The seventh wave
The purpose of this activity is to engage students in using mathematical strategies to solve a problem involving a sequence.
About this resource
This activity assumes the students have experience in the following areas:
- Continuing a sequential pattern.
- Expressing the rule for a sequential or growing pattern.
- Find an unknown value or figure in a sequence.
The problem is sufficiently open-ended to allow the students freedom of choice in their approach. It may be scaffolded with guidance that leads to a solution, and/or the students might be given the opportunity to solve the problem independently.
The example responses at the end of the resource give an indication of the kind of response to expect from students who approach the problem in particular ways.
The seventh wave
Achievement objectives
NA3-3: Know counting sequences for whole numbers.
NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
Required materials
See Materials that come with this resource to download:
- The seventh wave activity (.pdf)
Activity
Sam, the surfer, knows that the seventh wave in each set is the biggest. It is also the middle wave in each set.
Sam is surfing with five other surfers. To be safe, they each take a wave in turn. Sam takes the very first wave of the first set.
They are in the water long enough to surf every wave of ten sets. How many "seventh waves" does Sam get?
The following prompts illustrate how this activity can be structured around the phases of the Mathematics Investigation Cycle.
Make sense
Introduce the problem. Allow students time to read it and discuss it in pairs or small groups.
- Do I understand the situation and the words? (Students may need to act out how the allocation of surfers to waves occurs.)
- How do I expect there to be a pattern in who gets the seventh wave? Why?
- What will my solution look like? The solution will be how many 7th waves Sam gets in 10 sets. (The result will be justified by a diagram and calculations.)
Plan approach
Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.
- What strategies will be useful to solve a problem like this? (A table or diagram might be useful, though the problem can be solved abstractly with numbers.)
- What maths do I know that is likely to be helpful? (Predicting a sequential pattern and finding multiples are useful skills.)
- How many surfers are there? How many waves are there in total? How many waves does each surfer get in a set?
- How does that knowledge help to solve the problem?
- How will I record my thinking so I can see any patterns?
Take action
Allow students time to work through their strategy and find a solution to the problem.
- What is my first strategy? Why did I choose to do that?
- Am I recording my workings in an organised way that will help me see patterns?
- What patterns am I seeing? How can I describe the patterns?
- How are the patterns helping me to predict when Sam gets a 7th wave?
- How do my results look different or different to others? Why could this be?
- Do others have better ways to solve the problem? How might I adjust what I am doing?
Convince yourself and others
Allow students time to check their answers, and then either have them pair share with other groups or ask for volunteers to share their solution with the class.
- What is my solution?
- Are my workings clear for someone else to follow?
- Did I account for all the waves in my workings?
- How would I convince someone else that I am correct? Have I justified my answer?
- Is my rule expressed in a mathematical way?
- What strategies worked well in this situation? What other situation might it work for next time?
- Is there some mathematics I need to learn to solve similar problems?
Examples of work
The student creates a 10 x 13 array to represent the ten sets of 13 waves. They mark the waves that Sam catches using an x then check how many x’s are in the 7th row.
The student uses a table representation to match wave numbers to surfers in the first set. They adjust the sequence by one for the next nine sets and recognise when Sam will next get a 7th wave.
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