Algebra in Action: Patterns, Generalisations and Relationships

# 1. Rules, Rules, Rules

Suggested Learning Intentions

- To use a pattern to find an unknown
- To determine an unknown using pronumerals
- To explore, construct and explain generalisations between two variables

Sample Success Criteria

- I can describe a pattern
- I can determine an unknown using pronumerals
- I can create a rule to describe a relationship between two variables
- I can model and justify my thinking using a range of manipulatives

- Matchsticks for activities
- Max’s Matchsticks activity
- Counters to simulate the wheels on a monorail

To tune students in, write "What is my pattern?" on the board. Then proceed to walk around the class enunciating a pattern, for example counting by 5s. After a short while, ask a student to explain what they notice about the numbers you are calling out. Encourage students to use words such as 'the pattern goes up by 5s' or 'each time you are adding 5' rather than 'you are counting by fives'. Proceed to call out a new pattern, again after a few minutes ask a student to describe the pattern for what you are doing.

Discuss with students that you are going to be exploring number patterns and ways to describe them. Show students the following diagram:

Students form pairs and draw the next two terms (triangles) in the pattern. Have students record the number of sticks per term in a table.

Ask students to predict how many sticks are needed for the 7th term. Students describe how they came up with their response. Encourage students to identify the relationship between the number of triangles and the number of sticks.

**Enable **students by providing them with match sticks to model the triangles prior to formally recording their response.

**Extend** students by asking them to formalise the pattern as an algebraic expression.

**Step 1.**

Students work in pairs to draw or build the next three terms in the pattern below. Students then complete a table of the first 6 terms. When using the word ‘terms’ some students might become confused. Point to each term in the diagram as you say ‘this is considered Term 1, this is considered Term 2 and so on.

Read the following scenario to students:

"My sister works for a monorail company. She is in charge of buying the wheels. She sent me the following drawing and asked me to help her calculate how many wheels she will need to order depending on the number of carriages. If she needed to order enough wheels for 6 carriages how many wheels does she need to order?"

Draw this pattern on the board. The algebraic equation for this pattern is: *y* = 4*x* + 2.

**Enable** students by providing them with the following simplified pattern. Provide students with square and round counters to physically build each term in the pattern. The building of the pattern supports some students to identify the repeating components of a pattern.

The algebraic equation for this pattern is *y *= 4*x*.

**Extend** students by providing the following two patterns. Encourage students to use the following pronumerals in their table Carriages (*x*) and Wheels (*y*)

The algebraic equations for these patterns are *y* = 4*x* + 2 and *y* = 4*x*.

Add to the previous scenario to students:

"As I said before my sister works for a monorail company, and she is in charge of buying the wheels for the trains. Now we helped her out before. We helped her order the wheels for 6 carriages but now she needs our help to come up with a rule that she can use for 10, 35 and an unlimited number of carriages. Can you help me come up with a rule for her?"

Using the information contained within the tables, students describe the pattern using words and numbers. Students determine how many wheels are required for ten carriages using their generalisation.

Students graph their tabulated information, extrapolating the line to confirm that their calculations for the tenth term is correct. Once students confirm their rule, have students calculate how many wheels are needed for 35 carriages.

**Enable** students to make generalisation and define the rule by guiding them with targeted questioning. See below for an example:

Teacher: ‘How many wheels do you need each time you add a carriage?’

Students ‘4’

Teacher: ‘So is that the pattern?’

Student: ‘Yes. It goes up by 4’

Teacher: ‘Great, that is how we say the pattern with words, but how can we say that with maths?’

Student: ‘Plus 4?’

Teacher: ‘That’s correct and if we want to be even more mathematical, what can we say…?’

Student: ‘Add 4?’

Teacher: ‘Great, can you tell me how many wheels a 7-carriage train would need?’

Student: ’30. I just added 4 on to the 6-carriage train, 26 plus 4 is 30.’

Teacher: ‘Well done, that is the best way to do it when you need to know how many counters for the next size up. But what would you do if you needed to build a 20-carriage train? Would you keep adding 4? Can you think of a quicker way?’

Some students may now “see” the rule and be able to describe it but for those who cannot………

Teacher: ‘How many wheels do you need for EVERY carriage?’

Student: ‘4’

Teacher: ‘‘To find the rule you need to decide how many wheels you need for every carriage’

Student: ‘I need 4….’

Teacher: ‘Great. Have a go at writing the rule’.

