Carrying out Investigations: 4.5
Supporting materials
- Related Progression Points
- Developmental Overview of Working Mathematically (PDF - 31Kb)
- Strategies to approach mathematical problems (PDF - 30Kb)
Indicator of Progress
Students' growing experience in conducting investigations with mathematics leads them to:
- becoming more independent in posing questions, planning what to do and in carrying out investigations
- designing more sophisticated investigations of greater complexity
- identifying the mathematical content of the investigation more clearly
- gathering and organising data more systematically
- reporting the method and results with increasing clarity and precise use of mathematical language
- evaluating the results of their investigation more thoughtfully
Mathematics teaching should include authentic problems arising in real life situations because:
- it motivates many students if they see that they are learning skills useful in everyday life and future employment
- students learn to apply mathematics, and
- it widens students' appreciation of the role of mathematics in society
Illustration 1: Identifying the mathematics involved
Students may need help in identifying the mathematics involved in any real situation.
Students often have difficulty with Maths Talent Quest and similar projects because they have a good idea but do not identify the mathematics involved. For example, as part of a project to design an energy efficient building, students made a scale drawing, in preparation for making a large three-dimensional model. Some students did not realise that they could then use their scale drawing to get measurements when they were making the model.
Good opportunities for substantial mathematics are often missed. For example, students often only find out how much it costs to make something, instead of identifying the Space, Measurement and Number skills that could be used in its construction.
Illustration 2: Organising data systematically - use labelled tables
Students' skills in organising data has a substantial impact on the likelihood that they can see patterns and trends.
Here is an example. Some students were asked to find some perfect numbers. A perfect number is a number whose factors (excluding itself) add to itself. For example, 6 is a perfect number, because its factors are 1, 2, 3 and 6 and 1 + 2 + 3 = 6. The students tried many numbers, and they were not perfect. As they worked, the students erased these wrong numbers and their working. After a while, they had forgotten which numbers they had tried, so they tried some more than once. They were also unable to check their calculations. A much better approach would have been to make a list of numbers tried, and their factors and the sum of their factors in a labelled table.
NOTE: 28, 496 and 8128 are the next three perfect numbers. There is a lovely pattern in these numbers, involving the powers of 2.
Illustration 3: Evaluating the results of the investigation
A thoughtful evaluation compares the results of an investigation with the original aims, to see how adequately the questions are answered, and makes sure the claims are reasonable.
For example, if you wanted to see if the hand-spans of girls and boys were different, you would gather data and probably find averages for each set of people. However if the results were quite close, and there was a lot of overlap of the data you cannot claim that one was bigger than the other.
Thoughtful evaluation also looks beyond the results. For example, students carried out an investigation into painting the school. Later in the investigation, it became clear that they did not consider that different surfaces require very different amounts of paint per square metre. In evaluating the outcome of their investigation, they should consider how much difference this could make to the results.
Teaching Strategies
The best way to learn to investigate is to do it, but during and after each one, it is valuable to reflect on the experience using some general principles. The relevant teaching strategies are:
- to provide rich experiences for students to practise investigations in a supportive environment
- to have focussed discussions on the problem solving strategies that may be useful
- to create opportunities that strongly encourage students to reflect on what they have learned (eg by evaluating and writing about their results)
Activity 1: Pizza value combines data collection with analysis of data to determine value for money. (Investigation contexts are Data and Number.)
Activity 2: Vertices, faces, edges combines spatial data collection with finding patterns and rules. (Investigation contexts are Space and Structure.)
Activity 3: Fermi Problems provides a list of questions that encourage students to think creatively about real world problems. (Investigation contexts are mainly in Measurement.)
Activity 4: Pick’s rule involves an intriguing discovery about areas with patterns and rules. (Investigation contexts are Measurement and Structure.)
Activity 5: Statistical investigations provides a shell for many statistical projects. (Investigation context is Chance and Data).
Activity 6: Posing questions from a data set suggests websites providing data sets that can be used to focus students' attention on posing questions, an important part of the investigative process.
Activity 7: Design a ... suggests an important class of real world investigations that capture the interest of many students, and can be easily linked into school and community life.
The following problem solving and investigation strategies for students to use are from the Problem Solving Task Centre project.
- Play with the problem to collect and organise data about it.
- Discuss and record notes and diagrams.
- Seek and see patterns or connections in the organised data.
- Make and test hypotheses based on the patterns or connections.
- Look in their strategy toolbox for problem-solving strategies which could help.
- Look in their skill toolbox for mathematical skills which could help.
- Check their answer and think about what else they can learn from it.
- Publish their results.
Activity 1: Pizza value
Investigation: Which pizza is best value for money?
The emphasis here is on four aspects of working mathematically:
- determining what data to collect
- systematically organising data (e.g. finding the areas of the pizzas)
- using mathematical ways of analysing data to answer the specific question posed (finding value as a rate)
- reporting the results.
What data to collect?
If pizza value is related to the size, we need to collect data on sizes and prices. But what other variables should we consider?
