text:
SA NC Doing Investigations/Chapter 3

The wisdom of the winners: hints and ideas for science and mathematics educators
"Wisdom is not about saying things that no-one has ever said before but rather about saying truths clearly that everyone else says confusedly." (Anonymous)

"Knowledge comes, but wisdom lingers." (Alfred Lord Tennyson, the English poet)

In the Mathematics and Science Teacher of the Year competitions, finalists are asked to list classroom management tips, projects they have found successful and general suggestions for educators that they have found personally useful. Many of these pieces of teaching wisdom come from finalists in the retired educator category. It is wonderful to meet older educators who are still enthusiastic about teaching and willing to sift through their years of experience to pass on good advice to their younger colleagues.

In a workshop attended by all the finalists, a brainstorming session produced still more ideas. Here are some pearls from the brainstorm and the portfolios of the 2003 Mathematics, Science and Technology Educator of the Year winners and runners up. They provide ideas, suggestions and food for thought. Bear in mind that the list is not meant to be exhaustive Â it is just a resource to spark better ideas in your own mind. If you like anything you find here don't copy it blindly. Assess it carefully and think how it could be adapted for your circumstances, your school and your classes. On the other hand, don't just dismiss it saying "that could never work for me because my learners are too poor and my school is too rural and the roof is leaking and I am too overworked ... ." Good ideas apply everywhere and anywhere if you have a will to make them work. Development for the professional educator means constantly trying new ideas and strategies and making them work. The educator who doesn't try new ideas will never develop and isn't very professional.

One can see from the suggestions and hints that follow why their authors are amongst the top science and mathematics educators in the country. Most of them, it must be said, also teach in the most disadvantaged communities in the country. The lists are categorized alphabetically for easy reference:

A. Assisting memory

B. Change and the dynamic classroom environment

C. Classroom atmosphere

D. Classroom management

E. Developing an interest in science, technology and mathematics

F. Developing a relationship with your learners

G. Enlivening your presentation style

H. Games

I. Groups work; working in groups

J. Language issues and development

K. Links and continuity in science and mathematics teaching

L. Motivating learners

M. Policies, preparation and planning

N. Professional relationships amongst colleagues

O. Projects in and around the school

P. Respect for others

A. Assisting memory
positive in first quadrant, only the S ine is positive in second, only the C osine is positive in third and only the T angent is positive in fourth) help learners to recall and link content.
 * Mnemonics (eg: " A ll S tudents T ake C offee" = In trigonometry A ll functions are

 * Use music to memorize geometry theorems (and to relieve stress!).

 * Compose rhymes and songs; reciting and singing them to memorize lists and orders.

 * Dramatize events or ideas.

B. Change and the dynamic classroom environment
with opportunities for learners to sit where they want to.
 * Assign seating in the classroom according to the nature of the activity. Balance this

key or difficult concepts. a bit confusing- which educators
 * Do block tutorial where all educators co-operate in setting tutorials for learners on

stereotyped in your approach.
 * Vary your teaching and use different techniques. Don't become dull, stale and

aware of e.g. did you know that the fuel gauge in a car uses an electric circuit with a variable resistor to detect the amount of fuel in the tank? This encourages learners to start asking themselves "how?, stimulating inquiry and hence learning in science.
 * The element of surprise: surprise your learners; give them information they were not

C. Classroom atmosphere

 * Make the classroom welcoming and enjoyable to be in.

are like movies of actors standing still. Activities stimulate learner interest; improve understanding; stimulate inquiry; and make learners more aware of their environment.
 * Every lesson should be planned with an activity in mind. Lessons without activities

 * Use photographs and other visuals and maintain a notice board.

(and boring) display.
 * Change the notice board's contents regularly so that it does not become a static

especially those that relate to problems in their community.
 * Encourage learners to bring science articles from magazines and newspapers etc.

 * Collect resource books and keep these handy in your classroom.

D. Classroom management
this theme is "don't smile before Easter"!)
 * Be friendly to learners but avoid the familiarity that breeds contempt. (A variation on

 * Know and take into consideration any problems specific learners may have.

allowed and make sure that learners stay "on task."
 * Allow learners the freedom to discuss ideas amongst themselves. Specify the time

security; have teams of learners act as "lab assistants" to keep things clean, report breakages and maintain equipment.
 * Keep equipment and apparatus in containers or cabinets in the classroom for

bottles made from 2-litre cold drink bottles.
 * Use waste material available in the community for teaching purposes e.g. volume

occasions to lapse into aimless (and noisy) debates based on untested assumptions rather than logically set out arguments based on well-founded information or knowledge.
 * Allow learners to explain their thoughts or answers to their peers. Do not allow these

E. Developing an interest in science, technology and mathematics
they can be learned and understood.
 * Have a positive attitude to science and mathematics because like any other subject,

science.
 * Stress that every learner has the potential to learn and understand mathematics and

learn from others and improve your skills and abilities.
 * Take part in science and mathematics competitions (Expos and Olympiads). You

 * Make learning mathematics and science fun.

motion of two moving bodies etc.
 * Let learners enact moving or bonding molecules, the motion of planets or the relative

 * Form a science and a mathematics club at your school.

they can coincide with major events.
 * Hold regular science awareness weeks (posters, presentations etc.) especially when

(for example as part of environmental education).
 * Hold demonstrations (with appropriate permission) to protest important local issues

 * Hold mock conferences to resolve environmental problems.

start playing around with numbers to make your own patterns. You will love it!
 * Identify patterns formed by numbers and marvel at what numbers can do. Then

 * Make discoveries with numbers.

classroom's dimensions; now calculate requirements for tiling the whole room. How many off-cuts etc.?
 * Let learners do practical science and mathematics e.g. measure a tile and the whole

F. Developing a relationship with your learners

 * Know your learners and not just the subject.

 * Try to find out what learners' own expectations are.

go / what they want to be.
 * Encourage learners to set goals and aims and have a vision for where they want to

 * Tell learners in advance what your expectations are and set them high.

class. Explain their right to free, quality, public education but also their responsibility as learners to learn. Explain other rights like privacy, clean water, shelter etc.
 * Plan a session to explain rights, duties and responsibilities of the learners in the

learning still continues in the rest of the class.
 * Individual attention to slower learners to make sure they are not left behind but

Rather ask probing questions so that learners can see for themselves where their thinking is illogical or incorrect. In other words work with learners towards a correct answer or understanding.
 * Don't say outright that an answer is wrong as it might discourage sensitive learners.

G. Enlivening your presentation style

 * Do little demonstrations throughout a lesson. (Bite-sized chunks!)

 * Use color to highlight important issues/facts and to make associations.

makes learning more meaningful.
 * Use teaching media e.g. flip charts etc. as it enhances the teaching atmosphere and

 * Using the media (magazines, newspapers, radio, TV etc.) in class wherever possible.

H. Games

 * Games play a role in learning even if the learning is not conscious.

