MATHEMATICAL MODELING IN PROBLEM SOLVING Problem solving strategies

MATHEMATICAL MODELING IN PROBLEM SOLVING Problem solving strategies

Mathematical Modeling q Mathematical modeling is generally understood as the process of applying mathematics to a real world problem with a view of understanding the latter. q A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (such as computer science, artificial intelligence), as well as in the social sciences (such as economics, psychology, sociology, political science).

Mathematical Modeling Process

Mathematical Modeling Strategies 1. Draw a Diagram, Picture or Model 2. Make a List 3. Make a Table 4. Write a Number Sentence 5. Working Backwards

STRATEGY 1: DRAW A DIAGRAM, PICTURE/ MODEL The draw a picture strategy is a problem-solving technique in which students make a visual representation of the problem. Pictures and diagrams are also good ways of describing solutions to problems

Example 1: Draw a Diagram, Picture/ Model The Boy Scouts were planning a breakfast in the school gym. There were 6 round tables and 4 sq uare tables. 6 people can sit at each round table and 4 people can sit at each square table. How many people can sit at all the tables?

Example 1: Draw a Diagram, Picture/ Model Solution: 52 people can sit at all the tables.

STRATEGY 2: MAKE A LIST q For problems that have multiple solutions, the best way to solve them is to write down all the combinations or possibilities in an organized list. That way, you can clearly see the answer and be sure not to forget any parts. q Making an organized list helps problem solvers organize their thinking about a problem. Recording work in an organized list makes it easy to review what has been done and to identify important steps that must yet be completed.

Example 2: Make a list Mr. Thomas is making three layered candles. He is using red, green, and blue wax. How many different ways can he arrange three colors in the candles?

Example 2: Make a list There are two ways to arrange colors when red is first. Red, green, blue Red, blue, green There are two ways to arrange colors when green is first. Green, blue, red Green, red blue There are two ways to arrange colors when Blue is first. Blue, green, red Blue, red, green

Example 2: Make a list Solution: There are 6 different ways to arrange the colors in the candles.

STRATEGY 3: MAKE A TABLE q Make a Table is a problem-solving strategy that students can use to solve mathematical word problems by writing the information in a more organized format. q This problem-solving strategy allows students to discover relationships and patterns among data. It encourages students to organize information in a logical way and to look critically at the data to find patterns and develop a solution.

Example 3: Make a table There are six swimmers on each relay-race team. The first team member swims 300 meters. Each team member swims 50 meters less than the swimmer before. How many meters did the last team member swim in the relay race?

Example 3: Make a table SWIMMERS 1 2 3 4 5 6 DISTANCE 300 meters 250 meters 200 meters 150 meters 100 meters 50 meters Solution: The last swimmer swam 50 meters.

STRATEGY 4: WRITE A NUMBER SEQUENCE q. Number Sequence Problems are word problems that involves a number sequence. Sometimes you may be asked to obtain the value of a particular term of the sequence or you may be asked to determine the pattern of a sequence.

STRATEGY 4: WRITE A NUMBER SEQUENCE q. You can Write A Number Sentence to solve most problems. Use the Write a Number Strategy when ØThere is only one possible answer. ØYou can add, subtract, multiply, or divide to solve the problem. ØYou can use a formula to solve the problem.

Example 4: Write a number sequence The first term in a sequence of number is 2. Each even-numbered term is 3 more than the previous term and each odd-numbered term, excluding the first, is – 1 times the previous term. What is the 45 th term of the sequence?

Example 4: Write a number sequence Step 1: Write down the terms until you notice a repetition 2, 5, - 2, 2, 5, -2, . . . The sequence repeats after the fourth term. Step 2: To find the 45 th term, get the remainder for 45 ÷ 4, which is 1

Example 4: Write a number sequence Step 3: The 45 th term is the same as the 1 st term, which is 2 Solution: The 45 th term is 2.

STRATEGY 5: WORKING BACKWARDS q. It is a particularly useful problem-solving strategy when you can clearly define the goal or end state of the problem, and you know a sequence of operations that were used in the problem.

Example 5: Working backwards John took a drive to town at an average rate of 40 mph. In the evening, he drove back at 30 mph. If he spent a total of 7 hours traveling, what is the distance traveled by John?

Example 5: Working backwards Step 1: Set up a rtd (rate, time, distance) table. r Case 1 Case 2 t d

Example 5: Working backwards Step 2: Fill in the table with information given in the question. John took a drive to town at an average rate of 40 mph. In the evening, he drove back at 30 mph. If he spent a total of 7 hours traveling, what is the distance traveled by John? Let t = time to travel to town. 7 – t = time to return from town.

Example 5: Working backwards r t Case 1 40 t Case 2 30 7 -t d

Example 5: Working backwards r t d Case 1 40 t 40 t Case 2 30 7 -t 30(7 –t)

Example 5: Working backwards •

Example 5: Working backwards Step 5: The distance traveled by John to town is 40 t = 120 The distance traveled by John to go back is also 120. So, the total distance traveled by John is 240. Solution: The distance traveled by John is 240 miles.