The shape of the demand curve depends on two forces: the substitution effect and the income effect.  A typical treatment:

When the price of q1, p1, changes there are two effects on the consumer. First, the price of q1 relative to the other products (q2, q3, . . . qn) has changed. Second, due to the change in p1, the consumer’s real income changes.

In case that’s not instantly clear, intermediate microeconomics textbooks graphically decompose the two effects for consumers choosing between two goods:

While there’s nothing incorrect about the preceding text or figure, they’re unlikely to make the typical undergraduate say, “Aha!  I get it.”  Most students readily grasp the substitution effect: If the price of x goes up, you’ll naturally cut back on x.  How can teachers make the income effect just as obvious?

My answer: Instead of poring over the standard two-good choice problem to decompose the substitution and income effects, teachers should start with a one-good choice problem.  In this one-good setting, there is plainly no substitution effect; there is nothing to substitute to.  Yet quantity demanded still clearly depends on price.  If you have \$50 to spend, and the price of the only good jumps from \$10/unit to \$25/unit, consumers reduce their consumption from 5 units to 2 units.  Why?  There’s only one possibility: When the price goes up, the consumer’s real income automatically goes down.  That, students, is an unadulterated income effect.

Once students see an income effect without a substitution effect, many professors will want to show them a substitution effect without an income effect.  Soon they’ll be patiently explaining the income-compensated demand curve – and once again losing most of their audience.  A better approach: Posit a consumer choosing between a thousand different goods, none of them a large fraction of the budget.  If the price of a single good rises, the effect on the consumer’s real income is negligible.  But consumers will still respond to the rising price of x by buying less x.

My one-good and thousand-good problems lead straight to the meaty question: When is the income effect important, and when can it be safely ignored?  With only one good, the income effect is all-important.  With many goods, each a small share of the budget, the income effect is trivial.  So when is the income effect important without being all-important?  When at least one good is a sizable chunk of the budget, without being the whole tamale.

Housing is a great example.  Most students will instantly see that a 10% rise in their rent will make them feel poorer across the board.  And once students concede this point, it’s fairly easy to convince them that leisure fits the mold at well.  Backward-bending labor supply curves, here we come.

What about the math and the graphs?  Frankly, if your students aren’t going to graduate school, I’d skip both.  But if that’s too radical, you should still begin with my approach.  Make sure your students have the underlying intuitions firmly in mind.  Then show them the traditional way of formalizing those intuitions.  If you must bore your students into a stupor, at least teach them some economics first!