The purpose of this activity is to engage students in applying place value and combinations to work out the number of possible number plates.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The procedural approach (show more)
- 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.
The conceptual approach (show more)
- 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.