First, students will learn how to list the multiples of a number. For example, the multiples of 8 are

$$0,\,8,\,16,\,24,\,32,\,40,\,48,\,56,\,64,\,72,\,\ldots$$After listing the multiples of several different numbers, ask students if they notice any sort of pattern in the multiples of some number. For example, if you asked about the multiples of 8, we hope students will notice the pattern

$$0,\,8,\,6,\,4,\,2,\,0,\,8,\,6,\,4,\,2,\,\ldots$$in the units digit of each number. If students don't find the pattern on their own, you can ask them more directly if they see a pattern when looking at only the units digit of each number. You could even cover up the tens digit to make the pattern really obvious. Next, students will learn how to list the multiples of some number plus some other number. For example, list the numbers which are 2 more than a multiple of 6:

$$2,\,8,\,14,\,20,\,26,\,32,\,38,\,44,\,50,\,56,\,\ldots$$Ask students what the pattern is now. They should respond with something like

$$2,\,8,\,4,\,0,\,6,\,2,\,8,\,4,\,0,\,6,\,\ldots$$If you asked students to list the numbers which are 2 more than a multiple of 6, you should now ask them to list the numbers which are 4 less than a multiple of 6. Students may be surprised that these two lists are equivalent. After that, students should be asked to list the numbers which have a remainder of 2 when divided by 6. Students now have three different ways to think about this sequence of numbers.

Conclude by leading this investigation:

The Broken Sword (addition, division, mini-mathematical universe)

by MathPickle