Car number plates

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Purpose

The purpose of this activity is to engage students in applying place value and combinations to work out the number of possible 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.
Description of Mathematics

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.

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 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 a 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 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/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 got 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 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 to 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 to 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? 
  • Is my working clear for someone else to follow?
  • How would I convince someone else 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, why would this be?

Examples of work

Work sample 1

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.

Click on the image to enlarge it. Click again to close. 

Multiplicative thinking is needed to work out the number of possible combinations without the need to list them all.

Click on the image to enlarge it. Click again to close. 

Recognition that A is one of 26 letters that can be used to ‘start’ a plate allows student to anticipate that the 259 974 possible plates with A is multiplied by 26 to get the number of plates possible from AA to ZZ.

Click on the image to enlarge it. Click again to close. 

Work sample 2

The student uses multiplication to calculate the number of possible plates without any need for listing.

Students first need to recognise that order of letters and numbers matters, so the problem is about permutations not combinations. AZ is a different permutation to 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.

Click on the image to enlarge it. Click again to close. 

Recognising the effect on the number of possible plates of exchanging a digit for a letter is a sign of multiplicative understanding. 6760000 x 26/10 = 17 576 000 show 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.

Click on the image to enlarge it. Click again to close. 

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.

Click on the image to enlarge it. Click again to close. 

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Level Four