Ainsley and the Planting Problem
Years 3 - 9

Preparation

  • A large piece of paper or four A4 pieces and sticky tape
  • At least ten (10) small objects like pebbles, pasta, blocks or...
  • Write the title of this challenge and today's date on a fresh page in your maths journal.

The Story

Ainsley was asked to try this problem when she was home schooling with Grampa in Year 3.
The story on the teacher's slides went something like this:
  • Dad's three (3) children liked to help in the garden.
  • They were 7 years old, 6 years old and 5 years old.
  • Dad thought they would learn more about gardening by looking after their own plants.
  • He said he would buy them one (1) plant for each year of their age. The children thought that was a fun idea.
  • So they packed a picnic lunch and went to a big plant nursery.
  • At the nursery Dad discovered he only had enough money for ten plants.
"Mmm" said Dad. "Let's sit under that big gum tree over there and have our picnic while I think about what we can do."

First Thoughts

  • Think about Dad's situation.
  • He doesn't want to disappoint the kids.
  • In your journal write or draw any suggestions that might help Dad.

Aha! Moment

At the picnic spot the oldest child could see that Dad was a bit sad because he couldn't afford to keep his promise.
"It's okay Dad. We don't mind. Just get ten plants and we can share looking after them."
That helped Dad relax a bit.
He started thinking as he opened up the picnic backpack and began passing out the lunch.

A few minutes later, in the middle of a cheese and Vegemite sandwich, Dad suddenly jumped up, grabbed a stick and drew this picture in the dirt.
"You're right!" he almost shouted. "I can buy ten plants and you can share ... and if I design the garden like this you can still be caring for the same number of plants as your age."
Then he picked up ten gum nuts and put four (4) of them in the centre...

Challenge 1

Ainsley's teacher asked the class to find out what Dad did with the rest of the gum nuts.
That's your challenge too.
How did Dad place the ten gum nuts so that each child was caring for the same number as their age?
Have fun exploring Ainsley and the Planting Problem.

Hint

  • Who looks after the plants in the middle section?
  • Who looks after each of the other sections?

When you find out how Dad did it, draw and write in your journal to explain.
After you try your own way, you can look at Ainsley's answer below.

Challenge 2

Grampa suggested to Ainsley (the real one) that a mathematician might start wondering if there was another way to do it.
"What do you mean?"
"Could there be a different number of blocks in the middle, so that the ones left in your hand still work out?" Grampa answered.
Ainsley started moving the blocks around on the drawing.
She tried other numbers of blocks in the middle.
Each time the middle ones were the only ones shared.

Your challenge is to:

  • Try 1, 2 and 3 in the middle as the only ones shared.
  • Explain in your journal what Ainsley found out.
When she finished Grampa said, "But you only tried numbers less than four. What happens if you try numbers bigger than four in the middle and they are the only ones shared?"
So she tried this one. Then she stopped.

Your challenge is to:
  • Explain why Ainsley stopped at five.

Challenge 3

When Ainsley explained to Grampa why she thought the only answer was four in the middle, he told he was proud of her for trying every possible way, just like a mathematician would.

Then he asked if she had used all of her drawing.

"What do you mean?"
"You used the centre. Point to it. Now point to all the other sections you used."
"Oh, I get it," said Ainsley, "I left out the bits of the garden shared by two children."
"Yep. So what would happen if you tried using those sections as well?"
Ainsley thought for a minute. Then she surprised Grampa by saying,
"I know. I'll put the differences in those places first."
Ainsley did find an answer when she started this way.
This time she remembered to use all the sections of the garden if she needed to. Your challenge is to:
Start the way Ainsley did and find her solution.
Record your answer and your thinking.
You can see Ainsley's solution below.
Don't look until you try for yourself first.

  • Mathematician's are never finished with a problem.
  • They are only finished with it for now - until someone asks the next question.
  • Then they can come back to it.

Ainsley decided to stop here.
She made a good journal record of her work, then did other Year 3 stuff.
The next questions are waiting for her when she comes back.
You can start them now.

If you decide to finish here, go to the Just Before You Finish section below and do that first.

Challenge 4

Ainsley's solution in Challenge 3 has two plants in the middle.
There is another way to do it with two in the middle.
Find a different solution with two in the middle.

Challenge 5

Now we have a problem with three correct answers so far:
  • 4 plants in the middle ... 1 answer
  • 2 plants in the middle ... 2 answers
Mathematicians love a problem like this. It means they have to ask:
  • How many solutions are there?
  • How do I know that I have found them all?

Mathematicians don't expect to answer questions like this in one day. They would probably leave their drawing, blocks and some scribble paper on the table and come back to look for more answers when they could.

Your challenge is to keep on looking for solutions until you know that there can't be more.

At least you know there can't be more than four in the middle.
But what is the lowest number that might be in the middle?

It might help if you record your solutions in a table like this which shows the seven (7) sections of the drawing:

ABC A B C AB AC BC
4 3 2 1
... ... ... ... ... ... ...
2 2 2 1 2 1
2 2 1 1 2 1 1
... ... ... ... ... ... ...

Extra Challenges

Most of mathematics has developed when someone asked a What happens if...? question.
  • What happens if Dad had thought of the planting idea one year sooner when the children were aged 6, 5 and 4 and he could still only afford ten plants?
  • What about if he thought of it two years earlier?
  • Is there a lowest age limit if ten plants are all Dad can ever afford?
  • What about if he thought of it one year later, when the children were 8, 7 and 6 and he could still only afford ten plants?
  • Is there a highest age limit if ten plants is all Dad can ever afford?

Just Before You Finish

The only reason for tackling this problem is to practise working like a mathematician. And you can prove that you have been.
It doesn't matter how many solutions you found, even if it was zero, follow these steps to discover how you were working like a mathematician.
  • Print this Working Like A Mathematician page.
  • Read every line carefully and tick anything on the page that you were doing or thinking during the problem.
  • When you have finished stick it in your journal with the heading: I worked like a mathematician when I...

 

Answers & Discussion

Challenge 1

  • Ainsley joined some paper with sticky tape and drew the picture.
  • She gave the children names.
    She thought oldest child should have her name.
  • She got blocks from the toy box and put four in the middle like Dad did.
  • Then she had six (6) left in her hand.

This is what she told Grampa.

Christine is already sharing four, so she only needs one (1) more to make five (5).
So I'll put one in her bit.
Now I've got five left.
Bill needs two (2) and Ainsley needs three.
Hey, that makes five and that's what I've got.
It works!
And she put down the last blocks with a big smile.

Challenge 3

After she put in the differences, Ainsley put two in the middle so Christine would be finished.
Then the rest was easy. And she didn't need to use all the areas.

Send any comments or photos about this activity and we can start a gallery here.

 

Maths At Home is a division of Mathematics Centre