 * Play games to illustrate principles e.g. Monopoly in EMS.

terminology.
 * Play mathematical and number games to do remedial work and assist with

I. Groups work; working in groups
as a group and not just as individuals sitting together in a group.)
 * Let class work in groups or pairs and encourage team work. (Make sure they work

 * Let groups set their own rules to make the group work effectively

these tasks. Make tags with responsibilities written on them.
 * Encourage learner flexibility by assigning tasks within a group and then rotating

 * Composition of learner groups should be varied.

share knowledge.
 * Group work (cooperative learning; whole group participation) encourages learners to

J. Language issues and development
the differences between the formal, colloquial and mathematical/scientific meanings of words and phrases e.g. "at least" which, to some learners, means "the least" or "the minimum."
 * Impact of language on the teaching of science and mathematics; always point out

 * Because of the language we use, some concepts can be misinterpreted.

a learner understands by them and to develop their ability to express themselves.
 * Write poetry and short passages on scientific or mathematical ideas to assess what

 * Do sessions of reading aloud.

understand terminology and the particular form of language usage from the onset.
 * Use the language of the subject and of the examinations so that learners can

K. Links and continuity in science and mathematics teaching
concept behind constructivist teaching.) Brainstorming can be useful.
 * Before you teach something find out what learners already know. (This is the

continuity.
 * Link work being done currently with work done previously to make links and ensure

 * Relate subject matter to real life experiences.

from learners' experience e.g. applications of gas behaviour in car tires and combustion engines etc. Science becomes meaningful because it is linked to the world beyond the classroom walls.
 * Make science a living subject, relevant to real life situations. Take relevant examples

teaching.
 * Use the school's local environment and community as much as possible in your

debaters research their topics beforehand. Great outpourings of passion based on incorrect facts and poor assumptions are counter-productive to science and technology education.)
 * Rational debate on important scientific and technological issues. (Make sure that

arts, culture and technology through investigation of local technologies (A&C and Technology) or survey to determine how cell phones have influenced local social and business practices (SS, EMS & Technology).
 * Integrate learning areas through events and/or joint investigations and surveys e.g.

mathematics, engineering and technology.
 * Spend time regularly on keeping up with current events that relate to science,

and further education.
 * Relate any particular section of the work to its applications in commerce, industry

L. Motivating learners

 * Motivational talks from successful former students.

improvement e.g. a pen to a learner who does not have one. (Do not overdo this kind of incentive otherwise it loses its value and significance.)
 * Offer small prizes and incentives from time to time for a significant piece of work or

them to read books on "how it works" as many scientific applications are found in them. In general encourage them to find answers themselves.
 * Encourage learners to ask the question "how?" (how does it work?) and encourage

 * Find or devise activities that involve all learners.

 * During class allow time for Independent learning

 * Publicly or privately appreciate and acknowledge good work done by learners.

 * Point out that science and mathematics underpin the world's technological advance.

 * Demonstration of mathematical content by real life examples.

good assessment tool.)
 * Do geometry orals where learners solve geometric problems orally. (This is also a

 * Focus on study skills by holding short but regular seminars. Start early in the year.

"how"?
 * Encourage inquiry by constantly asking the two critical questions: "why?" and

range of scientific applications in almost every item we use in our daily lives.
 * Look for books in the library on the subject "How it works" and discover a wide

concept you learn (or teach) in your science class. This way the understanding of science and the importance of science in your life becomes real.
 * Always make an effort to find relevant, practical, everyday applications of every

everyone involved and focused.
 * Active participation by learners is essential. Ask directed questions to keep

solutions. (You will be surprised at how fulfilling completing even a small project is.)
 * Do informal investigations which challenge learners to come up with different

 * Encourage learners to do self assessment after each lesson.

 * Let the learners evaluate you.

of work.
 * Encourage learners to make their own mind maps, especially at the end of a section

 * Provide materials and opportunity for learners to do their own research.

M. Policies, preparation and planning
willingness to control homework.
 * Draft school homework policy and include parent body to get their "buy-in" and

 * Establish a school academic support program based on enrichment and diagnosis.

to work towards the outcomes of the lesson. She can spend time figuring out the best method to use, the materials and teaching aids needed, the type of questions to ask and generally how to get learners to where she wants them to be by the end of the lesson.
 * Prepare lessons. A well prepared lesson gives an educator direction and helps her

 * Always be prepared for the unforeseen.

is going.
 * Always select a variety of examples to use and select according to the way a lesson

 * Plan lessons to accommodate for special needs.

 * Thorough planning is vital for success.

 * Do long term planning (broken into weeks, days, hours).

N. Professional relationships amongst colleagues
during their teaching recently. The strategy of class visits by educators to colleagues also helps in that one sees problems that other educators encounter and how they remedy them e.g. learners with disabilities and measures for coping.
 * Hold educators' phase and grade meetings to discuss what educators have learnt

 * Hold phase, grade and subject meetings for strategic- and year planning.

one is doing.
 * Team teaching helps all to move at the same pace; all know more or less what every

keep all abreast of latest developments.
 * Continuous reporting back from workshops and other in-service training helps to

 * Evaluate and reflect on one's successes, failures and ways of improving

O. Projects in and around the school
Department of Education, non-governmental organizations (NGOs) and an international funding scheme. We exchanged views about getting across the message of HIV/AIDS awareness and prevention. The community was sensitized to the scourge of HIV/AIDS by children placing cards with their writings on the school walls and other places where the community usually congregates. People in the community started talking and advising others on the epidemic. An HIV/AIDS centre has been established as a result.
 * We organized an HIV/Aids awareness campaign in the community by involving the

can be used in the science, mathematics and technology classroom.
 * Train learners to collect and obtain equipment and items (often thrown away) that

even without knowing it. You then have a chance of highlighting the principles when you assess their projects or when they explain them. A project like building a model of a house with an electrical light system lets learners discover how, in practice, bulbs are connected in a house.
 * Give your learners projects to work on. They will use scientific principles, at times

situations from doing things themselves and making informed judgments.
 * Allow a reasonable time frame to complete a project. Children deduce real-life

the area to be paved, choose the tiles, their color and suitable texture/finish of the material. Do calculations.
 * Calculate the cost of materials and labour for paving the school courtyard. Sketch

P. Respect for others
would like their rights to be respected. be ridiculed.
 * Encourage punctuality as a mark of respect for educators and others learners.
 * Explain the importance of respecting the rights of others, just as every individual
 * Respect the rights of each other and their property.
 * Establish a classroom code of conduct with regard to mutual respect.
 * Learners should feel free to participate in lessons in the knowledge that they won't
 * Respect each other's language.

title: SA NC Doing Investigations/Chapter 3


text:
SA NC Doing Investigations/Chapter 4

Ideas for investigations
Ideally learners should choose their own questions because a true investigation will follow a topic that is of interest to and stimulates the imagination of the investigator. In reality the educator may have to prompt learners or even assign questions. One hopes that by doing investigations learners will start developing a more critical eye for their own questions.

At the MSTotY workshop in March 2004, the participants were asked to brainstorm ideas for investigations and then to classify them according to a classification scheme. The purpose of the exercise was to gain a better understanding of what one might expect from investigations on these topics. The process of classifying ideas for investigations also gave the participants the opportunity to imagine the directions these ideas might take a learner and the potential pitfalls and complexities hidden in the topics. One can never determine these exactly because by the very nature of an investigation that we cannot, nor should not, be precise about the end point. However, if an investigation activity is to be well managed, educators had better try to gain some insight into the possibilities presented by a topic for investigation before agreeing to it. Obviously one's knowledge of particular learners must be taken into account.

A classification scheme for investigation topics
Is the topic from one or some of these disciplines?

What is the phase level of the topic?

Which of these categories does the topic fit into?

... and which of these?

... and which of these?

Do I have any concerns about the topic itself of where it may lead?

