Set Theory Topic 1: Logical Reasoning

I can determine and explain a strategy to solve a puzzle. I can verify a strategy to win a game. I can analyze puzzles and games that involve numerical and logical reasoning.

Explore… Consider the following game that uses three different shapes (triangle, square and circle) in three different colours (yellow, pink and brown) to fill a game board. Each of the figures must be used exactly once. Colours Y – Yellow P – Pink B - Black

Explore… Use the two clues below to fill in the game board with all nine pieces in their correct positions. Clue 1: Clue 2: an open shape indicates shape with the colour undetermined. a shaded square of the grid indicates the colour with an undetermined shape

Try this explore activity in your workbook before looking at the answers on the next slide. Explore… Game board to be completed

You should notice… Game board to be completed

Information There are many different strategies that could be used to solve puzzles or to determine how to win a game. Some strategies may include: Guess and check Look for a pattern Eliminate possibilities Simplify problems Work backwards Develop alternate approaches Use manipulatives

Example 1 The goal of a particular two-player game is to create a line of four adjacent squares using the same letter. To play, each player takes turns placing their first initial somewhere on a six-by-six grid. Margaret and Gerda have started playing this game, as shown on the grid on the next slide. Determine a strategy to solve a puzzle

Example 1 Column Row G 2 G 3 GG 4 MMG 5 M 6MMG It is Margaret’s turn, and she determines that she can guarantee a win by placing the letter M in Row ______ Column ________ Determine a strategy to solve a puzzle Try this example in your workbook before looking at the answers on the next slide.

Example 1: Solution Column Row G 2 G 3 GG 4 MMMG 5 M 6MMG It is Margaret’s turn, and she determines that she can guarantee a win by placing the letter M in Row __4____ Column ___2_____ Determine a strategy to solve a puzzle

Example 2 In the game Hoppers the object is to have only the Red frog left on a lily pad. The rules are: The frogs may hop in the directions indicated by the lines between the lily pads. You may only move by jumping over a frog on an adjacent lily pad, and landing on the next lily pad. When a frog is jumped over it is removed from the pond. On each turn, any one of the frogs may be moved as long as the above rules are followed. To win you must have the RED frog as the only remaining frog in the pond. Identify a strategy to win a game

Example 2 Identify a strategy to win a game a) What strategies could you use to help you win this game? (ie, How could you track your moves? How can you determine if you will win? Etc.) Try this example in your workbook before looking at the answers on the next slide.

Example 2: Solution Identify a strategy to win a game a) Create a method to track your moves, refer to the 2 nd column.

Example 2: Solution Identify a strategy to win a game b) Create a method to track your moves, refer to the 2 nd column. A-K, H-F, K-G, F-H

Example 3 The goal of a particular puzzle is to fill the circles in a grid with the letters A, B, C, and D so that no letters are repeated in any row, column, or set of connected circles. The three entries in the grey circles were given to start the puzzle. Jerome has already completed three entries shown in the white circles, but he has made an error. Correct errors in a puzzle

Example 3 Correct errors in a puzzle a)Identify the error that Jerome made in his solution to the puzzle. b)Explain why his entry is incorrect. Try this example in your workbook before looking at the answers on the next slide.

Example 3: Solution Correct errors in a puzzle a)Identify the error that Jerome made in his solution to the puzzle. b)Explain why his entry is incorrect. The error that Jerome made is in row 3, column 3. The entry in row 3, column 1 must be an A, and the entry in row 4, column 4 must be a D. However, since the last three circles in the last row are connected to the circle in row 3, column 4, one of the rules of the game is violated because a set of connected circles has repeated letters.

Example 3 Correct errors in a puzzle Try this example in your workbook before looking at the answers on the next slide. c) Correct the error that Jerome made and complete the puzzle.

Example 3: Solution Correct errors in a puzzle c) Correct the error that Jerome made and complete the puzzle.

Need to Know Logical reasoning can be used to explain and verify a strategy to solve a puzzle. A set can be represented by:

Need to Know Possible strategies: Guess and check Look for a pattern Eliminate possibilities Simplify problems Work backwards Develop alternate approaches Use manipulatives You’re ready! Try the homework from this section.