Product: Does the topping come into the question? It is hard to put a value on people’s favourite toppings, so for now focus just on the area of the base and the price, for any one topping. (Many classes vote on the topping to choose.)
Price: What about the fact that different shops have different prices, for the same size and topping? Some students might want to find the cheapest, and sample different products, but a mathematical comparison is only worthwhile if the areas are similar or the same.
Size: How do we determine the size of a pizza in the shop? Is there a standard? (Yes, there is, but it is in inches!) Can we get permission to measure the diameters? (A shop may give you the cardboard bases)
Choosing the variables to consider requires some careful thought.
Estimating the area of a pizza base
This activity does not require students to have learned the formula for calculating the area of the pizza. The purpose of the activity is to devise a method for estimating the area. Possible methods that students could use include:
- Counting squares
- Arranging pizza slices to form an approximate parallelogram or rectangle
- Approximating the area of the pizza by using the area of the pizza square box
- Cutting out a cardboard circle to match each size of pizza and comparing the areas (eg dividing into squares, or weighing them)
- Weighing real pizzas with the same topping or pizza bases.
Students may find other creative ways of estimating the area and some may already know how to calculate the area.
Analysing the data to measure value for money
To compare the values set up a table in a book or on a spreadsheet. Discuss the appropriate method of calculation. Value is defined as what you get for your money, so centimetres squared per dollar is an appropriate unit. The results are obtained by division of area by cost.
Reporting the results
When students come to report, they should clearly state:
- the question they were investigating
- what they did to get the results
- the mathematics involved
- what the result were
- what the results actually mean.
The medium of presentation should be efficient. Computer graphics might be helpful, and a PowerPoint presentation can quickly show others.
They should also evaluate the adequacy of the investigation, discussing for example whether the identified differences are important, whether important factors were omitted and sources of possible errors.
Activity 2: Vertices, faces, edges
The emphasis in this activity is on collecting and coordinating data.
The question is: "How can you predict the number of edges of a polyhedron from the number of vertices and faces?"
This investigation involves collecting data from as many different polyhedra as possible to find relationships between their numbers of vertices, edges and faces. The investigation is more efficient if different groups share data on different types of polyhedra. A nice starting point for this activity is to ask how many drinking straws and joiners are required to make a polyhedron with a given number of faces (e.g. a cube, tetrahedron, etc.).
Students collect and categorise various polyhedra, e.g. prisms, pyramids. They may need to construct some from nets but it is quicker to collect real three-dimensional examples. Students might bring examples to class. Start with simple polyhedra. The investigation is enriched by including composite examples such as a pyramid on top of a cube (or pyramids on several faces), tetrahedron with a triangular prism on one face etc. Just ensure that when two polyhedra are joined together, they join on a whole face. Note: if they do not join on a whole face, then it is not a polyhedron and the results below do not hold.
Students analyse and tabulate their data and make conjectures about the relationship between faces, edges and vertices. They then test the conjecture on other types of polyhedra.
The result for a very wide class of polyhedron is Euler’s rule:
f + v = e + 2 (where f = faces, v = vertices and e = edges).
For example, for a cube f = 6, v = 8 and e = 12, so 6 + 8 = 12 + 2.
However there are interesting results for the sets of prisms and for pyramids, taken separately. The numbers of faces, vertices and edges depend on the number of sides in the base, n.
For prisms the formulas are: f = n + 2, v = 2 × n , and e = 3 × n.
Example of cube: n = 4, f = 4 + 2 = 6, v = 2 × 4 = 8, and e = 3 × 4 = 12.
For pyramids the formulas are: f = n + 1, v = n + 1, and e = 2 × n.
Example of square pyramid: n = 4, f = 4 + 1 = 5, v = 4 + 1 = 5 and e = 2 × 4 = 8.
Activity 3: Fermi questions - surprising estimations
Fermi questions are estimation questions that are very open-ended and that may initially seem to be impossible to answer. By making some sensible assumptions and estimates, however, we may be able to come up with a moderately good approximation of the real answer quite quickly. Note: They are named for Enrico Fermi, an Italian physicist who liked to ask these kinds of questions.
Here are some examples, with quick solutions to show how they might be answered. Mental arithmetic can be encouraged for these questions.