Classifying science and mathematics investigations
Imagine learners are asked to come up with questions for their own investigations. You collect the questions in the class and the list looks like the example below. By classifying the ideas you can determine whether they are appropriate for the age and phase of the learners. For the purposes of managing the investigation you can determine what resources the learners will need and what sort of inputs you may have to make. The final judgment you must make &Acirc &shy and the analysis should help you make it - is whether the topic is feasible. To classify an intended investigation accurately you must discuss it with the learner. A discussion will help you to advise the learner if his/her intentions seem too ambitious and together you can agree on appropriate modifications or a complete re-think. This negotiation places you, the educator, in a much stronger position to monitor the investigation's progress.

As an exercise try to classify some of the topics suggested here. Most of the topics are appropriate to more than one phase but this will depend on the conceptual level at which learners attack the problems and analyse their results.

These topics have already been classified according to their phase-appropriateness. Do you agree with the classification? Are any of the topics too advanced for the senior phase or too simple for the intermediate phase? Which topics could be investigated at either phase? Are any of these topics suitable for the FET band? How would you modify the topics for FET? What aspects of any topics would you ask an FET learner to concentrate on?

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: SA NC Doing Investigations/Chapter 4


text:
SA NC Doing Investigations/Chapter 5

Managing and assessing investigations
''This chapter is based on work by Maryna de Lange, a primary school science educator from Saldanha Bay and the 2002 Maths and Science Educator of the Year. She is passionate about investigations and was the prime mover behind making "investigations" the theme of this year's MSTotY workshop.''

A suggested procedure for managing a scientific or mathematical investigation for school or "Expo" purposes
In order to get the most out of learner investigations the process must be managed carefully. It is important to ensure that the learners (and sometimes their parents too) are well-informed about the process and exactly what is expected of them. The following points will be useful for planning your investigations.

1. Clearly state: "This investigation will be for ... e.g. final year assessment / regional competition etc. and will count 50% towards the year mark in science / may, thereafter, be submitted for the national competition etc. ... ."

2. Clearly state and publicise timelines and deadlines.

3. Within the first week of the announcement, the CRITERIA (see the assessment pro forma below) must be fully discussed with participants / learners by organizers / educators.

4. The learner's choice of the topic is the most important part. These must be finalized within a week of the announcement.

5. Educator reviews possible topics with each learner and tests the topic against the criteria provided for the investigation. Get clarification of and agreement on the "focus question."

6. Learners must start work on the literature review / research immediately and get as much outside help as possible on this preliminary stage of the investigation.

7. Within one month of the announcement (or less, depending on overall timeline and deadlines), the educator must review each learner's topic, focus question(s), hypotheses, plans and "research design" in a one-to-one interview with the learner.

8. The investigation report must be handed in within the agreed period. On the deadline for the investigation, all reports must be handed in regardless of their state of readiness. (No extensions can be allowed without seriously disrupting the year programme, so careful monitoring of progress by the educator is essential.)

Doing independent assignments / projects / investigations develop extremely important attitudes, life skills and scientific and mathematical skills e.g. time management self discipline perseverance  careful, thoughtful and insightful work  reflection  accuracy etc. These skills and attitudes will all be assessed at the end of the senior phase.

In order that learners realize the importance of meeting deadlines etc., they can be made to sign a "contract"

Declaration: I understand and agree to abide by all conditions and

deadlines as explained to me by: __________________________________

Learner's signature: __________________________________

Dated: 			 			 ____/____/200____

Parent's signature:		 __________________________________

Dated:				 		 ____/____/200____

Publication of notice and criteria: Dated: ____/____/200____

Educator's signature: 	 	 __________________________________

Dated: 			 ____/____/200____

Obviously investigations take a lot of time to set up. Therefore the management of the process must be meticulous and starting on time and keeping to your schedules is critical. Each learner must be allocated time for discussion and consultation if investigations are to serve their educational purpose. It is possible to manage up to 35 learners. For classes larger than this it may be advisable to work in pairs. The problem with doing investigations is that the experience we want the learners to have is really a private one and it is most powerful when it is a personal learning experience. When learners are in pairs there is no guarantee that each will get the full benefit of their experience. It becomes important to choose well-balanced pairs in which both individuals will contribute equally and communicate well, so that they learn from each other.

ASSESSMENT CRITERIA FOR A SCIENTIFIC INVESTIGATION / PROJECT (INCLUDING REPORT AND PRESENTATION OF FINDINGS)
NAME: ___________________________________

TOPIC:___________________________________

Tips on assessing investigations.
When all is said and done, the amount of effort put into an investigation by educators and learners is considerable and therefore deserves meticulously thought out and planned assessment. Investigations provide a wonderful platform to assess scientific and mathematical thinking in action. But it also means that the assessment must be continuous in order to extract the most value possible from the process. Assessment can take on different forms, depending on the situation at a particular time. Here are some ideas.

discuss class work or homework, especially when moving to the next topic. It helps you to recap on work done, provides a basis for moving ahead, saves time and provides immediate feedback. Its drawbacks include not getting to see individual learners' problems. Learners tend to mark themselves correct even when they are not. Sometimes they mark an answer correct when the steps leading to the answer are not.
 * Self-assessment. Learners assess themselves. This method works well when you

Possibly members of a group assess other members. The method poses the problem of learners giving in to peer pressure and awarding marks for incorrect answers. Control is essential when peer assessing and learners' books must be checked for the accuracy and honesty. It becomes easier when a class gets used to the method and sees the value of accurate assessment.
 * Peer assessment. Like self-assessment but here learners assess each other.

learners. Rubrics are particularly useful when learners present the findings from their investigations. Key items to be assessed are listed clearly with their rubrics alongside them and learners' strengths and weaknesses can be "pegged" at a glance.
 * Rubrics. work well and if well designed they are one of the best tools for assessing

In order to assess individual strengths and weaknesses there is no substitute for the educator reviewing and marking each learner's work. The purpose must be not only to allocate a mark but to find the basis for remedial work where it is needed. The other forms of assessment add value, but only if they add to the educator's own review of learner's written work. It may be time consuming but it gets you to know your learners. At least half of a learner's homework and class work must be assessed by the educator. Formal tests must be marked by the educator and ideally even informal tests must at least be reviewed.
 * Educator assessment. is the most common form because it is the most important.

Assessing presentations and products
There are certain standard types of assessment instrument but apart from the general type, instruments are seldom "one size fits all". So we usually construct instruments for specific activities. An instrument like the one below might be used when individual learners or groups are presenting their "research" findings on a task assigned to them. This example would be used when they are explaining the idea of "systems" in the context of the human body. Assigning a "maximum possible" mark for each item provides the educator with a benchmark or standard by which to judge performance on that item. An important principle of assessment is to ensure that learners know well beforehand, exactly what the criteria are against which they are being judged. When learners know this they begin to develop their own "internal" standards.

A similar instrument might be used to assess a model that has been constructed. In this case the model is of the lungs inside the thorax. It shows how, by using the diaphragm to increase the volume of the thorax the lungs can be inflated and deflated. This model could be made from a two litre cold drink bottle, a balloon, a plastic drinking straw and a cork with a hole bored in it. In this activity, the learners would have been given clear instructions to follow.

The score on an instrument like this can be related to an overall, 5-point scale e.g. excellent (5/5) good (4/5)  satisfactory (3/5)  unsatisfactory (2/5)  poor (0-1/5).

The words used to describe the different levels are a matter of choice. If, for instance, it is a 5-point scale, the five words or phrases used must describe a logical sequence of performance levels, communicate to learners how their educator will judge their performance and must indicate to learners where they have performed well or poorly on the task. With the assessment instrument in front of them the educator and learner can decide how to remedy shortcomings and build on strengths.

