Tiểu luận - Critical Reading - đề tài - Problem Solving ( Đọc Phân Tích - đề tài - Giải Quyết Vấn Đề

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 8

1 Nguyễn Quỳnh Anh

2 Nguyễn Thị Lan Anh

Problem ?

 Achieve the task may need considerable thought and planning

 Systematic or involve trial and error

 Three clearly defined :

1 Identifying which pieces of data are relevant when faced with a

mass of data, most of which is irrelevant

2 Combining pieces of information that may not appear to be related

to give new information

3 Relating one set of information to another in a different form – this involves using experience: relating new problems to ones we have

previously solved

Problem

Phương Thảo

Function:

How do we solve problems?

Problem information question

Quỳnh Anh

Data Proces s Solutio n

How do we solve problems?

Quỳnh Anh

Ways to solve

‘heuristic’

How do we solve problems?

Quỳnh Anh

How do we solve problems?

Frist involves thinking carefully about when you last came into the house and what you did

Second involves searching every room of the house thoroughly until they are found

Quỳnh Anh

Selecting and using information

• Problem solving involves understanding and making use of

information

• The problem to solve is to select the correct pieces of

information and to use them in an appropriate manner

• Information can come in 5 main forms

Nguyễn Phương Thảo

5 forms that sets of information can take:

Tables: these could include summaries of surveys, specification sheets

Diagrammatic: diagrams come in a wide range of forms: flow charts, maps, schedules, decision trees and many other types can summarise numerical and spatial information.

Selecting and using

information

Nguyễn Phương Thảo

The pie charts illustrate the change that the introduction of the

CD in 1985 had on the recorded-music market Total annual

sales of all types of recording in 1984 were 170 million and in

1994 they were 234 million

Selecting and using

information

Nguyễn Phương Thảo

* What, approximately, happened to the actual annual sales of vinyl singles between 1984 and 1994?

• A They fell by 14 million

• B They fell by 5 million

• C They were unchanged

• D They rose by 17 million

• E They rose by 64 million

Selecting and using

information

Nguyễn Phương Thảo

In 1984, vinyl single sales were 44% of 170 million, or 74.8

million; in 1994, they were 26% of 234 million, or 60.8 million

=> So A is correct – they fell by 14 million

Selecting and using

information

Nguyễn Phương Thảo

Selecting and using

information

Summary

• In this part, we have seen how data can be presented in

several different forms

• We have also seen the importance of reading the

question carefully to ensure that the correct pieces of data are extracted from the information given and used correctly

Nguyễn Phương Thảo

1.Finding methods of solution:

• The way of proceeding to an answer in some problems may not be clear:

- either because it is necessary to find an intermediate solution first,

- or because we need to work simultaneously forward from the data and backwards from the required answer

Processing Data

Huyền

• One strategy that can help to solve problems when you are not clear how to proceed is to analyze the problem:

- organize the information

- write down or underline those pieces of information which are important

- simplify

- look at the question and decide what pieces of information

could lead to the answer

- make a sketch, list or table

Processing Data

Huyền

Processing Data

Answer

required Information required

Intermediate result

Solution Data

Huyền

2.Solving problems by searching:

Three main types of solving problems by searching:

- Exhaustive search

- Directed search

- Selective search

=>> With all these searches, the important thing is to be

systematic in carrying out the search so that no possibilities are missed and the method leads to the goal

Processing Data

Huyền

Recognising pattern

• Introduction

You will learn how to develop and use the methods make a table and look for a pattern

• Read and Understand Given Problem Situations

The most difficult parts of problem-solving are most often the first two steps in our problem-solving plan You need to read the problem and make sure you understand what you are being asked Once you understand the problem, you can

devise a strategy to solve it

• Develop and Use the Strategy: Make a Table

The method “Make a Table” is helpful when

solving problems involving numerical relationships When data is organized in a table, it is easier to

recognize patterns and relationships between

numbers Let’s apply this strategy to the following example.

Recognising pattern

• Example 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?

• Solution

• Understand

• 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.

• A good strategy is to list the data we have been given in a table and use the information we have been given to find new

information

• 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

• You 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

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 3

• You arrange tennis balls in triangular shapes as shown How

many balls will there be in a triangle that has 8 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 8 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

• 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

• 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

 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

• 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

Hypotheses

Lan Anh

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 5 minutes early On all the other days he finds that he gets there 5 minutes late Which one of the following could explain the

times that Nikul arrives at the gym?

A The buses leave at 5 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.

Hypotheses

Lan Anh

* The correct answer is D It is also illustrative to see why the wrong answers do not work:

A Buses at 5 and 35 past the hour would always get Nikul to work on a quarter hour which could not be 5 minutes early or late.

B Buses at 15 and 45 past the hour would mean that Nikul was always 5 minutes

early.

C Buses at 25 and 55 past the hour would mean that Nikul was always 5 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 5 minutes early nor 5 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

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 4 more half-crowns than guineas

B There are 5 more pennies than guineas

C There are 3 more pennies than half-crowns

D There is one fewer penny than guineas and half-crowns together.

Necessity and sufficiency

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.

Necessity and sufficiency

Thank You!