Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Identify patterns in addition and multiplication. For example, the many properties of adding and multiplying even and odd numbers. Lead them through the addition properties before the multiplication properties. For either, students can try a bunch of random examples to discover the patterns. For the addition properties, some students may want to use counters. A demonstration is here. If none of your students thinks to create an addition or multiplication table, after investigating both the patterns for addition and multiplication, have students make such tables. Then have them highlight the even and odd numbers. This doesn't prove the patterns but may help convince students. Conclude by giving your students the challenges found here and here, in any order.
TODO: Can manipulatives be used to discover patterns for subtraction or multiplication of odds and evens?