From 2014-2019 while my children were in elementary school, I ran a weekly lunchtime math enrichment program for students at the school. We had about 35-40 minutes per session, with students from grades 4-7. The goal was to introduce some fun and interesting extracurricular math topics; the sessions were almost entirely activity-based, usually in the form of some game or competition. Some of the students participated in the Caribou Contest (6x per year, see resources below) and the grade 6 and 7 students participated in the University of Waterloo Gauss contest (see resources section below), though preparing for these contests was not the focus.

The worksheets below were used during guided activities with elementary school students between grades 4 and 7, but they would also be appropriate for older students. They are most appropriate for an interactive teacher or parent led session with someone who is familiar with the topic in advance. In some cases, I don't have the original worksheet, but noted the topic.

Most of the sessions didn't have specific worksheets, and were based around solving a set of problems on some topic in the context of a game or activity that made the session more exciting. I have summarized a few of the popular activities that I used below. I've also included a list of resources that I used to find problems and get ideas.

Worksheets:

Basic arithmetic:

Building the numbers from 1 to 40 using basic operations

Adding the numbers from 1 to N using geometry (motivate by how many games would need to be played between N people if everyone plays everyone else exactly once).

Basic combinatorics (counting):

A set of counting problems

Sequences and patterns:

Guessing the next number in a sequence

Some problems with pattern finding

Geometry:

Perimeter

Questions on area and perimeter

A few more area questions

The mathematics of cutting

Introduction to angles

Pythagorus's theorem (elementary proof by rearrangement of shapes).

Area optimization (the mathematics of chicken farming)

Basic geometric constructions (see e.g. www.euclidea.xyz).

Probability:

Basics of probability

Probability practice questions

A game with dice - You can make use of this simulation

Game theory:

Nim

The Josephus problem

Number theory:

The Sieve of Eratosthenes and some questions about primes

Deciphering codes with prime factorization

Binary numbers:

An introduction to binary numbers

A different introduction to binary and other number systems

Basics of computation with binary numbers and logic gates (followed first worksheet above)

Math with computers:

Using Scratch to calculate things

Graph theory:

The seven bridges of Konigsberg

The travelling salesman

The four colour theorem

Weighing problems:

Finding the heavy/light coin

Functions:

worksheet with "function machines" (original examples were lost, but you can fill in each "machine" with a rule e.g. N -> N + 4 or A,B -> A + B and some input values. The goal is to fill in the table)

Algebra:

Some algebra problems

A list of algebra problems

Ideas for activities:

1) The "Math Computer": Divide the students into small groups. Choose a set of problems, number them and print them, and cut up the sheets so there is one problem on each piece of paper.

Give the first problem to each group. When the students have an answer, they can check it by entering the answer on the "Math Computer" (running on a laptop at the front). If they get it right, the math computer will tell them how many points they get. Sometimes, for each point, the group got a raffle ticket placed in a bin (e.g. if group A got 3 points, they put three tickets marked with an A in the bin). At the end of the session, I did a draw for prizes. If students get a wrong answer, you can help them figure out what went wrong.

Example: here is a Scratch program for students to check answers on the "set of counting problems" above in the Combinatorics section. The answers in this program can be modified to fit any other set of problems.

2) Envelopes: Divide the students into small groups. Choose a set of problems on individual slips of paper, and order them. Label an envelope with the answer to the first problem and put enough copies of problem 2 for each group in this envelope. Label a second envelope with the answer to problem 2 and put copies of question 3 in that envelope and so forth. Continue until all the problems are in envelopes. You can label an envelope with the answer to the last problem and fill it with a congratulatory message. Finally, put a bunch of decoy answers on empty envelopes and place all the envelopes around the room. Give a copy of the first problem to each group, and tell them that when they get the answer to a question, they should find the next question in the envelope with that answer written on it.

3) Treasure hunt: The worksheets below required the students to work through a series of problems in order to find the coordinates to a hidden prize on a treasure map. They could then go and search for the prize using the map.

For example, the following hunts made use of this Treasure Map

Hunt 1 Hunt 2 Hunt 3

Two more hunts with their own map:

Hunt 4 Hunt 5

Resources:

There were a variety of resources that I used to get ideas while running the sessions.

Books: These are mostly appropriate for the upper elementary school level (or higher):

Math for Smarty Pants

The I Hate Mathematics Book

Math Circles for Elementary School Students

Mathematical Circles Diaries

Moscow Math Circles

Online resources:

The Caribou Contest runs fun online math contests for students of all ages. Their website has many past contests, games, etc... The past contests on this site were an excellent source of problems for all grade levels, though they are now charging a fee to access all of the past contests.

Another math contest for elementary school students is the Kangaroo Contest. They have a bunch of freely available past contests for different grade levels.

For older grades (5+), there are lots of excellent problems available at the University of Waterloo math and computer contests website. They even have a problem set generator that allows you to specify a topic and difficulty level and get a set of problems.

A lot of reseources on more advanced topics are available at the Art of Problem Solving website. They have a free online learning system called Alcumus with lots of good problems. This site is more oriented for high school students.