Book 9: Puzzles and Games

Transcription

1 Math 11 Recreation and Wellness Book 9: Puzzles and Games Teacher Version Assessments and Answers Included

2 Year Overview: Earning and Spending Money Home Travel and Transportation Recreation and Wellness 1. Earning Money 2. Pay Statements and Deductions 3. Responsible Spending Habits 4. Data in Your Life 5. Measurement 6. Angles and Triangles 7. Let s Travel Project 8. Personal Health and Wellness 9. Puzzles and Games Topic Overview Recreational activities such as playing games, solving puzzles, and participating in sporting events as well as activities connected to personal wellness will be used to teach problem solving strategies and reasoning skills. Students will use data collection and analysis to make predictions and inform decisions in order to effect changes in their own lives in terms of recreation and personal wellness. Outcomes Overlapping Outcomes M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. Theme Specific Outcomes M11.2 Demonstrate understanding of reasoning by analyzing puzzles and games. 1

3 Contents Topic Overview... 1 Outcomes... 1 Overlapping Outcomes... 1 Theme Specific Outcomes... 1 Contents... 2 Glossary of Terms Analyze and Strategize... 5 Discuss the Ideas Problem Solving Strategy: Drawing a Diagram A Problem: Planting Grapes A Practice Your Skills Problem Solving Strategy: Guess and Check A Problem: Chicken and Goats Practice Your Skills Problem Solving Strategy: Make a List A Problem: ABCD Order Practice Your Skills Problem Solving Strategy: Finding a Pattern A Fibonacci Series Practice Your Skills Online Puzzles National Library of Virtual Manipulatives (NLVM) Illuminations NCTM The Handcuffs Puzzle Additional Activities Puzzle It Out Stations Organized by Teacher What s the Name of the Game? Discuss the Ideas Practice Your Skills Game Design Activity Student Evaluation Learning Log Answers

4 9.1 Analyze and Strategize Problem A Practice Your Skills Practice Your Skills Practice Your Skills Practice Your Skills

5 Glossary of Terms problem solving strategy the steps one would use to solve a problem strategies a plan to achieve one or more goals under conditions of uncertainty, 2, 5, 6, 4

6 9.1 Analyze and Strategize Introduction: Many people have an interest in puzzles and games. Sometimes, it is difficult to determine how puzzles work. However, with a few simple strategies you can usually figure them out. Discuss the Ideas If there are other students around you working at the same pace, do a Think, Pair, Share with the problem below: Think: Take some time to think about and answer this question on your own without discussing it with anyone. Pair: Share your solution with a partner and discuss the strategies that you each used to solve the puzzle. Share: As a pair, share the strategies that you used to solve the puzzle with another pair or with the whole class so that your teacher can record them on the board. Problem: On a shelf, there are 10 books with 100 pages each. If a bookworm starts at the first page of the first book and eats through the last page of the last book, how many pages does the bookworm eat through (excluding covers)? 5

7 Examples of strategies that may have been used are: guess and check look for a pattern make a systematic list draw or model eliminate possibilities solve a simpler problem work backwards develop alternative approaches Sections highlight some common problem solving strategies. There are many different ways to solve problems. There is no one right way to solve a problem but sometimes one strategy makes more sense or is more efficient in a given situation. (Sections (explanations, examples and questions have been taken from: 6

8 9.2 Problem Solving Strategy: Drawing a Diagram Drawing a diagram is the most common problem solving strategy. Very often, a problem solver needs to draw a diagram just to understand the meaning of the problem. The diagram represents the problem in a way we can see it, understand it, and think about it while we look for the next step. Tell your students that they will use the strategy of drawing a diagram again and again. First they need some practice to learn how to interpret a problem and draw a useful diagram. Give them an easy problem to solve, i.e. a problem for which it is easy to draw a diagram, and then, once the diagram is drawn, to see the next step. If the next step is a calculation, your students should already know the formula or the algorithm they will need to do the calculation. 9.2A Problem: Planting Grapes Matthew has decided to plant grapes in the garden behind his house. His neighbor Vera has grown grapes successfully for a long time and has given Matthew advice about how to plant vines. Vera told him to plant them three meters apart in rows that are three meters apart. He also told him to leave at least three meters between each vine and the edge of the garden. Matthew has measured the size of his garden and learned that it s a rectangle with sides of 25 meters and 35 meters. How many vines should Matthew buy? Here is one way to solve this problem by drawing a diagram. If you answered this question by drawing a diagram, yours may be more or less precise than the one below but you should arrive at the same answer. 7

