Programming Patterns to Make a Conjecture Questions

2
To solve these
problems,
you will pull
together ideas
about inductive
and deductive
reasoning.
Pull It All Together
Reasoning and Proof
You can observe patterns to make a conjecture; you can prove a conjecture is true by
using given information, definitions, properties, postulates, and theorems.
Task 1
You have the yellow game piece, your friend has the red
game piece, and your brother has the blue game piece. Read
the rules of the board game and then answer the questions.
ROLL
AGAIN
Rules
• You play counterclockwise.
• If you land on red, then you go back 1.
• If you land on green, then you advance 1.
• If you land on yellow, then you pick a card.
a. You roll 3. What must you do next? How do you know?
b. Your brother picks a card at the end of his turn.
On what colors might he have landed? Explain.
FINISH
RT
STA
c. Your friend rolls 2. What else must your friend do? How do you know?
d. Based on the colors already shown on the board, what color should
the roll-again box be? Justify your answer.
Task 2
Consider the number pattern at the right.
a. What is the sum of the numbers 31–40?
b. What is the sum of the numbers 101–110?
The sum of the numbers 1–10 is 55.
The sum of the numbers 11–20 is 155.
The sum of the numbers 21–30 is 255.
c. What kind of reasoning did you use in parts (a) and (b)?
d. Following is the development of a formula for the sum of n consecutive integers.
S 5 x 1 (x 1 1) 1 (x 1 2) 1 c 1 (y 2 2) 1 (y 2 1) 1 y
The sum of n integers from x to y
1 S 5 y 1 (y 2 1) 1 (y 2 2) 1 c 1 (x 1 2) 1 (x 1 1) 1 x
The same sum in reverse order
2S 5 (x 1 y) 1 (x 1 y) 1 (x 1 y) 1 c 1 (x 1 y) 1 (x 1 y) 1 (x 1 y) Add the equations.
2S 5 n(x 1 y)
S5
There are n terms of (x 1 y).
n(x 1 y)
2
Divide each side by 2.
Use the formula to find the sum of the numbers 101–110.
e. What kind of reasoning did you use in part (d)?
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Chapter 2
Pull It All Together
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