MATHEMATICS
The Mathematics component of the General Education portion of the Licensure Examination for Teachers include Fundamentals of Math, Plane Geometry, Elementary Algebra, Statistics and Probability
DEVELOPING PROBLEM SOLVING SKILLS
1. Understand the problem
2. Plan what to do
3. Do it
4. Look back
In planning what to do, there are a number of strategies that
can be selected
from.
Following is Partial list.
1. Draw a diag ram 6. Look
for a pattern
2. Guess and check 7. Solve
a Simpler problem
3. Work Backwards 8. Apply
logical reasoning
4. Organize a list 9. Simulation or experimentation
5. Make a table 10. Write and solve equations (s)
While the above strategies have bearing on mathematics, they
are not particular to mathematics alone. One may not need to solve any mathematical
problems in his adult life, but he may apply the strategies and have generality
and power.
Conversational mathematics, being concerned more with content and algorithms, focuses on the last two strategies. Unfortunately, conventional mathematics does not give adequate attention to the other problem-solving strategies, and therefore fails to five a realistic flavor to the problem-solving process. Experience with a variety of problem-solving strategies promotes the improvement of many of the processes involed in mathematical/scientific thinking.
Experience in looking for patterns helps students develop their ability to make generalizations on the basis of examples. Experience in drawing diagrams improves the student’s ability to visualize ability to formulate hypotheses and conjectures. Many of the strategies enable students to recognize and remedy errors. There are no set rules or algorithms for solving problems that involve these processes, therefore experience in problem solving develops the student’s capability to cope with situations about which they have not been taught.
This collection of mathematical problems has been obtained from many sources, has been classified for the benefit of the teacher, according to different strategies in the list given previously. This collection should not give the impression that there is only one way to solve a problem. A problem with sucj cataloging in mind will develop in the teacher/supervisor some confidence in the use of each strategy and some appreciation of its power.
Ultimately, focusing on these problem-solving strategies may
convince you that teaching mathematics is not just imparting knowledge and
skills but also inculcalaing way of looking at things, a desirable attitude of
mind.
Problem
A snail is at the bottom of a 10-meter well. Each day it
crawls up 3 meters, but at night it slips down 2 meters. How many days will it
take the snail to get out of the well?
Understanding the problem
1. What is the snail doing?
2. How deep is the well? How does the depth compare to the height in this room?
Planning the solution
The drawing if the well is divided into 10 equal parts. Each part representing a meter. The drawing shows that in 2 days the snail reaches a height of 4 meter.
1. How far up the well is the snail in 3 days? 4 days? 5days?
Guess and Check
Often when no direct procedure for a solution comes to our minds the guess-and- check strategy keeps us from losing hope – the strategy is always applicable. This strategy involves guessing, checking, guessing again, checking again repeating until a reasonable answer is arrived. The first guess may be a random one, but successive guesses should become better and better, based on results from the precious guesses. It is in these successive “educated” guesses where careful thinking becomes a necessary ingredient.
Problem
Ticket were sold for a school activity. Adult tickets cost P12
while student tickets cost P8. Ana sold 14 tickets for P132. How many tickets for
each type did she sell?
Understanding the Problem
1.
What did Ana do?
2. Could all the tickets Ana sold have been adult ticket? All student’s tickets?
Planning the solution
We try 10 adult tickets.
This leaves 3 students ticket
(i.e, 14-10). 10 adult tickets cost P120 (i.e, 8x4)
This gives a total cost of
P152, more than Anas collection.
1.
We make a second guess. Shall we try more than
10 adult tickets or less than 10?
2.
Make a second guess. Check your guess.
3. If you still didmt get the right answer you third guess should be better. Continue guessing, but don’t forget to check each time.
The guessing strategy may suggest an algebraic solution as shown below:
X adult ticket.
This leaves (14-x) student
tickets. X adult tickets cost 12x pesos.
(14-x) student ticket cost
8(14-x)pesos.
This gives a total cost of
12x+ 8 (14 – x) pesos.
Therefore 12x+8(14 – x) = 132
Make an organized list
The make an organized list strategy often facilitates a
systematic approach to certain types of problems. Emphasis is on the word
organized, as apposed to “Random”
Problem
The five disks shown are placed in a box and mixed. Three disks at a time are then drawn out. The score is the sum of the numbers drawn. How many different scores are possible?
What are the possible scores?
3 6 2 5 1
Understanding the problem
Here are pictures of few draws.
2 6 1 2 6 5 1 5 3
1. What is the score in each draw shown above?
2. The picture show 3 different draws. Are the 3 different?
3.
What are the possible scores?
