Mathematical Reasoning MTES3023 Problem Solving PDF

MATHEMATICAL REASONING MTES3023 2.0 PROBLEM SOLVING Prepared by : Muhammad Saffuan bin Jaffar  2.0 PROBLEM SOLVING 2.0 PROBLEM SOLVING 2.1 MATHEMATICAL...

MATHEMATICAL REASONING MTES3023 2.0 PROBLEM SOLVING Prepared by : Muhammad Saffuan bin Jaffar  2.0 PROBLEM SOLVING 2.0 PROBLEM SOLVING 2.1 MATHEMATICAL 2.3 PROBLEM SOLVING 2.2 POLYA MODEL PROBLEM STRATEGIES 2.3.1 Guessing 2.3.4 Drawing a 2.1.1 Routine Problem and checking diagram 2.3.2 Looking 2.3.5 Working 2.1.2 Non-routine Problem for a pattern Backward 2.3.3 Making a 2.3.6 Solve a table simpler problem 2.3.7 Use algebra  2.0 PROBLEM SOLVING 2.1 Mathematical Problem 1. Mathematical problem-solving skills are just as important as high-level thinking skills in guiding students to creatively and critically address the problems they face. 2. Gagne's Learning Theory (1970) states that students need to master and understand mathematical concepts before applying them to solve mathematical problems. 3. Problem solving is an activity requiring an individual (or group) to engage in a variety of cognitive actions, each of which requires some knowledge and skill, and some of it not routine. (Lester, 2013, p249) Lester, F. K. (2013). Thoughts About Research On Mathematical Problem- Solving Instruction. The Mathematics Enthusiast, 10(1), 245-278  2.0 PROBLEM SOLVING 2.0 PROBLEM SOLVING 2.1 MATHEMATICAL 2.3 PROBLEM SOLVING 2.2 POLYA MODEL PROBLEM STRATEGIES 2.3.1 Guessing 2.3.4 Drawing a 2.1.1 Routine Problem and checking diagram 2.3.2 Looking 2.3.5 Working 2.1.2 Non-routine Problem for a pattern Backward 2.3.3 Making a 2.3.6 Solve a table simpler problem 2.3.7 Use algebra  2.0 PROBLEM SOLVING 2.1.1 Routine Problem There are two main types of word problems used in teaching mathematics: routine problems and non-routine problems (Wong & Matore, 2020). ROUTINE PROBLEMS 1. These are problems that can be solved using methods that students are familiar with, by replicating the steps they have learned before. 2. Solving routine problems emphasizes the use of a known or prescribed set of procedures (algorithms) to solve the problem. 3. Routine word problems in mathematics are the type of questions that have a straightforward solution, where students only need to apply basic operations to solve the problem.  2.0 PROBLEM SOLVING 2.1.2 Non-routine Problem There are two main types of word problems used in teaching mathematics: routine problems and non-routine problems (Wong & Matore, 2020). NON-ROUTINE PROBLEMS 1. These are problems that require mathematical analysis and reasoning: many non-routine problems can be solved in more than one way and may have more than one solution. 2. For non-routine word problems in mathematics, students are required to think deeply and apply more than one method or step to solve the problem (Ariffin & Aziz, 2016).  2.0 PROBLEM SOLVING 2.1.1 & 2.1.2 Routine and non-routine Problem Characteristic Routine Problems Non-Routine Problems Can be solved using familiar, step-by-step Require deeper analysis and reasoning, may Solution Method procedures or algorithms have multiple solution methods More complex, may not have an obvious Complexity Relatively simple, straightforward problems solution path Focus on applying previously learned Require applying concepts in new or Application of Concepts concepts and skills unfamiliar ways Number of Solutions Usually have a single, correct solution May have multiple valid solutions Lower-level thinking skills, such as recalling Higher-order thinking skills, such as Cognitive Demand and applying analyzing, evaluating, and creating Word problems involving basic arithmetic Problems that require logical reasoning, Examples operations pattern recognition, or creative thinking  2.0 PROBLEM SOLVING 2.1.1 & 2.1.2 Routine and non-routine Problem Characteristic Routine Problems Non-Routine Problems Jill has 12 ice cream cones. She eats 5 of The school canteen sells ice cream cones for them. How many ice cream cones does Jill RM1.50 each. Jill has RM10 to spend. How Question have left? many ice cream cones can she buy, and how much change will she receive? This is a straightforward problem involving This problem requires students to think Complexity basic subtraction. through multiple steps to find the solution. They need to calculate the number of ice Apply the familiar step-by-step procedure of Application of concept cream cones Jill can buy with her RM10, and subtracting 5 from 1 then determine the change she will receive. 