MATHEMATICS

INVESTIGATION

Mr. Mario M. RamonTeacher IIIJGMNHS

It is an open-ended problem which students may choose to pursue in quite different directions. It gives the students opportunity to make important mathematical explorations or discoveries that they will remember much longer than if they had been simply told to them by the teacher. It encourages students to use higher order thinking skills, which are far more important than the concept involved.

What is Mathematics Investigation?

Another important aspect of Mathematics investigation is the need to make guesses and to test them. What matters is not whether the guess is right or wrong, but the willingness and ability to deduce information from the data gathered and to try to forecast what will happen in other cases and then to learn from the extra information. The fear of being wrong needs to be overcome as it hampers exploration and understanding.

Situations to be investigated are generally open-ended but can be more restricted when a new concept is being introduced particularly in the lower section or if the teacher has a particular concept in mind that he/she wants to students to explore. And this activity becomes guided math investigation.

• It improves mathematical thinking.

• It causes better retention of concepts learned.

• It develops positive attitude towards mathematics.

• It helps students to be more creative and resourceful.

Why do we use Mathematics Investigations?

It helps students to express themselves freely.

It develops a feeling of satisfaction and fulfillment in mathematics achievement.

It develops cooperative learning.

It can be a source of enjoyment.

1. Preliminary Skirmishing – Getting Started At this stage the student needs to attain familiarity with the problem or situation. He starts from the simplest or whatever is interesting, often in an unorganized way. Different ways of interpreting the problem may be explored, one or more problems or aspects inherent in the situation may be identified and one or more productive lines of action may emerge.

What are the stages in Mathematics Investigation?

Unit fractions have a number of 1. ◦Investigate those unit fractions whose decimal equivalents recur.

The Question...

Interpretation of the question… What is being asked? How can we condense the question? What are the possible outcomes? How can we apply the new knowledge from

the investigation in the classroom context?

Preliminary Skirmishing

2. Generating Examples – Exploring systematicallyOnce the student has familiarized himself with the problem and decided a line of action to follow, he starts to generate examples and to explore. He can make use of any strategy like draw a diagram, make a model or make an organized list.

What is a recurring decimal?

A recurring decimal is one whose digits after the decimal point do not end, but repeat the same sequence forever.

They are also referred to as repeating or periodic decimals.

3. Organizing Data From the examples he generated, he collects necessary data or information and he organizes them in such a way that he can look for patterns. He can make use of a tabular presentation or an organized list of the necessary data.

Interesting discoveries…• Any denominator containing 2 or 5 give us a non-

recurring decimal. • Eg:

1/2 = 0.51/4 = 1/(2*2) = 0.251/8 = 1/ (2*2*2) = 0.1251/5 = 0.21/25 = 1/(5*5) = 0.04 and so on

Any multiple of 3 will give us a recurring decimal.

1/3 = 0.333333333…

1/6 = 0.166666666…

1/9 = 0.111111111...

Interesting Discoveries cont...

The number of digits in a repeating pattern is called the period.

Eg: 0.121212… has a period of 2.

0.74537453… has a period of 4.

4. Gestating – Taking a BreakThe student may take a break when his mind gets tired, he feels there is no progress in his investigation or he can not see a way to continue. This is the best time for him to relax, to gather other relevant information or to think of other line of actions necessary to pursue the investigation. There may be several such breaks during the investigation depending on the mode and ability of the student.

5. Making ConjecturesWhile data are already generated and organized, the student may notice some patterns or relationships. These patterns suggest generalization which appear to apply to other cases under consideration. Since the validity of such generalization which has been obtained inductively is not yet known to the student, it is in fact a conjecture. Conjectures are statements arrived at inductively or based on existing data. They may be in the form of a statement or just a number. At a later stage, they may look for a rule in symbols.

1. Any multiples of 2 and 5 are non-repeating decimal.

2. Any multiple of 3 will give us a recurring decimal.

3. Any multiple of 7 will give us a recurring decimal.

4. Any prime denominator will give us a recurring decimal.

6. Reorganizing As the investigation progresses, it may become necessary to reorganize the approach in order to make it more simplified and more systematic or more general or otherwise improved so that it is easier for the student to make conjectures. This can also be done after testing the conjecture especially if it does not hold true for some other cases.

TEST

Any power of 2 and 5 are non-repeating decimal.

JUSTIFY

7. Testing ConjecturesThis consists of checking the consistency of the conjecture against existing cases for which data are available, or predicting results of untried cases and then obtaining the relevant data. The data may support the conjecture or provide a counter-example indicating the need to revise or reject the conjecture.

Any multiples of 2 and 5 are non-repeating decimal.

Counter-examples:

BACK

8. Explaining or JustifyingOnce a conjecture has been tested against and supported by the data, students should be encourage to explain why the conjecture holds for new cases. Such an explanation should consist of more than the observation that the conjecture has held for the cases examined so far. As students become more sophisticated, more careful and complete explanations or justifications may be expected. It may also be possible for some students to provide deductive justification, or a formal proof, for a generalization.

9. Elaborating – Making ExtensionsDepending on the ability of the student, the investigation can be extended by considering other relevant situations that start with, “What if .” or “What would happen if…”

10. SummarizingAt this stage the student is encouraged to write a summary or a brief account of the investigation. This highlights the major ideas and the phases of the investigation including all conjectures, tests and proofs carried out.