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.

This cameo has a From The Classroom section which explains in detail how one primary school used this investigation as an assessment task across the school.

 

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)
Heads & Legs

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. Both the Maths300 lesson and the eTask Package provide print sheets of two types of animal.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 41, Heads & Legs, which includes an Investigation Guide and companion software.

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

From The Classroom

Bayswater Primary School

Sue Mugridge
Teaching & Learning Coach 		With Sue's guidance the school staff dared to use Heads & Legs as a focus task for a school wide professional learning experience based around developing common assessment tasks. Read about the way they used Heads & Legs from K to 6, see photos of student work and examine rubrics the teachers developed.

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