Rubrics
Rubrics use words to describe successive levels of performance. They are useful assessment tools but they are of no value unless they are carefully constructed and then used with fairness, insight and care. It doesn't matter how good a rubric is, if the educator does not use the rubric consistently appropriately learners will have no confidence in the method or the instrument. When learners lack confidence in an educator's judgments (about their performance) they miss the learning opportunity that good assessment practice can provide. The assessment of an investigation can teach almost as much about the process of investigation as the activity itself.

Here are some rubrics. In the first example we have a rubric to assess designs in a technology class. The second the rubric is for assessing actual design drawings. The third rubric is related to RNCS Learning Outcomes (LOs) and Assessment Standards (ASs) for the Natural Sciences.

A rubric for assessing designs in technology.

A similar rubric could be used for assessing learners' actual design drawings.

Any good rubric can be adapted or used as a template to produce others. The revised national curriculum statements (RNCS), scheduled for implementation in 2007, call for assessment of four level of performance in the GET band. Therefore particular tasks and activities can be assessed by designing rubrics from the learning outcomes and assessment standards appropriate to the task. In these examples (adapted from All Aboard Science 7 by Heinemann Publishers) various learning tasks are covered. The first example involves an investigation into the causes of corrosion in metals.

The investigation of sensitive issues can lead to debate that involves prejudices and emotion. In science it is important to recognize and keep these distractions in check. That is why scientists try &Acirc &shy not always successfully &Acirc &shy  to take account of how we accumulate knowledge and the peculiar role that personal feelings and prevailing ideas play in attempting to understand the world we live in. The third learning outcome (LO3) in the Natural Sciences deals with this issue and is notoriously difficult to assess. Rubrics can help when we discuss issues that bring out emotion and contradiction. But we should not allow difficulties like this to put us off because dealing with topics that call for the interpretation of historical evidence and differing points of view are important to our development as scientifically literate people. The rubrics for the Natural Sciences LO3 AS1 can be adapted, for instance, to issues of gold mining and processing of gold ore, the development of dynamite and the effect that refrigeration has had on the international trade of agricultural products.

Strategies: your own assessment policy
Maryna de Lange has set her assessment policy so that several assessable aspects of her learners' work count towards the final mark for the year. She writes: "I balance five forms of assessment so that every learner's preference can be accommodated. I use the assessments to build up learners' portfolios which I keep in my Master file. I regularly record the outcomes of my observations on a formative record sheet. Learners are allowed to choose the work they want to keep in their portfolios for promotion purposes. Assessments consist of educator-, peer- and self- assessments. The forms of assessment I use are:

(a) Four tests and two examinations. The tests count 30 marks each and the examinations are set for between 100 and 150 marks. The totals for the tests and the examinations are then converted to 60 marks each. [Total 60 + 60]

(b) Eight class- or homework activities count 15 marks each. 120 marks are converted to 45. [Total 45]

(c) Two projects of 40 and 35 marks respectively. [Total 40 + 35]

(d) Two assignments of 30 mark each. [Total 60]

(e) Two investigations of 30 mark each. [Total 60]

The total mark for Science for the year is 360. The pie-chart below gives us a picture of this policy. Remember that there is also a portfolio and a formative record that can be used when the learner's performance for the year is assessed. OBE calls for continuous assessment to ascertain a learner's development. Does this policy meet these requirements?

You will agree that this assessment policy gives a very comprehensive picture indeed. The pie chart also shows that investigations and projects (both involve learners working independently on problems of their own choice) make up more than a third of the total mark.

Proportions by Form of Assessment

title: SA NC Doing Investigations/Chapter 5


text:
SA NC Doing Investigations/Chapter 6

Examples of investigative activities in science and mathematics.
The investigations described so far are all projects and of the type where learners come up with an idea and then try to find out as much as they can about it. The purpose of investigations like these is to get learners into the processes of science (see Chapter 9 for a comprehensive list of the facets of scientific literacy). But the spirit of investigation can be found in much simpler activities and lives on in the saying of a great science educator of the past, Julius Sumner Miller: "more doing and less talking"! When introducing a new concept or simply when presenting learners with a remarkable phenomenon to stimulate their imaginative juices, investigations remain a very powerful tool in the hands of science and mathematics educators. Here are a few popular, investigative activities. Follow the instructions and work out what scientific concepts are being demonstrated. Investigative activities leading from each of these would be to find out where each phenomenon can be seen at work.

A useful acronym that summarises an approach to using these quick investigations is "OPE" or Observe, Predict and Explain. When doing them in class it is important to allow learners the time to observe carefully (possibly re-doing the demonstration in the process) to discuss their predictions and the theory behind their predictions  and then their explanations of what happened, particularly if their predictions were incorrect.

Lifting an ice cube without touching it.
You will need a glass of water, ice cubes, a piece of string and some salt. Step 1. Put an ice cube into a glass of water. It floats with about one tenth of the cube sticking out above the water.

Step 2. Put an end of the string on the exposed ice and sprinkle some salt on it. Let it stay there for a minute or two.

Step 3. Gently lift the string.

What do you expect? That the ice cube will remain in the water and the string will pull away. What actually happens? The string sticks to the ice cube strongly enough to lift it from the water. How does it work? Salt lowers the freezing point of water to below 0&deg C. When the ice melts where the end of the string lies, the melted ice dilutes the salt and the ice freezes again. The warmth of the string will first cause a small hole to form in the ice and the water-salt mixture then freezes around it again when the string is inside the ice cube.

Heavy paper.
You will need a double sheet of newspaper and a ruler. Step 1. Place the ruler on a table with a few centimetres sticking over the edge.

Step 2. Cover the ruler on the table with the sheet of newspaper.

Step 3. Hit the piece of ruler that extends over the edge of the table hard, with your hand.

What do you expect? You expect the paper to leap into the air as you hit the ruler. What actually happens? The paper remains stationary on the table. It does not even tear. How does it work? The weight of air on the palm of your hand, turned upwards, is about 45 kg. What is the weight of air on a double sheet of newspaper? Certainly more than a ruler can lift!

How strong are your lungs?
You will need a balloon and an empty cold drink bottle.

Step 1. Place the balloon in the neck of the bottle.

Step 2. Wrap the opening of the balloon around the opening of the bottle.

Step 3. Blow into the balloon which is inside the bottle.

What do you expect? The balloon should inflate and fill the bottle. What actually happens? The balloon inflates very slightly but then it is impossible to inflate it any more. How does it work? As the balloon inflates, the air in the bottle compresses into the remaining space. This means that the pressure back on the blower's lungs also increases. It is very difficult to inflate the balloon, even a little!

It's not so easy to move coins.
You will need a heavy water glass, a strip of newspaper, two large coins and a ruler.

Step 1. Put one end of a strip of paper over the rim of the glass.

Step 2. Balance the two coins on the edge of the glass.

Step 3. Make sure the coins are just touching the paper but are actually balancing on the rim and not on the paper.

Step 4. Lift the free end of the paper and hold it horizontal.

Step 5. Strike the paper with a ruler about 5 cm from the edge of the glass.

What do you expect? You expect the coins to drop into the glass. What actually happens? The paper is whipped out from under the coins and they remain on the lip of the glass. 'How does it work? All bodies resist being put into motion (i.e. being made to move). The more mass in a body, the more difficult it is to make it move. So the light paper moves but the heavier coins do not. (Find out what this property of matter is called.)

Paper scorch and char.
You will need a spirit burner, matches, a rod of wood and metal (make one with a dowel rod and a piece of aluminium foil wrapped around one half of it a few times) and a piece of writing paper.

