On the engage stage, I usually connect to previous lesson by asking some of them to remember and even explain some of the topics seen before. I also try to make them understand that we will build new knowledge from it.
On an EMI classroom, I definitively will start off with short games.
It depends on the topic, but my students usually expect to reflect on how the topic we are talking about affects their own lives and the lives of the people around them. In some cases, they expect to write an article.