If I'm teaching students about gravitational energy stores and how to use E=mgh, I basically have two options
Option 1:
[Assume prerequisite knowledge quiz has been done and results suggest we're good to go]
If I have two tennis balls, Ball A and Ball B, and I drop one from really high up and one from much lower, which one will make a louder sound when it hits the floor? Ball A, why? Because I dropped it from a greater height, exactly. Okay, if sound is an energy transfer, which ball transferred more energy? Ball A, awesome, how do we know? Because it made a louder sound. Perfect. If ball A can transfer more energy, must it have had more or less energy to begin with than ball B? More, why Leah? Because of the law of conservation of energy, well done. If the ball is high up, what store will that energy be in, Rachel? Gravitational, well done. Okay, so which ball had more energy in its gravitational store, Becky? Ball A, why? Because it is higher up, fantastic!
Okay, what if I had a tennis ball, ball C, and like, a bowling ball or something, something much heavier, I'll call it Ball D? I drop them from the same height, which one will make the louder sound, which one will transfer more energy? Ball D, exactly, why? Because it's heavier, well done, if it's heavier, what does that mean in terms of mass? If it's heavier than ball C, is its mass bigger or smaller than ball C's mass? Bigger, well done.
Okay. So, if the ball if the ball is higher up it can transfer more energy, if the ball has more mass it can transfer more energy, and if it's somewhere with a stronger gravitational field it can transfer more energy. What if I have a really heavy ball like the bowling ball from earlier, and I drop it from like, half a metre off the floor, and then I have the tennis ball and I go to the lab upstairs and drop that one out of the window, so like 5m, any idea which one will be able to transfer more energy? No? Isaac, you think the bowling ball? You disagree, Karen, you think the tennis ball? Why do you think the tennis ball? Cause it's higher up okay, but the bowling ball is heavier. Interesting! It's not so clear anymore is it! We're going to need an equation* aren't we! So, we've got mass, we've got gravitational field strength and we've got height from the floor, so energy in the gravitational store is equal to the height multiplied by the mass multiplied by the gravitational field strength
Option 2:
The amount of energy an object has in its gravitational store depends on its mass and the strength of the gravitational field that it's in (so its weight) and how high up it is. What does it depend on? The mass, the gravitational field strength and how high up it is. We can put this into an equation, E=mgh and the triangle looks like this:
Let's do some examples
[Worked examples, MWBs, independent practice]
Clearly Option 1 takes a lot longer. In fact with some classes I've had it take 40+ minutes, especially with groups that have a weaker understanding of conservation of energy and I have to really labour the point that if it can transfer that much energy, it must've had that much energy to start with. But the thing is, Option 1 is better. And I'm not just saying that because it's what I do.
Student motivation
Often students don't care about equations. They don't see the point. They're abstract things that you shove a bunch of random numbers in and get another random answer out and sometimes it's right and sometimes it's not. It's no wonder teachers rely on triangles to "teach equations". All too often that's what's happening - they're teaching equations - and not starting from the physics.
By starting from the physics, students see "the point" of the equation. They see it as a tool for solving a problem, as opposed to the problem itself.
For more on motivating equations in physics, I recommend Tom Chillimamp's blog of the same name.
Abstractness/abstraction/content demand
The more abstract a concept, the harder it is to understand. "E = mgh" is pretty darn abstract - at this point it's just letters that may or may not mean anything. Even when we bring in what they mean, it stays quite abstract - the idea of gravitational field strength is something the students just have to take your word for, the whole mass vs weight thing - they just stay as values you have to do something with. By starting with the physics and what the equation actually means, it becomes a little less abstract. A ball falling from a height is very concrete, and very familiar. It is something a student can hook their understanding onto.
Expectations
One of the things I hear a lot is that linear rearrangements are just too tricky for students to understand so just give them the triangle and practice it. Except it isn't.
At the end of KS1, students are expected to be able to answer questions like "🔺+ 2 = 3. What does 🔺 = ?" These kids are, what, 7? And that's the expectation of them. And yet, when they get to KS3, we act like they can't handle a question like "🔺 ÷ 12 s = 4 m/s. What does 🔺 = ?". Of course it says "d" not 🔺, but the concept is very much the same.
(To be honest, I think this is a problem that starts very early, and in maths. As Ed Southall put it in his rather brilliant article in SSR there is "an over-reliance on memorisation rather than an appreciation of the simple and logical laws of mathematics" and this "does not prepare students for further studies." I think the same can definitely be said for KS3+4 physics. There is an over-reliance on memorisation. Why else would exam boards be issuing physics equation sheets for the 2022 exams?)
TL;DR
Start with the physics, not the equation.
*Yeah yeah, technically it's a formula. Bite me ;)
NB: when I actually teach E = mgh, I would probably give them a couple of examples of things other than balls, so they know this equation doesn't apply only to balls. I'd also vary the amount of questioning depending on the class - some classes will need to be brought in more, with others I'll need to get to the end as quickly as possible before I start losing them, it varies.