Step 1. Wrap a single layer of paper around the wood-metal rod.

Step 2. Light the spirit burner.

Step 3. Play the flame lightly on the paper over the boundary between the metal and the wood on the rod.

What do you expect? The paper will burn. What actually happens? The paper chars on the half that covers the wood but not the aluminium foil. How does it work? The metal conducts the heat away from the paper while the wood does not. So the temperature of the paper over the wood soon rises to the point at which paper scorches and chars. This does not happen over the metal.

A balancing trick at the dining room table - if your mother lets you do it!
You will need a cork, a pin, two forks and a heavy glass.

Step 1. Place the pin in one end of the cork.

Step 2. Stick the forks into the side of the cork so that they hang downwards &shy their handles being below the bottom of the cork that has the pin in it.

Step 3. Balance the pin on the bottom of an upturned glass.

What do you expect? You expect the whole arrangement to fall over because it looks so unstable. What actually happens? The arrangement balances on the pin and although it may wobble at first it soon settles down. How does it work? Every body has a point called the Centre of Gravity (CoG) associated with it. It is the point at which the entire mass of the object appears to be concentrated. If the CoG is below the balance point the object will be very stable.

Superboy and Supergirl!!
You will need two broomsticks and a length of rope.

Step 1. Let Superboy/Supergirl hold two broomsticks vertically, one in each hand, and pull inwards while two other people try to pull the broomsticks (and the holder's arms) outwards.

Step 2. Now, while the two "pullers" each hold a broomstick vertically, tie one end of a piece of rope onto one broomstick (about one third of the way up) and then wind the rope about 5 times around both, working in an upwards direction.

Step 3. Let the two pullers try to pull the broomsticks outwards again while Superboy/Supergirl pulls on the loose end of the rope.

What do you expect? You expect the two people to pull the broomsticks apart with ease. What actually happens? The rope pulls the two people inwards quite easily. How does it work? This acts like a series of pulleys (or block and tackle). Every turn of rope almost doubles the pulling power of the person on the end of the rope.

Tight fit.
You will need a single sheet of A4 paper and a pair of scissors.

Step 1. Give each learner a pair of scissors and a piece of A4 paper.

Step 2. Set them the problem: "Can you cut the paper so that it fits over your body?"

Step 3. Fold the paper in half short end to short end.

Step 4. Cut out a narrow rectangle from the folded end as shown (X).

Step 5. Make an odd number of cuts according to this pattern:

What do you expect? Can't do it! What actually happens? It works!! How does it work? Look carefully at the arrangement and see why it works.

Step 6. You can experiment using another piece of paper. Try disobeying the instructions, for instance by making an even number of cuts or not cutting right to the edge of the paper!

Investigating Number Patterns
''This series of activities was prepared by Professor Otis Thompson of the University of Montana Western in his capacity as Principal Investigator and Mathematics Specialist on the Tlhatloga Project. The activities were presented at a series of workshops held in 2003. Professor Thompson has had a distinguished career as a mathematics educator and regards his three years working with South African educators as the highlight of his professional life.''

Educator's Guide
Mathematics has been defined as the study of patterns. Patterns are everywhere &shy on wallpaper, floors and wall tiles  in artistic works  in plant and animal growth and behavior in traffic and even in television schedules. Scientists look for patterns in order to isolate variables so that they can reach valid conclusions in their researches. Learners can gain experience with patterns by investigating numerical and geometric patterns and expressing these patterns mathematically in words or symbols. To analyze patterns we first determine the structure of the pattern and how the pattern changes. We then organize the information systematically and finally develop generalizations about the mathematical relationships in the patterns. In these activities learners can explore number patterns and learn some methods for analyzing them. They can then use their analyses to express the patterns mathematically. Learners can also investigate how pure number patterns apply to the real world.

Activity 1 In this activity, the learner will explore some familiar number patterns and will name the set of numbers described.

Activity 2 In this activity, the learner will explore arithmetic sequences, geometric sequences, and sequences of figurate numbers. An arithmetic sequence is one in which each successive term is obtained from the previous term by the addition or subtraction of a fixed number called the difference. A geometric sequence is one in which each successive term is obtained from the previous term by multiplying by a fixed number called the ratio. Figurate numbers are numbers that can be represented by dots arranged in the shape of certain geometric figures.

Activity 3 The Fibonacci sequence will be introduced using the original problem that Leonardo Fibonacci presented in 1202. Then the learner will explore some other problems that lead to the Fibonacci sequence. The learner will discover some of the interesting relationships within this sequence and express these relationships mathematically. The learner will explore how plant growth patterns are aligned with the Fibonacci numbers.

Activity 4 1+ \frac{1}{\frac{1}{1} + \frac{1}{1 + L} } Learners will explore the continued fraction and see that stopping at a given point always leads to a ratio of Fibonacci numbers. When converted to a decimal these ratios approach 1.61803..., the golden ratio. The learners will then experiment to find a "most pleasing rectangular shape" and learn that the ratio of length to width is approximately this golden ratio. The learners will then locate rectangles in their surroundings that are "golden rectangles."

Materials needed
 * Paper and pencil
 * Calculator (optional)
 * A whole pineapple (optional)
 * Compass
 * Straight edge
 * Scissors or cutting knife

Number Patterns Activity 1
Thembisa had to learn a large list of spelling words for her final spelling test for the term. The first day she learned two words, and each day thereafter she learned to spell two more words. If she had the list learned by the 15th day, how many words were on her list?

To solve this problem, complete the following table:

So we see that Thembisa's list contained _____________ words.

The list of numbers you created (called a sequence) is part of a very special set of whole numbers. This set, written as {0, 2, 4, 6, 8, 10, 12, 14, 16, ...}, where the ... (called ellipsis) means the pattern continues forever. What is the name of this set of whole numbers?

There are other sequences of whole numbers for which you probably have a name. Let's explore and see which ones you know. In each sequence below, fill in the next three terms and give a name to this sequence.

1. 1, 3, 5, 7, 9, ______, ______ , ______ , ...

Name: ________________________________

2. 0, 1, 4, 9, 16, 25, ______, ______ , ______ , ...

Name: ________________________________

3. 0, 3, 6, 9, 12, 15, ______, ______ , ______ , ...

Name: _______________________________

4. 2, 3, 5, 7, 11, 13, 17, _____, ______ , ______ , ...

Name: ______________________________

Number Patterns Activity 2
A large theatre is set up in such a way that there are 20 seats in the first row and 4 additional seats in each consecutive row. The last row has 144 seats. How many rows are there in the theatre?

If we try to solve this problem by creating a list as we did in Activity One, it might take us a while to finish the list. Let's explore some of the properties of a sequence such as this one to try to simplify our method of solution.

The sequence we need to create to solve this problem is called an arithmetic sequence. An arithmetic sequence is one in which each successive term is obtained from the previous term by the addition or subtraction of a fixed number called the difference. Write the first five terms of the sequence that correspond to the first five rows: What would be the tenth term in this sequence? What would be the twentieth term in this sequence? Write a rule for finding the nth term of this sequence?

Now use this rule to find the number of rows in the theatre. To determine if a sequence is an arithmetic sequence, one simply looks to see if the difference between consecutive terms is constant. Which of the following sequences are arithmetic sequences? If the sequence is an arithmetic sequence, write the 100th term of this sequence.

1. 5, 8, 11, 14, 17, ...

Arithmetic ? Yes/No

100th term = ________________

2. 4, 6, 9, 13, 18, ....

