How to use the Problems
These questions can be used by a whole class once a week or once a fortnight. Depending on students’ prior knowledge they might only complete a couple of parts of a question or make it all the way to producing a full solution to a “Further Exploration.” Note however that the further explorations are intentionally very open and generally require quite complex ideas to completely solve and
Alternatively, they could be used as an early finisher activity, or as the basis for an extension maths group at a lunch time or before school.
Answers and Solutions
The first thing we want to say is that the words ‘answer’ and ‘solution’ means two different things.
An ‘answer’ is a short reply that gives the answer to the question asked. For example, in question 1a, the answer is two. Sometimes we say a few extra words that roughly give some idea of how you might get the answer. Answers are included at the end of each question. Just click the text and they will appear.
On the other hand, a ‘solution’ requires more effort. It is where the story of how to get the answer is written out.
Solutions can be found at the top of each set of questions.
Problem Solving Techniques
By completing some (or all) of the questions below we hope to build students’ awareness of and competence with four specific problem-solving techniques. These are:
- Acting it Out
- Working Systematically
- Starting with simple cases and finding a pattern
- Extending and generalising*
*Note that Extending and generalising may not seem like a problem-solving technique, however often when we go beyond a problem, we find even nicer ways to solve the original one.
Acting it Out
‘Acting it Out’ involves representing the problem in some way. It could be with a written model, physical objects or even people. For instance, a written model can be used in Problem 1, physical objects can be used in Problem 2 and people are perfect as frogs for Problem 3.
For problems requiring diagrams, such as Problem 1, it is important to make them sufficiently large so all details can be clearly seen.
Working Systematically
When a problem involves lots of steps or raises many cases to consider, it is helpful to work through the steps or cases in some order to ensure none are missed. Doing this is called ‘working systematically.’ When working systematically we work in a methodical and efficient way which could clearly show others that we are using a pattern or system. (NRICH)
In fact, Problem 1 below is a perfect example of where working systematically is beneficial. The image below shows how to approach the problem systematically.
Even with only 5 steps we had to work systemically to ensure all cases were considered. In this problem, working systematically involved considering three separate cases: zero two-hops (1 way), one two-hops (4 ways) and two two-hops (3 ways). In each of these cases we work systematically again, for example when considering the one two-hops we start with it being in the last position, then the second last position etc.
When working through the following questions, think carefully about how you will work through the cases to ensure you don’t miss anything.
Start with simple cases and finding a pattern
Mathematicians love to look for patterns to help them solve problems. To support students to do this too, in many of the problems we use scaffolding questions to expose students to a sufficient number of smaller cases that will help them see a pattern. However, students can still see questions in isolation so encouraging them to document their findings in a table can be really helpful.
Extend and generalise
Mathematical knowledge is forever expanding. Almost every idea can be extended in some way. Generalising is the process of trying to describe what happens “in general.” If we can create a formula or rule to describe what happens in all cases we’ve done a really great job!
A note on checking as we go
As you or your students work through the problems, check, check and check some more! When we check our work and see that we are correct, we can celebrate the fact that we are moving forward. When we check and find ourselves to be incorrect, we potentially save hours of unnecessarily travelling down the wrong path.