| How many days would it take to walk from Melbourne to Sydney? |
It is approximately 900km from Melbourne to Sydney (student can find a a more accurate distance if required). I can walk at about 5km/hr, for maybe 8 hours a day which means 40 km in a day. Consequently it will take me close to 23 days to walk there. |
| How many soccer balls could fit in our classroom? |
A soccer ball is about 30cm in diameter. If the classroom is 9m wide and 12m long then I can fit 30 soccer balls across the width, and 40 down the length, which means there is room for 30 × 40 or 1200 soccer balls on the floor. Then, if the room is 3m high, there is room for 10 layers of soccer balls, or 12000 soccer balls altogether. (In fact, there is room for even more. Can you think why?) |
| Could all the people in the world fit in Victoria? |
There are between 6 and 7 billion people in the world; let's make it 7 billion. Let's give each person one square metre, which is plenty of room to stand up in. Since there are 1000 × 1000 (ie, 1 million) square metres in a square kilometre this means that 7 billion square metres is the same as 7000 square kilometres, and so all the people of the world could fit into 7000 square kilometres. How big is this? Well, let's try to think of making a rectangle with an area of 7000 square kilometres. A rectangle that is 70km × 100km has an area of 7000 square kilometres, so we can fit all the people in the world into a rectangle 70km × 100km. Since this rectangle is quite small compared to the size of Victoria (imagine a rectangle using Melbourne to Geelong as one side and going north-west to Castlemaine and Maryborough), then this means that the world's population will fit into Victoria easily. But what assumptions have been made? Should we have clarified the question first? |
Fermi questions are good for:
- helping students develop the ability to choose the right combinations of operations to use (or 'see how all the pieces fit together' or, more technically, 'do mathematical modelling')
- developing multiplicative thinking
- developing estimation skills (both estimating measurements and doing estimation when calculating)
- developing facility with large numbers
- using assumptions and understanding their implications
- helping students think about whether answers are reasonable or not
- allowing students to use a variety of approaches to answer the problem
- helping students make use of benchmarks (e.g., my soccer ball diameter was determined from my hand span, which I know is 20 cm)
- gain greater familiarity with 'social number facts' (such as the population of the world, the size of Victoria)
The attached set of Fermi questions (PDF - 24Kb) provide some starter problems.
You should also take advantage of any Fermi questions that arise naturally in the classroom (e.g., if someone is flying overseas, think about how far it is and ask how long it would take to get there if you were driving), or get students to pose and solve their own Fermi problems. Some useful 'social number facts' are also included on the sheet.
Activity 4: Pick’s rule
Pick's rule can be discovered by careful and systematic recording of correct data.
A mathematician named Pick discovered that the area of any polygon drawn with its vertices on a square grid can be found from counting the number of grid intersections inside the shape and the number actually on the edge. The task is to work out his formula.
Area = half the number of intersections on the boundary + the number of intersections inside - 1
Example: For this kite, there are 9 intersections inside, and 8 on the edge. The area is 12 square units.
To discover the formula, students need to systematically vary only the number of intersections inside, and then only the number on the edge. This will enable them to discover how the formula works.
Students should not be satisfied that Pick's rule will work in all cases from the examples that they have found, but should appreciate the need for a proof, even if they cannot prove it at this level.
Activity 5: Statistical investigations
The emphasis here is on the normal process of a statistical investigation.
- Determine the questions to be answered.
- Determine what data needs to be collected to answer the questions, and from what source (for example, how you will choose the sample to make it represent the population).
- Determine in advance how the data will be used, to answer the questions (averages, graphs etc.). If a comparison is to be made, look at the warning in Illustration 3.
- Prepare the questionnaire, or list of data to collect.
- Collect the data and record it carefully, being especially careful about data mishandling. Keep checking that all the data is present, make lists, run simple checks, double check typing, etc.
- Graph the data and look for outliers or errors.
- Calculate the required statistics to answer the questions.
- Make a note of other interesting things that are discovered along the way.
- Write a careful report of the question, the methods, the calculations and the results. Make sure the results answer the question as clearly as possible. Remember that with samples you can only make a prediction about the population from which the sample comes. You can never be quite sure, as the sample might be biased by accident.
Activity 6: Posing questions from a data set
This activity helps to develop students' ability to consider a data set and then write appropriate questions to form the basis of a mathematical investigation. The teaching approach is to provide students with real life data sets or graphs and then have them pose questions to be answered.
Students need to make decisions about the questions that can be answered using the supplied data set and the questions that might require different or additional data and hence cannot be answered fully.
The following websites provide examples of some of the many useful resources now available on the internet:
- The Australian Bureau of Statistics has a section called Education Resources which is useful for schools.
- The Australian Consumers' Association Choice has some interesting statistics related to fast food
Activity 7: Design a ....
Design problems allow students to use their creative and mathematical abilities to solve real or potentially real-world problems. Some tasks involve spatial concepts as well as computational/numerical concepts.
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Some general ideas
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Some related principles of investigation
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Property developer This is an open-ended design task that can be made into a challenge to see which student or group can make the most profit. Teachers may want to change some of the values or parameters; you could even ask students to design their own property developer problem!
You are developing an apartment block, made out of centicubes, and you have a budget of $1 000 000 and planning approval to build something up to 4 storeys high. Rooms on higher storeys cost more to build but can be sold for more than rooms on lower floors. Rooms have costs associated with painting the external walls and there are tiling costs of $20 000 for every square of external roof. The other costs are given in the table above. You cannot have overhanging rooms (ie, every room must have a room underneath or be on the ground), and every room must have at least one external wall. [Use centicubes or similar, so that each cube represents a room, and the vertical square faces on the sides are walls. Use calculators.] Find the cost and profit for your apartment block. Identify your assumptions. |
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References
Dodge, B.D. (2003) First Steps in Board Game Design Accessed 2 June 2006
Williams, D and Lovitt, C (1998) Problem Solving Task Centre project, Melbourne, Curriculum Corporation.