Arithmetic ? Yes/No

100th term = ________________

3. 1040, 1032, 1024, 1016, 1008, ....

Arithmetic? Yes/No

100th term = _____________

4. The set of even whole numbers

Arithmetic? Yes/No

100th term = _____________

Now the question arises regarding the introductory problem in Activity 2: How many seats are there in the theatre?

To answer this question, first let's consider a simpler problem. What is the sum of these numbers: 1 + 2 + 3 + 4 + ... + 100? (Note this is an arithmetic sequence with a difference of 1.)

Write out the sum: 1 + 2 + 3 + ... + 98 + 99 + 100

Below it write the Sequence "backwards.": 100 + 99 + 98 + ... + 3 + 2 + 1

Now if we add the pairs above and below each other, the sum is 101, and since there are 100 such pairs, the total sum of the two rows is 100 x 101. But since this contains every number in the sequence twice, we will divide this product by 2 giving us 1 + 2 + 3 + ... + 99 + 100 = 5050.

(Note: To find the sum of a finite number of terms of an arithmetic sequence, you need to know the first term, the last term, and the number of terms. Write a rule using this information for finding such a sum and then use this rule to find the number of seats in the theatre.)

Find the sum the following sums: 1 + 2 + 3 + 4 + ... + 1000 = _______________

2 + 4 + 6 + 8 + ... + 72 = _______________

8 + 11 + 14 + 17 + ... + 305 = _______________

Not all sequences are arithmetic. Other types of sequences exist. A geometric sequence is a sequence in which each successive term is obtained from its predecessor by multiplying by a fixed number called the ratio. Find the next three terms in these geometric sequences:

1. 2, 4, 8, 16, ______, ______ , ______ , ...

2. 729 486  162, _____, _____ , _____ , ...

3. 1, 0.4, 0.16, 0.064, _____, _____ , _____, ...

Figurate numbers (numbers that can be represented by dots arranged in the shape of certain geometric figures) provide examples of sequences that are neither arithmetic nor geometric. Below is the first four terms of the sequence of triangular numbers.

Explain why the next triangular number is 15.

The sequence 1, 3, 6, 10, 15, ... is not an arithmetic sequence (since no common difference) nor a geometric sequence (since no constant multiplier.) But if we look at the difference between consecutive terms and then difference between consecutive differences, we see a pattern that should help us find the next three terms of this sequence.

Here are the first three pentagonal numbers. Find the next three pentagonal numbers using the method you used with the triangular numbers.

When asked to find a pattern for a given sequence, first look for some easily recognizable pattern and determine if the sequence is arithmetic or geometric. If a pattern is unclear, taking successive differences (as done with the figurate numbers) may help. It is possible that none of these methods will reveal a pattern.

Number patterns Activity 3
In 1202 Leonardo Fibonacci presented the following problem in his book Liber abaci (Book of the abacus): A certain man put a pair of baby rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year, if it is supposed that every month each pair begets a new pair which, from the second month on, becomes productive?

To solve this problem, let's make a table and keep track of how many pairs we have each month. Remember, we will start with one baby pair. At the start of the second month, they are now an adult pair of rabbits, so they breed. The gestation period for rabbits is one month so at the start of the third month, we now have an adult pair and a baby pair.

Now complete the table to arrive at the answer as to how many pairs we have at the end of the year. Carefully analyze the sequence of numbers you obtained for the total number of pairs. Describe how one can determine the next term in this sequence.

(Note: this sequence is neither an arithmetic nor a geometric sequence. The method of successive differences doesn't work like it did for the figurate numbers, but it may give a clue to how to build successive terms in this sequence.)

Now let's explore this sequence in greater detail and make some conjectures about it. 1. Complete the following table for the first 25 terms of the Fibonacci sequence: There is nothing anywhere here to explain what the Fibonacci numbers are or how they are obtained?

2. If your table was completed correctly, the twenty- fifth Fibonacci term, F25, should be 75025.

3. Find the following sums:

a. F1 + F2

b. F1 + F2 + F3

c. F1 + F2 + F3 + F4

d. F1 + F2 + F3 + F4 + F5

e. F1 + F2 + F3 + F4 + F5 + F6

f. F1 + F2 + F3 + F4 + F5 + F6 + F7

g. Write a rule that would let you give the answer to the sum of the first n terms in terms of a Fibonacci number.

h. Use your rule to find the sum of the first twenty terms of this sequence.

4. Find the following sums (note: we are just using the odd numbered terms):

a. F1 + F3

b. F1 + F3 + F5

c. F1 + F3 + F5 + F7

d. F1 + F3 + F5 + F7 + F9

e. F1 + F3 + F5 + F7 + F9 + F11

f. Write a rule that would let you give the answer to the sum of the first n odd numbered terms in terms of a Fibonacci number.

g. Use your rule to find the sum of the first ten odd numbered terms.

5. Find the following sums (note: here we are just using the even numbered terms):

a. F2 + F4

b. F2 + F4 + F6

c. F2 + F4 + F6 + F8

d. F2 + F4 + F6 + F8 + F10

e. F2 + F4 + F6 + F8 + F10 + F12

f.Write a rule that would let you give the answer to the sum of the first n even numbered terms in terms of a Fibonacci number.

g. Use your rule to find the sum of the first ten even numbered terms.

6. Find the sum of the square of two consecutive Fibonacci numbers -This is written symbolically as: (F_n)^2 + (F_{n+1})^2. The result should be another Fibonacci number. Which one is it?

7. a. Take any three consecutive Fibonacci numbers from your list. Multiply the first number by the third number and subtract the square of the middle number. -This is written symbolically as: F_{n-1} X F_{n+1} - {F_n}^2 Write a rule that would explain what the answer will be.)

b. Without actually doing the computations, what is the value of F24 x F26 &shy F25^2

8. a. Now compute F_{n-2} x F_{n+2} - {F_n}^2 for various values of n.

b. Write a rule that will explain what the answer should be each time.

9. (For the "experts.") Write a rule that will give the answer to F_{n-k} x F_{n+k} - {F_n}^2.

10. All the Fibonacci numbers are divisible by F_1 and F_2 (since F_1 = F_2 = 1 and every whole number is divisible by 1.)

a. Which Fibonacci numbers are divisible by F_3 ?

b. Which Fibonacci numbers are divisible by F_4 ?

c. Which Fibonacci numbers are divisible by F_5 ?

d. Describe the relationship between the Fibonacci numbers that are divisible by other Fibonacci numbers.

11. Take any 10 consecutive Fibonacci numbers and add them together. Divide this sum by 11. What was the result? Does this work every time?

12. Find the sum of the squares of the first n Fibonacci numbers. While the sums do not seem to be related to Fibonacci numbers directly, is it possible that they might have Fibonacci numbers as factors? If so, write the factors. Now write a rule for finding this sum of squares.

13. Every natural number is either a Fibonacci number or can be expressed uniquely as a sum of Fibonacci numbers whereby no two are Fibonacci numbers. Here is one way to do it:

a. Write down a natural number.

b. Find the largest Fibonacci number that does not exceed your number. That Fibonacci number is the first term of your sum.

c. Now subtract that Fibonacci number from your number and look at this new number.

d. Next find the largest Fibonacci number that does not exceed this new number. That Fibonacci number is the second number in your sum.

e. Continue this process until the remainder becomes a Fibonacci number (the last term of your sum.)

