Problem-Solving Part 2 PDF

Moment of Mindfulness Heavenly Father, We come before You today, united in spirit, even though we are physically apart. Thank You for the gift of technology that allows us to continue our learning despite the challenges brought by the typhoon. We are grateful for the opportunity to gather as a...

Moment of Mindfulness Heavenly Father, We come before You today, united in spirit, even though we are physically apart. Thank You for the gift of technology that allows us to continue our learning despite the challenges brought by the typhoon. We are grateful for the opportunity to gather as a class and for Your constant presence in our lives. Lord, we lift up to You those who are affected by the storm. Protect those in its path and provide shelter, comfort, and peace to those in need. Grant strength and wisdom to the responders and caregivers who are tirelessly working to assist and protect others. We pray for our class today. Despite the challenges around us, help us to stay focused and open to learning. Grant us patience and understanding as we navigate these circumstances. May our discussions and activities be fruitful and bring us closer to our goals. As we begin this session, fill our hearts with gratitude, hope, and determination. May Your light guide our thoughts and actions throughout the day. We ask this in Your most holy and loving name. Amen. Problem-Solving: Polya’s Approach  Intended Learning Outcomes: 1. recognize and describe various heuristic strategies (e.g., working backward, guess check),; 2. explain the heuristic approach and its importance in problem-solving; and 3. demonstrate the ability to apply Polya’s four-step process (Understand the Problem, Devise a Plan, Carry Out the Plan, and Look Back) to solve mathematical and real-life problems.  What is HEURISTICS Heuristic (Greek: "find" or "discover") refers to experience-based techniques for problem solving, learning, and discovery that gives a solution which is not guaranteed to be optimal POLYA’S STEPS IN PROBLEM SOLVING 1.Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back POLYA’S STEPS IN PROBLEM SOLVING https://www.competitivekids.org/blog/problem-solving-strategi es  What is HEURISTIC APPROACH solving strategies are techniques or methods that guide problem solvers toward solutions through a practical approach, often involving educated guesses, trial and error, or simplified steps.  Strategies: HEURISTIC APPROACH 1. Guess and check. 2. Solve a similar but simpler problem. 3. Make an organized list. 4. Make a table and look for a pattern. 5. Work backwards. 6. Draw a diagram.  Strategies: HEURISTIC APPROACH 1. Guess and check. 2. Solve a similar but simpler problem. 3. Make an organized list. 4. Make a table and look for a pattern. 5. Work backwards. 6. Draw a diagram.  Strategies: Devising a plan 1. Solve a similar but simpler problem. 2. Make a table and look for a pattern. 3. Work backwards. 4. Guess and check Solve a similar but simpler problem.  STEPS: Solve a similar but simpler problem. 1. Break apart or change the problem into ones that are easier to solve; 2. Solve the simpler problem; 3. Use the answers to the simpler problem to solve the original problem.  EXAMPLE 1: How many squares are there in the picture? Problem: Step 1. Break apart or change the problem into ones that are easier to solv Square with sides of 1 unit Square with sides of 2 units Square with sides of 3 units  EXAMPLE 1: How many squares are there in the picture? Step 2. Break apart or solve the simpler problem Square with sides of 1 unit 1 2 3 no.of 1 unit squares = 9 4 5 6 7 8 9  EXAMPLE 1: How many squares are there in the picture? Step 2. Break apart or solve the simpler problem Square with sides of 2 units 1 2 3 4 no.of 2 unit squares = 4  EXAMPLE 1: How many squares are there in the picture? Step 2. Break apart or solve the simpler problem Square with sides of 3 units no.of 3 unit squares = 1 1 Step 3. Usethe answers