These MATHCOUNTS Practice Plans are designed to help you prepare for competition. They cover a variety of topic areas and generally increase in difficulty as you progress through them. Each plan is intended to cover 45 minutes to 1 hour of practice time. The plans are divided into four sections:


To warm-up, Mathletes will start with a short problem set to practice related skills that will be expanded upon throughout the practice plan.

The Problems

To introduce a common type of MATHCOUNTS competition problem and/or a helpful problem-solving strategy, Mathletes will watch a video which solves and explains the approach to two or three problems. 

Piece it Together

Building on the warm-up and the video, Mathletes will combine their prior knowledge and the strategies they learned to solve another set of related problems.

Optional Extension

Each practice plan will have an activity, puzzle or game as an option to end with. The extension will be related to the concept the problems explored and give Mathletes an opportunity to have a little fun and/or be creative with the problem-solving skills!


MATHCOUNTS Practice Plans

Counting Paths Along a Grid

Explore combinatorics by looking at a common type of MATHCOUNTS counting problem – counting paths between two points. End with an extension that connects counting paths to another type of combinatoric problem.

Counting Shapes in a Complex Figure

This plan will help Mathletes to develop a strategic approach to counting the occurrences of a certain shape in a more complex figure made of multiple intersecting lines.

Difference of Squares

An important formula to know, the difference of squares identity is derived geometrically in the video for this practice plan. Mathletes will then try to recognize the difference of squares structure in various expressions and use the identity find the value.

Distance = Rate x Time

Explore the formula d = rt by starting with unit conversion problems. Mathletes will solve for distance, rate and time by paying attention to the units given in the problem and using the appropriate equivalent version of the formula: d = rt, r = d/t or t = d/r.

Divisibility Rules

Students will apply divisibility rules of various integers to simplify computation, better understand number composition and aid in problem solving. In the extension, Mathletes can prove why each of these rules work!

Faster Arithmetic Methods

Using the commutative, associative and distributive properties, Mathletes will arrange arithmetic problems in a different order that allows them to be solved more readily.

More MATHCOUNTS Practice Plans

The Fundamental Counting Principle

This plan will introduce students to The Fundamental Counting Principle – a faster method to determining the total number of possible outcomes of an event without listing them all out!

Least Common Multiple

Calculating the least common multiple is something many students are asked to do, but in this plan they will use their understanding of the least common multiple to stretch themselves to solve more complex problems.

Order of Operations & Defining New Rules

After refreshing Mathletes on the order of operations, the video will then focus on how to solve problems where an unfamiliar symbol is defined to be a new type of operations that follows given rules.

Ratios & Simple Probability

This plan builds on what students already know about ratios to introduce the definition of probability as a ratio of desired outcomes to total outcomes. Mathletes will then practice calculating the likelihood of single events.

Representing Patterns Numerically

In this practice plan, Mathletes will recognize visual patterns and practice defining them numerically in order to find the number of elements in the pattern after a large number of repetitions.

Special Right Triangles

Mathletes will become familiar with properties of 45-45-90 and 30-60-90 triangles. In this plan, the relationships between the sides of these two special right triangles will be derived. Then, Mathletes will apply these to solve for unknown lengths in geometric figures. 

Even More MATHCOUNTS Practice Plans & Additional Coach Resources

You Don’t Have to Solve for x!

Often the immediate reaction when Mathletes see an algebraic equation is to solve for the unknown but depending on what you are looking for it might be easier to manipulate the equation without solving it.


Need more practice? Check out the similarly structured MATHCOUNTS Minis from Art of Problem Solving's Richard Rusczyk! These minis explore even more topics and go in depth on areas touched on in these practice plans.

LinkedIn Coaches Group

Connect with fellow coaches and MATHCOUNTS staff on the LinkedIn MATHCOUNTS Competition Coaches page. This is a great opportunity for dialogue with the coaching community about best practices, share resources and to get answers to questions plus additional support!