Car number plates
The purpose of this activity is to engage students in applying place value and combinations to work out the number of possible number plates.
About this resource
This activity assumes the students have experience in the following areas:
- Multiplying and dividing whole numbers.
- Calculating the number of possible combinations using cartesian products.
- Applying the place value structure of large whole numbers.
The problem is sufficiently open-ended to allow the students freedom of choice in their approach. It may be scaffolded with guidance that leads to a solution, and/or the students might be given the opportunity to solve the problem independently.
The example responses at the end of the resource give an indication of the kind of response to expect from students who approach the problem in particular ways.
Car number plates
Achievement objectives
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
NA4-6: Know the relative size and place value structure of positive and negative integers and decimals to three places.
Required materials
See Materials that come with this resource to download:
Car number plates activity (.pdf)
Activity
In New Zealand, each car has an individual number plate, so it can be identified.
Number plates were first introduced in 1964 using a system with two letters of the alphabet and four digits.
In 2001, Waka Kotahi Transport New Zealand ran out of combinations when ZZ9999 was issued. A new system began using three letters and three digits.
- How many plates were possible under the old system, and how many are possible under the new system?
In New Zealand, the number of new plates issued each year keeps increasing. Let’s say that about 250 000 new plates are issued each year.
- How long will the new system last?
- What would be an option for the next "new system"?
The following prompts illustrate how this activity can be structured around the phases of the mathematics investigation cycle.
Make sense
Introduce the problem. Allow students time to read it and discuss it in pairs or small groups.
- Do I understand the situation and the words? (Students may need to look at actual number plates to see how the registration plates are made up. An important assumption is that personalised plates are not included in this problem.)
- Is there an order to the way letters and numbers are used? What is fixed, and what can vary?
- Does this look or sound like a problem I have worked on before? (Students might see that they have solved problems with combinations before.)
- Could I write some examples of plates under the old system and the new system?
Plan approach
Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.
- What will I need to find out to solve this problem? Have I gotten all the information I need?
- How will I systematically find all the possible plates available in the new system?
- What representation will help me be systematic and not miss any possibilities? (A table might be useful or a tree diagram if the student recognises the connection to probability.)
- What are the maths skills I need to work this out? What operations might I use to save work?
- Will I need to write down all the combinations, or is there an efficient strategy to show all the possible outcomes?
- What tools (digital or physical) could help my investigation?
Take action
Allow students time to work through their strategy and find a solution to the problem.
- Am I approaching this problem in a systematic way so I do not miss any possible number plates?
- Have I recorded my ideas in a way that helps me see patterns?
- Are there any patterns that save me work?
- How might I describe the pattern?
- Does the pattern help me to answer the question?
- Can I use multiplication to find how many different plates are possible?
- Does my solution make sense? Is the number of plates sensible for the situation? Why?
Convince yourself and others
Allow students time to check their answers, and then either have them pair share with other groups or ask for volunteers to share their solution with the class.
- What is the solution?
- Are my workings clear for someone else to follow?
- How would I convince someone else that I am correct? Can I justify my strategy for finding all the possible plates?
- Have I considered all possible cases (plates)? How can I be sure that no possibilities were missed?
- How can my strategy be applied to finding the number of possible plates for other "numbering" schemes? How?
- What representations were helpful to solving this problem? Why were they helpful?
- What connections can I see to other situations, and why would this be?
Examples of work
The student uses a listing system leading to multiplication to work out how many plates are possible.
Students may begin by listing some plates. Under the old system, AA0001 was the first plate issued, followed by AA0002 through to AA9999. Listing is laborious, and students are likely to look for short cuts.
AA1, AA2, ..., AA9999.
There must be 9999 AA plates.
AB1, AB2, ..., AB 9999.
Multiplicative thinking is needed to work out the number of possible combinations without the need to list them all.
There must be 26 x 9999 plates that start with A.
AA1 |
AA2 |
AA3 |
… |
AA9999 |
AB1 |
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AC1 |
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AZ1 |
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AZ9999 |
That's 26 x 9999 = 25 9974 plates.
Recognition that A is one of 26 letters that can be used to "start" a plate allows students to anticipate that the 259 974 possible plates with A are multiplied by 26 to get the number of plates possible from AA to ZZ.
260 000 ÷ 26 = 259 974
You can have 26 different starting letters: A, B, C, ... 2.
So, the AA-A2 number needs to be multiplied by 26.
26 x 25 = 9974
The student uses multiplication to calculate the number of possible plates without any need for listing.
Students first need to recognise that the order of letters and numbers matters, so the problem is about permutations, not combinations. AZ is a different permutation from ZA, although the same letters are involved. Students should be able to justify their use of multiplication and why the new system produces more possible plates.
Recognising the effect on the number of possible plates of exchanging a digit for a letter is a sign of multiplicative understanding. 6 760 000 x 26/10 = 17 576 000 shows that the exchange creates 2.6 times more plates. A new system that has four letters and two digits will result in 17 576 000 x 2.6 permutations.
Look for students to recognise that 250 000 new plates each year is one quarter of a million. The new system has about 17.5 million permutations.
250 000 is a quarter of one million.
That is 1 000 000 new plates every four years.
17 576 000 is about 17.5 million.
17.5 x 4 = 70 years
The new system will last until 2072.
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