The scientific method includes a hypothesis, creating an experiment to prove the hypothesis and collecting observations of the experiment to either prove it or disprove it. Similarly, engineers have a model. It includes asking, imagining, planning, creating and improving in order to achieve a goal. For young students, using this model introduces how engineering problems are solved. Use this with age-appropriate and practical problems like how to balance a wobbly classroom desk or how to reach something on a high shelf. This introduces the students to the engineering process.
Math and science are the easiest subjects to connect to engineering, but other subjects can connect to engineering, too. In physical education, students can be challenged to explore a particular physical disability and come up with a way around it by building something. For instance, how can someone climb who has a disability affecting one arm? How can someone participate in a basketball game when in a wheelchair? In art, students can be challenged to draw something as an engineer would draw it rather than an artist. In reading students can read stories about engineering projects like the building of the Brooklyn Bridge.
Out of their natural curiosity, young students like to build things and take things apart. What they don't know is that they are actually working like engineers. Notice when students are working on building something or trying to figure out how something works. Make the point to the students that their activities are really "engineering." Encourage them to find other ways to explore, build and take apart. This can be as easy as model-making or even playing with blocks for very young children.
There are almost limitless projects that young people can take on to learn about engineering. Consider finding one or two projects as extra curricula work for a science class or math class. In a geometry lesson, consider introducing a lesson about the building of a skyscraper or the ancient pyramids. In biology or chemistry, introduce the bio-engineering involved in the cloning process or from stem cell research. You can chose from numerous actual engineering projects and find ways to bring them back to a particular subject.