The above prompts are based on the enable pattern of *y* = 4*x*. When supporting students to create a rule for the *y* = 4*x* + 2 pattern encourage them to move from additive to multiplicative thinking.

Teacher: “I have a 10-carriage train how many wheels do I need?’

Student: ‘I would need 10 lots of 4, which is 40, then two more wheels for the starting of the train. So I would need 42 wheels altogether.’

Teacher: ‘And if I had a 100-carriage train?’

Student: ‘100 times 4 and 2 more.’

Teacher: ‘Great. Have a go at writing the rule’.

To **extend** students, have them record the rule of both lines in the form of *y* = *mx* + *c*.

**Step 2.**

Students graph their table of data and extrapolate the line to determine if their rule is correct. Ask: “Did you predict the correct number of wheels for 10 carriages? “

Extend students who tabulated two patterns *y* = 4*x* + 2 and *y* = 4*x* by inviting them to compare both lines and identify their x- and y-intercepts. Have students extrapolate both lines in each direction.

Ask:

- What do you notice about the gradients of each line?
- Does each line increase by the same amount?
- Are the lines parallel?
- Does each line go in the same direction?
- Does each line start at the same point?

Encourage students to see that the slope of each line is identical however, the *y* = 4*x* + 2 line crosses the *y*-axis of the graph at (0,2) whereas *y* = 4*x* crosses the *y*-axis at (0,0).

When graphing, observe student responses to note any common errors or misconceptions and use 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 guiding them to check both their coordinates and the labelling of their axes.

**Further Exploration**

Additional teacher support information can be found on page 3 of Introducing Targeted Interventions.

**Max's Matchsticks**

This activity (available in the Materials and texts section above) provides an additional opportunity for students work in small groups to investigate the use of rules to describe a pattern. The following diagram and four different correct solutions are provided to students. Students determine how each person reached their rule.

**Enable** students to visualise each solution by providing them with matchsticks to build each pattern. Working in small groups, students articulate their ideas aloud, working together they identify how each strategy was derived.

For example, Di’s strategy is 10 x 3 + 1. Ask students to identify one group of three.

Once they have clearly identified one group of three have them identify the ten groups of three. Then have students point to the single remaining + 1 from the end of the equation.

This stage supports students to recognise key language phrases and translate these into mathematical symbols. Students describe patterns and relationships using algebraic expressions. Assess student understanding by asking them to create algebraic equation from the following scenario.

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

Have students write an expression to describe the relationship and then solve the equation. Students should use manipulatives to justify their solution.

To further develop your professional understanding and to support you to guide students to describe mathematical relationships using symbols read Translating from words to symbols, from the Literacy in Mathematics section of the Literacy Teaching Toolkit.

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), 2008. *List of Targeted Interventions. *[Online]

Available at: https://www.education.vic.gov.au/Documents/school/teachers/teachingresources/discipline/maths/assessment/lafzone7intro.pdf

[Accessed 15 March 2022].

State Government of Victoria, (Department of Education and Training), 2013. *Max's matchsticks. *[Online]

Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=e0968a22-ad38-4aa2-a368-9777a72913ba&SearchScope=All

[Accessed 15 March 2022].

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].

Other stages

2. Garden Beds

EXPLORESuggested 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

3. The Snail Problem

EXPLORESuggested Learning Intentions

- To use a pattern to find an unknown
- To describe an unknown using pronumerals
- To solve linear and non-linear equations using algebraic and graphical techniques
- To compare and contrast the elements of linear and non-linear equations
- To successfully apply a range of problem-solving strategies to a given task

Sample Success Criteria

- I can describe a pattern
- I can model a pattern using concrete materials
- I can determine an unknown using pronumerals
- I can graph algebraic expressions
- I can identify the elements of a graph
- I can use a range of problem-solving strategies
- I can explain and justify my solutions using a variety of manipulatives

4. Chocolate Boxes

EXPLORESuggested 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 complete a mathematical investigation involving variables
- To successfully apply a range of problem-solving strategies to a given task

Sample Success Criteria

- I can use a pattern to find an unknown
- I can model a pattern using concrete materials
- I can use pronumerals to describe an unknown
- I can describe a relationship between two variables using an algebraic expression
- I can complete a mathematical investigation involving variables
- I can select an appropriate problem-solving strategy from a range of strategies and successfully apply it to a task
- I can use a range of manipulatives to explain and justify my solutions