# Garden Beds Years 4 - 10

### Preparation

• Print this square tiles template
Cut off the white strip and the half / half strip beside it.
You might need the other tiles later.
• Cut the two strips into separate square tiles.
• Print this Square Line Paper to record the shapes you make.
• Write the title of this challenge and today's date on a fresh page in your maths journal.
When the activity is over your recording sheet should be put into your journal beside any other notes you make.

### Getting Started

• Open this Garden Beds Starter. You can read it on screen or print it.
• Use the questions and the Challenge on the starter to begin your exploration of Garden Beds.

Have fun exploring Garden Beds.

Teachers learning about Garden Beds

Note: If your school is a member of Maths300, your teacher can provide software that will help you explore Garden Beds.

### Digging Deeper

• In a moment you will start a slide show called a Picture Puzzle.
It will guide you through more of this activity.
• Slide shows don't have to be 'finished' in one lesson.
They can be revisited and continued at another time.
• Use the arrows on your keyboard to change slides.
• Do all that each slide asks before you move to the next one.
Start your Picture Puzzle by clicking the image.
Say yes to 'Full Screen' when asked.
Use 'Esc' to move out of full screen.
You can also get to full screen in the View menu.

### Changing The Picture

Extra Investigation
What Happens If...?
• What happens if ... I tell you any number of plants in a row and any number of those rows?
• What happens if ... I tell you the number of rows is always the same as the number of plants?
• What happens if ... I tell you the plants are always planted in an L shape with equal arms.
Investigate how to calculate the tiles.
• What happens if ... I tell you the gardener always puts a double border of tiles around any garden?
Investigate how to calculate the tiles.
Make up your own rule for laying out a garden and discover how to calculate the tiles.
Special Note: It really doesn't matter whether you try any of the 'What happens if...?' challenges (although we would like you to try one). What really matters is that you learn there is always another question a mathematician can ask.

### Just Before You Finish

For this part you need your maths journal and your Working Like A Mathematician page.
• Look at your notes for this activity. Think back through what you did.
• Draw an oval in your journal.
• Change it into a face that shows how you feel about Garden Beds.
• Add a speech bubble if you wish.
• How did you work like a mathematician through this activity? Record at least 2 ways.
• What do you know now that you didn't know when you started Garden Beds?

### Answers & Discussion

These notes were originally written for teachers. We have included them to support parents to help their child learn from Garden Beds.

### Garden Beds Gallery

Send any comments or photos about this activity and we can add them to this gallery.

30th March 2022
 Hi Doug, Just popping into your Inbox to say thanks for another great activity. The children and I had fun with Garden Beds today. We had great discussions about how we could determine the tiles for any number of plants and my 12 year old extended her learning with the 'Ordered Pairs' graph activity.  I've attached a photo of the children's work for your activity gallery.  I've also shared the link to the activity on my Instagram site so hopefully other homeschooling families will take advantage of these great lessons! Blessings, Jo Click the photo to access a high resolution version, then use the zoom tool in your software to read the detail of the children's recording and the Ordered Pairs Investigation Guide. You will be impressed.

31st March
Following a question asking how she had been able to encourage her children to achieve so much with Garden Beds, Johanna sent this description of her learning objectives, environment and teaching craft.

Hi Doug,
After we had spent time working with tiles and plants and talked about our thinking process, we used the Picture Puzzle resource. I wanted the children to understand what their thinking was before encouraging them to see that the same problem could be looked at from different angles and still provide an answer.

For example:
When we did the '1 plant, how many tiles?' my 7yr old responded straight away.
"We'll need 4 tiles."
I said, "How do you know?"
"No, wait a minute, I forgot the corners, we actually need eight."
"Ok, what about 2 plants?"
"Well, the ends have 3 tiles and then the middle, well, they have a tile above and below them. We need to count 2 for each plant and then add on the ends."
My 10yr old jumped in and said, "However many plants there are we need to multiply that by two and then add 6 for the ends."

We spent a bit of time working with the tiles. I would call out a random number and the children would try to work out the tiles before modelling it with the materials to see if they were correct. We then worked through the Picture Puzzle trying to see and understand how others might work out a solution to the problem. This also took time. I asked the children what they thought was being illustrated. With some of the pictures they couldn't work out the thinking. It seemed strange to them and one of my girls commented "That's just complicating a simple thing.". But I tried to help them understand that there are many different ways to 'see' a problem. Just because it's not the same way that they would do it doesn't make it wrong ... just different.

We did this over a morning, unrushed, I would say we spent about 1 1/2 hours.

Then in the afternoon I asked the children to stick some garden beds in their math journals and write down their thinking for how to work out a garden bed of any number of plants. For my 12 year old, I asked her to also do the order pair activity. I went through it with her to make sure she understood what was required. After they brought me their finished work I had them tell me about what they had written.

Over all it was a day of work. But in a relaxed way, not rushing the process.

Blessings,
Jo

Those different ways of seeing the problem in the Picture Puzzle, all of which have originated from real learners, are important for at least two reasons:
1. School mathematics is about learning to work like a mathematician and one of the mathematician's key questions is Can I check this another way?. Exploring the ways other learners have thought about this problem validates the developing confidence that in maths there is always another way. It's the perceived existence of 'another way' that helps a learner help themselves to learn mathematics.
2. The different ways of explaining the 'any number of plants' problem are a perfect example of equivalent algebraic expressions, a topic in any secondary school text book. Is there learning advantage in coming at this usually rule-based, symbolic topic from what learners can confidently see, touch and say??

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