2.1. Conjectures and Counterexamples http://www.ck12.org
Concept Problem Revisited
Your older brother made the conjecture that all guys must play sports. You could prove him wrong, or disprove his
conjecture, by offering a counterexample.
Say your friend, John, is a guy and does not play sports. Just one counterexample is enough to disprove your
brother’s conjecture.
Vocabulary
Aconjectureis an “educated guess” that is based on examples in a pattern. Acounterexampleis an example that
disproves a conjecture.
Guided Practice
A car salesman sold 5 used cars to five different couples. He noticed that each couple was under 30 years old. The
following day, he sold a new, luxury car to a couple in their 60’s. The salesman determined that only younger couples
by used cars.
- Is the salesman’s conjecture logical? Why or why not?
- Can you think of a counterexample?
Answers:
- It is logical based on his experiences, but is not true.
- A counterexample would be a couple that is 30 years old or older buying a used car.
Explore More
Read the following examples of reasoning in the real world. Do you think the conjectures are true or can you give a
counterexample?
- For the last three days Tommy has gone for a walk in the woods near his house at the same time of day. Each
time he has seen at least one deer. Tommy reasons that if he goes for a walk tomorrow at the same time, he
will see deer again. - Maddie likes to bake. She especially likes to take recipes and make substitutions to try to make them healthier.
She might substitute applesauce for butter or oat flour for white flour. She has noticed that she needs to add
more baking powder or baking soda than the recipe indicates in these situations in order for the baked goods
to rise appropriately. - One evening Juan saw a chipmunk in his backyard. He decided to leave a slice of bread with peanut butter on
it for the creature to eat. The next morning the bread was gone. Juan concluded that chipmunks like to eat
bread with peanut butter. - Sarah noticed that all her friends were in geometry class and reasoned that every 10th grade student is in
geometry. - Describe an instance when you observed someone using invalid reasoning skills.
Give a counterexample for each of the following statements.
- Ifnis an integer, thenn^2 >n.
- All numbers that end in 1 are prime numbers.
- All positive fractions are between 0 and 1.