6.EE.A.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 properties of operations to y + y + y to produce the equivalent expression 3y.*

Students will learn what the distributive property is. Then students will learn the distributive property works because multiplication is repeated addition. Consider that \(2 \cdot 3 + 2 \cdot 4\) is the sum of 3 2's and 4 2's, which is the same as 7 2's. Mathematically, we say \(2 \cdot 3 + 2\cdot 4 = 2 \cdot 7.\) More generally, but using the same reasoning, we find \(x \cdot y + x \cdot z = x(y + z).\) After students understand why the distributive property is true, they will be asked to simplify expressions such as 3(2 + x) or 6(4x + 3y). Conclude by showing your students this card trick, and explaining the math behind it.

Currently, algebra tiles can be used to show the simplification of expressions like 3(2 + x), but not 6(4x + 3y). That's because algebra tiles don't have a \(y\) or \(y^2\) tile. However, if you make your own tiles, you could choose the unit tile to have a width of \(10\) mm, the \(x\) tile to have a width of \(10\sqrt{5}\) mm, and \(y\) tile to have a width of \(10\sqrt{10}\) mm. You could choose any two distinct non-square numbers to scale by. In this case, we used \(\sqrt{5}\) and \(\sqrt{10}.\) To understand the benefit of choosing such lengths, imagine choosing the unit tile to have a width of \(10\) mm, and choosing the \(x\) tile to have a width of \(20\) mm. This could cause students to assume \(x = 2,\) which is not our intent. Here's a fantastic blog post on how to make your own algebra tiles.