Planning the solution
We can make a list of draw as follows. First we list all draws that start with 3 and
6. Then we list all draws that start with 3 and 2.
The draw “ 3,2,6” was crossed
out because it gives the same score as 3,6,2 So also, “3,2,5” was crossed out since
it gives the same score as , “3,6,1”
Draw | Score
_______________________________________
3 6 2 | 11
3 6 5 | 14
3 6 1 | 10
3 2 6 | 11
3 2 5 | 6
Draw | Score
________________________________________
3 5 …
We continue to list all the
other draws beginning with 3. In the same manner we list all the drawings
beginning with 2. Complete the table.
Make a table
The making a ble strategy
often helps to organize a data of a problem. It may be useful in relation to
other strategies such as guess-and-check or look for a pattern
Problem
Lita read every 100g of sea
water contains 3g of salt. Using this information, how much sea water would be
needed to obtain 10g of salt?
Understanding the problem
1. How much a salt can be obtained from 200g sea water?
2. Is the relation 100g sea water to 3g salt the same as:
a) 50g sea water to 2g salt
b) 50g sea water to 1g salt
c) 30g sea water to 1g salt
d) 300g sea water to 9g salt
Planning the solution
Let us make a table
Sea water | 100 | 200 | 300 |
400 | 500
_____________________________________
Salt | 3 | 6 | ? | ? | ?
Then use the information to arrive at an answer.
Look for a pattern
Look for patterns helps students develop their inductive
reasoning ability –i.e
Making tentative generalizations
on the basis of examples. As a start, the make a list or make a table
strategies may be used to bring up some examples; the list is not exhaustive.
Problem
A rich neighbor gave jerry a choice of P600 for a 16 day
painting job, or 1 centavo the first day, twice as much the second day, and so
on, doubling the amount each day to the 16th day. Which arrangement should
jerry choose?
Understanding the problem
1.
According to the double the pay plan, how much
will jerry make on the second day? The third day?
2. According to the double the pay plan, what will be jerrys total pay after two days? After 3 days?
Planning solution
We need to find out how much jerry will make in 16 days according to the double the pay plan. We can make a table of the pay Jerry gets each day.
1. Continue the table a few more days
a. Do you see a patter?
b.
How do we find out how much jerry will make on
the 16th day?
DAY |1 | 2 | 3 | 4 | 5 | 6 | ….
_________________________________
Pay | 1 | 2 | 3 | 4 | 5 | 6 |
2. We add another row to our table to record the total pay.
a. Continue the table a few more days, Do you see a pattern?
b.
How do we find the total pay for 16 days?
Day | 1 | 2 | 3 |
4 |5 |
6 |
_______________________________________
Pay | 1 | 2 | 3 |
4 | 5 |
6 |
_______________________________________
Total Pay |
1 | 3 |
7 | | |
Solve a simple problem
If you can’t solve the problem posed, try to solve a simpler related problem.
Basically, the strategy consist of breaking up a problem.
The strategy could be used together with other strategies,
e.g., draw-a-pic; Make an organized list, make a table, look for pattern.
Problem
Bener and 8 friends decided to play chess. If each of them
plays one and only with each of others, how many chess games will be played?
Understanding the problem
1. How many people are in the group?
2. How many people are involved in one chess game?
3. Will Ben play in all the games?
4. How many games will Ben play?
Planning the solution (1)
We can make the problem sumpler by reducing the number of people in the group.
What if there are only 2 people in the group? 3 people? 4 people?
Diagrams may help us. Here p1, p2, p3, Indicate the players and the loops indicate the games.
P1 P2 P1 P2 P3 P1P2P3P4
2 players 3Players 4Players
1 game 3 games ?games
1. Draw a diagram for the P1
P2 P3 P4 P5 number of games among 5 players.
2. Look at the diagrams. Counting the loops gives us the number of games. Let us arrange our information in table.
Players | 2 | 3 | 4 | 5 | 6 |
7 | 8 | 9
________________________________
Games | 1 | 3 | 6 |? | |
| | |
| | | |
__ __ __
2 3 ?
Look at the successive differences (indicated by | |).
What is the pattern to find in the successive differences?—
3. Continue the pattern to find the number of games among 9 people.
Planning the solution (2)
Another way to analyze the
problem is to focus on the number of games added each time one more person is
added to the group-indicated in the diagram below by hold loops.
P1 P2 P1 P2 P3 P1 P2 P3 P4 P1 P2 P3 P4 P5
2players 3Players 4 Players 5Layers
1 game 2 games added 3 games added 4 games added
By this thinking process, each diagram builds on the one immediately preceding, therefore the eventually of forgetting a loop is lessened
We can then rearrange the
table like this :
Players | 2 | 3 | 4 | 5 | . . . . . . .
