I'm always experimenting with different ways to explain things, and each year, I am happy if I can just find one or two "biggies" that really make a big difference for my students. Here are the gems I gathered this year. There are a lot of them this year, since I taught all the same classes as I did the year before. It has been a good year to refine my teaching techniques a bit...
1. Visualizing solving equations as "unwrapping the onion" to get back to x. The kids who practiced this were awesome at re-arranging equations in terms of variables, even if the equation looks like this and they're asked to solve for Q:
R = S + 3Q - 5W
2. Drilling equations of the form ax + b = c on mini whiteboards until every Grade 7 kid can do it in their sleep (including equations with fractional solutions). This really helped in the long run, because afterwards, when we moved on to much more complex equations, once they got down to an equation of this form, they could essentially do the rest while sleeping, and they wouldn't make any careless mistakes as their attention started to drift from the problem. It also helped them drill integer skills in algebra context, in the early stages of equation-solving.
Incidentally, I always read an equation like 5x = 3 as "5 copies of x is worth 3 in total. How much is each x worth?" I think this has helped my kids stay away from being confused about what to divide by what. If they can sound out the equation in their heads, the same way that I do, then it's not hard...
3. Always pronouncing fraction multiplication as "of" instead of "multiplied by", when reading an equation or expression out loud. For example, if the kids are simplifying (1/3)(x + 12), I say, "what is one-third of the quantity 'x plus 12'? Let's do it in steps. What is one-third of x? What is one-third of 12?" This helped to reinforce both the meaning of fractions and its connection to the multiplication operation.
4. Similarly, never let the kids get away with telling you that they don't know how to find a non-unit fraction, such as four-fifths of something. Always rephrase the question as, "What is one-fifth of ________? Then, how would you find four-fifths?" Push them, push them to do more in their heads. Don't let them think that fractions are harder than they are, or let them think that not knowing the meaning of fractions is OK.
5. Highlighting matching descriptions or units inside a proportion. Worked superbly to help kids set up proportions correct, consistently!
6. Teaching log by slowing down and focusing on its definition. This is a tried-and-true method for me this year; my students never had any weird issues at any point with logs this year, and I was able to repeat the same success with different students (some new transfers, some from other grades) at a later point. They were calm, independent, and their work all made sense from the start to finish during the entire unit. This had never happened to me before while teaching logs in any other way!!
7. Teaching sequences by making kids make a table of values (index vs. actual value) every single time, prior to setting up any equations. For some reason, this really slowed them down to thinking about what the word problem is giving them to work with, and they were consistently successful at tackling a variety of problems without getting confused.
8. Teaching Calculus by making kids sketch f'(x), f''(x), or f(x) graphs, given related graphs. They must do this consistently at the start of each class before moving on to work on anything else. The graphical understanding will underpin their entire algebraic understanding of Calculus, and help to bring everything together.
9. As soon as the kids differentiate a function via algebra, they must write down next to f(x) and f'(x) some word descriptions, such as "Height" for f(x) and "Gradient" for f'(x). This will build their independence in choosing the correct function to plug x-values into, and free them from having to ask you what to do at the next step of their analysis. Nag them while supervising/going over every problem, to write down these descriptions. Eventually, they won't need this anymore and they can visualize the descriptions in their heads. But, this builds their independence -- fast.
10. Repetitive quiz practice, on a complex topic, until you feel that it is quiz-worthy. This builds their confidence, while focusing their attention on a key skill, integrated with other skills they've seen before or that are nice to have.
I'm not done with the school year yet (still doing things that I'm pretty excited about, for the last few weeks of school), but I think that these are the little things that have made the most impact on my students' achievements/understanding this year. I hope that they will help you as well as they have helped me!!