Algebra in Action: Patterns, Generalisations and Relationships

# 2. Garden Beds Suggested Learning Intentions

• To use a pattern to find an unknown
• To describe an unknown using pronumerals
• To explore, construct and explain generalisations between two variables
• To investigate, interpret and analyse graphs
• To connect algebraic expressions and their graphical representations
• To understand and justify that there is more than one way to describe a generalisation

Sample Success Criteria

• I can describe a pattern
• I can use a pattern to find an unknown
• I can determine an unknown using pronumerals
• I can construct generalisations between two variables
• I can create a rule to describe a relationship between two variables
• I can describe the elements of graphs
• I can match algebraic expressions with their graphical representation
• I can represent a generalisation in more than one way
• I can use manipulatives to model and justify my solutions
• Allow at least 6 counters and 20 tiles per group of students.
• Get started - card-match activity: docx PDF

Throughout this stage students will explore algebraic expressions and equations further. Students will generate rules and graph linear equations.

Begin by presenting the following scenario to students:

"My friend Elijah is an artist and he recently started taking commissions for his paintings. He charges \$500 for an initial design consultation and then charges \$250 per hour to complete the painting. Can you write a rule for the cost of hiring Elijah to paint a picture?"

Encourage students to check their rule by substituting in several different values for hours. Ask: Does your rule makes sense? As you increase the number of hours does the total cost for each painting increase or decrease? Would you expect this?

Extend students by asking them to rearrange the equation in as many ways as possible so that it is still true. Have students pair up and take it in turns to write the other pair a scenario. Students must first write the expression and then solve it. Throughout this activity probe students to assess their understanding. Using the painting question as an example, consider the following questions:

• If Elijah worked 4 hours on one painting last week how much money did he make?
• Can you tell me how many hours he worked on a painting if he charged \$1250 in total?
• Can you tell me how many hours he worked on a painting if he charged \$1125 in total?
• Elijah’s painting costs have increased, and he now must set aside 10% of his hourly rate. What effect would this have on your original rule?

Card Match

Card match activities are a great way to support students to see the connection between the real-life situation, the algebraic expression and equation. One set of card match activities have been supplied.

Either laminate the cards to make a permanent set for students to work in pairs to match up or provide individual sheets for students to match up and paste in their workbooks. Enable and extend students by creating a range of card match activities that suit student learning needs.

Create a range of cards which can include all students in the Getting started activity. Use the supplied set (available in Materials and texts section above) as a template to guide the development of more card sets.

For example, consider modifying the original problem of: to The success of this learning activity lies in the accessibility of the cards. Ensure that the problems will meet the learning needs of students in your class. When creating the cards consider the operations and the numbers used. In general, decimal and fractional numbers add a layer of difficulty which can often mean the ‘algebra’ gets lost. Students may become confused by these numbers and become distracted from the written algebraic expression. Instead allow students an opportunity to first develop a level of confidence with writing an algebraic expression before introducing decimal or fractional numbers. Enable students by providing them with a range of manipulatives such as counters to support them to model each relationship to support their problem-solving.

Garden Beds

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.

Garden Beds supports students to identify relationships and construct simple algebraic expressions to describe these relationships. Students investigate substitution and sketch simple linear equations and plot points on Cartesian plane and interpret and analyse graphs of relations from real data.

Students develop simple linear models for situations, make predictions based on these models, solve related equations and check their solutions.

Part 1.

Provide students with the following scenario:

"My uncle and aunt own a nursery. They sell beautiful golden plants, quite rare and fragile.  When people buy a plant, they also buy tiles to surround the plant so that the plant is protected."

Display the following example to students. "This is the first garden bed." "Some customers like to buy more than one plant, and plant them in a single straight line. The more plants they buy the more tiles they need. Each time a customer buys multiple plants, my uncle and aunt have to race into the back room and build a model and count the tiles needed."

"Is there a way of calculating this more easily rather than counting them each time?"

Show students the second and third garden beds.   Students work in pairs to draw or build the first six terms in the pattern above. Students then complete a table of the first 6 terms. Using the information contained within the table, ask students to describe the relationship between the number of plants and the number of tiles. Students may choose to respond to this in multiple ways. Some may write a worded explanation, while others work towards forming expressions or rules. All these various ways of solving the problem reflect the concept of ‘multiple representations’ – and they all qualify as forms of algebraic reasoning.

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. Below is a range of different generalised rules that students may create, written in ‘student speak’.  Enable students by modelling each generalisation above to support them to see how each was derived. Use two different sets of coloured counters to physically demonstrate how each generalisation is derived. For example, when demonstrating generalisation 4, start by laying three rows of one coloured tile, and then physically remove part of the middle layer and replace with different coloured counters to represent the plants being added. Throughout the demonstration, build in questioning so that students begin to derive the generalisations for themselves. Encourage students to ‘think aloud’ while modelling each generalisation.

