The Farmer's Puzzle
Years 4 - 10

A mathematician's work begins with an interesting problem.
The focus of this activity is on learning the steps a mathematician takes when they find a problem that interests them.

Preparation

G'day young mathematician.
You are just starting this problem, so you really don't know what you need yet.
Some stuff you could have around just in case might be:
  • Objects like a collection of plastic screw caps that can be used to represent things in the problem.
  • Blank paper and markers for drawing and doodling as you think through the problem.
  • A piece or two of this Square Line Paper.
Two (2) things you know you always need when you are learning to work like a mathematician are:
  • The Working Mathematically page. Printing it for this activity is probably best.
  • Your Maths Journal with the name of the activity and today's date.
Mathematicians never solve a problem straight away. So, they need to keep a diary of their explorations.
That's what happens in your Maths Journal.

These are the steps in solving all problems:

1. READ & UNDERSTAND
  • What is the problem asking?
  • Read, cover, write, check.
2. PLAN A STRATEGY
  • Decide which strategies on the Working Mathematically page might be useful.
  • Choose one to start with.
  • Plan what you will need and how you will start.
3. DO IT
  • Play around with the problem using the top part of the Working Mathematically page as a guide.
  • When you get to 'looking in your Strategy Toolbox' you know which one you might start with.
  • Be ready to modify it as you learn more about the problem.
  • If it doesn't work you can go back and choose another strategy.
4. CHECK & REFLECT
  • Did it work?
    - Can you check it another way?
    - Could there be another solution?
    - How will you know when you have found them all?
  • What happens if...?
    - Change something in the problem and you might learn more.
  • Use your journal notes to prepare a report for your colleagues.
    - If they know what you have done, they don't need to solve this problem all over again.
    Your report could be written, or a poster, or a power point, or a video, or a comic strip or ....

Have fun exploring The Farmer's Puzzle

while you are learning to work like a mathematician.

1. Read & Understand

  • In a moment you are going to click a link to see the farmer's puzzle.
  • Read it through once or twice - no more ... then close the file so you can't see the writing.
  • Use a marker on blank paper to write or draw or doodle things you remember from the puzzle.
  • Then open the link again and check and change what you remembered.
Ready?
Here is The Farmer's Puzzle ...

click the drawing to find it.

Artist: Rob Mullarvey
Hint

  • What did she do?
  • She bought some animals. Right?
  • Sure, but what was special about how she spent her money?
Copy and complete this sentence in your journal.
I have to find out...

2. Plan A Strategy

NO PENS FOR THIS PART.
If you have a pen in your hand you will start doing something before you have planned what to do. Sit on your hands if you have to.
  • Read through the strategy toolbox on the Working Mathematically page.
  • Which strategies do you think might work? - guess & check or make a model or draw a diagram or make a table or write an equation or ...?
  • When you choose one, what will be the first thing you do when you are allowed to pick up a pen?
... (Pause for thinking) ...

Okay. If you know where you are going to start...

3. Do It

Pick up a pen or anything else you need and start.
  • Starting is all a mathematician can do.
  • They know what they are aiming for
    - buy 100 animals and spend $100
    - but they don't know how they will get there or how long it will take to get there.
  • If they did ... it wouldn't be a problem.
Enjoy the journey. You will know when you have the answer because you will know how many of each animal the farmer bought.

There is one hint at the bottom of the page, but don't look unless you really, really, have to.

4. Check & Reflect

Check
So you spent $100 and you know how many of each animal and there are a total of 100 animals. You should be right, but...
  • Can you check your answer another way?

Reflect

  • Is there another solution? If not, how can you be sure? If so, how can you be sure you have found them all?
  • What happens if the problem is still 100 animals and 100 dollars, but the prices are different?
  • Is 100 such a special number? Could a similar problem be constructed with say 80 animals for $80. How about for other numbers?
One class decided to try explore this last one. These three (3) slides will show you something about what happened. If you have the time, you could try to solve at least one of their puzzles.

Just Before You Finish

Prepare a report for your colleagues to explain what the problem is and how you solved it. Perhaps you could make a video with your phone.

 

Answers & Discussion

Hint: Is there anything special that must be true about the number of hens?
That's it. That's all you get. If you haven't figured it out yet come back another time and have another go. Or work with a partner. Almost all the mathematics invented since the middle of last century has been by a team of two or more.
Send any comments or photos about this activity and we can start a gallery here.

 

Maths At Home is a division of Mathematics Centre