Happy Friday! Welcome to another edition of It All Adds Up! Once more, my Geometry class had gone further in content in two weeks than I can cover here. Below, you’ll find a bulleted recap of what we learned in Geometry this week. In red are the topics I will cover here, and in black are additional topics we’ve learned that you may want to research on your own:
- Writing Conjectures using inductive and deductive reasoning
- Writing the inverse, converse, and contrapositive of conditional statements
- Structure of a conditional statement
- Laws of Detachment and Syllogism
- An introduction to everyday postulates and theorems that deal with geometry
Let’s start with some vocabulary:
Conjecture: an unproven statement based on observations
Inductive Reasoning: the process of finding a pattern for specific cases and then writing a conjecture for the general case.
Deductive Reasoning: to process of using facts and logic to write a conditional statement
Counterexample: a specific case for which the conjecture is false.
Here’s an example. What if I were to ask you to make a conjecture about the sum of 3 consecutive odd numbers? Answers can always vary- which is perfectly fine, as long as your conjecture can’t be proved wrong with a counterexample. To answer this, you might say The sum of any 3 consecutive odd numbers will always equal a multiple of 9.
Yes, it’s true in some cases, but I can find a counterexample to prove it wrong. Just one counterexample is enough- for example, 3, 5, and 7 equal a multiple of 3. That’d be your revised conjecture.
By the way, how would you get to that conjecture in the first place? By using inductive reasoning, through the process of observing patterns (like noticing how 19 + 21 +23 is equal to 63, and 31 + 33 + 35 is equal to 99), and then coming to a conclusion about it.
You can find patterns in anything- remember the patterns of shapes and numbers you learned in first grade?
A conjecture is made using patterns- but a conditional statement is made using deductive reasoning, or logic and facts. A conditional statement is made up of 2 parts: the hypothesis and conclusion, and is written as an if/then statement. The hypothesis is the If something were to happen, and the conclusion is the Then this would happen part. It doesn’t have to be about math, though!
As you know, I live in Connecticut. For this example, I’ll use our capital, Hartford. To write a conditional statement, you might write: If you live in Hartford, you live in Connecticut. That’s absolutely true (unless another Hartford exists somewhere else- *counterexample!*).
In class, though, this was only the beginning. There are three main other ways to write this statement. Here they are:
Inverse: negates both the hypothesis and the conclusion. If you don’t live in Hartford, you don’t live in Connecticut. That’s the right inverse, but the facts behind it are completely false. When writing forms of a conjecture, it’s always very important to evaluate the validity of your statements.
Converse: switching the order of the original hypothesis and the conclusion. If you live in Connecticut, you live in Hartford. Also false. Here’s a pattern that we discussed: if the inverse is false, the converse will also be false, or vice versa. The same goes for the If/Then and Contrapositive.
Contrapositive: taking the converse and negating it. If you don’t live in Connecticut, you don’t live in Hartford. True.
To clarify confusion, and inverse will always be the inverse, no matter if it is true or false. You might be asked to evaluate the validity of that inverse, is all.
Here’s something else to add: If both the conditional AND converse are true, your conditional statement is a definition. For example: If an angle measures 90 degrees, it is a right angle. If an angle is a right angle, it measures 90 degrees. Both are true, so you have a definition.
One more vocabulary word: Equivalent Statements: two statements that are both either true or false.
Biconditional Statement: a statement that uses “if and only if” in the middle, in which both the conditional and converse are true.
For example: Two lines are perpendicular if and only if they intersect to form a right angle.
For the final part of the lesson, I wanted to go over the two laws of reasoning.
Law of Detachment. This law takes place when you have a three-part statement. If the 3rd part logically follows the first two, it shows this law. If not, it is invalid.
Here’s an example:
- If you have a drivers license, then you can drive a car.
- Gabe doesn’t have a drivers license.
- Gabe cannot drive a car.
Also, here is an example of when it is invalid:
- Adjacent angles share a common interior side.
- Two angles share a common side.
- The angles are adjacent.
Nope. #1 is true, but it leaves out that adjacent angles also need to share a common vertex. Because we don’t know that the angles share a common vertex, we can’t assume that they are adjacent. This would be considered invalid.
Law of Syllogism. This is a four part statement, written as two if/then statements. The interior hypothesis/conclusion (in red) are the same, while the exterior two are different. If both statements are true, we can write a conclusion that combines the exterior parts. Here’s an example:
If John buys a pair of glasses, then he will see better. If John will see better, than he can see the board during class.
Conclusion: If John buys a pair of glasses, than he can see the board during class. We exclude the interior parts, since they apply to both statements.
One other thing to mention: a statement might not always be set up in exactly this way- as long as there is a common hypothesis/conclusion, you can rearrange the statements into this format.
Here’s one where it isn’t valid:
If Gabe buys glasses, he will see the board better. If he does better in class, he will make his parent happy.
There aren’t any common interior parts, so we can’t make a valid conclusion about this (we can’t assume that buying glasses will help Gabe do better in class).
Who knew that Geometry would turn into English class!
Thanks so much for reading! Look out for a new French Connection this Monday, and a new It All Adds Up in two weeks. Have a great weekend!