Bell Work
Number Theory is a branch of mathematics that explores patterns in numbers. The
mathematician Carl Friedrich Gauss (1777-1855) wrote, “Mathematics is the queen of the
sciences and number theory the queen of mathematics.” A story about Gauss’s childhood
illustrates his mathematical ability at an early age. In order to keep his students occupied,
Gauss’s teacher instructed them to add the numbers from one to one hundred and place
their slates on his desk as soon as the task was completed. Right away, Gauss had the
correct answer. How did he complete this seemingly difficult task so quickly?
1. Examine the numbers from one to ten below.
1
2
3
4
5
6
7
8
9
10
To find their sum, you could simply add from left to right, or you could pair the
numbers as illustrated and add each pair.
1
2
3
4
5
10
9
8
7
6
a. What is the sum of each pair?
b. How many pairs are there?
2. Using the information in Exercise 1, what is the sum of the numbers from one to
ten?
3. Explain how you could quickly compute the sum of the numbers from one to
twenty. What is the sum?
4. Generate a formula to compute the sum of the numbers from one to any number n.
(Hint: Think of what each pair would add up to and how many pairs there would
be.)
5. What was the answer to Gauss’s problem?
Suppose you only wanted to add the even numbers from two to twenty. How would that
change the formula you generated in Exercis
Interdisciplinary Application
Baseball
You are learning the basic rules of baseball in gym class.
Your gym teacher explains that teams consist of nine people
who use a hard ball, a wooden or aluminum bat, and gloves.
Teams alternate turns in the field and at bat. One turn at
each constitutes an inning, and nine innings constitute a
game. In the field there is a pitcher, catcher, four infielders,
and three outfielders. The pitcher throws overhand from the
pitching rubber to home plate, while an opposing batter
attempts to hit the pitches and safely reach base. If the
batter hits the ball, the fielders attempt to put the batter out
through various plays.
A run is scored every time a batter crosses home plate after
touching each base in the prescribed order. When the
fielding team puts out three batters, the two teams exchange
places. A team wins when it scores more than the opposing
team at the end of nine innings. Extra innings are played if
the score is tied.
In Exercises 1–3, use the following conditional
statement.
If the batter makes an out, then the batter does not become
a base runner.
1. Write the converse of the conditional statement.
2. Write the contrapositive of the conditional statement.
3. Write the inverse of the conditional statement.
In Exercises 4–6, use the following contrapositive.
If play does not continue into extra innings, then the score is
not tied at the end of nine innings.
4. Write the conditional statement.
5. Write the converse.
6. Write the inverse.
7. If your team scores a run, then your team will win the
game. Using the Law of Detachment, what can you
conclude when your team scores a run?
8. Write two conditional statements that can be made from
the following true statements using the Law of Syllogism.
a. If a batter crosses home plate, then a run is scored.
b. If there are three outs, then the inning is over.
c. If the outfielder makes the catch, then there will be
three outs.
d. If a run is scored, then the game will be tied.
Answer Key
Lesson 2.3
Interdisciplinary Application
1. If the batter does not become a base runner, then the batter makes an out.
2. If the batter becomes a base runner, then the batter does not make an out.
3. If the batter does not make an out, then the batter becomes a base runner.
4. If the score is tied at the end of nine innings, then play continues into extra innings.
5. If play continues into extra innings, then the score is tied at the end of nine innings.
6. If the score is not tied at the end of nine innings, then play does not continue into extra
innings.
7. Your team wins the game.
8. (1) If a batter crosses home plate, then the game will be tied. (2) If the outfielder makes the
catch, then the inning is over.