I am a regular substitute for the last 14 yrs in elementary with a lot of 1st and 3rd graders.
We don't push too much work in the summer with the exception of reading. They are no longer learning to read... they are expected to read. They have weekly reading logs that are given out on Monday and turned in on Fridays for a small grade and they are expected to read at least 15 minutes each night.. In reality, that is not much at all. I encourage more.
Our 3rd graders are introduced to cursive writing which is a part of daily warmups and practice. Multiplication charts are introduced as well and there are a lot of practice games and sheets we do with multiplication.
It is 3rd grade when students get the first "real" report card and they are also are required to take the state tests for placement. We do a lot of practice with reading and comprehension and multiple step math problems.
My personal thoughts are that the 2 most critical points are being a good reader and knowing basic concepts of math. Third grade expands from those points and if a student is not good at those, it makes the year harder.
The students will be expanding on research and writing reports. They amaze me at the computer skill they have at this age as far as power point presentations. I love seeing them enjoy that part of class.
FREE tutoring is offered after school once all of the fall placement testing to see where students are in the grade level.
Bottom line, don't overdo anything in the summer but Encourage reading. I wouldn't start workbooks, etc. Let your 3rd grader help you in the grocery and get math lessons that are fun such as pricing and weighing produce, etc.