Suggested 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

- Concrete materials used to help student exploration of the snail trail problem (chalk, number lines, blu-tack, blocks or counters)
- 'Find my triplet' card matching activity - Worksheet 1: docx PDF
- 'Find my triplet' card matching activity - Worksheet 2: docx PDF
- 'Find my triplet' card matching activity - Worksheet 3: docx PDF

**Find my triplet**

Stages 1 and 2 of this sequence support students to identify linear relationships, and to describe them by formulating a rule and an expression or equation. This stage introduces students to the concept of describing a non-linear relationship. When graphed, linear equations form a straight line, whereas when graphed non-linear relationships form a curved line. This curved line of the non-linear relationship reflects that the change in the *x *variable does not result in the same change in the *y* variable. The 'Find my triplet' activity (available in Materials and texts) requires students to match a graph with a table of values and an equation.

Card match activities allow students to trial multiple solutions in a short period of time. Three different sets of card match activities have been included in the Materials and texts section above. **Enable** and **extend** students by customising the cards further. Create a range of cards which can include all students in the Getting started activity.

- Students working at level 8 in algebra should probably select Worksheet 2 cards.
- Students working below level 8 should probably select Worksheet 1.
- Students working above level 8 should probably select Worksheet 3.

Either laminate each set of cards to create a permanent set or provide individual sheets for students to match up and paste in their workbooks.

At the conclusion of the Find my triplet activity, students reflect on the observations they have noticed. Some observations might include:

- the difference between a linear equation and a non-linear equation when graphed
- the difference between the slope of a positive gradient and a negative gradient
- the difference between two gradients (steepness)
- that the constant represents the
*y*-intercept value.

When creating a set of 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. Provide students an opportunity to first develop a level of confidence with writing an algebraic expression before introducing decimal or fractional numbers.

**Snail Trail**

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

This lesson supports students to develop their pattern making skills and their ability to identify relationships and substitute into a formula.

Provide students with the following scenario:

A snail starts at the bottom of a well that is eleven metres deep. It starts crawling upwards to escape from the well.

The snail can crawl upwards five metres in an hour, but then it gets tired. Once it gets tired it needs to rest for one hour. During the resting time the snail slips back down by three metres.

The snail then crawls upwards for another hour, and then rests for an hour. This pattern of crawling and sliding continues until the snail escapes from the well.

During which hour will the snail finally escape the well?

Before commencing the task, have students predict how many hours it will take for the snail to escape. Predicting and estimating are essential skills to develop in mathematics as they allow students to check for the reasonableness of their answers. Collect predictions on the board and tell students you will come back to them later.

A common (incorrect) prediction, with an accompanying explanation, may be:

“If the snail crawls for 5 metres and then slides back 3, it is gaining 2 metres every two hours, which is the same as one metre per hour. Therefore, it will take the snail 11 hours to go 11 metres upwards, so it will escape after 11 hours.”

Students work in pairs to solve the presented problem. Encourage students to present their findings in a table.

The ‘secret’ to this puzzle is determining how long it takes for the snail to get to a point from which it can escape the well after the next ‘climb’. This realisation is crucial for students to form generalisations.

**Enable **students by providing them with opportunities to explore this problem using concrete materials. Suggested strategies include:

- use chalk to draw number lines on a pavement and ask students to ‘be the snail’
- stick a number line on the wall or window and use a token or marker (post-it note, peg etc) on the number line to represent the snail
- use concrete materials such as blocks or counters to represent the movement of the snail.

**Extend** students by asking them to create a rule for this relationship. The rule for this relationship is:

Not all students will be able to create the generalised rule from their table of values and therefore you may decide to provide it to the whole class using the following phrase (or similar): "Last year I taught this same activity to one of my older classes and they came up with the following rule."

Ask students what they think each pronumeral in the rule stands for. Suggest they use their tables of values as prompts.

Have all students check their answers using the rule. Depending on the level of understanding in your class you may decide to lead a whole class discussion detailing how to substitute into the rule.

Ask students to compare their initial prediction to their answer in the beginning. Were they close? Can they identify the initial misconception they had at the beginning which allowed them to over/underestimate the amount of time required?

There are many examples of these types of problems available online. You may like to show students different forms of the problem that are readily available online – try searching for ‘frog in a well’ or ‘snail climbs a wall’.

