The X Factor: Using Algebra to Solve Problems

# 2. Garden Beds Suggested Learning Intentions

• To explore, construct and explain generalisations representing relationships between two variables
• To understand and justify that there can be different ways of writing a generalisation

Sample Success Criteria

• I can model a problem using manipulatives and find a solution
• I can construct at least two generalisations to represent a pattern or relationship
• I can explain why these are valid in the context of the problem
• I can draw a graph of the relationship
• I can justify my solution using a variety of manipulatives
• Blocks or square counters of two different colours (to represent the tiles and the plants). Allow at least 5 of one colour and 16 of the other colour per student, or group of students.
• Garden beds – recording and reflecting sheet: docx PDF

This stage has been inspired by the Garden Beds problem on Maths300, and is reproduced with permission. Access to the problem on Maths300 requires a subscription.

Tell students a story:

'My uncle and auntie own a nursery. They sell beautiful golden plants, that are quite rare and fragile. People who buy a plant also buy tiles to surround the plant with, so the plants can be protected'.

Using the blocks or square counters, show students how the first garden bed is constructed (see diagram below). Ask them how many tiles were needed. 'Some customers like to buy more than one plant, and plant them in a straight line, so they need some more tiles. Every time a customer buys plants, my uncle and auntie have to race into the back room and build a model and count the tiles needed. Perhaps there is a way of calculating this more easily rather than counting them each time?'

Show students how the second, and subsequent garden beds might be constructed. Again, ask them how many tiles were needed. Ensure you do not use any particular strategy or pattern to ‘place’ the blocks, as it is imperative that students develop their own ways of constructing the garden beds.  Give students time to construct several different garden beds using the blocks, so that they can investigate any relationship between the number of plants and number of tiles.  Tell students they will be recording their findings and discuss how they might do this. Prompt students to use a table of values if needed.

Enable students who require further support to organise their thinking for this and subsequent parts of the task by using the Recording and Reflecting sheet.

After some time, pose a problem to students:

'A customer came in and bought 100 plants to plant beside her long driveway. How many tiles did she need?'

Observe the way that students interact with this problem. Some will continue to use the concrete materials, while others will look for patterns and shortcuts. Allow students time to experiment and discuss their varied approaches.

After some time, pose a second problem for students to solve:

'How could you work out the number of tiles required for a garden bed of any size?'

Students may choose to respond to this in multiple ways. Some may write a worded explanation, while others may begin to form expressions or rules, with or without the use of variables, or reason from patterns in the table. Yet others may draw a diagram. Importantly, these various ways of solving the problem reflect the concept of ‘multiple representations’ – and they all qualify as forms of algebraic reasoning. Encourage students to consider the strategy they used to determine how many tiles were needed for 100 plants, and how these strategies could be used to solve this second problem.

A common response from students to the generalised problem above is likely to be:

'You need two tiles for each plant, and then six for the ends', or

'T = 2p + 6, where T represents the number of tiles, and p represents the number of plants.'

However, students who view the construction of the garden beds in different ways will draw and describe the pattern differently, which can lead to different generalisations.

A key feature of this activity, when compared to Paths in the Park, is the broader range of generalisations possible. While there may be a single common response that most students will give, it is valuable to expose students to a variety of generalisations, to develop their abilities to explain, justify and make sense of algebra in the context of a problem.

Some different generalised rules that students may create, written in ‘student speak’ are listed below.

Generalisation 1: ‘I saw that each plant needs two tiles – one ‘on top’ and one ‘on the bottom’. Then, I need six tiles to fill in the ends.’

Rule: Generalisation 2: ‘I saw that the top layer and one end form an “L shape”, which means you need one tile for each plant, and then three extras. You need two of these L shapes to finish the garden bed.’

Rule: Generalisation 3: ‘I saw that you needed 5 at each end to ‘surround’ the end plants, so that’s 10 in total. After that you need two for the remaining plants – one on top, one on the bottom.’

Rule: Generalisation 4: 'I saw that for the whole top layer, you needed one per plant and then two extras (one for each end).  You need the same number of tiles for the bottom layer, and then two more tiles to fill in the ends.’

Rule: Generalisation 5: ‘If you built the whole bed out of tiles, it would be made up of three rows, each row would have the same number of tiles as the plants, plus two.  Then, you would remove the middle section of tiles so you could fit in the plants.’

Rule: Generalisation 6: ‘You need 8 tiles for the first plant, and then an extra two tiles for every plant after that.’

