Lesson 2-1 Key Terms

A conjecture is an educated guess based on known information

Inductive reasoning is reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction.

Counterexample- a false example that shows that a conjecture is not true.

Make a conjecture about the next number based on the pattern.2, 4, 12, 48, 240

Answer: 1440

Find a pattern:

2 4 12 48 240

×2

The numbers are multiplied by 2, 3, 4, and 5.

Conjecture: The next number will be multiplied by 6. So, it will be or 1440.

×3 ×4 ×5

Make a conjecture about the next number based on the pattern.

Answer: The next number will be

Given: points L, M, and N; Examine the measures of the segments. Since the points can be collinear with point N between points L and M.

Answer:

Conjecture: L, M, and N are collinear.

For points L, M, and N, LM = 20, MN = 6, and LN = 14 make a conjecture and draw a figure to illustrate your conjecture.

ACE is a right triangle with Make a conjecture and draw a figure to illustrate your conjecture.

Answer:

Conjecture: In ACE, C is a right angle and is the hypotenuse.

UNEMPLOYMENT Based on the table showing unemployment rates for various cities in Kansas, find a counterexample for the following statement. The unemployment rate is highest in the cities with the most people.

County Civilian Labor Force Rate

Shawnee 90,254 3.1%

Jefferson 9,937 3.0%

Jackson 8,915 2.8%

Douglas 55,730 3.2%

Osage 10,182 4.0%

Wabaunsee 3,575 3.0%

Pottawatomie 11,025 2.1%

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Examine the data in the table. Find two cities such that the population of the first is greater than the population of the second while the unemployment rate of the first is less than the unemployment rate of the second. Shawnee has a greater population than Osage while Shawnee has a lower unemployment rate than Osage.

Answer: Osage has only 10,182 people on its civilian labor force, and it has a higher rate of unemployment than Shawnee, which has 90,254 people on its civilian labor force.

DRIVING The table on the next screen shows selected states, the 2000 population of each state, and the number of people per 1000 residents who are licensed drivers in each state. Based on the table, find a counterexample for the following statement. The greater the population of a state, the lower the number of drivers per 1000 residents.

State Population Licensed Drivers per 1000

Alabama 4,447,100 792

California 33,871,648 627

Texas 20,851,820 646

Vermont 608,827 831

West Virginia 1,808,344 745

Wisconsin 5,363,675 703

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Answer: Alabama has a greater population than West Virginia, and it has more drivers per 1000 than West Virginia.