Solving the problem in this activity gives us the chance to define some terminology related to number theory, such as even, odd, prime, factor, multiple, and so on, as the students use these ideas to explain their thinking about the problem.

The idea of this activity is to engage students in discussion about conclusions that can be made about divisibility relationships.  We typically use this activity after discussing divisibility tests for the numbers $2$ and $3$, and so this activity can feel like a more general case of divisibility testing.

The goal of this activity is to introduce the prime numbers, and to discuss the ``Sieve of Eratosthenes'' for finding prime numbers.

This activity introduces the content of the Unique Factorization Theorem.

The purpose of this activity is to give students more practice with the definition of factors, and more practice with exponential notation.

This is usually the first activity we do towards introducing the Least Common Multiple and Greatest Common Factor.  The problems make use of a common factor or a common multiple, but do not necessarily ask for the greatest or least.

These problems are intended to help to solidify students' understanding of GCF and LCM through the use of prime factorization.  To encourage reasoning, the numbers are generally too large for a calculator to handle.  Students should attempt to use the definitions of these terms to solve the problems, rather than just computing.

This activity has students make connections between our two languages of ``partial numbers'' (fractions and decimals) and then create methods of translating between them.  At the end of this activity (which takes several class periods), students should be able to translate from fractions to decimals and decimals to fractions, and be able to justify that all fractions must either terminate or repeat as decimals.  Finally, students should be able to make the conclusion that not all decimals can be represented by fractions.