# Animal Farm

### Task 229 ... Years 2 - 7

#### Summary

In essence, Animal Farm is a more visual form of Arithmagons (Tasks 188 & 194) which has also been embedded in a simple story shell. This allows younger or less experienced students to enter the family of problems. As one teacher commented, I love this task. It helps them to understand what's going on.. The problems are based around three known numbers, three unknown numbers and addition relationships between them. In the first part of the task students create problems where all the numbers are known. Then come the questions with missing information where the unknown numbers have to be found. It is not necessary to use algebra to find them, even though this will probably be the tool of choice for many teachers. Instead working like a mathematician to play with the problems, collect and organise data, look for patterns or connections and make and check hypotheses is the first principles approach structured into the task.

#### Materials

• 25 'animals'
• One board, marker pen and wiping cloth

#### Content

• equations, creating
• equations, substitution & solution
• mental arithmetic
• patterns, number
• recording mathematics

#### Iceberg

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

Questions 1 and 2 are intended to get students playing with the problem and collecting data about how they are constructed. Once a problem is successfully created on the board it is best if it is transcribed into the students' journals, or onto this Recording Sheet. The numbers involved in each creation become the data for the missing number problems which begin with Question 3. If a permanent record isn't kept this data isn't accessible.

In preparation for the next part of the task, the double line on the card is the signal to discuss the work so far with the teacher. You might ask questions like:

• In the drawing on the card which circle number is the horse counted in? Ans: Two of them, 6 & 7.
• What can you tell me about where the other animals are counted? Ans: They are each counted in two circle numbers?
• Now you have some data about these animal farm problems. See if you can find any connections between the numbers before you start the next section.
One way to work to answer Question 3 is by guess and check. Put some animals in one of the fields and see if you make the numbers work. If you can't, try again. An improvement on this is to try every possible case in an organised way.
• Start in the top field (for example) with the largest possible number.
• In this case, that must be 9 (unless you are willing to accept negative numbers in the right field).
• 9 tells you that left field must be 1 and right field must be 0. Do these two numbers add to 13? No.
• 8 tells you that left field must be 2 and right field must be 1. Do these two number add to 13? No.
• 7 tells you that left field must be 3 and right field must be 2. Do these two number add to 13? No.
• ...
• ...
• 3 tells you that left field must be 7 and right field must be 6. Do these two number add to 13? Yes!!!
• Continue this process to show that there is only one answer (for positive numbers of animals or zero)
There is definitely a pattern in this search. It could be expressed as:
Beginning with the smaller of the left and right circle numbers...
• Put a number in the top field.
• Subtract this field number from the left circle number and the right circle number.
• Add the two answers and if their sum equals the bottom circle number you have found the top field number.
A shorthand way of writing this is for Question 3 is:
• T is the top field number.
• Find 10 - T and 9 - T.
• You know T when (10 - T) + (9 - T) = 13
So now finding T (and hence the numbers in the other two fields) only involves an organised substitution of every possible value of T into the equation. Or, for the mad keen algebraists, solving:
(10 - T) + (9 - T) = 13 <=> 19 - 2T = 13 <=> T must be 3 because 19 - 6 = 13.

Other observations of the data from Questions 1 & 2 that can be lead to other methods of solution are:

• The total of the circle numbers must be twice the total of the animals because every animal is counted twice. So...
• Divide by 2.
• Look for three numbers which sum to this total.
• Choose two field numbers. The difference between them is the same as the difference between the circle numbers nearest the opposite corner. So...
• Put your fingers in two fields.
• Work out the difference between these two field numbers using the difference between the two circle numbers nearest the opposite corner.
• Find two numbers with this difference which also add to the third circle number
• Put them in the correct way around in the 'finger' fields so they work with all the other numbers.

#### Challenge

The challenge asks What happens if the farmer's map is missing one circle number?. The short answer is you get multiple solutions.

Using the example on the card, suppose the 13 is missing. To show that let's call the missing number X. The equations above then become:

• (10 - T) + (9 - T) = X <=> 19 - 2T = X
And we can have fun choosing any number we like for X (as long as it's less than or equal to 19) and calculating the related value of T. In this case there will be 10 solutions for T (all matching odd numbers for X) and one will be T = 3 solution matching X = 13.

#### 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.

To convert this task to a whole class investigation you need lots of objects to be animals and an A4 size Animal Farm for each pair. The farm is fairly easy. Sketch on the board as you say:

• Draw the largest triangle you can on the paper.
• Put a dot in the middle.
• Join the dot to the middle of each side.
• Add a circle near each midpoint.
Or you can print off this Animal Farm Board. If you have Poly Plug, the red board provides 25 'animals', otherwise you need counters or small blocks.

The lesson will develop along the lines of the iceberg above and you will have the opportunity to (a) pause to share individual insights with the class and (b) highlight the various elements of working like a mathematician.

At this stage, Animal Farm does not have a matching lesson on Maths300. However Lesson 63, Arithmagons, extends the problem for older, or more experienced, students. Heads & Legs, a related problem, is explored in Maths300 Lesson 41, which also includes 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.

Animal Farms is not in any MWA kit. However it can be used to enrich the Pattern & Algebra kit at Years 3/4.