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10 9.2A Practice Your Skills Draw a diagram to answer this question. 1. Climbing Snail: A snail is climbing up a drainpipe. Each day the snail climbs 8.5 meters. Every night it rains, however, and when it rains the snail slips back down 1 meter. How many days will it take the snail to reach the top of the drainpipe, which is 45 meters high? 9

11 9.3 Problem Solving Strategy: Guess and Check Often when presented with a problem, you are encouraged to guess the answer. For example, "What's the largest city in the world? Take a guess." You guess Istanbul. "No, it's Tokyo," you are told. "How many people live in Tokyo? Take a guess." You say you don't know. "Go on, take a guess," you're told again. You guess 15 million. "No, 34 million!" Guessing often produces the wrong answer. But the strategy called "Guess and Check" often produces the right answer. It should probably be called "Guess and Check and Guess Again," because the process of checking the accuracy of each guess and then making another, more informed guess is an essential part of the strategy. 9.3A Problem: Chicken and Goats Belle went to her grandfather's farm. Her grandfather has chickens and goats on his farm. She asked him how many chickens and how many goats. Because her grandfather likes mathematical puzzles, he told her that his animals had 26 heads and 68 legs and from that information she could calculate the number of chickens and the number of goats. If you were Belle, how would you solve the problem? To use the Guess and Check strategy, you think about the problem and start by making a guess. You expect your first guess to be wrong, but it will give you some information to help you make a better guess next time. You could start by guessing 13 chickens and 13 goats. It's a good idea to keep a record of your guesses, like this: Guess Chickens Goats Number of heads Number of legs

12 You see that the number of legs you guessed is too high, because Belle s grandfather said that there are 68 legs. So you guess again-you have to add more chickens and subtract some goats. Guess Chickens Goats Number of heads Number of legs Now you have 64 legs and you need four more. But you can't add any more heads, since 26 is the correct number of heads. So you take away two chickens (two heads and four legs) and add two goats (two heads and eight legs). Guess Chickens Goats Number of heads Number of legs Now you have the correct answer: 18 chickens and 8 goats 11

13 9.3 Practice Your Skills Use Guess and Check to answer the following problem. 1. Grandpa Ian decided to take some of his children and grandchildren to see a movie in the mall. The tickets cost $5 for children and $7for adults. He spent $116. How many children and how many adults went to the cinema? 12

14 9.4 Problem Solving Strategy: Make a List There is a common saying, "Don't just stand there, do something!" When solving a problem, don't just think, write something! Draw a diagram or make a list! It is often obvious when you should draw a diagram. And it is often obvious when you should make a list. Take a very simple example: putting things in order. 9.4A Problem: ABCD Order The letters ABCD, can be put into a different order for example: DCBA or BADC. How many different combinations of the letters ABCD can you make? To answer this question, obviously, you have to make a list. It is best of you can make a SYSTEMATIC list so that you know you have all possible combinations: For example: ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA There are 24 different combinations. 13

15 9.4 Practice Your Skills Make a list to answer the following problem. 1. A linen shop sells sheets, duvet covers, and pillow cases. Linens from the Easirest company offer a choice in each color: a plain sheet or a sheet with a pattern, a plain duvet cover or a duvet cover with a pattern, plain pillow cases or pillow cases with a pattern. Customers may mix and match the plain or patterned sheets, duvet covers, and pillow cases in any combination they wish. To help the shop make its next order from the Easirest company, you have been asked to make a record of the different combinations that customers have chosen. How many different combinations are possible? 14

16 9.5 Problem Solving Strategy: Finding a Pattern 9.5A Fibonacci Series One of the most famous patterns in mathematics is known as the Fibonacci series, named for a mathematician who lived in Italy in the 13th Century. This pattern is often seen in nature. Fibonacci introduced this pattern by posing a problem: A pair of rabbits, one male and one female, are put into a pen. After two months they have two offspring, one male and one female. They continue to have an additional two offspring every month thereafter, always a pair, one male and one female. This pattern continues: after two months, every pair of rabbits start to reproduce and every month thereafter they have a pair of offspring. After one year, how many pairs will there be? 15

