Heads & Legs

Task 14 ... Years 2 - 8

Summary

You are told the number of heads and the number of legs which can be counted in a paddock of buffalo and hens. The problem is to work out how many of each animal are in the paddock.

Materials

1 farmyard
8 buffalo cards and 8 hen cards

Content

basic number skills
'groups of', multiplication, division
number patterns
algebra including:
- generalisation
- simultaneous equations
- linear programming (constraints)

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Some of the iceberg of this task is already presented on the card. The initial 'Sarah & Sam' problem is closed (8 heads & 22 legs), but the card goes on to ask What happens if...? and suggest that it is possible to find a general way of solving the problem which would work for all appropriate numbers. However, there is more.

Focus on strategy
How many ways can students work out the answers? This is an application of the mathematician's question: Can I check it another way?. They could:
- Use the cards to make a model.
- Draw a diagram.
- Reason from legs:
  Everything has to have at least 2 legs so that uses up 16 legs for the 8 heads. There are 6 legs left, so put 2 extra legs on three heads to make buffalo. That gives 3 buffalo and 5 hens.
- Reason from heads:
  Pretend all the 8 heads are buffalo. That means there are 32 legs. That is 10 too much, so take 2 legs off 5 heads to make hens. That gives 5 hens and 3 buffalo.
- Try all possibilities in a table.
  For example:
  
  Problem: 8 Heads, 22 Legs
  Buffalo Hens Legs Yes/No?
  
  1 7 18 No
  
  2 6 20 No
  
  3 5 22 Yes
  
  Many students will appreciate the developing pattern.
Transfer the learning
Suppose that under a fence you could only see the wheels of a parking area for bikes and trikes. Suppose too that someone told you the number of machines and the number of wheels. Create and solve some problems.
Symbolic representation
Assuming students have sufficient expertise in symbol manipulation it is possible to introduce the skill of solving simultaneous equations.
- Represent the known information in symbols, eg:
  B + H = 8 (heads)
  4B + 2H = 22 (legs)
- Transform one equation so that one variable can be represented by the other, eg:
  B = 8 - H.
- Relate this transformation to the problem:
  If you knew the number of hens you could subtract this from 8 to find the number of buffalo.
- Substitute into the second equation and solve for the other variable:
  
  4B + 2H = 22
  4(8 - H) + 2H = 22
  32 - 4H + 2H = 22
  10 - 2H = 0
  H = 5
- Substitute into the first equation to find the number of Buffalo.
The original equations can also be represented graphically and the solution will be the intersection of the two graphs.

Whole Class Investigation
Tasks 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.
The whole class lesson for this task is detailed in Maths300 Lesson 41, Heads & Legs. This lesson also extends the investigation with software. You will need multiple copies of the animal cards, or many plastic farm animals, or objects such as bottle caps and sticks which students can use to make representative animals. The lesson supplies a printable sheet of animals.
Visit Heads & Legs on Maths300:
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Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Heads & Legs task is an integral part of:

MWA Pattern & Algebra Years 3 & 4

The Heads & Legs lesson is an integral part of:

MWA Pattern & Algebra Years 3 & 4

MWA Pattern & Algebra Years 7 & 8

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