to the simpler problem to solve the original problem. Square with sides of 1 unit 9 Square with sides of 2 units 4 Square with sides of 3 units 1 Number of square in the picture = 9 + 4 + 1 = 14  EXAMPLE 2: Probability of Rolling a Sum of 7 with Two Dice Problem: What is the probability of rolling a sum of 7 when two six-sided dice are rolled? Step 1. Break apart or change the problem into ones that are easier to solve Step 2. Solve the simpler problem solve a simpler problem: What is the probability of rolling a sum of 4? Step 2. continuation (1,3), (2,2), (3,1) *List all pairs of dice rolls that sum to 4: 6 x 6 = 36 Total number of outcomes for two dice: Probability of sum 4: Step 3. Use the answers to the simpler problem to solve the original problem. List all pairs of dice rolls that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) Total number of outcomes is still 36 Probability of Sum 7 FINAL ANSWER:  EXAMPLE 3: Solving an Exponential Equation Problem: Step 1. Break apart or change the problem into ones that are easier to solve Step 2. Solve the simpler problem Step 2. continuation Solution to Simpler Problem: Since the bases are equal, set the exponents equal: Step 3. Use the answers to the simpler problem to solve the original problem. Rewrite 81 as a power of 3: Substitute: Set the exponents equal: Solve for x: Answer: CHECK: Make a table and look for a pattern.  EXAMPLE 1 Problem: 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? Step 1. Understand the problem We know in the first week Josie jogs 10 minutes per day for six days. We know in the second week Josie jogs 12 minutes per day for six days. Each week, she increases her jogging time by 2 minutes per day and she jogs 6 days per week. We want to find her total jogging time in week six. Step 2. Devise a plan (make a table) We are told that Josie jogs 10 minutes per day for six days in the first week and 12 minutes per day for six days in the second week. We can enter this information in a table: Week Minutes Per Day Minutes Per Week 1 10 60 2 12 72  We are told that each week Josie increases her jogging time by 2 minutes per day and jogs 6 times per week. We can use this information to continue filling in the table until we get to week six. Week Minutes Per Day Minutes Per Week 1 10 60 2 12 72 3 14 84 4 16 96 5 18 108 6 20 120  Week Minutes Per Day Minutes Per Week 1 10 60 2 12 72 3 14 84 4 16 96 5 18 108 6 20 120 Apply strategy/solve To get the answer we read the entry for week six.  Week Minutes Per Day Minutes Per Week 1 10 60 2 12 72 3 14 84 4 16 96 5 18 108 6 20 120 ANSWER: Answer: In week six Josie jogs a total of 120 minutes. CHECK Josie increases her jogging time by two minutes per day. She jogs six days per week. This means that she increases her jogging time by 12 minutes per week. Josie starts at 60 minutes per week and she increases by 12 minutes per week for five weeks. That means the total jogging time is 60+12×5=120 minutes.  EXAMPLE 2: Problem: You arrange tennis balls in triangular shapes as shown. How many balls will there be in a triangle that has 8 rows? Figure 1  EXAMPLE 2: Step 1. Understand the problem We know that we arrange tennis balls in triangles as shown. We want to know how many balls there are in a triangle that has 8 rows. Step 2. Devise a plan make a table and list how many balls are in triangles of different rows. One row: It is simple to see that a triangle with one row has only one ball. Two rows: For a triangle with two rows, we add the balls from the top row to the balls from the bottom row. It is useful to make a sketch of the separate rows in the triangle. Step 2. Devise a plan Three rows: We add the balls from the top triangle to the balls from the bottom row. 6=3+3 Step 2. Devise a plan Now we can fill in the first three rows of a table. No. of Rows No. of Balls 1 1 2 3 3 6 Step 2. Devise a plan We can see a pattern. To create the next triangle, we add a new bottom row to the existing triangle. The