17 Month Pairs of Rabbits 1 1 The original pair (A) 2 1 The original pair (A) 3 2 After two months the original pair produce a pair of offspring (B1) 4 3 The first pair of rabbits produce a second pair (B2) 5 5 The first pair of rabbits produce a third pair (B3). The first pair of offspring (B1) produce a pair of offspring (C1) 6 8 And so on And so on.... If you look closely at the pattern that is developing in the column of the Pairs of Rabbits, you will see that each succeeding number in the series is the sum of the previous two numbers. When you see a pattern you can make a prediction, and that is the essence of the problem solving strategy: see the pattern, make a prediction. Using this pattern you should be able to predict how many pairs there will be after 12 months (144) and after 13 months (233) by continuing the table and the pattern. Month Pairs of Rabbits The solution produces a series of numbers known as the Fibonacci numbers. The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. 16

18 9.5 Practice Your Skills 1. In their biology class, Alice and Michael learned how to count a population of yeast cells. Using a special counting microscope, they counted the cells every hour and entered their data in a table. Time Yeast Cells 9: : : :00 75 Their teacher told them that the population would stop growing and remain stable at about 500 cells. At what time would Alice and Michael discover that the population had stopped growing? 17

19 9.6 Online Puzzles You can find many online puzzles and games. Try out some of these online games and puzzles on the following sites. National Library of Virtual Manipulatives (NLVM) Try: Circle Zero (Number and Operation Gr. 9 12) Towers Of Hanoi (Algebra Gr. 9 12) Peg Puzzle (Algebra Gr. 9 12) Coin Problem (Algebra Gr. 9 12) Illuminations NCTM Try: Brainteasers and Calculation Nation 18

20 9.7 The Handcuffs Puzzle The Handcuffs Puzzle This is a classic puzzle that has been around for at least 250 years. It is very challenging, but it does give students a chance to get students up and moving. It depends on lateral thinking and topology (rubber geometry or the mathematics of distortion). It can also make a good party game (for adults too). The Set-Up For this puzzle you need two people, some rope and some empty space to do the puzzle in. Each person will need a piece of rope with a loop tied in both ends, so it can be worn as handcuffs. The rope should be reasonably long, so that the person wearing it can easily step over it if they want. Each person puts on a complete set of handcuffs. Before putting them on, they loop their handcuffs around each other so they are tied together. Each person should wear a complete set of handcuffs. They then have to get themselves apart while following these rules: The Handcuff Puzzle Rules The handcuffs cannot be removed. Do not break, cut, saw through, bite through or in any other way damage the rope. Damaging each other is probably a bad idea too. If you are doing this puzzle with a class, make certain you tell them they need to be able to show you their solution. Otherwise it won't be so enjoyable. 19

21 The Solution Imagine you are one of the people doing the puzzle. Start by moving the other persons rope along yours until it is lying on your arm. Make sure that the other person's rope is not wrapped around your rope, it should only be touching your arm. You can see now that if you could pull the rope through your arm, the puzzle would be solved. However, this would be very messy and not have a lot of repeat potential. Instead, you need to move it around your arm. Reach in through your handcuff with a thumb and finger; and grab the other person's rope. Now pull it through your handcuff and over your hand. It should now be on the other side of your arm. Now let it go back through the handcuff and you are separated. Tips If you are showing this one to a class, some students will say that the puzzle is easy and they know how to do it. Don't worry, they don't. The first thing they will do is either step over the other person's rope or duck under it. This will not work. People very rarely solve this one without assistance. Some hints that may help is to get them to stand within arm s reach of each other, then tell them that they do not need to move their feet. Alternatively, explain to them that it is impossible to unlink two linked rings, but they are not two linked rings, there is a gap at each wrist. If you are working with a single pair, you can lead them to the solution using a rubber band. First have one student take off a handcuff, put the rubber band on their arm and get them to put the handcuff back on. Ask them to take the rubber band off their wrist without getting it caught on the rope. They will take it under the handcuff. Next put the rubber band back on their arm but this time hold onto one point on the band. They will take it under their handcuff again, but this time they will have to stretch it over their hand and let it go back again. Now use their partner's rope in the place of the rubber band. 20

