CT EXAM 2018 MATHEMATICS FULL DESCRITION HCF and LCM

MATHEMATICS

  HCF and LCM
§ Square and square root
§ Percentage and its application (profit and loss, simple and compound interest)
§ Ratio and Proportion
§ Algebraic simple equation and its solution
§ Set- its elements, operations on set
§ Geometry of Triangles, quadrilaterals and circles
§ Basic Geometrical Concept

 

HCF and LCM

Highest Common Factor and Lowest Common Multiple

I've identified six methods for finding the HCF and LCM of two numbers. I'll explain each method here and identify any pros and cons.

1. Listing

Source: mathx.net

There's no harm in the listing method. It's brilliant in terms of underlying conceptual understanding - students can see exactly what they're trying to achieve here. List all the factors of the two numbers and find the biggest number that's in both lists - that's your HCF. List all the multiples of two numbers and find the smallest number that's in both lists - that's your LCM. Simple! Shame it's so time-consuming. And, in my experience, students sometimes miss factors from their list. One way to avoid this is by listing factors using a pairing method like this:

Factor rainbows are a pretty alternative (see this article from the NCTM).

2. The Venn Method

This is a popular method in the UK. First, we need to do a prime factor breakdown. By the way, if you're teaching prime factorisation then you might like these lovely factor tree activities from Don Steward.

Once you have the prime factors of each number, draw a Venn diagram and place the common factors in the intersection of two sets, as shown in the example below.

tutorvista.com

The HCF is the product of the elements in A intersection B (ie 2 x 2 x 2 x 2 in the example above) and the LCM is the the product of all the elements in A union B (ie 2 x 2 x 2 x 2 x 2 x 2 x 5). Note that UK GCSE students are not yet familiar with this terminology (ie intersection and union), but they will be under the new GCSE syllabus.

Even though I taught the Venn method for years, I'm not a huge fan of it. In my experience, students are ok with filling in the Venn diagram but then they often can't remember which 'bit' is the HCF. If they do remember the method then they probably don't have a clue why it works.

Confusingly, it seems that some people use a different Venn method which involves putting all factors (not prime factors) into a Venn diagram and identifying the highest factor in the intersection (see example below). This is another form of the listing method described above - it's just a different way of organising the list. Let's call this Lenn Method - it's a hybrid of Venn and Listing.

http://edtech2.boisestate.edu/brianroska/506/finalproject/gcf.html

3. Prime Factor Pairing

An alternative to the Venn Method is to do the prime factorisation but then skip the Venn. Write the prime factors of each number out as shown in the example below so it's easy to see which factors appear in both number - the product of these is the HCF. This method is featured in this post by Don Steward.

4. Euclidean Algorithm

I love the subtraction-based Euclidean Algorithm. It sounds complicated but it's incredibly easy. Try a few examples yourself to see how straightforward it is.

The method (including why it works) is explained in James Tanton's video below. I really like this method but for some reason I'm hesitant to use it with students... Would you?

Note that this method doesn't give you the Lowest Common Multiple, but it's easily found once you've got the Highest Common Factor. 

This looks like a pain, but cancelling helps - in the example above I found the HCF of 60 and 84, so to find the LCM I multiply 60 by 84 and divide by the HCF.

5. The Indian Method

I've stopped using the Venn Method at school and now use this instead. I'm not sure it's really called the Indian Method, I'm only calling it that because of this video. At my school we call it the Korean Method because a Korean student introduced it to us! I've found that my students really like it. It's hard to go wrong. It's easy to explore why it works too.

Say we want to find the Highest Common Factor and Lowest Common Multiple of 315 and 420. Write down the two numbers, then (to the left, as in my example above) write down any common factor. I've chosen 5. Now divide 315 and 420 by 5 and write the answers underneath (63 and 84 in this case). Keep repeating this process until the two numbers have no common factors (ie 3 and 4 above). Now, your Highest Common Factor is simply the product of numbers on the left. And for the Lowest Common Multiple, find the product of the numbers on the left and the numbers in the bottom row (to find the LCM, look for the L shape).

6. The 'Upside Down Birthday Cake Method'

I should mention this method because I keep spotting it online (video here). The only difference between this and the Indian Method is that here we can only remove prime factors. This seems unnecessary - the Indian Method is quicker. In the example below, why divide by 2 if you spot larger common factors? Why not start by dividing by 4, 6 or 12?

Source: https://uk.pinterest.com/pin/13229392628819982/

So that's it - six methods for finding a HCF and LCM.

Integers

I really like this set of questions from Don Steward. The following question confused one of my brightest Year 10s:

"A person has a rectangular plot of land measuring 8.4m by 5.6m. To survey the number of dandelions they want to divide it equally into the minimum number of square plots. What is the size of each square plot and how many such squares will there be?"

My student's approach to this question was to attempt to find the HCF of 8.4 and 5.6 using the Indian Method. This is what she did:

But she realised that her answer made no sense. 28 can't be the HCF of 8.4 and 5.6. Can you see where she went wrong? Although you can divide 8.4 by 7 (indeed, you can divide 8.4 by anything), it doesn't mean than 7 is a factor of 8.4. The numbers on the left must be factors of the numbers at the top. Non-integers don't have factors. A better approach would have been to convert the measurements to centimetres as shown below. 

Factors vs Multiples

If your students struggle to remember the difference between factors and multiples then they might find this helpful: for factors, think of a factory (where separate parts are put together). For multiples, think of multi-packs eg if cokes are sold in multi-packs of 6 then I can buy 6, 12, 18 etc. These ideas are taken from this resource from the thechalkface.net.

Conclusion

Did you know all of these methods? Will you try something new? Please let me know of any good methods that I've missed.

Even if you decide to stick with the method you're currently using, at least you've now reflected on how you teach this topic. Teachers rarely have the opportunity to pause and reflect.

This post should be read alongside Ed Southall's post Complements #9 LCM and HCF which explains the underlying concepts.

The whole presentation from my workshop is here. In my next post (give me a few days to write it!) I'll cover methods for teaching sequences, linear graphs and surds.

 

 

VIDEO LINK

 

https://www.youtube.com/watch?v=zwBhKZezXXg