Students will learn how to use the distributive property to evaluate expressions. 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 3 + 4 2's. Mathematically, \(2 \cdot 3 + 2 \cdot 4 = 2(3 + 4).\) Students will also use the distributive property to mentally evaluate certain expressions. For example, when looking at \(16 \cdot 43 + 16 \cdot 57,\) I don't want to figure out \(16 \cdot 43,\) nor \(16 \cdot 57.\) But I notice that \(43 + 57 = 100,\) so mentally I replace \(16 \cdot 43 + 16 \cdot 57\) with \(16 \cdot 100,\) which is obviously \(1{,}600.\) Thus, my original expression \(16 \cdot 43 + 16 \cdot 57\) must be equal to \(1{,}600.\) As another example, suppose I want to know \(9 \cdot 16.\) This isn't something I know off the top of my head, so I mentally replace this with \(9 \cdot 10 + 9 \cdot 6.\) That's \(90 + 54,\) I think to myself. And \(90 + 54 = 144.\) Thus, my original expression, \(9 \cdot 16,\) must be equal to \(144.\)

The Common Core includes this as a grade 3 topic, but without understanding parentheses, the distributive property doesn't make sense. Parentheses are first mentioned in 5.OA.A.1, so I've placed this distributive property without variables in the 5th grade curriculum instead of the 3rd. Khan Academy includes this topic as part of their 6th grade curriculum, and mistakenly cites 6.EE.A.3, which is about the distributive property with variables.

Conclude by leading this investigation:

Full Integer Collatz

by MathPickle