The X Factor: Using Algebra to Solve Problems

# 1. Paths in the Park Suggested Learning Intentions

• To explore, construct and explain generalisations representing a relationship between two variables

Sample Success Criteria

• I can use manipulatives to model a pattern or relationship
• I can construct my own generalisation to represent a pattern or relationship
• I can explain my generalisation to another person
• I can write my generalisation using mathematical language
• I can use my generalisation to solve problems
• I can justify my solutions using a range of manipulatives
• Square tiles – at least 20 per student or group of students
• Paths in the park – Recording and reflecting: docx PDF
• Card sort activity: docx PDF

This stage has been inspired by the 4 Arm Shapes problem on Maths300, and is reproduced with permission. Access to the problem on Maths300 requires a subscription.

Tell students a story:

'A council gardener is designing a new pathway for people to walk on in a local park.  She wants to build four equal paths using square tiles that will all meet ‘in the middle’. She calls them 'four-arm paths' and the pathway will look something like a cross.'

Show students a sample design, with an ‘arm length’ of 4 tiles. 'The gardener starts thinking about building pathways like this in different parks in her council. Some of the parks she works in are quite small, while others are much bigger. She wonders if there is a way of quickly working out how many tiles she will need for designs with different arm lengths.'

Give students a set of coloured square tiles and ask them to design some different pathways with different arm lengths. Through the process of ‘doing’ the designing, and moving the tiles around, students will begin to recognise the relationship. Using a different colour for the central tile also helps.

Ask students to record some different combinations of arm length and total number of tiles. You may wish to provide students with the Recording and Reflecting sheet from Materials and texts section for this purpose.

Depending on their prior exposure, students may independently use a table of values, or you may like to introduce this concept to students now as a way of organising information. Alternatively, you could wait until the ‘review’ section to explicitly address this concept.

After some time, present students with this problem:

'The gardener is asked to build a four-arm path in a large park. The 'arm' will be 80 tiles long. How many tiles will she need?' Give students time to discuss this problem. Once they have an answer, ask them to explain how they worked it out.  This may be done in pairs – students completing the task individually, and then explaining to a partner. Alternatively, it could be done in small groups, or as part of a whole class discussion.

The choice of a large number such as 80 is deliberate, as it promotes visualisation, and prompts students to find another way of answering the problem other than using the concrete materials.

Enable students by asking them to show you a design that they have made themselves. Once they have identified how many tiles are needed for their design, ask them how many tiles would be needed for a design where the arm length is one tile longer. Through a series of scaffolded prompts, such as the samples provided below, students can begin to better understand the relationship between the arm length and the number of tiles required. An example of a scaffolded prompt:

Teacher: 'I can see that your design has an arm length of two. How many tiles did you need?'

Student: 'Nine tiles. Two per arm, that’s eight in total, and then one for the middle.'

Teacher: 'Right, so what if the arm length was three? How many tiles would you need then?'

Student: 'Thirteen. This time it would be three per arm, that makes twelve, and then another one in the middle.'

Teacher: 'OK, great. How about if the arm length was 4?'

Next, ask students the original problem from the beginning of the lesson:

'Can you explain how the gardener could quickly work out how many tiles she needs, based on any arm length?'

It is important that students use their concrete experiences with the tiles to help ‘see’ the explanation that makes sense to them. Encourage students to use their response to the ’80 tiles’ question to help develop their generalisation.

Oral and written language are the genesis of the symbolic representation of this problem, so it is important to begin with these before moving to a symbolic representation.

Ask students to begin with an oral explanation and then transfer this to a written explanation. Model a ‘think-aloud’ strategy to help students move from a written explanation to one using a variable (the example provided is using one of the generalisations, but the approach could be replicated for other generalisations). For example:

We can see that to find the number of tiles, we can multiply the arm length by four and then add one for the centre tile.

Symbolically, this could be expressed as: We usually write this as: Ask students to clearly explain and justify the rule to someone else.  A sample explanation may be:

'The 4 is there because there are 4 arms, and the extra one on the end is for the central tile.'

Note that this explanation of the rule is in terms of the context of this problem, and not just derived from seeing a pattern in a table. Students are ‘making sense’ and demonstrating understanding.

### Areas for Further Exploration

This sequence addresses a range of mathematical skills and concepts that could be explored with students. Some of these skills and concepts are also explored and reviewed in other stages in this sequence. These skills and concepts include:

1. Substitution

Ask students to use their generalisation to work out how many tiles are needed for certain arm lengths.

2. Solving Equations

Present students with problems such as:

• The gardener can fit 85 tiles in the back of her car. What is the largest design she could make?
• Can you describe a way of working out the arm length of a design, if we know how many tiles the gardener used to make it?

3. Equivalent Expressions

There are multiple ways that students may describe the generalisation. Four such ways are listed below. Generalisation 1 is arguably the most common that students will develop.

