Students: 1. Analyze problems by identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing, and observing patterns. 1. Students make decisions about how to approach problems. Steve, Allan, Cheryl, Carl, and Jason want to play a video game. They have 40 minutes. Only two can play the game at one time. Is there enough time for each to have at least 10 minutes? For four days the sports store sold the same number of skateboards each day. They sold a total of 68 skateboards. How many did they sell the second day? 2. Students use strategies, skills, and concepts in finding solutions. 2. Determine when and how to break a problem into simpler parts. Students: 1. Use estimation to verify the reasonableness of calculated results. In a round-robin tournament, each team plays against each of the other teams once. If there are 5 teams, how many games will there be? One of my answers was wrong on the quiz. Which answer looks unreasonable? a. 47 x9 423 b. 98 x16 1,516 c. 38 x12 3,516 d. 27 x32 874 47
2. Apply strategies and results from simpler problems to more complex problems. 3. Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models to explain mathematical reasoning. 4. Express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and support solutions with evidence, in both verbal and symbolic work. How many diagonals can you draw inside a nine-sided figure? Look at very simple figures for a pattern. Ray has to decorate his backyard for a birthday party. There are 8 trees in the yard arranged in a circle, and Ray has decided that each tree must be connected to all other 7 seven trees with a streamer. How many streamers will Ray need? Use a diagram to help solve the problem. Suppose we have two positive integers and we know that: a) the sum is odd b) exactly one number is prime c) one number is larger than 15 Can we tell if each of the following is definitely true, possibly true, definitely, false? a) both numbers are odd b) the prime number is odd c) the sum is at least 19 48
3. Students move beyond a particular problem by generalizing to other situations. 5. Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. 6. Make precise calculations and check the validity of the results from the context of the problem. Students: 1. Evaluate the reasonableness of the solution in the context of the original situation. Shaun needs $25. He is offered lawn work at $3.65 per hour for 8 hours. Is an estimate enough when: a) Shawn decides if he he ll make enough money? b) The boss figures out how much to pay Shawn? There are several different fish in Hammer s Pond. Each fish eats at least one type of algae, and no two fish eat the same combination of algae. There are four types of algae: red, blue, yellow, and green. What s the largest number of fish that could be in the pond? During the last race at the Belldish Race Track, Happy Boy finished 2 yards ahead of Diana s Pride. Sky Rocket finished 8 yards ahead of Light Bulb. Light Bulb finished 3 yards behind Happy Boy. Tootie Fruity finished 6 yards ahead of Diana s Pride. Who won the race, and how far behind the winner were each of the other horses? 49
2. Note method of deriving the solution and demonstrate conceptual understanding of the derivation by solving similar problems. 3. Develop generalizations of the results obtained and extend them to other circumstances. Use a logic grid to solve the problem. Julie, Leroy, Doris, and Bruno visited four different museum exhibits. Doris asked Bruno to come see the Peruvian Paintings with her, but Bruno decided not to visit an exhibit that started with the letter p. Julie had just broke up with her boyfriend, Yorick, and she didn t want to see an exhibit that had the letter y anywhere in it. Leroy wanted to see the Irish Tapestry, the Japanese Armor, and the Persian Pottery, but he only had time to see one of them. Who saw each exhibit? The ancient Greeks studied oblong numbers, which were made up by arranging pebbles into rectangles. For every oblong number, one side had one more row in it than the other. 50
The first oblong number is 2 (a rectangle that s 1 by 2): * * The second oblong number is 6 (a rectangle that s 2 by 3): * * * * * * The third oblong number is 12 (a rectangle that s 3 by 4) * * * * * * * * * * * * What s the biggest oblong number that is less than or equal to 100? 51
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