___________________________________________________
Games | |1+2 | 1+2+3 | 1 + 2 + 3 + 4 |
Working backwards
Sometimes it is easier to start at an end result and work backward to an initial condition, taking nite if the steps on the working backwards process.
Problem
A student obtain a grade of
75% in each of 3 tests. He needs to take one more test, and he wants an average
of 80% for the four tests.
What grade should he work for in the fourth test?
Understanding the problem
1.
What is meant by “average”?
2.
Can the grade in the 4th test be lessen than 75%
equal to 75%?
3. What should be the total grade of the 4 tests?
Planning ether solution
To obtain the average we proceed as follows.
Grade in 4th test = add grades in first 3 tests minus 4
Working backwards
81x 4 then subtract grades in first 3 tests = grade in 4th test.
Logical Reasoning
Problems solved using logical reasoning are usually problems “to prove” or those requiring little computations but uses mostly reasoning in their solutions. Many puzzles can come under this strategy.
Problem
There are three boxes, one containing 2 black marbles, one containing 2 white marbles and another one white and one black marble. The boxes were labeled for their contents – BB, WW and BW – but someone has switched the labels so that every box now is incorrectly labeled. If you draw one marble from particular box without looking inside, how will you know the contents of all the boxes?
Understanding the problem
1. How can the boxes be mis-labeled?
2. From which box should you pick a marble?
Planning the solution
There are two ways in which the boxes could have veen mis-labeled.
If we select a box labeled BB, it might contain either WB or a WW box. Thus if you pick marble and it turned to be white, you will not be able to tell wither it is indeed a WB box or a WW box. The same thing is true if you select a box labeled WW.
However, you select the box labeled WB, then you know that is either a WW box or BB box. Thus, upon picking a marble and nothing its color, one can tell the correct label of the box. You can then easily correct the labels of the two remaining boxes.
Write and solve equations
Many of the usual “word problems” can be solved using this strategy. After determining what is the “unknown”, we represent it by some letters ( or variable) and then set up the equation which can be obtained from the conditions given in the problem. The final solution then depends on whether the resulting equations (s) can be solved or not.
Problem
Today, Allan spent 10 more minutes asleep in class then he spent awake. If the class period is one hour, how long was Allan asleep?
Understanding the problem
1.
What is the unknown?
2.
What are the conditions in the problem?
3. What quantities are equal?
Planning the solution
If we let x represent the time Allan was awake in class then x + 10 is the time he was asleep. The total time he was awake and asleep was the total class period. Hence,
X + x + 10 = 60
Time awake time asleep Class Period
Continue to solve the equation
to get the answer.
Simulations and experiment
Sometimes, the solution of a problem involves setting up and carrying out an experiment, gathering data and making a decision based on an analysis of the data. When the undertaking of an experiment is too unrealistic or oo costly, simulation is an appropriate and powerful problem-solving strategy
Problem
Form a square the segment shown as a side
Understanding the problem
In this problem, we have to “experiment” actually the instruction given. In how many ways can this be done?
Planning the solution
First, draw the square with one of its side at the top of the left most nails. Then translate the square one nail to the right. How many times can you translate to the right? Down?
Activity 1. Problem Solving: Learning to Use a Strategy
Problem: Juan, Mario, Gary and Ralph each read four magazines. They read a different magazine each week, always choosing from a group of four magazines. During Week 2, Juan read Travel ang Gary read Auto Guide. During Week 3, Mario read Update and Ralph read Listen. During Week 4, Gary read Listen and Ralph read Auto Guide. What week did each boy read each magazine?
Situation:
1. Each boy read _______- magazines.
Did any of them read the same
magazines more than once a week? _________
2. What are you asked to find? __________
Data:
3. During Week, Juan read ___ and Gary read __________
Note how it is shown in the table below.
4. During Week 3, which magazine did Mario read? Which magazine did Ralph read?
Record that information in the
table below:
5. During Week 4, which magazine did Gary read? Which magazine did Ralph read?
Plan:
6. What strategy is being used to solve the problem? ________
Boy Week 1 Week 2 Week 3 Week
4
________________________________________________
Duke Travel
Rene
Franklin Auto
Guide
Allan
7. During Week 4, Gary read Listen and Ralph read Auto Guide. Since
Juan read Travel during Week 2, which magazines in the table.
8. During Week 3, Mario read Update and Ralph read Listen. Since
Juan read Travel during Week 2, which magazine did he read third? So which
magazine did Gary read third?
Record the magazine in the
table.
9. Use logical reasoning to find which boy read which magazine.
Record the magazines in the table.
Answer:
10. Read across in the table to tell which magazine each boy read in each week Check:
11. Make sure that no magazine appears more than once in each column and that no magazine appears more than once in each row.