I like to take “real-world” math questions and unpack them for content, then look at other solutions, and then finally think of bigger and better questions that could be asked instead. For example, here is the classic tree problem:
The students here are forced to use the information given, not to go out and get their own, and they are usually asked to use one particular method for doing it. They look at the problem and just accept that the 40 degrees and 75 feet appeared out of nowhere with no measurement. That is what textbooks always do, magical formulas and measurements appear, kids accept their authority and look for the one correct answer using the one right method. Never mind that this is supposed to be an estimation technique.
What do I like to do instead? Take them outside. Have them get in groups of three and pick a tree. Have them brainstorm possible methods to estimate the height of the tree. Don’t “teach” them any methods before taking them out, let them create stuff. When you go back inside or the next day, depending on your schedule, have each group make a little poster with their method to share out to the class. Pick a couple of the methods that kids like and develop them further. At this point you might teach them some stuff about trigonometry to develop a method. Do they need angles? Teach them about clinometers and have them make one. When they have a few methods down, have them go back out and estimate using three methods. Then have them answer the following quesitons: 1) How close were your estimates using the three methods? 2) What could you do with your three estimates to help you decide on the best estimate for the height of the tree? 3) How many leaves do you think are on that tree? Come up with an idea for how to estimate that.
This can lead to all kinds of things. If you want them to understand similarity, you could help them use that shadow method for finding the height as one of their methods. You could then talk about the self-similarity of a tree. Maybe look at the branch of a tree. What is the relationship to the length of the branch and the number of leaves on it? How does this relate to the height of the tree and the number of branches, etc.
But this all leads to the question of how we count stuff without explicitly counting each object. Think census, predictions of who is going to win the next presidential election. But wait! Is that before the primaries or after? This can lead to an understanding of the mathematics of ranking in voting systems and just how bad our plurality system is for such elections. We could compare our system to other countries that have proportional representation (more similarity!).
Before going into all that though, I would really like to have an arborist come out with us and look at the trees and find out how she estimates their heights. It wouldn’t surprise me if she could just do it by looking at them without writing anything down and get better estimates than the rest of us with all of our calculations. This kind of real-life experience is what really invalidates these tests but that is another conversation.
I could go on and on but hope you get the idea. Math should be less like a test-prep center and more like a walk in the park.