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Understanding Reversible Operations and Backtracking Methods**

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12 diapositives · 282 Views · Sep 22, 2014

This guide explores the concept of reversible operations, which involve a second operation that returns a situation to its original state. It discusses examples such as setting clocks, reversing actions like taking off shoes, and why certain operations, like multiplying by zero, are not reversible. We delve into backtracking as a method for solving equations by reversing operations and comparing it to guess-and-check strategies. Understanding these concepts can enhance problem-solving skills in both mathematics and everyday scenarios.

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  1. Reversing operations 2.7 in your book

  2. Actions you can reverse • Do the directions backwards • Close the window • Take off your shoes, then your socks • Set the clock forward 1 hour or behind by 11 hours • Walking to a location in the city (like yesterday’s problem) • Opening a window • Putting on your socks and then your shoes • Set a 12-hour clock forward one hour

  3. REVERSIBLE OPERATION • A “reversible operation” is an operation if there is a second operation that ALWAYS brings you back to the situation you had before the first operation • Not all operations are reversible. How do you reverse an earthquake? A baby being born? Multiplying by zero?

  4. Either give the reverse operation or explain why there isn’t one • Adding 5 to a number • Multiplying a number by 3 • Multiplying a number by 0 • Adding 7 hours to the current time * (there are two answers) • Adding the digits of a number together

  5. Disprove the statement that multiplying by zero is reversible! • To do so: find two numbers that, when you multiply them by zero, you get the same number. • This is proving by counterexample. In order to disprove a statement, you only need to find ONE example of it not working. • Try disproving: any number plus 1 equals 14.

  6. Reversible Operations

  7. Operations that aren’t reversible

  8. Operations that are SOMETIMES reversible • “I picked a number. When you square it, you get 225. What’s the number?” • “15 – obvi.” • “Nope. Guess again.” • WHAT’S THE NUMBER?

  9. Backtracking • Backtracking is used to solve equations. • You do it by reversing all the operations in the equation.

  10. Use backtracking to solve an equation • Solve the equation: 3x – 14 = 37 • Write out the steps that show how to get from the input variable, x, to the output value, 37: • Make a list that reverses the order and undoes the operations:

  11. Compare backtracking to guess-and-check • Some people solve equations like 3x-14=37 by guessing values for x, plugging them into the equation, and checking if it works, until they guess the correct number. • What advantages does backtracking have? • What disadvantages does backtracking have?

  12. CHECK YOUR COMPREHENSION