Garden Beds

Task 147 ... Years 4 - 10

Summary

Plants are sown in a row and a path of tiles is built to enclose them. If you are told any number of plants, can you calculate the number of tiles needed?

This is only the start of an investigation which can illustrate most of the elements of pattern and algebra. Trying different numbers of tiles to check the hypothesised rule is substituting into an equation. Asking the backwards question, If I tell you the number of tiles, can you tell me the number of plants? is solving an equation. Application of the mathematician's question, Can I check it another way?, leads to equivalent algebraic expressions. Realising that plants and tiles form number pairs with an order - the number of plants determines the number of tiles - leads to linear graphs.

In addition, the From The Classroom section shows that the problem is a meaningful investigation for Year 2 children, who get quite excited about the numbers they are challenged to work with and the patterns they find.

Garden Beds also appears on the Picture Puzzles Pattern & Algebra B menu where the problem is presented using one screen, two learners, concrete materials and a challenge. The teaching notes for the Garden Beds Picture Puzzle are available here.

School Mathematics is Learning to Work Like a Mathematician (PDF) is both professional reading and the framework for a DIY professional development workshop. Garden Beds is the example investigation in this professional challenge to reconsider the context in which content is delivered and the value of making deliberate teaching craft choices.

Materials

About 5 tiles in one colour and 16 in a different colour

Content

algebra including:
- concept of a variable
- generalisation
- substitution
- solving linear functions
- domain and range
- equivalent algebraic expressions
arithmetic, addition & subtraction
arithmetic, multiplication & division
equations, substitution & solution
graphical representation
mental arithmetic
multiplication, array concept
multiplication, calculations / times tables
patterns, number
patterns, visual

Iceberg

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

In true textbook style the card first asks students to work out the number of tiles for 1, 2, 5 and 10 tiles. The first three can be found using the tiles if necessary, but the fourth encourages further visualisation, either with a sketch or imagined. The answers are: 8, 10, 16 & 26. However, correct answers does not reveal the method for working them out. Perhaps the students 'see' a construction pattern such as 2 tiles for each plant and 3 more for each end. But perhaps they have used a form of tabulation and discovered a number pattern such as this...

Plants	Tiles
1	8
2	10
3	12
4	14
5	16
...	...

...which is sufficient data to suggest continuing the counting by twos pattern down the Tiles column to obtain the answer for 10 plants.

Question 2 encourages students to explain. Either of the approaches above will lead to a generalisation that will provide the answer for 100 plants (206 tiles).

To find the number of tiles, double the number of plants and add six ... which can lead to T = 2xP + 6
To find the number of tiles, start with 8, then add twice the number of plants less 1 ... which can lead to T = 8 + 2x(P - 1)

The challenge suggests there are at least two more ways to work out the number of tiles for any number of plants. How would you use the tiles to interpret these alternatives which have been suggested by students in the past?

To find the number of tiles, add 3 to the number of plants and double that ... which can lead to T = 2x(P + 3)
To find the number of tiles, add 2 to the number of plants, multiply that by 3, then subtract the number of plants ... which can lead to T = 3x(P + 2) - P

Are there other ways?

Extensions

The challenge opens the door to equivalent algebraic expressions and several other extensions are suggested in the summary above. There are also the What happens if..? questions such as:

What happens if there is more than one row of plants?
What happens if the garden bed is a different shape, for example, an L shape or a square?

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

Visit the Home Page for more Background.

For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

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.

Garden Beds can be introduced away from the front of the room on the floor or sufficiently large table using squares of card or coloured paper similar to the tiles in the task. You will also need sets of tiles or blocks for each pair. Present the story shell and involve students in placing the cards. Record data from various lengths of garden bed on the white board and set the main challenge of If I tell you any number of plants can you tell me the number of tiles?. Invite students to return to their tables with a partner to explore this question with the tiles. Poly Plug can also be used if you adjust the story to be stepping stones bordering the garden beds in the same way.

Develop the lesson into pattern, tables, equivalent algebraic expressions, backwards questions, ordered pairs or graphing as appropriate. There is sufficient in this investigation to spend several lessons with it

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 16, Garden Beds, which also includes software and an investigation guide with answers and discussion. You will also find some fantastic student work in the Classroom Contributions, including some amazing PowerPoint presentations from students at Settlebeck High School, UK.

Visit Garden Beds on Poly Plug & Tasks.

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 Garden Beds task is an integral part of:

MWA Pattern & Algebra Years 5 & 6

MWA Pattern & Algebra Years 7 & 8

The Garden Beds lesson is an integral part of:

MWA Pattern & Algebra Years 5 & 6

MWA Pattern & Algebra Years 7 & 8

From The Classroom

Svartedalsskolan
Hisingen, Sweden

Jakob Norrby
Year 2

Jakob's school organised a half day Discussion Lesson program with a visitor from Mathematics Centre. Jakob observed a Year 9 class exploring Garden Beds. He was thrilled with the way the learning unfolded with these teenagers. As the teachers' discussion after the lesson unpacked the teaching craft, Jakob decided that his Year 2 children could understand the problem and learn from it. After lunch he invited the visitor to observe these much younger children at work on the problem and both were excited by the children's enthusiasm for the problem and willingness to extend it to 'big numbers'.

Telling the Story

My Grampa planted a garden with three plants.
He made a path of stones around them like this...

They went all the way around.
Can you show me how you think he made the path?

Extending the Story

Grampa liked his plants so much that the next year he planted 5.
How many stones do you think he will need to go around this garden?

Will you help me make the path so we can check your guesses?

Great so if there are five plants he will need 16 stones.

The next phase of the lesson encouraged exploring different size gardens - first 6 plants ... then 10 ... then 20. Lots of work with materials and lots of recording. Some children gave up modelling and began drawing, especially when the number of plants got bigger.

Can you see the children's concentration and enthusiasm for the problem and pride in their work? Some began to sense that because of the building pattern, they didn't have to count all the way around to know the answer.