14. The game, Fibonacci nim, is played with two people: Person 1 and Person 2. All that is needed is a pile of sticks. Person 1 moves first by taking any number of sticks (at least one but not all) away from the pile. After Person 1 moves, it is Person 2's move, and the moves continue to alternate between them. Each person (after the first move) may take away as many sticks as he or she wishes with the restrictions that he or she must take at least one stick but no more than two times the number of sticks the previous person took. The player who takes the last stick wins the game. Player 1 can always win using the following strategy. Make sure that the initial number of sticks is not a Fibonacci number. Write the number of sticks in the pile as a sum of nonconsecutive Fibonacci numbers (see problem 13.) Figure out the smallest Fibonacci number occurring in the sum and as Player 1, remove that many sticks from the pile. Now your opponent (Player 2) plays. No matter what Player 2 does, you repeat the preceding procedure. That is, once Player 2 is done, Player 1 expresses the number of remaining sticks as a sum of nonconsecutive Fibonacci numbers, and then removes the number of sticks that equal the smallest Fibonacci number in the sum. It is a fact that, no matter what Player 2 does, Player 1 will always be able to remove that number of sticks without breaking the rules and eventually Player 1 will be able to remove the last stick and win. Now let's explore some other problems.

1. A child is trying to climb a staircase. The maximum number of steps he climbs at a time is two that is, he can climb either one step or two steps at a time. If there are n steps in total, in how many different ways can he climb the staircase?

If there is only one step (n = 1), clearly there is only one way to climb it. If there are two steps, the child can either climb the two steps at once or take them one at a time thus there are two ways. What if there were 3, 4, 5, ... steps?

2. Suppose we have two glass plates made of slightly different types of glass mounted face to face. If we shine light through the plates, the light rays can (in principle) reflect internally at four reflective surfaces before emerging. More specifically, they can pass through without reflecting at all, or they can undergo one internal reflection, two internal reflections, three internal reflections, and so on before emerging (see the diagrams below.) Count the number of beams that emerge when there are n internal reflections.

3. In a honeybee colony a drone (male) bee gets his start from an unfertilized egg and thus has just one parent (the queen.) The queen bee gets her start from a fertilized egg and thus has two parents (a queen and a drone.) If we went back 10 generations in a drone bee's "family tree," how many great-great-great-great-great-great-great-great grandparents would this drone bee have?

4. Here is an algorithm for creating a sequence known as the Golden Sequence. Start with the number 1, and then replace 1 by 10. From then on, replace each 1 by 10 and each 0 by 1. How many 1's are there in the sequence of 0's and 1's after you have done this iteration process 12 times?

The above problems all lead to the Fibonacci sequence. The Fibonacci sequence is everywhere. It even occurs in plants. Petal counts and petal arrangements of some flowers harbor Fibonacci numbers. Most field daisies have thirteen, twenty-one, or thirty-four petals (all Fibonacci numbers.) The leaves along a twig of a plant or the stems along a branch tend to grow in positions that would optimize their exposure to the sun, rain, and air. The passage from one leaf to the next (or from one stem to the next along a branch) is characterized by a screw-type displacement around the stem. For example, on the pear tree it takes three complete turns to pass through eight stems, a pair of alternate members of the Fibonacci sequence. The hexagonal scale on the surface of a pineapple is a part of three different spirals. Most pineapples have five, eight, thirteen, or twenty-one spirals of increasing steepness on their surface. All of these are Fibonacci numbers. When you look at the head of a sunflower, you will notice both clockwise and counterclockwise spiral pattern. In the pictures below, there are 21 clockwise spirals and 34 counterclockwise spirals, again Fibonacci numbers.

A pinecone's exterior is composed of two sets of interlocking spirals, usually 5 spirals in one direction and 8 in the other. What examples of Fibonacci numbers can you find in the plants in this location?

Number Patterns Activity 4
Now let us look at a very different type of never-ending expression, this time involving fractions: 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac {1}{1 + \frac {1}{...}}}}}

This is a special case of mathematical entities known as continued fractions. How would we compute the value of this fraction? One way is to "truncate" the continued fraction as various points, simplify the fraction, and look for a pattern in our results.

1. Express as an improper fraction. 1 + \frac{1}{1}

2. Express as an improper fraction. 1 + \frac{1}{1 + \frac{1}{1}}

3. Express as an improper fraction. 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1}}}

4. By this time you may have discovered a "short cut" for doing this. So let's do one more. Express as an improper fraction. 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac {1}{1}}}}

5. Have you seen the numbers that appear in the numerator and denominator of these improper fractions before? What are they? (Fn+1)/Fn

6. Use a calculator to get a decimal approximation to the fraction for larger and larger values of n where Fn is the nth Fibonacci number. This number is known as the Golden Ratio and is usually identified by, the Greek letter phi. It can be shown mathematically that this number is exactly.

Now let's try a little experiment. You will be given a piece of paper with several rectangles drawn on it. You are to select the one rectangle that is the most "pleasing, attractive, and elegant" of the set. The results for the group will be tabulated. Let us now construct a rectangle that has some special appeal to many people.

1. Draw square ABCD.

2. Locate the midpoint of segment DC.

3. Place the compass point at M and using radius MB, draw arc BE.

4. Point E is the point where DC extended intersects the arc.

5. Locate point F so that AFED is a rectangle.

It can be shown mathematically that in rectangle AFED, the ratios of sides frac{AF}/{DE} and {BC}/{BF} are both equal to, the Golden Ratio. Hence rectangle AFED is known as the Golden Rectangle. Much has been written about how various artists and architects have allegedly used the Golden Rectangle in designing their various works. Near the center of a sheet of paper, construct a Golden Rectangle. Carefully cut the rectangle from the sheet leaving the outside of the sheet intact. Now use this "template" to find objects around you that are close to being "Golden Rectangles."

Investigating Combinatorial Mathematics
''This activity is based on an idea derived from "The Man Who Loved Only Numbers: the story of Paul Erdös and the search for mathematical truth," by Paul Hoffman (1998) and "The Combinatorial Mathematics of Scheduling," Scientific American, March 1978. The activity was written by Dr Peter Glover and was first used in June 2001 during the Tlhatloga Project's Summer Institute in Dillon, Montana, USA.''

PROBLEM 1
Divide a group of 5 weights (2, 2, 2, 3 and 3) into two piles so that each pile is as close as possible in total weight.

'SOLUTION 1-1' By inspection (i.e. just by looking at the weights) we can arrange them (3, 3) and (2, 2, 2). When dealing with a few weights the problem can be solved by "trial and error" i.e. by trying all the variations you can until you get the best answer. This is a problem in the branch of mathematics called "combinatorics" &shy the mathematics of making the best decision about what to put where and how to combine quantities in an optimal way to get a desired result. It is a relatively new branch of mathematics and it is finding applications everywhere from defense to cutting lengths of piping to packing of delivery trucks.

SOLUTION 1-2 Ronald Graham is one of the best combinatorial mathematicians today. Here is his algorithm* for packing weights most efficiently: "starting with the heaviest weight and working down to the lightest, put each weight into the pile that tends, at each step of the way, to keep the weights of the piles as equal as possible."

[* The word algorithm is taken from the ninth century Persian number theorist called Abu Ja'far Mohammed ibn Musa al-Khowarizmi, who also gave his name to "algebra." "An algorithm is ... a step-by-step procedure in which every step is explicitly stated so that a problem can be blindly solved by man or machine." (Hoffman. 1998).]