1. 12 cones - 5 cones = 7 cones 1. Find the number of cones Jill can buy, 2. Therefore, Jill has 7 ice cream cones RM10 ÷ RM1.50 = 6.67 cones (6 left. cones) 2. Find the change Jill will receive Solutions 6 cones × RM1.50 = RM9 3. RM10 - RM9 = RM1 4. Therefore, Jill can buy 6 ice cream and will receive RM1 in change.  2.0 PROBLEM SOLVING 2.0 PROBLEM SOLVING 2.1 MATHEMATICAL 2.3 PROBLEM SOLVING 2.2 POLYA MODEL PROBLEM STRATEGIES 2.3.1 Guessing 2.3.4 Drawing a 2.1.1 Routine Problem and checking diagram 2.3.2 Looking 2.3.5 Working 2.1.2 Non-routine Problem for a pattern Backward 2.3.3 Making a 2.3.6 Solve a table simpler problem 2.3.7 Use algebra  2.0 PROBLEM SOLVING 2.2 POLYA MODEL Polya's Model (1957) Polya's model has become a primary reference source for researchers and teachers. It was introduced by George Polya (1887-1985) in his book "How To Solve It". This model focuses on problem-solving techniques and principles of effective mathematics learning. Polya proposed a problem-solving instructional model based on heuristics. This model consists of four steps arranged in a sequence.  2.0 PROBLEM SOLVING 2.2 POLYA MODEL Polya's Model 1. Understand the problem - Identify the given information - Determine what the problem is asking - Clarify any unclear parts of the problem 2. Plan a strategy - Decide on the appropriate mathematical operations or procedures to use - Consider any related problems you have solved before - Outline the steps you will take to solve the problem 3. Carry out the plan - Implement the strategy you have chosen - Show your work step-by-step - Perform the necessary calculations or operations 4. Look back and reflect - Check your work to ensure the solution is correct - Consider alternative ways to solve the problem - Think about what you have learned and how you can apply it to similar problems  2.0 PROBLEM SOLVING 2.0 PROBLEM SOLVING 2.1 MATHEMATICAL 2.3 PROBLEM SOLVING 2.2 POLYA MODEL PROBLEM STRATEGIES 2.3.1 Guessing 2.3.4 Drawing a 2.1.1 Routine Problem and checking diagram 2.3.2 Looking 2.3.5 Working 2.1.2 Non-routine Problem for a pattern Backward 2.3.3 Making a 2.3.6 Solve a table simpler problem 2.3.7 Use algebra  2.0 PROBLEM SOLVING 2.3.1 Guessing and Check The Guess and Check problem solving strategy is a fairly easy way of solving problems. Think of it as a 3-step-approach: 1. Guess 2. Check 3. Repeat if needed While we are guessing the numbers, we’ll need to learn how to make smart guesses. Knowing how to do that helps us minimize the number of guessing, making the process more efficient. We’ll see how to do that in a while.  2.0 PROBLEM SOLVING 2.3.1 Guessing and Check (Question 1) Choose and fill in only 6 numbers into the circles below so that both mathematical statements are true 1 2 3 4 5 6 7 + = - =  2.0 PROBLEM SOLVING 2.3.1 Guessing and Check (Question 2) Jay did 20 Math questions during his math practice. He received 5 marks for every correct answer and he got 2 marks deducted for every wrong answer. If Jay earned 37 experience points in total, how many questions did he answer wrongly? True True False False Total Check Question Marks Question Marks True + False 20 20 x 5 = 100 20 - 0 = 20 0 x -2 = 0 100 Too far 10 10 x 5 = 50 20 - 10 = 10 10 x -2 = 20 50 - 20 = 30 Too low 12 12 x 5 = 60 20 - 12 = 8 8 x -2 = -16 60 - 16 = 44 Too far 11 11 x 5 = 55 20 - 11 = 9 9 x -2 = -18 55 - 18 = 37 That's right  2.0 PROBLEM SOLVING 2.3.1 Guessing and Check (Question 3) Jay did 20 Math questions during his math practice. He received 5 marks for every correct answer and he got 2 marks deducted for every wrong answer. If Jay earned 37 experience points in total, how many questions did he answer wrongly? True True False False Total Check Question Marks Question Marks True + False 20 20 x 5 = 100 20 - 0 = 20 0 x -2 = 0 100 Too far 10 10 x 5 = 