BIG idea: 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.

Rules

Image Long Description

• 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.
  1. You roll 3. What must you do next? How do you know?
  2. Your brother picks a card at the end of his turn. On what colors might he have landed? Explain.
  3. Your friend rolls 2. What else must your friend do? How do you know?
  4. 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 below.

The sum of the numbers 1 − 10 1 minus 10  is 55.

The sum of the numbers 11 − 20 11 minus 20  is 155.

The sum of the numbers 21 − 30 21 minus 30  is 255.

1. What is the sum of the numbers 31 − 40 ? 31 minus 40 question mark
2. What is the sum of the numbers 101 − 110 ? 101 minus 110 question mark
3. What kind of reasoning did you use in parts (a) and (b)?

4. Following is the development of a formula for the sum of n consecutive integers.

   S = x + ( x + 1 ) + ( x + 2 ) + … + ( y − 2 ) + ( y − 1 ) + y + S = y + ( y − 1 ) + ( y − 2 ) + … + ( x + 2 ) + ( x + 1 ) + x 2 S = ( x + y ) + ( x + y ) + ( x + y ) + … + ( x + y ) + ( x + y ) + ( x + y ) 2 S = n ( x + y ) S = n ( x + y ) 2 The sum of  n  integers from  x  to  y The same sum in reverse order Add the equations . There are  n  terms of ( x + y ) . Divide each side by  2. table with 1 row and 2 columns , row1 column 1 , table with 5 rows and 2 columns , row1 column 1 , s , column 2 equals x plus . open , x plus 1 , close . plus . open , x plus 2 , close . plus dot dot dot plus . open , y minus 2 , close . plus . open , y minus 1 , close . plus y , row2 column 1 , plus s , column 2 equals y plus . open , y minus 1 , close . plus . open , y minus 2 , close . plus dot dot dot plus . open , x plus 2 , close . plus . open , x plus 1 , close . plus x , row3 column 1 , 2 s , column 2 equals . open , x plus y , close . plus . open , x plus y , close . plus . open , x plus y , close . plus dot dot dot plus . open , x plus y , close . plus . open , x plus y , close . plus . open , x plus y , close , row4 column 1 , 2 s , column 2 equals n . open , x plus y , close , row5 column 1 , s , column 2 equals . fraction n . open , x plus y , close , over 2 end fraction , end table , column 2 table with 5 rows and 1 column , row1 column 1 , cap thesumof . n . integersfrom . x , to , y , row2 column 1 , cap thesamesuminreverseorder , row3 column 1 , cap addtheequations . . , row4 column 1 , cap thereare . n . termsof . open , x plus y , close . . , row5 column 1 , cap divideeachsideby . 2. , end table , end table

   Use the formula to find the sum of the numbers 101 − 110 . 101 minus 110 .

5. What kind of reasoning did you use in part (d)?