Rule: Show students how these generalisations work using the blocks or counters, but without saying anything yourself. For example, if you are demonstrating Generalisation 5, start by laying three rows of tiles, and then physically remove part of the middle layer, so the plants can be added.

Ask students to describe the generalisation in words, and then help them move towards the algebraic rule (using a ‘think-aloud’ strategy similar to that presented in Paths in the Park). This is a powerful illustration of equivalence and benefits greatly if the explanations involve showing the reasoning using the concrete materials.

Pretend you are using one of these generalisations mentally to work out how many tiles are needed for a given number of plants. Use this ‘think-aloud’ to represent the use of Generalisation 4: ‘So, if I had 12 plants, I’d need 14, then another 14, then 2, so that’s 30 tiles in total’.

Ask students if they can ‘make sense’ of this thought process and turn this into a generalisation.

Contrast this with the ‘think-aloud’ used for Generalisation 1, which would sound something like ‘If I had 12 plants, I’d need two lots of 12, that’s 24, and another 6, so that’s 30 in total’. Present students with one of the rules they have not developed themselves. Ask students to make sense of it by writing an explanation about how the rule relates to the actual problem of constructing a garden bed.

### Areas for Further Exploration

There are a range of mathematical skills and concepts that can be explored with students. Some of these skills and concepts will also be explored and reviewed in other stages within this learning sequence. These skills and concepts include:

1. Substitution

Ask students to use their generalisation, or rule, to determine the number of tiles required for a certain number of plants.

2. Solving Equations

Present students with a problem such as:

‘Harry packed his plants in his car, and then came back in and asked for 68 tiles. How many plants had he purchased?’

Repeat this problem with different values for T.

3. Equivalent Expressions

Ask students who have developed different generalisations to try to determine why two expressions may be equivalent.

4. Graphical representation of relationships

Ask students to plot the relationship between the number of tiles and the number of plants on the set of axes on the Recording and reflecting sheet and complete the subsequent questions. Extend students by asking them to explore different designs of garden beds; for example, two or three rows of plants, rather than a single row, or garden beds in an L-shape or square shape, with a view to being able to write a generalised rule for these designs.

You may like to use the ideas presented on the Visual Patterns website to help extend your students. The site contains over three hundred patterns that can be written as algebraic rules.

Encourage students to use the strategies already utilised in this task: trialling different designs using concrete materials, tabulate different values, establish a range of generalisations both in written and symbolic form, and use these generalisations to solve equations and draw graphs.

A challenging extension would be a design whereby there is a tile between each plant, as shown in this diagram: One way of determining the generalised rule for this design is:

Top Row: Two tiles for each plant, plus one for the ‘end'. Middle Row: One tile for each plant, plus one for the ‘end’. Bottom Row: Two tiles for each plant, plus one for the ‘end'. Hence, the total number of tiles could be written as:  Another way of ‘seeing’ this design would be recognising that 3 tiles are needed for the first ‘column’, and then each plant after this needs an additional 5 tiles to surround it. Therefore, the rule could be written as: These two rules, while looking very different, are in fact equivalent expressions.

Use these two rules with students, as well as any other different versions they develop themselves, to further explore concepts of equivalence and simplification.

Garden Beds exposes students to a range of mathematical skills and concepts which you may wish to review and consolidate with students. Consider the progress made by students when determining which skills and concepts you select to review.

Three suggested concepts for review are presented below.

1. Solving an equation involves finding a value for a variable that will make the equation correct.

Emphasise that we can use a range of strategies to help solve equations, including guess and check, backtracking and the use of inverse operations. In Garden Beds, we solved equations when we were told the number of tiles and had to work out how many plants had been purchased.

Support students to consolidate their understanding of solving equations by emphasising the relationship between the formal algebraic process of solving an equation, and the verbal understanding of the process, using the context of this task.

For example, to work out how many plants had been bought by someone who needed 52 tiles, a student working with the rule may say: ‘Take away 6 for the tiles at the end, so that leaves you with 46, and then divide that by two, which equals 23. There must have been 23 plants.’

Model this verbal explanation side-by-side with the more formal approach to solving equations. Present students with a range of problems to solve, using different strategies, including those modelled above. For example:

My parents paid me \$8 per hour in pocket money to tidy up the house. I already had \$35 in my wallet, and after I added my pocket money, I had \$59.  How long did I take to clean up the house?

2. Expressions are mathematical statements made up of numbers, variables, and operators.

Emphasise the following concepts to students:

• The parts of the expression that are separated by addition or subtraction signs are called 'terms'.
• Terms that include the same variable are called 'like terms'.
• Numbers can be called 'constant terms' or 'constants' because they always have the same value.
• Expressions that represent the same amount are called 'equivalent expressions'
• There are useful processes, such as 'collecting like terms' to show that two expressions are equivalent, that help to simplify expressions and equations.

