Suggested Learning Intentions

- 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 range of problem-solving strategies to help me solve a problem
- I can use materials to model a problem
- I can identify and justify which strategies are most useful
- I can complete a mathematical investigation involving variables
- I can justify my solution using a range of manipulatives

- Concrete materials used to help student exploration (chalk, number lines, blu-tack, blocks or counters)
- Problem Solving Strategies poster

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 puzzle has a long and interesting history and is used here as the basis of a rich mathematical investigation.

'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. It rests for an hour, but during this time it slips back down by three metres. Then, it crawls upwards for another hour, and rests for another hour. This pattern of crawling and sliding continues until it escapes the well. During which hour will the snail escape from the well?'

Give students some time to consider the problem before asking them for some predictions.

Predictions are valuable for a couple of reasons:

- They encourage student engagement with the problem, as students naturally want to know if their prediction is correct.
- Student predictions are often a window into their current understanding and any misconceptions they may have.

Accept a range of predictions from students, writing each on the board without comment.

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

“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. So, it will take the snail 11 hours to go 11 metres upwards, and it will escape after 11 hours.”

Once you have collected a range of responses, provide opportunities for students 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 using blu-tack and stick something to the number line to represent the snail.
- Use concrete materials such as blocks or counters to represent the movement of the snail.

The problem-solving strategies being used here include ‘act it out’ and ‘make a model’.

Through exploration, students should recognise that after 6 hours, the snail has travelled a total of 6 metres, and is now only 5 metres from the top. Therefore, only one more hour of crawling upwards is required for it to escape. The snail escapes after 7 hours.

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 be able to form generalisations.

Explain to the students that this is a variation on a famous riddle that appears in different formats. 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’.

Ask the students what aspects of this problem they could change if they were to write a similar problem. The purpose of this question is for students to recognise that this problem can involve variables, which provide a transition from a closed puzzle to an open investigation.

Responses could include:

- the depth of the well – it could be deeper than eleven metres, or shallower
- the speed of the snail – it could crawl upwards at a faster rate, or a slower rate, than five metres per hour
- the amount the snail slides back – 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.

All of these are fertile areas for investigation, but some are potentially quite challenging for students, particularly at this level. In this sequence, two areas will be investigated – changing the depth of the well and changing the speed of the snail.

Pose this question to students:

Can you predict how long it will take for the snail to escape the well, for any depth of the well?

Encourage students to use similar strategies to those used with the initial problem.

**Enable** students to engage with this problem by suggesting some different depths to explore (such as 7 m, 10 m, and 14 m), and to record their findings in a table of values.

Students could present their findings in a table of values, such as this:

The problem-solving strategies being used here include ‘make a list or table’ and ‘look for a pattern’.

The focus of this stage of the sequence is on the proficiencies of reasoning and problem solving. While students may try to write a generalised rule, this will be quite challenging for many students, as the relationship between the depth and the time is not linear for this problem. Some students may be able to develop a symbolic rule, but this does not need to be your immediate focus for all students.

Encourage students to notice the patterns in the table of values and use these patterns to help make predictions about escape times for other heights. For example, students could notice that the time taken goes up in increments of 2 hours, and all the times are odd values.

Ask students: “Why does it always take an odd number of hours for the snail to escape?” This will encourage them to reason about this problem; they should realise 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 of values for higher values of the 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.

The generalisation is more challenging for students, as it is different for odd depths and even depths.

Ask a series of scaffolded questions such that students will notice the relationship between the depth and the escape time for odd depths and even depths. Ask questions such as:

- What do you notice about how long it takes for the snail to escape when the depth is an even number, and an odd number?
- Why do you think this is different?
- Could you work out how long it would take the snail to escape from a very deep well that was an even depth? What if it was an odd depth?
- Could you explain the method you used? Could you write it as a rule? What values would your rule work for? Why?

The generalised rules can be written like this:

- Let
*T*represent the time, in hours, and*d*represent the depth of the well in metres. - For wells that have an odd depth,
*T*= (*d*- 5) + 1 - For wells that have an even depth,
*T*= (*d*- 4) + 1

By exploring a range of depths, students can discover several things:

- A climbing speed of 5 m/hr and a sliding speed of 3 m/hr results in a net gain of two metres every two hours. Proportionally, the snail is effectively moving up one metre in one hour.
- For wells with odd depths, the snail will reach a point where it has climbed (
*d*- 5) metres and will be able to escape the well after one more climb. Reaching this point takes (*d*- 5) hours, using the proportional ideas from the previous dot point. One more hour of climbing is required for the snail to escape. Hence,*T*= (*d*- 5) + 1. - For wells with even depths, the snail will reach a point where it has climbed (
*d*- 4) metres and will be able to escape the well after one more climb. Reaching this point takes (*d*- 4) hours. One more hour of climbing is required for the snail to escape. Hence,*T*= (*d*- 4) + 1.

