Topic 1: The Fundamental Counting Principle

Set TheoryTopic 1: Logical ReasoningI 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.ExploreConsider 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. ColoursY YellowP PinkB - Black

ExploreUse 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.ExploreGame board to be completed

You should noticeGame board to be completed

InformationThere 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 checkLook for a patternEliminate possibilitiesSimplify problemsWork backwardsDevelop alternate approachesUse manipulatives

Example 1The 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 puzzleExample 1Column

Row1234561G2G3GG4MMG5M6MMGRowIt is Margarets turn, and she determines that she can guarantee a win by placing the letter M in Row ______ Column ________ Determine a strategy to solve a puzzleTry this example in your workbook before looking at the answers on the next slide.Example 1: SolutionColumn

Row1234561G2G3GG4MMMG5M6MMGRowIt is Margarets 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 puzzleExample 2In 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 gameExample 2Identify 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: SolutionIdentify a strategy to win a gamea) Create a method to track your moves, refer to the 2nd column.

Example 2: SolutionIdentify a strategy to win a gameb) Create a method to track your moves, refer to the 2nd column.

A-K, H-F, K-G, F-HExample 3The 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 3Correct errors in a puzzle

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

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

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

Need to KnowPossible strategies:Guess and checkLook for a patternEliminate possibilitiesSimplify problemsWork backwardsDevelop alternate approachesUse manipulatives

Youre ready! Try the homework from this section.