I like to talk to students about patterns in general prior to having them look for the specific numerical patterns found in the addition table. This is a video that can be used with students and shows the expected patterns they've come to associate with math:
These are also math patterns, though students may not have thought of them as such before. I ask them to think about how math is involved in some of the patterns in nature, patterns with people, and patterns in behavior and actions.
I like to ask students to come up with a pattern from either their everyday life (bus routes, soccer practice schedule, price of games at store) or an area of interest (days it takes different planets to revolve around the sun, number of challenges required to pass a certain level in a video game). I ask them to write down what the pattern is, and how it is/might be connected to math.
I remind students that while today we are working with patterns we see in addition problems, I want them to keep the larger theme of patterns in the back of their mind. Are there ways in which the patterns they observe in addition can help them understand other parts of math, or other operations? Are there corollaries between addition patterns and other types of patterns?
Their job is to work with an addition chart, preferably a blank addition chart but, depending on students' unique needs, possibly an addition chart that is already filled in. Students also generate a written list of math facts to identify and describe as many different patterns as they can. I typically have different students write out different lists, so some may write out +2, others +3, +4 and so on up until +9 and possibly +11 or +12. This should man ways be a review of concepts that were explored in second grade but they will come to it with new ideas and understandings.
Once they have identified and named some patterns, they can make their numbers into multiples of ten and one hundred and determine if the pattern persists, and how it persists.
Example: Instead of 2 + 9, 3 + 9 and so on the student works with 2 + 90, 3 + 90 and so on.
It is not enough for a student to say, for example, I see that there are nines in a diagonal. They need to explain where the nines are in a diagonal on the number chart, why they are in a diagonal, and what does it mean? How does it help them understand addition to recognize that pattern?
I gather the students at the carpet to share the patterns they discovered and we count how many unique patterns students observed as well as noting the importance of certain properties of addition (zero/additive, commutative).