Critical reading Critical reading Problem solving L/O/G/O Group 8 1 Nguyễn Quỳnh Anh 2 Nguyễn Thị Lan Anh 3 Nguyễn Thị Huyền 4 Nguyễn Thị Kiều Mai 5 Đỗ Thị Mười 6 Đỗ Thị Phương Thảo 7 Nguyễn Phương Th[.]
Critical reading Problem solving L/O/G/O Group Nguyễn Quỳnh Anh Nguyễn Thị Lan Anh Nguyễn Thị Huyền Nguyễn Thị Kiều Mai Đỗ Thị Mười Đỗ Thị Phương Thảo Nguyễn Phương Thảo Contents What you mean a “problem” How we solve problems? Selecting and using information Processing data: Recognising pattern Hypotheses, reasons, explanations and inference Necessity and sufficiency Problem ? Problem Function: ✓ Achieve the task may need considerable thought and planning ✓ Systematic or involve trial and error ✓ Three clearly defined : Identifying which pieces of data are relevant when faced with a mass of data, most of which is irrelevant Combining pieces of information that may not appear to be related to give new information Relating one set of information to another in a different form – this involves using experience: relating new problems to ones we have previously solved Phương Thảo Problem Summary: Look at what a problem is and how the word can be used in different ways See how information is used to contribute to the solution of a problem Look at how various methods of using information can lead to effective solutions Phương Thảo How we solve problems? Problem information question Quỳnh Anh How we solve problems? Data Process Solution Quỳnh Anh How we solve problems? Ways to solve ‘heuristic’ method systematic method Quỳnh Anh How we solve problems? ‘heuristic’ method systematic method Quỳnh Anh • • • • • • • Apply strategy/solve To get the answer we read the entry for week six 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 The answer checks out You can see that making a table helped us organize and clarify the information we were given, and helped guide us in the next steps of the problem We solved this problem solely by making a table; in many situations, we would combine this strategy with others to get a solution • Develop and Use the Strategy: Look for a Pattern • 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 • Example • You arrange tennis balls in triangular shapes as shown How many balls will there be in a triangle that has rows? • • • • Solution Understand 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 rows • • Strategy A good strategy is to 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 • • 3=1+2 Three rows: We add the balls from the top triangle to the balls from the bottom row • 6=3+3 • Now we can fill in the first three rows of a table Number of Rows Number of Balls 1 3 • Apply strategy/solve: • 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 Number of Rows Number of Balls 1 3 6+4=10 10+5=15 15+6=21 21+7=28 28+8=36 Answer There are 36 balls in a triangle arrangement with rows • Each row of the triangle has one more ball than the previous one In a triangle with rows, • row has ball, row has balls, row has balls, row has balls, row has balls, row has balls, row has balls, row has balls • When we add these we get: 1+2+3+4+5+6+7+8=36 balls • The answer checks out • Notice that in this example we made tables and drew diagrams to help us organize our information and find a pattern Using several methods together is a very common practice and is very useful in solving word problems Hypotheses ✓ A hypothesis is a statement, which if true, solves the problem ✓ Research hypotheses specify a possible relationship between different aspects of the problem ✓ Research hypotheses are assessed by two criteria: a Does the hypothesis state a relationship between the variables? It should also serve to narrow the problem down to specific variables and/or contexts b Is the hypothesis testable? Careful wording is important, and the terms should be definable (operationally), observable, and measureable Lan Anh Hypotheses • E.g “A fair coin tossed 100 times, will show an equal number of heads and tails.” • ➔ We have encountered examples where we are required to suggest a hypothesis or a reason for the nature of variation in data Lan Anh Hypotheses Practice: Nikul runs exercise classes at his local gym, and gets there each day by train and bus Classes start at different times each day, but always either on the hour or at half past the hour He always gets to the railway station 45 minutes before he is due to start teaching and the train journey takes 20 minutes, after which he takes a bus to the gym, which takes 10 minutes Trains leave every 20 minutes, starting on the hour Some days Nikul finds that he gets to work minutes early On all the other days he finds that he gets there minutes late Which one of the following could explain the times that Nikul arrives at the gym? A The buses leave at and 35 past each hour B The buses leave at 15 and 45 past each hour C The buses leave at 25 and 55 past each hour D The buses leave at 5, 25 and 45 past each hour E The buses leave at 15, 35 and 55 past each hour Lan Anh The correct answer is D It is also illustrative to see why the wrong answers not work: A Buses at and 35 past the hour would always get Nikul to work on a quarter hour which could not be minutes early or late B Buses at 15 and 45 past the hour would mean that Nikul was always minutes early C Buses at 25 and 55 past the hour would mean that Nikul was always minutes late E Because Nikul arrives on the train at 20 or 40 past the hour, he would be getting the 35 or 55 past the hour bus The bus at 35 past the hour would get Nikul to work at 15 or 45 past the hour which is neither minutes early nor minutes late Lan Anh Necessity and sufficiency Function: ➢ A type of problem involves identifying whether there is enough data to solve the problem ➢ Useful building block in problem solving ➢ Save effort in finding unnecessary data or in making unnecessary calculations Mười Necessity and sufficiency Activity: I have a small collection of three types of old coin The collection contains 15 coins in total There are more pennies than half-crowns and more half-crowns than guineas →Which one of the following single pieces of information would enable you to know exactly how many of each type of coin there was? A There are more half-crowns than guineas B There are more pennies than guineas C There are more pennies than half-crowns D There is one fewer penny than guineas and half-crowns together Mười Necessity and sufficiency Guineas Half-crowns Pennies 12 11 10 10 8 Key: C Mười Necessity and sufficiency Summary: ➢ Meet a new type of problem where, rather than being asked to find a solution ➢ Identify an additional piece of information which is necessary either to help us with the method of solution or to choose between different possible solutions ➢ Learned the meaning, in this context, of the words ‘necessary’ and ‘sufficient’ ➢ See various types of problem which require extra data: some needing mathematical solutions; some only requiring us to establish a logical method of solution Thank You!