Does anyone have any suggestions for teaching students how to do high-school level proofs involving greatest common divisors and least common multiples. I suggest to them to start by doing a few numerical examples so it makes sense. I also mention translating statements such as, if gcd(a,b) = d, then a=md and b=nd.
Any suggestions on how can I help them with their thinking?
I just had this topic yesterday with my extra math group. We started with prime factorisation and noted that for $gcd$ we need the intersection of the two sets (the sets being the set of respective prime factors) and for $lcm$ we need the union. A Venn diagram helps a lot. We even used a more abstract representaion of numbers that is different colours for the different prime factors (and by colours I mean only a colour not writing the respective prime number in a specific colour). We ususally use colours this way, or just simply shapes, since prime numbers have this "unique" look in a way. We also talked about simplifying fractions since one can us the same idea with shapes and colours. A concrete example was what we can say about the numbers involved in an equation of the form $$ a\cdot b=c\cdot d, \qquad a,b,c,d\in\mathbb{N} $$ which we can rewrite as $$ \frac{a\cdot b}{c\cdot d}=1 $$ and the questions we asked is what we can say about whether $$ c\mid a \ \text{or} \ c\mid b? $$ or can we at all have any conlcusions like this?
We also proved that $$ (a,b)\cdot[a,b]=a\cdot b $$ using the Venn diagram approach. $(*,*)$ being $gcd$ and $[*,*]$ being the $lmc$. We finally looked at Euclids algorithm, since I wanted to show them that we can find $gcd$ without knowledge of unique prime factorisation.
Now I don't know if this will be useful for you I just figured I tell you my experience since this was the main topic for us yesterday and I have fresh experience with this area at this level.
What kinda proofs were you thinking about?