**Areas for further Exploration**

Snail Trail provides opportunities to explore decimal representations of time which requires students to demonstrate proportional thinking and reasoning.

Explore this concept with students by presenting the following scenario:

If the snail climbs at five metres per hour, and slides at three metres per hour, and starts at the bottom of a ten metre well, at what exact time will it escape?

Through modelling the problem, students will see that the snail reaches a height of 6 metres after 6 hours. It will then need to climb 4 metres to escape. The key question now becomes: ‘how long will it take the snail to travel 4 metres if it climbs at 5 metres per hour’. There are several ways of answering this. One possible student response might be:

- The snail climbs 5 metres in 1 hour
- Therefore, the snail climbs 1 metre in one fifth of an hour, which is equal to 12 minutes.
- So, the snail will climb 4 metres in 48 minutes (12 x 4)
- The total escape time is 6 hours and 48 minutes.

Consider varying the problem by changing the climbing speed or providing actual starting times for the snail escape challenge. For example:

- The snail started climbing at 7:40 am. What time did it escape?
- The snail escaped a fifteen metre well at 12:44 pm. Present at least three different possibilities for the depth of the well and the snail’s climbing and sliding speeds.

This stage provides an opportunity for students to develop their problem solving and reasoning proficiencies. Students investigate, apply existing strategies, and seek to hypothesise and find solutions. They develop these skills alongside the patterns and algebra achievement standards and can be formally assessed by examining their responses to the following scenarios.

These suggestions could form the basis of an assessment task, focussing on problem solving and reasoning. Students could select an area for exploration and write and submit an investigative report.

Have students work in pairs to discuss which aspects of the Snail Trail problem they could change to create a new but similar problem.

Responses could include:

- the depth of the well – it could be deeper than eleven metres, or more shallow
- the speed of the snail – it could crawl upwards at a faster rate, or a slower rate, than five metres per hour
- the sliding of the snail – it could slide more, or less, than three metres per hour
- the units of time used – the snail could crawl upwards, and rest, for half an hour each. Or the snail could crawl for an hour and rest for half an hour.

Collect the class responses on the board. Have students select one or two responses and alter the rule so that it is reflective of the change within the question. Have students check their changes by solving the problem for different time intervals. Ask students to compare the two new snails with the initial snail and determine which snail can cover the greatest distance in the shortest amount of time.

**Enable **students by nominating a change in question for them to investigate. For example, ask them to calculate how long it will take for the snail to escape if the climbing and sliding speed is changed to a climbing speed of 6-metres per hour and sliding pace of 3 metres per hour. Have students act out the scenario and then record their answers in a table. Ask students to identify which of the two snails is likely to escape the well first.

**Extend** students by asking them to determine how long it would take for the snail to escape if the well depths were changed to either 5 m, 6 m, 7 m, 8 m, 9 m and 10 m. Have students tabulate their data.

Ask students if they can see a pattern with the escape times. For example, students may notice that the time taken goes up in increments of 2 hours, and all the times are odd values. Ask: “Why does it always take an odd number of hours for the snail to escape?” Encourage students to reason that the snail will escape only by crawling up, not sliding down, which occurs in the ‘odd’ hours.

Once students have noticed the pattern, extend the table with higher values for depth, such as 20 m, 45 m and 100 m. Ask students to explain how they could easily determine the escape time for these depths. This generalisation is more challenging for students, as it is different for odd depths and even depths. Some suggested questions may include:

- What do you notice about how long it takes for the snail to escape when the depth is an even number or an odd number?
- Why do you think this might be different?
- Can you calculate how long it would take the snail to escape from a really deep well that was an even depth? An odd depth?
- Could you explain the method you used? Could you write it as a rule? For what values would your rule work for? Why?

The generalised rules can be written as:

*T*= (*d*– 5) + 1 for wells that have an odd depth, and*T*= (*d*– 5) + 2 for wells that have an even depth.

*T* represents the time in hours, and *d* represents the depth of the well in metres.

Addison, L., n.d. *Strategy Toolbox poster. *[Online]

Available at: mathematicscentre.com/taskcentre/stratool.pdf

[Accessed 15 March 2022].

Australian Association of Mathematics Teachers, n.d. *Snail Trail. *[Online]

Available at: maths300.com/members/m300full/064lsnai.htm

[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Other stages

1. Rules, Rules, Rules

EXPLORESuggested 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

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

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