22 Further Information If there was a gap in one handcuff or in your arm, the puzzle is easy to solve. You just need to move the other person's rope through the gap. To a topologist, this is no different to doing it when your wrist is in the loop. There is a gap between your rope and you; and you move the other persons rope through that gap. The only practical difference is that with the rope on your wrist there is a hand-shaped obstacle you need to move the rope around. Further Activities A single person version is to tie a knot in the rope, then put the handcuff on. They need to untie it, following the same rules. Another single person version is tying someone to a tall pole or something similar. The knot for this one is important. You need to take a loop of rope and wrap it around the pole. Then stick one handcuff through the loop before putting the handcuffs on. (See image) Tied to a Pole You can also use more than two people. 21

23 9.8 Additional Activities Puzzle It Out Stations Organized by Teacher Task: In groups of 3 or 4, provide students with a variety of puzzles that require the different strategies. Have each group work on one puzzle that is different from each of the other groups and ask that they find their solution using two different strategies. When each group member is confident with finding the solution and with the two strategies used, have the students form a jigsaw. One member from each group joins a new group. In the newly formed groups, each member will present their puzzle, provide time for the other members of the group to solve the puzzle, assisting when necessary and then providing the solution along with strategies as well. Activity: Provide students with puzzles that have incorrect solutions, so they can analyze them for errors. Games: Create stations with different games such as Tetris, Rubik s cube, Blokus, chess, checkers, Backgammon, Mastermind, Tic-Tac-Toe, Connect Four or Five, Battleship, Cathedral World, and Mancala. Provide students with YouTube videos if they are uncertain of how to play. Ask them to play and describe strategies of how they win each game. Award prizes for the student(s) that win the most games. 22

24 What s the Name of the Game? Introduction: When you think of your favorite game, what comes to mind? If may be a computer game or video game. You may also enjoy playing games that have been around a lot longer. These may include the use of a game board and may involve cards, dice, or specially designed playing pieces. Examples of these games include chess, checkers, dominoes, euchre, bridge, Monopoly, and Scrabble (MathLinks 9, p. 42). Discuss the Ideas 1. What kind of games do you play? (e.g. computer, video, board, card) 2. What are your favorite games? Why? 3. What makes a great game? 23

25 9.8 Practice Your Skills Game Design Activity Task: Individually or in pairs, create your own game. You must provide a materials list, provide the materials, and make the game. You must be able to describe the rules of the game, including how the winner is decided, provide written instructions, and provide a demonstration to their classmates. Extension: Have your teacher pair you up with a younger grade. You will then teach the younger students how to play your game. Observe the younger students playing, noting any strategies used in a math journal. Discuss these strategies with your classmates afterwards. 24

26 Student Evaluation Insufficient Evidence (IE) Student has not demonstrated the criteria below. Developing (D) Growing (G) Proficient (P) Exceptional (E) Student has rarely demonstrated the criteria below. Student has inconsistently demonstrated the criteria below. Student has consistently demonstrated the criteria below. Student has consistently demonstrated the criteria below. In addition they have shown their understanding in novel situations or at a higher level of thinking than what is expected by the criteria. Proficient Level Criteria IE D G P E M11.1 Extend understanding of arithmetic operations to rational numbers to solve problems within the home, money, recreation, and travel themes. b. I can use whole numbers, integers, fractions, decimals, and percent to solve puzzles and problems. f. I can determine if my answer is reasonable. M11.2 [WA 10.2] Demonstrate understanding of reasoning by analyzing puzzles and games. a. I can pick a strategy and explain how to use it to solve a puzzle or win a game such as: guess and check, look for a pattern, or make a systematic list. b. I can analyze puzzles or games for patterns, describe the properties of a given pattern, and identify if a set of objects fits the pattern or not and explain why. c. I can locate and correct errors in a solution to a puzzle or in a strategy for winning a game and explain the reasoning. d. I can create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning the game. 25

27 Learning Log Date Starting Point Ending Point 26

28 Answers 9.1 Analyze and Strategize Problem 802 pages 9.2A Practice Your Skills 1. 6 days Sample solution: 27

29 9.3 Practice Your Skills 1. 3 adults and 19 children Sample Solution: 28

30 9.4 Practice Your Skills 1. 8 possibilities Sample Solution: 9.5 Practice Your Skills The pattern Alice and Michale see a pattern in their data. The population doubles (approximately) every hour. The prediction At 13:00 there will be (approximately)150 cells; at 14:00 there will be (approximately) 300 cell. At 15:00, if the teacher is correct, there will be (approximately) 500 cells. They will know the population has stopped growing at 16:00 if there are still (approximately) 500 cells. 29