Once all students have investigated changing the depth of the well, ask:

- Why do you think the people who wrote this original puzzle used the speeds of 5 metres/hour for crawling and 3 metres/hour for sliding, and the well depth of 11 metres? Guide students to an understanding that these values meant that the snail escaped at an ‘exact time’ (7 hours).
- What would happen if we kept the well depth at 11 m, and changed the speeds? Would the snail still escape at an exact time, or not?

Provide students with opportunity to explore this new problem, using similar strategies to those used previously.

**Enable** students to engage with this problem by providing some sample crawling and sliding speeds, with varying differences between them.

For example, you may initially ask different students, or groups of students, to explore a given pair of climbing and sliding speeds and record the movement of the snail in a table of values.

The results of one of these explorations may be:

Climbing speed: 6 metres per hour.

Sliding speed: 3 metres per hour.

Conclusion: Snail 'did not' escape at an exact time.

After each student or group has explored two or three different combinations of crawling and sliding speeds, collate some results on the board.

Sample responses may include:

Ask students to use these results to determine if there are any combinations of crawling and sliding speeds that will result in the snail escaping at an ‘exact time’.

Through a careful inspection of the table, guide students to recognise that if both the crawling and sliding speeds are odd, and differ by 2 m/hr, the snail will escape at an exact time.* *This can be seen with the examples of crawling by 3 m/hr and sliding by 1 m/hr; crawling by 5 m/hr and sliding by 3 m/hr; and crawling by 7 m/hr and sliding by 5 m/hr. (This can be further tested with crawling by 9 m/hr and sliding by 7 m/hr).

Encourage students to turn this recognition into a generalised statement, such as:

'For a well that is 11 metres deep, if the crawling speed and sliding speed are both odd, and differ by 2, then the snail will escape at an exact time.'

This is an excellent example of both the problem solving and reasoning proficiencies, for which this specific task is designed. Students have designed an investigation, applied existing strategies to seek solutions, hypothesised and deduced.

Another possible generalisation is:

'For a well that is 11 metres deep, if the crawling speed and sliding speed differ by 1, then the snail will escape at an exact time.'

Ask students to complete their own investigation or provide them with an idea for an investigation. Sample investigations include:

- Set a different fixed well depth (or ask students to choose their own) and ask students to trial different climbing and sliding speeds, and then prepare generalised statements for a well of this depth.
- Explore climbing and sliding speeds with a set difference (e.g., 2, or 5) and explore the well depths that result in the snail escaping at an exact time.
- Explore different well depths that have a common feature (e.g., even, odd, multiples of 4) and asking students to trial different climbing and sliding speeds to form their own generalisations.

The purpose of this investigation is for students to explore further generalisations from strategic data collection and testing of ideas. Students may work individually or collaboratively on this task.

Any of these sample investigations 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.

**Areas for Further Exploration**

Given that many of the combinations of climbing speeds, sliding speeds and well depths will result in ‘inexact’ escape times, Snail Trail provides opportunities to investigate decimal representation of times, which requires students to demonstrate proportional thinking and reasoning.

Explore this concept with students by presenting them this problem:

'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 way is presented below:

- The snail climbs 5 metres in 1 hour.
- Therefore, the snail climbs 1 metre in ¹/₅ 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.

Vary problems such as these by changing the climbing speed or providing actual starting times. Sample problems include:

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

Snail Trail exposes students to a famous puzzle and uses it as the basis of a mathematical investigation. The purpose of this part of the stage is for students to make links between the task and a set of recognised problem-solving strategies. You may have your own set of problem-solving strategies that you would like to use instead of the set referenced below.

Introduce students to the Problem Solving Strategies poster. The poster presents a range of strategies that students can apply to problem solving tasks.

Ask students to read the list of strategies in the toolbox and identify any that they used during this task. Ask students which strategies they felt were most useful for them during this task.

The specific problem-solving strategies that students have used in Snail Trail may include:

- Act it out
- Draw a diagram
- Make a model
- Make a table
- Look for a pattern
- Test all possible combinations
- Seek an exception
- Break the problem into manageable parts
- Work backwards

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. Paths in the Park

EXPLORESuggested 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

2. Garden Beds

EXPLORESuggested 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

4. Chocolate Boxes

EXPLORESuggested Learning Intentions

- 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 range of problem-solving strategies to help me solve a problem
- I can model a problem using materials
- I can identify and justify which strategies are most useful
- I can justify my solution using a range of manipulatives