|Start with decimal notation using fully positional notation as opposed to something like Roman numerals which uses different symbols for powers of 10 but also has some positional rules. Give examples of arithmetic rules.
Then drop to octal, since it's easier to understand using less symbols - counting with just your fingers and not using the thumbs. More arithmetic examples.
Play around with a few other bases - duodecimal (base-12) or sexagesimal (base-60 - hours, minutes, seconds) - finally arriving at hexadecimal. More arithmetic...
Maybe show base-4 or move straight to binary. Show the same arithmetic rules apply even when there's only 2 symbols.
Show that any number in binary representation can be easily rewritten as hexadecimal or octal (and vice-versa) just by grouping/expanding the digits. Mention that, in computing, the term bit is a contraction of binary digit.
Introduce Boolean algebra as a separate topic using English-like examples for AND, OR and NOT.
Somewhere along the way, introduce De Morgan's Law - "NOT (A OR B)" gives the same results as "(NOT A) AND (NOT B)"
Finally, show that if TRUE is one and FALSE is zero, then AND is the same as multiplying in binary and only looking at the last digit and OR is addition.
You can then introduce logic gates and show how to build an adder circuit for 2 single bit numbers with a 2-bit output - least significant bit is OR, most significant/carry is AND but it can all be done with just AND/NOT or OR/NOT, so you only need one 2-input gate and an inverter (NOT).