It Stands to Reason
Before reading on, take time to consider how you would explain convincingly your response(s) to this simple-but-rich question from Learning and Doing Mathematics by John Mason: if a number is a factor of the product of some numbers, is it necessarily a factor of one of those numbers? Is it always / sometimes / never a factor of one of those numbers? Can we predict when it will be and when it won’t?
Pupils are more adept at, and more confident at, finding ways of solving mathematical problems when they develop the habit of looking for, understanding, and then exploiting, the underlying mathematical structure of the problem. With numerical problems, sometimes the most useful approach is to express a number as the (unique) product of its prime factors: for example 12 as 22 × 3, and 20 as 22 × 5. Being able to identify fluently the prime factors of any number, and from that deduce all its factors (not just the prime factors), is an important skill that often enables pupils to see possible routes to the solution of a numerical problem. Given that skill, which we assume in the following examples and suggestions, pupils can be prompted to find, talk about and then use, the underlying structure of the numbers expressed in terms of their factors and multiples. This is what we look at in this article. Our premise is that we can help pupils notice and exploit this structure by prompting them to think about structures that they should be able to see in visual images.
Strips of squares
These provide very simple images in which pupils can see facts and relationships involving factors. For example, suppose that the numbers 12 and 20 are represented by strips of 12 and 20 identical squares:
Now ask your pupils to use two contrasting colours in an effective and sensible way to show, on separate copies of the two strips together, factors of these two numbers.
If they know that since 12 = 22 × 3, its factors are 1, 2, 3, 4, 6 and 12, and since 20 = 22 × 5 its factors are 1, 2, 4, 5, 10 and 20, hopefully they will colour the strips in twos, threes, fours, fives, sixes and tens, as we have done here:
Now ask them what they can deduce from their images, and when they articulate their conclusions encourage discussion. You want them to explain how their images show that the common factors of 12 and 20 (excluding 1) are 2 and 4, and therefore that the highest common factor of 12 and 20, HCF(12, 20), is 4.
A very important conclusion is that the common factors of 12 and 20 are also factors of the difference between 12 and 20. Again, ask how the images show this.
By exploring, in the same way, a variety of examples, including their own examples, such as 12 and 30 …
… it is likely that pupils will be able (with some support, perhaps) to conjecture that the common factors of two numbers, including the highest common factor, are necessarily factors of their difference. In this example, all numbers that are factors of both 12 and 30 are also factors of their difference, so all the common factors of 12 and 30, including HCF(12, 30), are factors of 18.)
Your pupils might now be ready to use algebraic notation to express, succinctly and generally, what they conjectured using their strips of squares about the HCF and the difference of two numbers. For example a clear summary using algebraic notation and supported by a sketch might look like this:
- Suppose HCF(p, q) = h, p = a × h and q = b × h, where a and b have no common factors.
- Then q – p = h(b – a), and so h is a factor of the difference of the two numbers:
They should be able to say in their own words why any number that is a common factor of two different numbers is also a factor of their sum and of their difference. It is one of the many facts about numbers that pupils with a deep understanding of relationships between numbers should be able to deduce and explain – but see how the pictorial approach makes this abstract idea much more accessible.
You want your pupils now to realise that they can USE this fact to help find the HCF of two numbers. For example, if they want to find the highest common factor of 738 and 750 they could now reason like this …
- 750 – 738 = 12.
- The factors of 12 are 12, 6, 4, 3, 2, 1.
- So HCF(738, 750) is 12 or 6 or 4 or 3 or 2 or 1.
- I want the highest of these numbers that is a factor of both 738 and 750.
- Starting by dividing 738 by 12, I find that 738 = 12 × 61 + 6, so HCF(738, 750) is not 12.
- I then try the next highest, which is 6, and find that 738 = 6 × 123.
- 750 is 12 more than 738, so 6 is also a factor of 750 (or, 750 = 6 × 125).
- Therefore HCF(738, 750) is 6.
A powerful extension of this is to reason that the HCF of, say, 12 and 30 is not only a factor of 30 – 12, but also a factor of ANY multiple of 30 – ANY non-equal multiple of 12, for example 30 – 2 × 12 = 6. This helps when the numbers are not close together: the HCF of 221 and 68 is the highest common factor of
221 – 68 = 153
221 – 2 × 68 = 85
221 – 3 × 68 = 17
and since 17 is prime, this tells us that the HCF of the two numbers must be 17: 221 = 17 × 13 and 68 = 17 × 4.
Square tiles in a rectangle
Your pupils will gain further insight into the structure of factors and multiples by exploring images of rectangles into which they try to fit identical square tiles. For example, they could investigate the common factors of 12 and 20 by trying to tile a 12-by-20 rectangle (drawn on a square grid) with identical square tiles. If they sketch diagrams such as these …
… they will see that, although within the rectangle they can draw squares of side-length 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, they can only tile the rectangle (without trimming any squares and so that there are no gaps) with squares of side-length 2 or 4. Ask them to explain why this is a consequence of the fact that the only common factors of 12 and 20 (excluding 1) are 2 and 4. The argument you want them to propose and refine is something like: If I can tile a rectangle with a square of a particular side-length, that is if I can completely cover the inside of the rectangle with identical copies of the square leaving no un-covered gaps, then the side-length of the square must be a factor of both the width and the height of the rectangle. So the side-lengths of the squares with which you can exactly tile a rectangle, leaving no gaps and without trimming any squares, are the common factors of the width and height of the rectangle - given in the same units of course.
Once pupils understand, through devising and discussing a variety of (their own) examples,
- why it is possible (although not necessarily desirable!) to find the HCF of two numbers such as 12 and 20 by drawing square tiles in a rectangle
- that the side-length of the largest square tile with which you can completely cover a p-by-q rectangle, with no gaps, is the highest common factor of p and q,
you can prompt their thinking to a much deeper level.
In order to do this, pupils need to think, and express ideas, generally for a while (as in the second ‘understanding’ above). You could display an image such as this …
… and tell pupils that it shows a p-by-q rectangle filled exactly and completely with identical tiles of the largest possible side-length, h. Challenge them with the following questions about this image, which they could discuss in the given order for as long as necessary, perhaps first in pairs and then as a whole class:
- What could p and q represent?
- What then would h represent?
- Why are there a tiles going downwards
- Why b tiles going across?
- What must be true about factors of a and b?
- How can you express the total number of tiles in the rectangle (using given letters)?
Once pupils have articulated, in their own (idiosyncratic) ways, correct answers about which they agree, such as …
- p and q could represent two different whole numbers
- h would then represent the highest common factor of p and q
- There are a tiles going downwards because the height of the rectangle is p, and p = a × h
- There are b tiles going across because the width of the rectangle is q, and q = b × h
- a and b have no common factors (otherwise square tiles of side-length h would not be the largest square tiles that will fit in the rectangle)
- The total number of tiles in the rectangle is a × b.
… it will be instructive to return to the previous numerical example by asking, and discussing pupils' answers to, the following question:
If you rearrange the largest tiles that will exactly cover a 12-by-20 rectangle so that they now form a new rectangle in which the height is the side-length of a single tile …
… what does the length of the rectangle represent? Why?
We’ll leave you to ponder this before the next issue! In the meantime your pupils might enjoy this assortment of problems from NRICH. If your pupils use reasoning along the lines we’ve developed in this article to solve these or other problems in ingenious ways please tell us about their methods, or send a photograph of their reasoning by email to email@example.com or tweet @NCETMsecondary.
Teaching mathematics at Secondary Level, Anthony D. Gardiner, The De Morgan Gazette 6 no 1, 2014
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