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Problem Solving Techniques FIRST HALF

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1. Find a Pattern If a pattern can be established, it becomes fairly simple to predict what comes next. Once a pattern is checked for accuracy, it can be used at any stage of the problem.

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Example to work through: ant, butterfly, caterpillar, _____, _____, ____ 1, 3, 5, 7, 9, _____, _____, _____ 1000, 520, 280, 160, ____, ____, ____ Step 1: Look at the information you have been given. Step 2: Try to see if the information is alike. Step 3: Try to see if the information is different. Step 4: Using the information, determine if there is a pattern and if it can be continued.

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Example to work through: ant, butterfly, caterpillar, _____, _____, ____ (all in Animal Kingdom; alphabetical order) 1, 3, 5, 7, 9, ___, ___, ____ (add 2 to all to get next number) 1000, 520, 280, 160, ____, ____, ____ (divide by 2 and add 20 to each)

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2. Make a Table or Chart Set out the information given in an orderly chart, graph, or table. Fill in the missing information.

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Example to work through: In training for the Pie Eating Contest, Bill eats five pies the first day, ten the next, and so on, adding five more each day as he becomes better trained. How many days would Bill train until he used up all his 330 practice pies? Step 1: Read the problem thoroughly. Step 2: Organize the information given into a grid to show up any patterns. Step 3: If possible, fill in the blanks.

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In training for the Pie Eating Contest, Bill eats five pies the first day, ten the next, and so on, adding five more each day as he becomes better trained. How many days would Bill train until he used up all his 330 practice pies? Day 1234567891011 Starting point 05101520253035404550 Increase05555555555 Total Daily510152025303540455055 Total Pies515305075105140180225275330

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3. Work Backwards Problems that have a range of events that have occurred can be solved using this technique. Start at the end point and work backwards to find the beginning point.

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Example to work through: At the ice cream factory, the quantities of each of the six flavors are decided mathematically. The total amount of chocolate, mint, and butterscotch is always half of the days production. They produce fifteen more tubs of chocolate than mint and fifteen more tubs of mint than butterscotch. Of the remaining ice cream, half is honey, with the rest being split 1/3 : 2/3 between caramel and bubblegum. If there are fifteen tubs of caramel ice cream, how many tubs of each of the other flavors are made?

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Given Caramel = 15 (represents 1/3) (together represent ½ of remaining) Bubblegum = 30 (represents 2/3) Honey = 45 (represents the other ½ of the remaining) 90 (represents total of this half of the ice cream) 90 (represents total of the other half of the ice cream) Butterscotch = 15 Mint = 30 (15 + 15) Chocolate = 45 (30 + 15) 90

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4. Guess and Check (2 variables) Guess a possible answer, check back to see if it makes sense, and change the possible answer according to the results of the check.

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Example to work through: Billy has 56 marbles in his collection. If he had 14 more cats eyes than bullets, how many of each marble did he have? Step 1: Decide what you need to find out. Step 2: Read the question for clues. Step 3: Make reasonable guesses and check for accuracy.

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Billy has 56 marbles in his collection. If he had 14 more cats eyes than bullets, how many of each marble did he have? Step 1: How many cats eyes does Billy have? Step 2: 14 more cats eyes than bullets Step 3: half of 56 = 28 guess 1 30 cats eyes – 14 = 16 30 + 16 = 46 (off by 10) guess 2 35 cats eyes – 14 =21 35 + 21 = 56

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Problem Solving Techniques SECOND HALF

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Guess and Check (3 variables) Guess a possible answer, check back to see if it makes sense, and change the possible answer according to the results of the check. More variables means more information needs to be checked.

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Example to work through: At the zoo there are 90 animals, including birds, reptiles, and marsupials. If there are eight more reptiles than marsupials, but two more marsupials than birds, how many of each are there?

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ReptilesMarsupialsBirdsTotal 3022 (8 less)20 (2 less) = 72 3527 (8 less)25 (2 less) = 87 3628 (8 less)26 (2 less) = 90 At the zoo there are 90 animals, including birds, reptiles, and marsupials. If there are eight more reptiles than marsupials, but two more marsupials than birds, how many of each are there?

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Draw a Diagram or a Picture There are two drawing techniques which can be useful: a) scaling where the information is converted into a scale diagram, b) sketching where the information is arranged visually to produce a solution.

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Example to work through: Scaling If this is 3 miles l-----l how long is this? l-----l-----l-----l Answer: 9 miles

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Example to work through: Sketching A plot of land has been divided into 16 squares. The land is to be equally divided among 4 children so that each children get a piece of land the same shape and size. Can you find and color five different ways to equally divide the land?

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One way

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A second way

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A third way

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A fourth way

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A fifth way

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Make a List This technique is used when there is much information present. The information needs to be set out to show the possibilities for solutions.

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Example to work through: When five friends meet, they each shake hands with each other. How many handshakes are exchanged? 1 with 22 with 33 with 44 with 5 1 with 32 with 43 with 5 1 with 42 with 5 1 with 5 10 handshakes

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Use Logical Reasoning Statements or information can be used to create the next part of the solution. Each piece of information should confirm the one before.

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Example to work through: Match the students with the sport, using the clues outlined below. Students: James, Karen, Charles, Terry, Lee Sports: Football, Basketball, Tennis, Softball, Baseball James likes to practice shooting, but kicking is not allowed in his sport. Passing and catching arent the only moves in the game that Charles plays. Lee often uses her own glove and can get out more than once each game. Karen finds it difficult to practice her sport without others. Terry has developed a great throw in his sport.

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Match the students with the sports. StudentSport James__________________ Karen__________________ Charles__________________ Terry__________________ Lee__________________ Jamesshooting, no kickingbasketball Charlespassing and catching and running and kickingfootball Leeown glove and more than one outsoftball Karendifficult to practice without otherstennis Terrygreat throwbaseball

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REVIEW Find a Pattern Make a Table or Chart Work Backwards Guess and Check (using 2 variables) Guess and Check (using 3 or more variables) Draw a Diagram or Picture Make a List Use Logical Reasoning

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