But using Graham's algorithm to solve PROBLEM 1 we get this solution:

We can see that this is not the best solution, which our trial-and-error solution was i.e. where each pile totals 6 weight units (i.e. 3, 3 and 2, 2, 2). But it is also not the worst combination, which it would have been if piles ranged in size from 2 to 10 (i.e. 3, 3, 2 and 2, 2 OR 3, 3, 2, 2 and 2).

Our computer programs are simply algorithms (many of them) strung together and written in a language that computers can decode, "understand" and act on. The best solution vs the algorithmic solution The best solution (which ours was and which was found by trial and error) is 3, 3 and 2, 2, 2 (i.e. 6 and 6). But Ron Graham's algorithm gives us 3, 2, 2 and 3, 2 (i.e. 7 and 5).

Therefore Graham's algorithm gives a solution that is out by 1/6 in each case, (i.e. 6 vs 7 and 6 vs 5) or &plusmn  16% off the best answer.

Graham was able to prove that for 2 piles and any distribution of any number of weights, his algorithm will never be off by more than 16%. In 1973, Jeffrey Ullman of Princeton University showed that the so-called "first-fit packing algorithm" &shy where you pack items in whatever order they arrive into whichever container you choose, as long as they fit &shy  can be off by as much as 70%!

PROBLEM 2
You have the 33 weights shown below and bins with a capacity of 524 weight units. Using Graham's "first-fit, heaviest-to-lightest" algorithm, divide up the blocks provided into as few bins as possible.

1. Show any calculations you make.

2. Show how you packed your weights in the bins:

PROBLEM 3
You still have 33 weights, but now each one is half the weight of those used in PROBLEM 2. Where halving gives a 0.5, round up as shown in the examples. Your new bins have a capacity of 262. Before using the "first-fit, heaviest-to-lightest" algorithm, see if you can make any calculations to predict how many bins you will need to pack the weights. When you have made your prediction, pack your blocks into as few bins as possible.

Your prediction: _____ bins.

Show how you would pack your weights.

PROBLEM 4
Take the 33 weights as used in PROBLEM 3 except that wherever there was a "rounding up" for a 0.5, you should now round down any halves. The bins still have a capacity of 262. When you have predicted the number of bins required, use Graham's algorithm to pack your blocks into as few bins as possible.

Your prediction: _____ bins.

Show how you would pack your weights.

PROBLEM 5
Use the 33 weights you used in Problem 2 but discard the 46 weight. Predict the number of bins you will need now. (a) Use Graham's algorithm to pack the blocks into the fewest bins possible and (b) using a trial-and-error approach, try to improve on the solution your "first-fit, heaviest-to-lightest" strategy gave.

Your prediction: _____ bins.

Show how you would pack your weights.

A Pipe Cutting Problem: another combinatorial investigation
Fiberglass insulation comes in 1-meter precut sections. A plumber must install insulation in a basement on piping that is often interrupted by joints. The distances (measured in cm) between the joints on the stretches of pipe that must be insulated are 30, 40, 45, 30, 20, 25, 40, 50, 30, 35, 50, 55, 60, 70, 95, 10, 70, 30, 80, 85, 40, 45, 30, 20, 15, 90, 60, and 50. How many precut sections would the plumber have to use to provide insulation. A stretch of pipe must be covered by one solid piece of insulation. No joining of two shorter pieces to make a longer piece is allowed.

This problem and the one presented earlier are known as bin-packing problems. Bin packing refers to finding the minimum number of bins of weight capacity W into which weights w1, w2, w3, ..., wn (each less than or equal to W) can be packed. Graham's algorithm, "first fit, heaviest to lightest" is also known as the First Fit Decreasing (FFD) algorithm. This algorithm works quite nicely if you know all the weights prior to assigning to bins so they can be arranged in order from largest to smallest. In some bin packing problems, we do not have this luxury. In these cases other algorithms must be used. Here are the definitions of the more commonly used algorithms for bin packing.

First Fit (FF)

The next weight to be packed is placed in the lowest-numbered bin already opened into which it will fit. If it fits in no open bin, a new bin is opened.

First Fit Decreasing (FFD)

The first fit algorithm is applied to the list of weights sorted so that they appear in decreasing order.

Next Fit (NF)

A new bin is opened if the weight to be packed next will not fit in the bin that is currently being filled. No previously opened bin can be used.

Next Fit Decreasing (NFD)

The next fit algorithm is applied to the list of weights sorted so that they appear in decreasing order.

Worst Fit (WF)

The next weight to be packed is placed into the open bin with the largest amount of room remaining. If the weight fits in no open bin, a new bin is opened.

Worst Fit Decreasing (WFD)

The worst fit algorithm is applied to the list of weights sorted so that they appear in decreasing order.

Best Fit (BF)

The next weight to be packed is placed into the open bin that will leave the least room left over after it is placed in that bin. If the weight fits in no open bin, a new bin is opened.

Best Fit Decreasing (BFD)

The best fit algorithm is applied to the list of weights sorted so that they appear in decreasing order.

Try to solve the pipe-cutting problems at the top of this page using the different bin packing algorithms. Study the different answers you get from the various algorithms and see if you can understand why the answers differ. Which of the algorithms appears to be most useful? Why? The chart of pipe lengths will be useful.

Pipes and Bubbles: An investigation
Regard this as an educator's guide for teaching the lesson. Use the pages provided but do not distribute descriptions to the learners before time. The activity only works if an element of surprise is retained.

 * Four farmers' houses are exactly at the corners of a large, square field that they share.

 * They decide to put in a windmill at the very centre of the field to pump water for the farmhouses.

 * Naturally they want to use the shortest length of piping possible to reduce their costs.

 * The farmers discuss the matter and come to the obvious conclusion:

four pipes leading directly from the pump to each farmhouse will give the shortest total length of piping &shy and therefore the lowest cost.

Do the measurements. Measure to the centres of the circles and the square and write down the total length of piping in '''ARRANGEMENT A is ............. cm.'''

Now ...

 * ... using wire and pliers make a cube from wire (a cubic frame). The wires at the corners can be twisted together using the pliers. Make a handle so that the frame can be dipped into soap solution.

 * A cube is a "block" shape with 6, square faces. In the case of the cube you are making, it is really a framework that is not filled in &shy like a small version of a climbing frame at a playground.

 * When you and/or the learners have made a cubic frame, dip it in soap solution. (A good recipe for soap solution: 1 part dishwashing detergent, 2 parts water, ? part glycerine. The glycerine makes the soap bubbles stronger. Try these proportions and make adjustments to ensure that good bubbles can be blown. If you have no glycerin, add more detergent.)

 * Each face has a thin film of soap on it. You will also notice that some soapy film forms inside the cube. Shake your cube gently until the soap film settles. Do this a few times until you get the soapy film settling into the same shape of film each time.

 * Now look carefully at how the soap film is arranged inside the cube.

 * If you wind all the edges of the wire frame with knitting wool you will find that the soap films last longer. Make sure that the wool is well soaked with soap solution.

 * The shape of the soap films in the cube gives a clue as to how to arrange the water pipes on the farm? Let learners draw an arrangement of pipes that is different from ARRANGEMENT A and which might solve the farmers' problem.

 * Measure the total length of the pipes in ARRANGEMENT B is ............. cm.

Could the soap bubbles have helped the farmers to save piping?

ARRANGEMENT A Measure the total length of the four pipes, each going directly from the windmill(the sqaure in the image) to each farmhouse (the sircles in the image).

ARRANGEMENT B Measure the total length of the four pipes going from each farmhouse to the centre and the piece of pipe laid crossways, through the position of the windwill at the centre of the field.

title: SA NC Doing Investigations/Chapter 6