A Towering Sequence: IM Alg2.1.1
[url=https://im.kendallhunt.com/HS/teachers/3/1/1/preparation.html]“A Towering Sequence”[/url] from IM Algebra II © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg2.1.1 Lesson
- IM Alg2.1.1 Lesson: A Towering Sequence
- Alg2.1.1 Practice
- IM Alg2.1.1 Practice: A Towering Sequence
IM Alg2.1.1 Lesson: A Towering Sequence
Here is a rule for making a list of numbers: Each number is 1 less than twice the previous number.
Pick a number to start with, then follow the rule to build a list of 5 numbers.
In the Tower of Hanoi puzzle, a set of discs sits on a peg, while there are 2 other empty pegs.
A [i]move[/i] in the Tower of Hanoi puzzle involves taking a disc and moving it to another peg. To move a disc, click on it, and then click on the peg where you want it to go. You cannot [i]drag[/i] a disc. There are two rules:[br][list][*]Only move 1 disc at a time.[/*][*]Never put a larger disc on top of a smaller one.[/*][/list]
You complete the puzzle by building the complete tower on a different peg than the starting peg.
Using 3 discs, complete the puzzle. What is the smallest number of moves you can find?[br]
Using 4 discs, complete the puzzle. What is the smallest number of moves you can find?[br]
Jada says she used the solution for 3 discs to help her solve the puzzle for 4 discs. Describe how this might happen.[br]
How many moves do you think it will take to complete a puzzle with 5 discs?[br]
How many moves do you think it will take to complete a puzzle with 7 discs?[br]
A legend says that a Tower of Hanoi puzzle with 64 discs is being solved, one move per second. How long will it take to solve this puzzle?
Some checkers are lined up, with blue on one side, red on the other, and with one empty space between them.
A [i]move[/i] in this checker game pushes any checker forward 1 space, or jumps over any 1 checker of the other color. Jumping the same color is not allowed, moving backwards is not allowed, and 2 checkers cannot occupy the same space.[br]You complete the puzzle by switching the colors completely: ending up with blue on the right, red on the left, and with 1 empty space between them.
Drag the checkers to move them.
Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?[br]
Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed?[br]
Make guesses about the number of moves for 2 and 4 checkers on each side, then test your guesses.[br]
Noah says he used the solution for 3 checkers on each side to help him solve the puzzle for 4 checkers. Describe how this might happen.[br]
How many moves do you think it will take to complete a puzzle with 7 checkers on each side?[br]
IM Alg2.1.1 Practice: A Towering Sequence
Here is a rule to make a list of numbers: Each number is the sum of the previous two numbers.
Start with the numbers 0 and 1, then follow the rule to build a sequence of 10 numbers.
[size=150]A sequence starts [/size][math]\frac{1}{2},\frac{1}{4},\frac{1}{8},\ldots[/math][br][br]Give a rule that the sequence could follow.[br]
Follow your rule to write the next 3 terms in the sequence.[br]
A sequence of numbers follows the rule: multiply the previous number by -2 and add 3. The fourth term in the sequence is -7.
Give the next 3 terms in the sequence.[br]
Give the 3 terms that came before -7 in the sequence.[br]
A sequence starts 0, 5, . . .
Give a rule the sequence could follow and the next 3 terms for that rule.[br]
Give a [i]different[/i] rule the sequence could follow and the next 3 terms for that rule. [br]