Support students to consolidate their understanding of equivalent expressions by using the range of expressions developed throughout Garden Beds to demonstrate the concept of two expressions that look different being equivalent.

For example, explain that the expression 2(p + 3) can be shown to be equivalent to 2p + 6 in this way:

 2(p + 3) represents 'two lots of p + 3'. This can be reinforced by the worded explanation developed earlier: the expression p + 3 represented the 'L shape' and we needed two of them.  Two lots of p + 3 can be written as p + 3 + p + 3. The two p's can be collected and written as 2p (two lots of p). The two 3's can be collected and written as 6. Hence, 2(p + 3) is an equivalent expression to 2p + 6.

After modelling one explanation, ask students to use a similar approach to show why other rules in this task are equivalent to one another.

You can use the expressions within this task to introduce the concept of expanding a set of brackets, or let students explore this algebraic technique themselves.

Provide students with further opportunities to work with equivalent expressions. This may include:

• provide students with cards or sets of expressions that include some that are equivalent. Students match up pairs of equivalent expressions
• provide students with an expression and asking students to write an equivalent expression for it.

Sample problems at varying degrees of difficulty include:

• Write an expression with three terms that is equivalent to 4e + 5.
• Write an expression with four terms that is equivalent to 6y - 7.
• Write an expression with five or more terms that is equivalent to 6gy - 7y + 7.

Understanding mathematical terms and notation, from the Literacy in Mathematics section of the Literacy Teaching Toolkit, will support you to scaffold students' understanding of various conventions used when working with numbers and pronumerals.

3. Relationships between two variables can be represented graphically.

Emphasise the following concepts to students:

• Each axis can be used to represent a variable. In Garden Beds, the horizontal axis was used to represent the number of plants, and the vertical axis was used to represent the number of tiles.
• Each plotted point can be written in co-ordinate form, which helps describe the location of the point in relation to the origin.

Support students to consolidate their understanding of the graphical representation of relationships by asking them to construct a graph to represent the generalisations developed in Paths in the Park, and compare this graph to the one constructed for Garden Beds.

Extend students' understanding of the graphical representation of relationships by introducing the concept of domain, using these suggested prompts:

• Ask students to examine their graph from Garden Beds and the patterns demonstrated (in terms of the points and the direction they take).
• Ask students to confirm that their graph reflects the correct number of tiles required for different number of plants (e.g., 12 tiles for three plants; 10 tiles for two plants; 8 tiles for one plant).

Present students with this scenario:

'My uncle drew a similar graph to help him work out how much to charge the customers. He kept it by the register. One day, a customer arrived at the checkout, but changed her mind and decided not to buy any plants. My uncle checked his graph and said ‘Right, well, you still owe us for 6 tiles!’ The customer was not impressed!'

This scenario can be used to explain that the rule only ‘makes sense’ when p > 0. Extend students by introducing the concepts of continuous and discrete relationships, using these suggested questions:

1. Is it appropriate to plot more points on either side of pattern you see on the graph?

Students should recognise that more points could be plotted to the right of what they have already graphed as, theoretically, the number of plants and tiles could go on forever and the relationship will continue to be linear, due to the nature of the problem. However, they couldn’t be plotted to the left of what has already been graphed, due to the domain of the relationship - there cannot be a negative number of plants.

2. Could more points go between the points that are already there? If so, what would this mean?

Students should recognise that they could plot a point between, for example, (1, 8) and (2,10), but contextually this would suggest you could have a decimal number of plants and tiles, which is impractical.

3. Is it appropriate to join up the points to form a line?

Students should recognise that if we joined up the points with a line, it implies that the relationship exists for all values of p, which is incorrect for this particular context. In this context, T = 2p + 6 is a discrete relationship. The Recording and reflecting sheet (available in the Materials and texts section) may be used to formatively assess student understanding.

Australian Association of Mathematics Teachers, n.d. Garden Beds. [Online]
Available at: maths300.com/members/m300full/016lgard.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

State Government of Victoria (Department of Education and Training), 2019. Understanding mathematical terms and notation. [Online]
Available at: www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/Pages/lim_understandingmathsterms.aspx
[Accessed 15 March 2022].

visualpatterns.org, 2019. Visual Patterns. [Online]
Available at: www.visualpatterns.org
[Accessed 15 March 2022].