50 20 - 10 = 10 10 x -2 = 20 50 - 20 = 30 Too low 12 12 x 5 = 60 20 - 12 = 8 8 x -2 = -16 60 - 16 = 44 Too far 11 11 x 5 = 55 20 - 11 = 9 9 x -2 = -18 55 - 18 = 37 That's right  2.0 PROBLEM SOLVING 2.3.2 Looking for a pattern (Question 1) 1. Looking for a Pattern: A strategy that involves observing and analyzing data or situations to identify consistent relationships or trends that can inform problem-solving 2. Looking for a pattern is another strategy that you can use to solve problems. The goal is to look for items or numbers that are repeated or a series of events that repeat. The following problem can be solved by finding a pattern.  2.0 PROBLEM SOLVING 2.3.2 Looking for a pattern (Question 1) What is the color of the 21st apple? 1 2 3 4 5 6 7 8  2.0 PROBLEM SOLVING 2.3.2 Looking for a pattern (Question 1) What is the color of the 21st apple? 1 2 3 4 5 6 7 8 Odd : Even : 21st is odd  2.0 PROBLEM SOLVING 2.3.2 Looking for a pattern (Question 2) Skip Counting Challenge: There are 12 jumps on a hopscotch course. The first jump is 1 foot, the second is 3 feet, the third is 5 feet, and so on. What distance will be covered after the 8th jump?  2.0 PROBLEM SOLVING 2.3.2 Looking for a pattern (Question 2) Skip Counting Challenge: There are 12 jumps on a hopscotch course. The first jump is 1 foot, the second is 3 feet, the third is 5 feet, and so on. What distance will be covered after the 8th jump? (Pattern: Adding 2 more feet with each jump)  2.0 PROBLEM SOLVING 2.3.3 Making a table 1. The method “Make a Table” is helpful when solving problems involving numerical relationships. 2. When data is organized in a table, it is easier to recognize patterns and relationships between numbers. https://www.ck12.org/book/ck-12-algebra-i-second-edition/section/1.8/  2.0 PROBLEM SOLVING 2.3.3 Making a table (Question 1) Josie takes up jogging. On the first week she jogs for 10 minutes per day, on the second week she jogs for 12 minutes per day. Each week, she wants to increase her jogging time by 2 minutes per day. If she jogs six days each week, what will be her total jogging time on the sixth week? https://www.ck12.org/book/ck-12-algebra-i-second-edition/section/1.8/  2.0 PROBLEM SOLVING 2.3.3 Making a table (Question 2) Ali, Bala, and Chong are learning to count coins. Each of them brought coins. - Ali has a total of RM0.80. - Bala has 2 coins. - Chong has one coin that is worth more than Bala's total amount. State the types of coins they have.  2.0 PROBLEM SOLVING 2.3.3 Making a table (Question 2) Ali, Bala, and Chong are learning to count coins. Each of them brought coins. Solution Name 50 SEN 20 SEN 10 SEN ALI 1 1 1 BALA 0 1 1 - Ali has a total of RM0.80. CHONG 1 0 0 - Bala has 2 coins. - Chong has one coin that is worth more than Bala's total amount. State the types of coins they have.  2.0 PROBLEM SOLVING 2.3.4 Drawing a diagram 1. In mathematics, diagrams are often a useful way of organising information and help us to see relationships. 2. A diagram can be a rough sketch, a number line, a tree diagram or two-way table, a Venn diagram, or any other drawing which helps us to tackle a problem.  2.0 PROBLEM SOLVING 2.3.4 Drawing a diagram (Question 1) The total age of 4 family members is 93 years. The father's age is 3 years older than the mother's. The daughter's age is 4 years older than the son's. The son's age is 6 years. Calculate the ages of the father, mother, daughter, and son.  2.0 PROBLEM SOLVING 2.3.4 Drawing a diagram (Question 1) Father 37 3 tahun 93 - (3 + 6 + 10) = 74 Mother 37 74 ÷ 2 = 37 Son 3 tahun 3 tahun Father 40 tahun Mother 37 tahun Son 6 tahun Daughter 3 tahun 3 tahun 4 tahun Daughter 10 tahun  2.0 PROBLEM SOLVING 2.3.5 Working backward 1. Working backward scenario occurs when the quantity data is insufficient to work from the beginning. 2. Working Backwards is a problem-solving strategy in which you start with the end goal and work backward to figure out the steps needed to get there. 3. In other words, instead of starting from the beginning and moving forward, you start from the end and move backward. 4. This strategy is commonly used in math problems that ask you to find a starting value or figure out what happened before a given situation.  