2. Liquorice Factory

• Unifix blocks, or similar blocks that can be attached to one another.
• The liquorice factory 1 to 100 sheet: docx PDF

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

This stage uses the context of a story to illustrate the concept of prime numbers. Stories can provide a powerful context for student to explore and understand new mathematical concepts.

Share a story with students, using some stretchy liquorice as a visual prop:

“There’s a famous liquorice factory not far from here that produces stretchy liquorice. People come from all over to buy liquorice of different lengths. The factory has dozens of specially numbered machines that can take a unit of liquorice and stretch it out into whatever length the customer wants.

For example, the 4-length machine will stretch a unit length of liquorice out until it is 4 units long.”

Use the Unifix blocks to demonstrate a unit length of liquorice ‘going in’ to the 4-length machine and ‘coming out’ as a 4-unit piece of liquorice.

"The factory was going along really well until, one day, the 6-length machine broke down. Before long, someone came in and asked for some 6-length liquorice. What could they do?

Someone came up with the idea of putting the unit pieces into the 2-length machine, so the pieces doubled. Then, they took those pieces and put them into the 3-length machine, which turned them into 6-length liquorice."

Demonstrate this for students.

“So the manager was happy – the 6-length machine didn’t have to be fixed!

But then, the 15-length machine broke down…”

Invite suggestions from students about how the factory could do without this machine. Use this discussion to identify any potential misconceptions about how the machines work. For example, students may think that the 3-length machine adds three units to a length of liquorice, rather than multiplying it by a factor of three.

Continue your story:

“The manager of the factory suddenly realised that they may be able to shut a few more machines down and save money. A lifetime’s supply of liquorice was offered to the employee who could identify all the machines that could be shut down, while ensuring that all lengths of liquorice from 1 to 100 could still be produced.”

Provide students with a 1 to 100 number sheet and explain that these represent the machines from 1 to 100.

Ask students:

'Which machines could be shut down?'

Suggest that students use the sheet to record their progress. As they identify a machine that can be shut down, ask them to cross it off the list, and provide some evidence or an explanation about how to make that length without that machine.

A sample explanation may be: 'We don’t need 16-length because we can use the 8-length machine, and then the 2-length machine.'

In this situation, students may cross off the 16-length, and leave the 8-length and 2-length machines ‘working’.

Enable students requiring further support by reducing the number of machines to investigate (for example, machines between 1 and 40). Ask students to write down a series of multiplication facts that they are familiar with, and then discuss with them how these facts might be used to help identify machines that can be shut down. For example, see the following discussion:

Student: “I know that 4 times 5 equals 20.”

Teacher: “Great! So what might that mean for the liquorice factory? How is that knowledge useful?”

Student: “Well, if I put a piece of liquorice into the 4-length machine, and then into the 5-length machine, it will be 20 units long. So, we won’t need the 20 machine anymore!”

Teacher: “OK, so what else might this help you realise?”

Student: “Well, I know that all the numbers that end with a five or a zero can be divided by 5. So, these numbers might be machines I can cross off.”

Ask students to update the explanations on their sheet as they cross off new machines. For example, students may realise they don’t need the 8-length machine, because they can use the 2 and the 4-length machines. Students would then have to go back to 16 and update their explanation of how they could make a 16 unit liquorice piece without the 8 machine.

Monitor student progress to help determine whether to model a couple of these instances (where students need to update their explanations) with the class or provide this level of direction and support individually.

After some time, students may realise that they can identify whether any particular machine can be shut down by looking for the factors of that machine and using these factors to write a multiplication statement. For example, the 40 machine can be shut down because we can use the 2-length machine three times, to make the liquorice 8 units long, and then the 5-length machine. Students can write this as 2 X 2 X 2 X 5 = 40.

Collate responses from across the class and reach agreement on the machines between 1 and 100 that are still needed.

Areas for further exploration

1. Index notation

As students begin to cross more and more machines off, they are likely to be writing repeated multiplications of the smaller prime numbers, particularly 2 and 3.

For example: ‘The 36-length machine can be replaced with 2, 2, 3 and 3. This could be rewritten as 36 = 2 X 2 X 3 X 3.’

Introduce index notation to students as a means of writing repeated multiplication.

For example, the equation:

can be rewritten as

2. Divisibility Tests

As students begin to recognise the role of factors and divisibility in this task, you may like to introduce divisibility tests as a way of quickly determining whether a larger number is divisible by a smaller number.

Students may already be aware of some common divisibility tests, such as ‘all even numbers are divisible by 2’ and ‘numbers that end with a zero are divisible by 5 and 10’.

Provide opportunities for students to share divisibility tests they know about with the class, and test these using a range of numbers, with and without digital technologies such as a calculator. Challenge students to develop their own divisibility tests for any numbers that are left over.

Extend students by challenging them to find the smallest possible number that meets a combination of divisibility tests.

For your reference, a list of common divisibility tests can be found here.

Provide the class with an explicit definition of a prime number and emphasise the link between the machines and the prime numbers. Guide students to the recognition that the machines that were still needed represented prime numbers, and that the other machines were replaced by prime factors of their number.

For example, the 60-length machine was replaced because 60-unit long liquorice could be manufactured using the 2, 3 and 5 machines; 60 = 2 X 2 X 3 X 5. The prime factors of 60 are 2, 3 and 5.

Highlighting that “the machines that were still needed were of prime importance, and represent the prime numbers" may give some students a useful memory aid as well as making another link between the machines and prime numbers.

Facilitate a class discussion about whether 1 can be classified as a prime number.

Ask students to consider a much larger liquorice factory. Challenge students to find, with justification, the largest possible machine they can that could not be replaced. Check for student understanding of prime numbers and divisibility tests by asking students to submit an assessment piece explaining the process they went through to identify their largest possible machine. The format of the assessment piece could be differentiated and take the form of a short report, a poster, a presentation or a video.

Australian Association of Mathematics Teachers, n.d. Licorice Factory. [Online]
Available at: maths300.com/members/m300full/172llico.htm
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Clarke, D. & Lovitt, C., 1992. The Mathematics Curriculum and Teaching Program (MCTP): Activity Bank: Vol. 2. Canberra: Curriculum Development Centre.

Pierce, R., 2017. Divisibility Rules (Tests). [Online]
Available at: www.mathsisfun.com/divisibility-rules.html
[Accessed 15 March 2022].