Extend students by asking them to prove that Student 1’s generalisation and rule are equivalent to Student 2’s generalisation and rule.

Prove:

T = 2p + 6 is equivalent to T = 2(p + 3)

Possible student solution:

Step 1: 2(p + 3) represents ‘two lots of p +3’.

Step 2: Two lots of p + 3 can be written as p + 3 + p + 3.

Step 3: The two p’s can be collected together and written as 2p (two lots of p).

Step 4: The two 3’s can be collected together and written as 6.

Step 5: Hence, 2(p + 3) is an equivalent expression to 2p + 6.

Once students have found the rule have them test their rule by solving the following question.

"My neighbour Harry bought some plants and tiles from my aunt and uncle’s nursery. Harry packed his plants in his car, and then came back in and asked for 68 tiles. How many plants did Harry purchase?"

Part 2.

Using the information contained within the table, students graph the relationship between the number of plants and tiles. Students graph their table of data and extrapolate the line to determine if their rule is correct. When graphing, observe students to identify any common errors or misconceptions. Address these as a teachable moment:

• When students create the graph ensure the Cartesian Plane is labelled correctly.
• Ensure there is an equal distance between each value on each axis. The axis should be clearly labelled. Be aware that some students mistakenly label the boxes (as they do with bar and column graphs). Numbers should be placed on the lines to allow for accurate placing of coordinates.
• If a line is not straight, encourage students to identify their own mistakes by asking them to check both their coordinates and the labelling of their axes.

Once students have constructed their graphs, they extrapolate the line in both directions.

Ask: 'While we can extend the line in both directions on this graph, does it make sense in real life? What does your rule, and the graph, suggest the value of T should be when p = 0? Does this make sense? Is it possible to have a negative number of tiles or pot plants?'

Ensure students understand the difference between zero tiles and a negative value of tiles.

Deepen 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 3 plants; 10 tiles for 2 plants; 8 tiles for one plant)

Present students with this scenario:

"My uncle drew a similar graph to you 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!'"

Ask "Why would the uncle be stating the customer owes them for 6 tiles?"

This scenario can be used to explain that the rule only ‘makes sense’ when p > 0. Further 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 the 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.

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

• Students should recognise that we 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.

Areas For further Exploration

1. Algebra Walk

Garden Bed introduces students to linear models and encourages students to make predictions based on these models, solve related equations and check their solutions.

The following activity provides you with an opportunity to physically expose your students to a range of linear equations. This activity has been inspired by the Algebra Walk problem on Maths300, and is reproduced with permission. Access to the problem on Maths300 requires a subscription.

Students work together to model different algebra equations on a Cartesian Plane drawn on the ground. To begin the activity, draw a cartesian plane on the ground using chalk or tape. Have a group of students line up along the x-axis. Call out a rule that students apply to the number they are standing on. An example might be the equation y = 3x + 1. Introduce the equation by saying 'multiply your number by three and add one'. Students move to the appropriate location on the cartesian plane. Discuss what they notice about the line that is formed.

Repeat using different equations, swapping groups of students so that all have an opportunity to participate. Enable students by selecting simple equations such as y = 2x or y = x + 3 as they build confidence. Alternatively, pair students with others who can offer guidance and support.

Extend students by having two rows of students. Give each row of students a different equation. The creation of two or more lines allows you to:

• identify a point of intersection
• compare the gradient of two (or more) lines
• identify the y- or x-intercept.

Depending on the level of understanding, consider occasionally adding non-linear equations for students to model. This will provide an opportunity for students to ‘notice’ the difference between linear and non-linear relationships.

For your future reference: this activity can be adapted to be used when graphing other rules. For example, you can use it to graph parabolas.

This stage exposes students to a range of mathematical skills and concepts. Consolidate students' understanding of this stage by altering the garden bed design. Have students adapt their previous rule or generalisation to the new design. For example, place a tile between each plant, as shown in this diagram: Have students develop a generalised rule for this design. Encourage students to compare this design with the previous one and then adapt their previous generalisation or rule.

One possible solution includes:

Top Row:         T = 2p + 1 (two tiles for each plant, plus one for the ‘end’)

Middle Row:    T = p + 1 (one tile for each plant, plus one for the ‘end’)

Bottom Row:   T = 2p + 1 (two tiles for each plant, plus one for the ‘end’)

Hence, the total number of tiles could be written as

T = 2(2p + 1) + (p + 1)

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

T = 3 + 5p

Extend students by asking them to prove that these two equations are equivalent.

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].

Australian Association of Mathematics Teachers, n.d. Maths 300. [Online]
Available at: https://maths300.com/lessons/MzZNYXRoc0xlc3NvbjMwMA==
[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].