2.0 PROBLEM SOLVING 2.3.5 Working backward (Question 1) Amira took a collection of colored tiles from a box. Ali took 13 tiles from Amira's collection. Kiko took half of the remaining tiles. There are still 11 tiles left with Amira. How many tiles did Amira have at the beginning?  2.0 PROBLEM SOLVING 2.3.5 Working backward (Question 1) Amira took a collection of colored tiles from a box. Ali took 13 tiles from Amira's collection. Kiko took half of the remaining tiles. There are still 11 tiles left with Amira. How many tiles did Amira have at the beginning? START END ? -13 ÷2 11 35 35 +13 22 ×2 11 11  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem Johann Carl Friedrich Gauss 1777 - 1855  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem (Question 1) When Gauss was in elementary school, he was punished by his strict math teacher, who asked him to Find the sum of all natural numbers from 1 to 100 Can you do that?  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem (Question 1) When Gauss was in elementary school, he was punished by his strict math teacher, who asked him to Find the sum of all natural numbers from 1 to 100 Gauss said : I took less than 10 seconds  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem (Question 1) Find the sum of all natural numbers from 1 to 100  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem (Question 1) Find the sum of all natural numbers from 1 to 100 1 + 2 + … + 99 + 100 100 + 99 + … + 2 + 1 101 101 101 101 (101 × 100) ÷ 2 = 5050  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem 1. Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. 2. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3? 3. Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)). https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematics_for_Elementary_T eachers_(Manes)/01%3A_Problem_Solving/1.03%3A_Problem_Solving_Strategies  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem (Question 2) How many numbers between 2 and 50 are divisible by 3? What is the strategy you can use to solve this problem?  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem (Question 2) How many numbers between 2 and 50 are divisible by 3? 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49  2.0 PROBLEM SOLVING 2.3.6 Solve a simpler problem (Question 2) How many numbers between 2 and 50 are divisible by 3? 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49  2.0 PROBLEM SOLVING 2.3.7 Use algebra (Question 1) The total age of 4 family members is 93 years. The father's age is 3 years older than the mother's. The daughter's age is 4 years older than the son's. The son's age is 6 years. Calculate the ages of the father, mother, daughter, and son.  2.0 PROBLEM SOLVING 2.3.7 Use algebra (Question 1) Father m + 3 (m + 3) + m + 6 + 10 = 93 Father : 37 + 3 = 40 years old Mother m 2m + 19 = 93 Mother : 37 years old 2m = 74 Son : 6 years old Son 6 m = 37 Daughter : 10 years old Daughter 6 + 4  2.0 PROBLEM SOLVING Can we use different strategy for the same question?  2.0 PROBLEM SOLVING Tutorial Question 1 What are the 3 consecutive numbers whose sum equals 63? Apply Polya’s Model to solve this question and state what is the appropriate strategy to solve this question?  2.0 PROBLEM SOLVING Tutorial Question 2 Solve this without using a calculator 232 Apply Polya’s Model to solve this question and state what is the appropriate strategy to solve this question?  2.0 PROBLEM SOLVING Tutorial Question 3 Rashman had some pens. He bought 25 pens. Then he threw away 14 defective pens. In the end, Rashman has 62 pens. How many pens did Rashman have at the beginning? Apply Polya’s Model to solve this question and state what is the appropriate strategy to solve this question?  2.0 PROBLEM SOLVING Tutorial Question 4 Sarah has hidden some apples in her room. She knows the number of apples is less than 10, but more than 3. If she shares all the apples equally with 2 of her friends, there will be no apples left over (they cannot be cut or sliced). Apply Polya’s Model to solve this question and state what is the appropriate strategy to solve this question?