Before class starts, I write the following quote along the top of the whiteboard (high enough that it's out of the way for the entire class):
"There are 10 types of people in this world, those that understand binary and those that don't."If anyone asks about it, I say that we'll cover it at the end of class.
[Note: This quote should only be used for the first class that you introduce binary, otherwise it won't be nearly as effective. I used this quote last spring with these students when I introduced binary, but did not use it in this review class.]
Powers of Two
At the start of each class, we begin with a simple exercise. For the first few classes, we cover the powers of 2. I first (jokingly) check to make sure that everyone knows how to multiply by 2 and then write:
1and then go around the room asking the students to multiply the previous value by 2:
8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536I usually stop at 65536 and then point out 28 (256) and 216 (65536) as values that we'll be encountering later as we learn more about computers.
Even though my goal is to teach binary and hexadecimal, I always start with octal since I believe that it's an easier transition from decimal: when counting in binary, new digits are added rapidly for the first few number and this can be confusing. And with hexadecimal, we're adding additional symbols into the mix, so it's easier to introduce that once they've already mastered an alternate number system.
So, what did we cover last week? Numbers, binary, octal. We're going to continue with that this week.
I briefly review place-value number systems and describe base-10 (decimal) and base-8 (octal).
Decimal is based on 10's. It has 10 digits (0-9), and the places are based on 10:
Octal is based on 8's. It has 8 digits (0-7), and the places are bases on 8:
[Sidenote: I use this opportunity to remind the students that any number raised to the 0th power is 1. If the students were thinking of powers as "how many times do I multiply this number by itself" then this result will seem odd (since you'll expect the answer to be 0). Hence, this is a great opportunity to reinforce the correct answer.]
To practice a bit with octal, we start with a few worksheets:
Worksheet: Octal counting (1 & 2)
These worksheets have the students count circles that are pre-arranged into groups of eight to make it easy to determine the octal number. In addition to writing the octal number, the students must also write the corresponding decimal number.
I hand out these worksheets in class and then give the students a few minutes to complete them before going over the answers. The purpose of this worksheet is to make the numbers concrete (by using the circles) since that helps break the association that (for example) the number '10' always means ten items (it means ten items in decimal, but eight items in octal). In addition, these worksheets also allow practice with the idea that ten items can be written in multiple ways: '10' in decimal, or '12' in octal.
Worksheet: Octal dots
This is similar to the previous worksheet except that they students are now required to draw the correct number of dots and then convert to decimal.
When reviewing the answers for these worksheets, I point out that the answers can be checked by simply multiplying out the numbers by the place values. So if you have octal 35, you can break it up as:
= 3 in the eights position, 5 in the ones positionThis is a direct result of how positional number systems work, and it parallels what we do in decimal, where 29 is really 2 x 10 + 9 x 1.
= 3 x 8 + 5 x 1
[Sidenote: Depending on how much you want your students to suffer, you can tell the following octal joke: "Why do computer programmers often confuse Christmas and Halloween?" Answer: Because Oct 31 = Dec 25 (Octal 31 = Decimal 25).]
Worksheet: Counting in octal
This is the last of the octal worksheets and has the students count from 0 to 63 in octal.
[Sidenote: I make sure that I hand out this worksheet out after I finish all of the octal-decimal conversions that I want to cover in class. One time I had the students do this worksheet first and some of them kept it around as a reference for the later work. This meant that they were thinking less about the number representation/conversion and doing simple table lookups to get the answer.]
For binary, it follows the same pattern that we saw with decimal and octal:
Here's the decimal information for comparison: Decimal is based on 10's. It has 10 digits (0-9), and the places are based on 10:
Binary is based on 2's. It has 2 digits (0 and 1), and the places are bases on 2:
At this point we practice converting numbers between binary and decimal. I describe the process on the whiteboard and the students practice:
- Converting a decimal number into binary by repeatedly subtracting powers of 2.
- Converting a binary number into decimal by adding the power-of-2 that corresponds to each '1' digit in the binary number.
[Sidenote: I created some worksheets for this, but didn't have them ready in time for this class, so I'll provide more details in my post for next week's class.]
If there is time, I also like to introduce the Binary Magic Trick just after I cover how to convert from binary to decimal. Be prepared for the students to be really disappointed when they learn how the trick works - the younger students especially seem to prefer to live a world where you were able to magically read their mind (rather than simply convert from binary to decimal).
Worksheet: Counting in binary
Similar to the "Counting in octal" worksheet, but this time for binary. The students must count from 0 to 63 in binary ('0' to '111111')
- Binary is for computers, decimal is for humans
- Converting between decimal and binary is not difficult, but it is tedious:
- Converting binary to decimal requires lots of adding
- Converting decimal to binary requires lots of repeated subtraction
Since it's annoying to do this conversion all the time (especially for large numbers), we'd like to have an easier way. And we'll cover that next week.
Binary Quote (part II)
At the very end of class, I go back to the binary quote written at the top of the board. I ask a student to read the quote aloud. Invariably, they will start off with "There are ten types of...", and I'll stop them at that point and write "ten" on the board?
"Does it say 'ten'? I don't see a 'ten' there - I see a '1' and a '0'."This is usually sufficient to make the connection, but if needed, you can ask the students what '10' is in binary.
I now congratulate the students on their transition from the "those that don't" group into the "those that understand binary" one.
[Sidenote: I once asked a student what he thought the quote was about when he first read it at the beginning of class (before he knew binary). He responded that he thought it was unfinished and that I would be adding 8 more items later.]
End of class. Next week we'll continue with converting between number systems and introduce hexadecimal.