new bottom row has the same number of balls as there are rows. (For example, a triangle with 3 rows has 3 balls in the bottom row.) To get the total number of balls for the new triangle, we add the number of balls in the old triangle to the number of balls in the new bottom row. We can complete the table by following the pattern we discovered. Number of balls = number of balls in previous triangle + number of rows in the new triangle No. of Rows No. of Balls 1 1 2 1+ 2=3 3 3 + 3 =6 4 6 + 4 =10 5 10 + 5 = 15 6 15 + 6 = 21 7 21 + 7 = 28 8 28 + 8 = 36  Answer: There are 36 balls in a triangle arrangement with 8 rows. Check: Each row of the triangle has one more ball than the previous one. In a triangle with 8 rows, row 1 has 1 ball, row 2 has 2 balls, row 3 has 3 balls, row 4 has 4 balls, row 5 has 5 balls, row 6 has 6 balls, row 7 has 7 balls, row 8 has 8 balls. When we add these we get: 1+2+3+4+5+6+7+8=36 balls Work Backwards  Example Problem: Paul has money in his piggy bank. His mom adds 500 pesos in the bank. Paul takes 200 pesos to buy an ice cream but adds 1000 pesos his grandmother gave him for his birthday. At the end of the day, Paul empty his piggy bank to count his money and finds he has a total of 2000 pesos. HOw much money did Paul have in his piggy bank at the start of the day?  Step 1. Understand the problem Step 2. Devise Plan (Work Backwards/ Reverse) Step 3. Carry Out the Plan 2000 - 1000 = 1000 pesos Money Paul has before buying the ice cream 1000 + 200 = 1200 pesos Money after buying the ice cream 1200 - 500 = 700 pesos FINAL ANSWER: Paul started 700 pesos in his piggy bank CHECK: Paul has money in the piggy bank 700 pesos His mom adds to his piggy bank for helping with the household 700 + 500 = 1200 chores He takes out money for 1200 - 200 = 1000 ice cream Added money given by 1000 + 1000= 2000 his grandma  Example 2 Problem: A number is doubled, then 4 is subtracted from it. The result is 10. What is the original number? Step 1. Understand the problem Step 2. Devise Plan (Work Backwards/ Reverse) A number is doubled, then 4 is subtracted from it. The result is 10. What is the original number? Step 3. Carry Out the Plan 10 Start from the result: 10 + 14 = 24 Work backward by undoing the subtraction Undo the doubling FINAL ANSWER: The original number is 7. CHECK: A number is doubled, then 4 is subtracted from it. The result is 10. What is the original number? The roriginal number 7 The number is 7 * 2 = 14 doubled Then 4 is 14 - 4 = 10 subtracted Guess and Check  EXAMPLE 1 Problem: I am a 2-digit number. I have a 3 in the ones place. I am greater than 45 but less than 63. What number am I? I am a 2 digit number ______ _______ There is a 3 in the ones place ______ 3 _______ The number is greater than 45 but less than 63.  EXAMPLE 1 Problem: I am a 2-digit number. I have a 3 in the ones place. I am greater than 45 but less than 63. What number am I? Tre number is greater than 45 but less than 63. Is it 43? Is it 73? Is it 63? Is it 53? This is the answer  EXAMPLE 2 Problem: Nadia takes a ribbon that is 48 inches long and cuts it in two pieces. One piece is three times as long as the other. How long is each piece? Step 1: Understand We need to find two numbers that add up to 48. One number is three times the other number.  EXAMPLE 2 Step 2. Devise a Plan: Guess and Check We guess two random numbers, one three times bigger than the other, and find the sum. If the sum is too small we guess larger numbers, and if the sum is too large we guess smaller numbers. Then, we see if any patterns develop from our guesses.  EXAMPLE 2 Step 3. Carry Out a Plan Guess: 5 and 15 5+15=20 sum is too small Guess: 6 and 18 6+18=24 sum is too small Our second guess gives us a sum that is exactly half of 48. What if we double that guess? 12 + 36 = 48 ANSWER: The pieces are 12 and 36 inches long.  EXAMPLE 2 CHECK: 12+36 =48 The pieces add up to 48 inches 36 =3(12) One piece is three times as long as the other.