used. The
second term is 9, so the point( 2 , 9 )can be used. Applying what they know from Algebra I, the slope of the line is
−4, and they−intercept is 17, so the rule is− 4 n+17.
True MeansAlways True –In mathematics a statement is said to be true if it is always true, no exceptions.
Sometimes students will think that a statement only has to hold once, or a few times to be considered true. Explain
to them that just one counterexample makes a statement false, even if there are a thousand cases where the statement
holds. Truth is a hard criterion to meet.
Sequences –A list of numbers is called a sequence. If the students are doing well with the number of vocabulary
words in the class, the term sequence can be introduced.
Additional Exercises:
- What is the next number in the following number pattern? 1, 1 , 2 , 3 , 5 , 8 , 13 ,...
Answer: This is the famous Fibonacci sequence. The next term in the sequence is the sum of the previous two terms.
8 + 13 = 21
- What is the missing number in the following number pattern? 25, 18 ,?, 10 , 9 ,...
Answer: Descending consecutive odd integers are being subtracted from each term, so the missing number is 13.
Conditional Statements
The Advantages and Disadvantages of Non-Math Examples –When first working with conditional statements,
using examples outside of mathematics can be very helpful for the students. Statements about the students’ daily
lives can be easily broken down into parts and evaluated for veracity. This gives the students a chance to work with
the logic, without having to use any mathematical knowledge. The problem is that there is almost always some crazy
exception or grey area that students will love to point out. This is a good time to remind students of how much more
precise math is compared to our daily language. Ask the students to look for the idea of what you are saying in the
non-math examples, and use their powerful minds to critically evaluate the math examples that will follow.
Chapter 2. Geometry TE - Common Errors