Pointy FencesTask 181 ... Years 4  10SummaryOne hopes students haven't had too much experience with climbing over walls protected by spikes, but nonetheless the story shell gives the task a context that helps them imagine how the mathematics works. So, this visual and kinaesthetic challenge leads into a number pattern and the pattern encourages generalisation. Therefore this is another in the visual algebra series which includes 4 Arm Shapes, Garden Beds, Staircase, Double Staircase, Fold Up Houses and others. As with all these tasks, the way students 'see' the construction can lead to different ways of describing it, which in turn gives rise to equivalent algebraic expressions.Materials

Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
It is important to realise that the task is asking for the total number of shapes (N) in each case, but to calculate that students need to consider the two parts of the fence top, that is, the pyramids made from triangles and the square 'pathway' around them. Fence tops 1 and 2 can be made with the equipment and students quickly succeed in discovering that if the number of points (P) is one, N = 12 and if P = 2, N = 18. The key to generalising the problem is visualising sizes of fence top when materials aren't provided to make them, so Question 2 encourages that visualisation with the support of the drawing on the card and the Recording Sheet for sketching. Then students will discover that if:
Question 4, which asks for N given P = 10, might now be worked out in a number of ways, such as sketching or seeing the plus six pattern along the bottom row. The answer is that N = 66. However neither of these approaches is very efficient for P = 100, so if they haven't begun to do so already, students will need to be verbalising how they see the construction in order to efficiently calculate that N = 606 in this case. So far, so good, but now comes the real challenge  explaining how to calculate N for any P and being able to do so in two different ways (at least) because this is what a mathematician has to do to check their work (see the Working Mathematically process). Possible generalisations are:
For teachers there are now several ways to extend:

Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 
It is important to have equipment for each pair to start this investigation, otherwise it becomes just another textbooklike exercise. Many schools have 3D Geoshapes or MiniGeofix which are perfect for making the pointy fences. The materials list above shows how many squares and triangles are needed for each pair. It is useful to have enough prepared packs in sandwich bags for each pair. Begin the lesson by gathering the students around a Size 3 pointy fence you have made in advance. Also in advance, sketch the Size 3 on recording paper as in the diagram on the card, and scan or photograph it for projection. Introduce the SharpAsATack company and explain that we have been hired to find a way to explain to their employees how to know the number of shapes to use to make any size pointy fence. Invite pairs to make Size 1 and Size 2. Display your Size 3 drawing and ask them to draw Sizes 1 and 2. Develop the lesson using the notes above as a guide. Round off by asking each person to produce a poster to be displayed in the factory so that the workers will always know how to calculate the number of shapes and why the rule works. The poster should also include how to calculate not just the number of shapes, but how many will be squares and how many will be triangles. (We would be happy to include photos of your students' posters in this cameo.)
At this stage, Pointy Fences does not have a matching lesson on Maths300, but Lesson 16, Garden Beds is very similar. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Pointy Fences task is an integral part of:
