Here are my thoughts on what you can do over the summer to prepare your student for next year’s math club. It’s based on a message I sent to families just before finishing 4th grade, to prepare them for what was coming in 5th grade. But many of these things apply at any level.
General Considerations
 No one summer plan will work for every student. Everyone will have individual things s/he wants to work on or understand better.
 The recommendations below are a suggested menu, not a comprehensive list or a set of requirements. I imagine most students will just pick one or two categories, based on time and interest.
 Summertime is great for individual pacing. During the school year, kids are required to move at the same pace as the class. Over the summer, they can work at their own pace. A student may take a minute to understand one problem, and a week to understand the next one. Within reason, they should have that freedom.
 If nothing else, summertime is a good time to reflect on goals for next year. Do you just want to have fun? learn neat tricks? help your team? not feel lost? try for trophies? advance to state level and perhaps beyond? Each of these requires a different amount of work commitment per week. Also consider that, in 5th grade, students get busier; they will be practicing a band or orchestra instrument, transitioning to more sophisticated homework assignments, and spending significant time on their science fair project.
OK, with those things in mind, here are some things that I think will really help students advance when math club resumes in the fall.
Memorization
Try to memorize the following special numbers, which will appear over and over in math contests:

First 10 prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
[Prime numbers have only two factors: 1, and themselves.]

First 15 square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
[Square numbers result from multiplying numbers by themselves. 1×1=1, 2×2=4, 3×3=9, etc.]
 First 10 triangle numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55
[When you add the first few counting numbers, you get triangle numbers. 1=1, 1+2=3, 1+2+3=6, 1+2+3+4=10, etc.]
For those students who want to try more: powers of 2 starting with 2 to the zero power (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024); decimal reciprocals of 1 through 10 (1, 0.5, 0.333…, 0.25, 0.2, 0.1666…, 0.142857142857142857…, 0.125, 0.111…, 0.1).
Concepts
For next year, there are a couple of concepts that it would really help to have nailed down. These occur over and over, not just in math club but in daily life. There can never be enough practice on this stuff. I can provide notes on these things if that would be helpful.
 Proportionality
Practice doing the following: Solving ratios (by crossmultiplication or some other method); converting from one unit to another (by proportional reasoning or by unitfactor method); solving rate problems (eg with distance, rate, and time, knowing two and finding the third).
 Advanced Mental Arithmetic
Practice doing the following mentally: 2digit addition and subtraction; 1digit x 2digit multiplication; divisibility tests for 2, 3, 4, 5, 9, 10, 11. Just pick numbers and try it out; use a calculator to verify.
Games
For now I just want to get the suggestions out there in a timely fashion. I can provide more details here if people are interested.
 Games we do in class Nim, smallest integer, sprouts, 21 x’s, …
 Board games Prime Climb, Monopoly or Monopoly Jr (kid should be banker), strategic card games (Hearts, Pinochle, Gin, Blackjack, …). Note: Games with dice or playing cards help to make probability, an abstract concept, feel more real. And many questions on math contests assume familiarity with how dice and playing cards work; so familiarity with them is helpful.
 Origami Good books (kid recommendations) are Scholastic’s Origami Magic (easy to follow) and Uber Origami (comprehensive origami/craft book).
Everyday activities
Basically, every time you make a practical decision (as opposed to a moral one) as an adult, there will be probably be some numbers involved. Try to get your kids involved in computing the numbers and understanding why you’re deciding on particular things.
 Travel Traveling is actually full of numbers: ticket prices, bulk discounts, miles per gallon, distance and time remaining, …. When there’s a calculation to be made, see if your student can make a mental estimate that is good enough to inform a decision. (For example, we have 4 people in our party; should we get museum tickets at $12 each, or the family pack for $40?)
 Shopping I didn’t like shopping as a kid, so I amused myself with computations. It was very insrtuctive! For example: This shirt costs $30 and it’s 50% off; what was the original price? These socks are $12 per pair, buyonegetone free; what is the percentage discount? Even if given the original price, percent off, and new price, do a quick mental check to see if it’s accurate. Also try calculating unit prices: should we get the 64oz bottle or the 6pack of cans? Which one would have the lower price per ounce? Can you tell which is cheaper without going through all the calculations?
 Sports statistics Those who like sports will find plenty of things to calculate. Batting averages, games out of first place (very appropriate for the Mariners), winning percentages, yards per pass attempt or carry, completion percentages…. It’s also interesting to look up the formulas for things like ERA or quarterback rating which sports commentators mention all the time but don’t give details about.
Resources
 Puzzle/challenge math books When we do Math Club problems, all we are doing is solving puzzles. And who doesn’t love puzzles?
For summertime, any puzzle books are good, but I would suggest trying two particular types: Sudoku and Kakuro. Sudoku builds skills in logic, visual thinking, and understanding of sets. Kakuro is something I’m starting to try for myself; it still requires logic, but it’s more calculation based so perhaps more relevant to the types of problems we do.
Books that I like: 1000 Brain Games, Balance Benders series (starting with this one). I grew up with books by Martin Gardner, which had great problems but are now mostly out of print; also there are some stereotypes in there that feel way out of date today.
 Khan Academy Just go to their site, follow their steps, and let the kids go! This works well if kids learn from videos; some kids may get impatient with the presentation style, which is detailed — if so, just let them work the problems. Usual caveats about screen time and online safety apply.
Contest practice
If your student seems really interested in practicing contests (eg on a rainy weekend), here are some websites where you can find the most crucial practice materials.
 AMC 8 This contest and Blaine are the two new ones we’ll be preparing for next year. But unlike Blaine, we have not practiced for AMC 8 at all. Here is a link to previous problems and solutions. While I haven’t tried them, I’ve heard that the video solutions on that site (Art of Problem Solving) are pretty good.
 Blaine The 2015 contest and answers are here and all other previous years are here. The best things to practice are the most recent 5thgrade contests.
 Math Is Cool Previous contests are here. Again, look for the most recent 5thgrade contests.