Mathematics Developmental Continuum
Carrying out Investigations: 4.5
Supporting materials
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 January 05, 2019