What can you learn from the book?
What is mathematical reasoning and investigation?
Mathematical reasoning is the ability to develop and apply techniques for solving a specialised problem and, where possible, to abstract a principle from the specialised problem which might form a general rule or approach to problem solving. In this unit, that includes the ability to do so with visual information (eg, geometry).
In other words, the ability to reason using mathematics means that you can take a problem, figure out some kind of solution that you feel confident that you can justify, and then see if that solution might be applicable in another situation, setting or context.
Take a well-known example.
Painting
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Alma can paint a house in 3 hours, Marvin can paint a house in 5 hours. How long does it take them to paint a house if they work together? |
If you can work out the solution to this, you can probably work out the solution when Alma takes 12 hours and Marvin takes 29 hours – or even when Alma takes A hours and Marvin takes B hours, but a solution that only helps us solve specific painting problems might not be too interesting. At the surface level, Alma and Marvin’s painting problem doesn’t look very similar to the “point of no return” problem.
The point of no return
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A plane that travels 300 km/h in still air takes off with 4 hours of fuel. On the outward journey, it is helped along by a 50km/h tail wind which increases its speed to 350km/h. After cruising for some time the pilot realises that, on the return journey, the plane will only be able to travel 250km/h. What is the maximum distance the pilot can travel from the airfield and still have enough fuel to return home? |
However, the mathematics underlying some approaches to both these problems is actually the same, because both can be tackled by calculating an average of rates. Once we start making these kinds of connections, we can become very agile problem solvers!
What will you learn?
How can we develop our mathematical reasoning?
We hope that, by engaging with the book, your ability to employ mathematical reasoning to solve problems will be extended. More than that, we want you to reflect on what it is about undertaking problem solving that leads to the development of mathematical reasoning, so that you can assist a similar development with other learners.
Here’s how we’ve broken it down into learning outcomes.
Learning outcomes
| Learning outcome | What this means for you |
| Apply results from measurement, geometry, networks, sequences and probability to solve routine problems involving written and visual information. | You’ll need to use what you learn in geometry, measurement, networks, sequences and probability to solve routine problems.
A routine problem is one with a clearly defined method. That is, there’s a set routine when it comes to solving that type of problem. |
| Apply techniques of mathematical reasoning and problem-solving to solve non-routine problems related to measurement, geometry, networks, sequences and probability. | You are going to see a lot of open-ended problems, where there’s no perfect fit for a routine solution. You’ll need to use mathematical reasoning to solve the problems, justify your solution and see if you can apply a similar method in new settings. |
| Write code to develop programming skills and use algorithms and systematic iteration to solve problems and explore data. | You’ll be using a program called Scratch to develop code that can solve mathematical problems and explore patterns in data. |
These are the overall learning outcomes for the book as a whole. Each section has more specific learning outcomes, as well.
