Four and Twenty Blackbirds

Task 62 ... Years 2 - 10

Summary

Twenty-four birds land on eight feeding tables that are mounted around the edge of the Royal Garden. The garden is a rectangle and the feeding tables are at the four corners and the four midpoints of the sides. The birds arrange themselves so that nine can be counted along each side of the rectangular garden. How many are on each feeding table?
  • How many solutions are there?
  • How do you know when you have them all?
Cube Tube offers two videos from the whole class investigation of this task in a Swedish classroom. The first illustrates how the students have become captivated by the problem. The second is an explanation by three students of how they believe they can find every solution.
 

Materials

  • 24 blackbirds and a Royal Garden

Content

  • basic operations with number
  • searching for, finding and explaining patterns
  • problem solving strategies such as:
    • breaking a problem into smaller parts
    • trying every possible case
  • concept of proof
  • investigating "What happens if...?"
Four and Twenty Blackbirds

Iceberg

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

As the task card suggests, there are many ways to arrange the 24 birds so there are 9 on each side of the rectangle. For younger students it will be a joy to find only some, but the big challenge for older, or more experienced, students is to find them all. To do this the students have to find sufficient solutions to be able to discover clues such as:

  • the corner numbers are important because they are counted in two directions.
  • the sum of the four corner numbers is 12 (a corollary is that the sum of the opposite middle numbers is 6).
  • the numbers that go in the corners are limited by the total along each side being 9.
  • some solutions are reflections or rotations of others, so if we count them as different we are going to have to find many more solutions.

One way to approach finding all the solutions is to focus on one corner, say the top left, and break the problem into parts by trying each number in turn in this corner. For example, if 9 is in the top left...

9    
   
     

then we automatically know the top row and the left column...

9 0 0
0  
0    

and there must be 15 birds left to place. Starting with the bottom right corner, let's explore the possibilities.
  • placing 9 would solve the problem of nine each side, but would not use all the birds.
  • placing 8 would mean 1 more bird only could be placed along the bottom and up the right, so only 10 of the fifteen remaining birds would be placed.
  • placing 7 would mean 4 more birds could be placed bottom and right, using only 11 birds.
  • placing 6 would mean 6 more birds could be placed bottom and right, using only 12 birds.
  • placing 5 would mean 8 more birds could be placed bottom and right, using only 13 birds.
  • placing 4 would mean 10 more birds could be placed bottom and right, using only 14 birds.
  • placing 3 would mean 12 more birds could be placed bottom and right, using 15 birds as required.
  • placing 2 would mean 14 more birds could be placed bottom and right, using 16 birds, which is too many.
  • placing 1 would also use too many birds.
So, with 9 in the top left corner, there is only one solution...

9 0 0
0 6
0 6 3

Carefully continuing this reasoning for 8, 7, 6, ... in the top left corner will lead to finding all solutions.

The card also suggests the What happens if..? questions:

  • What happens if there are a different number of blackbirds, but the side totals are still 9?
  • How many numbers of blackbirds can give a side total of 9?
  • What happens if there are only 24 blackbirds, but the side total is 8?
  • How many side totals are possible for 24 blackbirds?
  • What happens if the Royal Garden was designed like this...?

 
   
     

Could the 24 blackbirds be placed with 9 along each side of the triangle?

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.
   

As a whole class investigation, there are three parts. Firstly a class 'floorboard' where the story can be told and each of the students can contribute a large blackbird to an attempted class solution.

Floorboard model
This is followed by a duplicated Royal Garden sheet (the one shown, and the big birds shown, are available from Poly Plug & Tasks below, or from Maths300) and materials to represent the blackbirds so each student pair can explore solutions. If you have Poly Plug, the red plugs are perfect for the problem.
Tabletop model
In the third phase, students begin personal recording at the same time as a class record of solutions is developing on the whiteboard.
Floorboard model
The investigation then proceeds in any of the directions described above.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 62, 4 & 20 Blackbirds, which includes and Investigation Guide.

Visit 4 & 20 Blackbirds in Menu Maths Pack D (where it is called 4 & 20 Red Birds).

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 Four and Twenty Blackbirds task is an integral part of:

  • MWA Number & Computation Years 5 & 6
  • MWA Number & Computation Years 9 & 10

The Four & Twenty Blackbirds lesson is an integral part of:

  • MWA Number & Computation Years 5 & 6
  • MWA Number & Computation Years 9 & 10

Green Line
Follow this link to Task Centre Home page.