3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply the properties of operations to y + y + y to produce the equivalent expression 3y.

This standard is for all those students who feel that math doesn't given them enough freedom of expression. You know, those kids who complain they can't use their artist's palette in the classroom, and that algebra "imprisons their creative spirit." Well, here's their chance to shine.

For this standard, we expect students to play (and we mean really play) with the properties of mathematical operations. It's almost like playing with lasers and anti-gravity boots, but with fewer potential injuries. Not none, but certainly fewer.

In order to apply the properties of operations, students need to be familiar with them. If students don't know the associative, commutative, and distributive properties yet, now's the time to teach 'em. Combining like terms is also, you know, kind of important. Only after they can recognize and identify these properties will they be able to play around with them freely.

As students use these properties to create equivalent expressions, stress that these expressions are equivalent because no matter how we evaluate them, they'll always end up equaling each other. This is true of numerical expressions and variable expressions; if students don't believe you, have them plug in values for the variable and they'll see that the two expressions should evaluate to give the same result every single time. (We'll get more into that in the next standard.)

The beauty of generating equivalent expressions is that there is literally an infinite number of ways to write the same expression. There might be two or three obvious choices, like rewriting y + y + y as 3y, but we can also write the same expression as y(1 + 1 + 1) or 3y + 0 or even ½y + ½y + ½y + ½y +½y + ½y. As students get more and more comfortable, encourage them to explore less conventional ways of writing the same expressions and testing them for equivalence.

So next time your students think math is too limiting, give 'em an expression have 'em go nuts.