Generalisation 1: To find the number of tiles, multiply the arm length by four and then add one for the centre tile.

Rule: Generalisation 2: To find the number of tiles, double the arm length, add one for the centre tile, double this value, and then subtract one, because you have counted the centre tile twice.

Rule: Generalisation 3: To find the number of tiles, you need five for the first time around (one for each arm, and one for the centre) and then four extra tiles for each round after that, to ‘grow’ the arm.

Rule: Generalisation 4: To find the number of tiles, double the arm length, add one for the centre tile, and then add two more arm lengths.

Rule: Ask students who have developed different generalisations to check that they give the same answer for the number of tiles required for different arm lengths.

This will help confirm for students that there are multiple ways of writing this generalisation. There is no one ‘best’ way, other than the one that helps students make sense of the problem.

Prompt students to discover generalisations they haven’t yet discovered themselves by saying, (as an example, to help them to discover Generalisation 4):

'The gardener was thinking about how many tiles she needed for a design where the arm length was 16. She thought about it like this: ‘Arm length of 16… so I’ll need 32, plus one is 33, plus another 32 is 65’.

How was she ‘seeing’ the design? How could you write this as a rule?' 4. Changing the Design

Extend students by changing the design of the paths. This will allow students to create new generalisations and consolidate their understanding of the features of a generalisation written using algebraic language.

Select from these ideas:

a) Parks of a different shape.

The gardener is working in a hexagonal park. She places four tiles in the centre and then builds six ‘arms’ of equal length spreading out from the centre. How could you describe how many tiles she would need for a design like this?

b) The design in a different park produced this table of values. What might the park look like? Deleting some of the values from the table will make it more challenging.

c) Provide students with a rule such as T = 8a + 9 and ask them to draw a design that would match this rule.

d) The gardener wonders if there is a rule that she could use to work out the number of tiles, length of arms or number of arms for any park of any shape, provided she knows two of the variables. Challenge students to determine if any such generalisations exist.

Paths in the Park 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. A table of values can help us to organise data and represent relationships

Emphasise the following concepts:

a) A table of values helps us to see any pattern in the relationship between two variables. In Paths in the Park, the variables were the length of the arm and the number of tiles needed.

b) We can use words as the ‘headings’ in the table of values, or we can choose a letter to represent these words. In Paths in the Park, we chose the letter T to represent the number of tiles, and a to represent the length of the arm. You can support students to consolidate their understanding of tables of values by:

• Asking students to create a table of values using data they can collect themselves, or find online, such as the temperature of their suburb/town throughout the day. Students could graph this data as well, which has links to the Statistics and Probability strand of the Victorian Curriculum: Mathematics.
• Presenting students with a table of values and asking them to describe the information the table provides. For example, a table of values showing the runs scored and the deliveries faced by a group of batters from a cricket match. The table could show that the batters who faced more deliveries scored more runs.

2. Oral, written and symbolic language are multiple representations that can all be used to explain and represent algebraic relationships

Emphasise the following concepts:

a) Algebraic language allows us to represent generalisations in several different ways.

b) The numbers used in an algebraic relationship or rule tell us something about the relationship. In Paths in the Park, the relationship T = 4a + 1 tells us that the number of tiles we need is four times the length of the arm, plus one.

c) An algebraic convention is to write a multiplication such as 3 x m as 3m.

d) '4a + 1' is an example of an expression, while 'T = 4a+ 1' is an example of an equation. Both are useful tools when developing generalisations.

e) When using a variable, it is important to correctly define what the variable represents. In Paths in the park, the variable T was used to represent the number of tiles, not just tiles. The variable a was used to represent the length of the arm, not ‘arm’ or ‘number of arms’.

f) The rules developed are all mathematical models of some real scenario.

3. Substitution is the process of replacing a variable in an expression

Emphasise the following concepts:

a) Substituting a variable with a number in an expression allows us to evaluate the expression.

b) Correctly evaluating an expression requires us to understand the different parts of an expression and the relationship between these parts. For example, an expression such as 4a represents 'four multiplied by a' rather than 'forty-something'. You can support students to consolidate their understanding of the use of language to explain and represent algebraic relationships, and the process of substitution, by asking students to complete the card sort activity (see downloadable resources in the Materials and texts section). The card matching activity in the Materials and texts section can be used as a formative assessment task. The Recording and Reflecting sheet may also be used to formatively assess student understanding.

Translating from words to symbols, available from the Literacy in Mathematics section of the Literacy Teaching Toolkit, can support you to scaffold students to represent algebraic relationships in multiple ways.

Australian Association of Mathematics Teachers, n.d. 4 Arm Shapes. [Online]
Available at: maths300.com/members/m300full/040l4arm.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

State Government of Victoria (Department of Education and Training), 2019. Translating from words to symbols. [Online]
Available at: https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/Pages/lim_translatingwordstosymbols.aspx
[Accessed 15 March 2022].