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problem solving of mathematics

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mathematical question and answers. a good example for students. problem solving of polya model.

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problem solving of mathematics

  1. 1. INTRODUCING Mathematics learning involves the acquisition of knowledge and skills especially problem solving skills. In real life, problem solving becomes the focus while knowledge is only the accessory. This is because, not a single day passes without we having to solve problems. So, the need for the problem solving approach in teaching mathematics. WHAT IS A PROBLEM? A problem is a statement or a situation where there is an obstacle between us and what we want. Problems are generally classified as routine and non-routine. WHAT IS PROBLEM SOLVING? Problem solving is the ability to overcome or remove the obstacle so that we can get what we want. Problem solving is a process. It requires critical thinking, ability to make decisions, use the correct strategy to find the solution and check the result. MODEL FOR PROBLEM SOLVING The most commonly used model is that of George Polya (1973), who proposed 4 stages in problem solving, namely : 1. Understand the problem 2. Devise a strategy for solving it 3. Carry out the strategy 4. Check the result
  2. 2. QUESTION 1 In your words give the definition for : a) Routine problem b) Non-routine problem QUESTION 2 Solve the following problems using Polya’s model in 2 different strategies. A police station has 25 vehicles consisting of motorcycles and cars. The total number of tyres of both motorcycles and cars equal to 70. Find the number of motorcycles and cars the station has. QUESTION 3 A T-shirt costs RM 6.50. A pair of socks RM 0.85. Azmer bought three pairs of socks and a T-shirt. How much did he pay? Answer for question 1: a) Routine problem is definied as a problem in mathematic lesson that involves easy and simple problem solving. It present a question to be answered with out need certain strategies. It means, the routine problem can be solved by direct application of previously learned algorithms. Example: Ali eat 2 piece of cakes. 5 minutes later, he eat 1 more piece of cakes. How many piece of cakes that Ali eat? Solution: 2 piece of cakes + 1 pieces of cakes = 3 piece of cakes
  3. 3. b) Non-routine is definied as a problem in mathematic lesson that involves difficult problem solving. It means, solving the non-routine problem need us to think analytically based on the problem. It requires us to use our cognitive by using the critical and creative thinking skills. It also need a solution in which applying the skills, acquired knowledge and understanding to a new and unfamiliar situations in order to solve it. Answer for question 2 1. For this question, the first strategy that we use is the Drawing or Sketches’s strategy. Step 1: FIND OUT First, we must draw the vehicles with two tyres. Then, we must add the tyres until the number of tyres equal to 70. After that, we can see how much motorcycles and cars. Step 2: CHOOSE A STRATEGY How should we approach this problem? We can make skatches. Step 3: SOLVE IT Before we add 2 more tyres to make the number of tyres become 70: OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO
  4. 4. After we add 2 more tyres to make the number of tyres become 70: OOOO OOOO OOOO OOOO OOOO OOOO OOOO OOOO OOOO OOOO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO Sign: OOOO – Car OO - Motorcycle From this sketches, we can see how much number of motorcycles and cars at the police station. There are 15 motorcycles and 10 cars in the police station. Step 4: LOOK BACK Did we answer the correct question, and does our answer seem reasonable? Yes. (If you want to know our answer is correct or not, you must count the number of vehicle’s tyres). 2. The second strategy that we use is Make a Chart’s strategy. Step 1: FIND OUT
  5. 5. What is the question we have to answer? How many motorcycles and cars in the police station. How many vehicles in the police station? 25 vehicles. How many number of vehicle’s tyres in police station? 70 tyres. How many tyres that motorcycles have? 2 tyres. How many tyres that cars have? 4 tyres. Step 2: CHOOSE A STRATEGY What strategy will help here? We could model this on paper, but accuracy would suffer. We could also use equations. But, let’s make a table. Step 3: SOLVE IT Firstly, we make a table with 5 rows and 3 columns. Then, we choose our target. For example, in the police station have 6 cars and 19 motorcycles. So we can see the total of vehicles in the police station is 62 vehicles. Then, we try and error with the same ratio until we get the answer which are 15 motorcycles and 10 cars: CARS ( 4 TYRES) MOTORCYCLES (2 TYRES) TOTAL OF VEHICLES (70 TYRES) 6 19 62 7 18 64 8 17 66 9 16 68 10 15 70 Calculation: 1 car have 4 tyres. 1 motorcycles have 2 tyres. So, if there has 6 cars and 19 motorcycles...
  6. 6. 6 x 4 = 24 and 19 x 2 = 38 And the total of vehicles are 62 (24+38). But, the answer is wrong. So, we try and error with the same ratio until we get the correct answer which are 10 cars and 15 motocycles: 10 x 4 = 40 and 15 x 2 = 30. The total of vehicles are 70 (40 + 30). The answer is correct. Step 4: LOOK BACK Did we answer the question asked? Yes. Does our answer seem reasonable? Yes. Answer for question 3: Step 1 Understanding the problems - Given : The price of a T-shirt = RM 6.50 - The price of pair of socks = RM 0.85 - Find the total costs of three pairs of socks and a T-shirt. Step 2 Devising a plan - Find the price of three pairs of socks by using addition and then add with a T-shirt. Step 3 Carry out the plan First strategy
  7. 7. Price of three of socks RM 0.85 + RM 0.85 + RM 0.85 = RM 2.55 Add with the price of a T-shirt = RM 2.55 + RM 6.50 = RM 9.05 Second strategy Find the price of three pairs of socks by using multiplication and add with the price of the T- shirt. Price of three of socks - RM 0.85 x 3 = RM 2.55 - Add with the price of a T-shirt = RM 2.55 + RM 6.50 = RM 9.05 Step 4 Looking back Check: By using an addition, check the answer. Three socks: RM 0.85 + RM 0.85 + RM 0.85 + RM 6.50 = RM 9.05 Suggested strategy The most efficient strategy to get the answer of this question is the second strategy. The second strategy is using multiplication strategy then adds with the other price.
  8. 8. A T-shirt costs RM 6.50. A pair of socks is cost RM 0.85. Azmer bought three pairs of socks and a T-shirt. How much did he pay? - RM 0.85 x 3 = RM 2.55 - Add with the price of a T-shirt = RM 2.55 + RM 6.50 = RM 9.05 This strategy is efficient because we can find the answer quickly. The step is also easier and no need to use many step. We just use to times the price of 3 pair of socks and then add with the T-shirt price.

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