California Career and Technical Education Standards

• IT.D.D2.3 Understand digital logic, machine-level representation of data, memory-system or...

• Beginning Activity 40min - Hooks / Set

  Ask the students this question: “What would it be like if you had no thumbs?”

  Use duct tape to tape their thumbs to their hands so they can only use their fingers. Pass out common household objects and challenge students to try doing everyday tasks without their thumbs like taking off a screw-on top from a bottle or paperclip some papers together. Ask them to brainstorm and come up with things that they think would be the most difficult to do without thumbs. If possible, have them try some of them. You could even have a competition!

  Show the video “Thumbless” where some students accepted the challenge to go a day with their thumbs disabled. If you wish to go a little further, show the video of “Charity”, a girl who was born without thumbs and with her fingers deformed. You can then show the video “Charity's Hands” which shows the difficulty the handicapped can have using a computer. This would be a good segue into a discussion of the need for accessibility options in an operating system.

  If you wish to make a historical connection, talk about how a common method in ancient times to humiliate a defeated opponent would be to cut off their thumbs and sometimes their big toes. This is referred to in the Bible in the book of Judges 1:6-7. Outside of the obvious pain, why would this be a humiliation?

  There is also a good place here for a biological connection if you have time. The existence of the opposable thumb is a major turning point in biology as it allows the use of tools.

  (Note: if this hook seems a little too graphic or you feel it distracts from the main point of the lesson, you can make the same point by using Mickey Mouse as an example. Mickey Mouse, as many of the older cartoons, had only three fingers due to the animators wanting to make their jobs simpler with fewer fingers to draw. Therefore, cartoon characters probably count in octal!)

  (Another historical connection would be to mention that some Native American tribes, like the Yuki from the Mendocino area, counted in octal. They counted using  the spaces between the fingers rather than counting the fingers themselves.)

  • Thumbless [ Watch Video ] [ Download Original Video ] Thumbless- Students spend a day without thumbs
  • Charity [ Watch Video ] [ Download Original Video ] Charity- the girls born with no thumbs
  • Charity's Hands [ Watch Video ] [ Download Original Video ] Charity's Hands
• Lecture 20min - Lecture

  But there is one small way that having no thumbs would make using a computer easier. What if we had no thumbs and only counted on our fingers? Then we would be counting more like computers do! When we count using all our fingers and thumbs, we count using a base of ten digits, because we have a total of ten fingers and thumbs. If we only counted with our fingers, we would count with only eight digits, which is much more like computers do. Counting with only eight digits is called “octal” and is very common in many computer systems, especially older ones. Unix uses octal values to calculate permissions..

  So what would it be like to count with only eight fingers? Well, you would only be able to use eight digits, 0,1,2,3,4,5,6,7. When you got to seven, you would treat it as you would a nine in decimal and the next number would be a “ten”, meaning a one and a zero, the one indicating one group of eights with none left over.

  So the place values in the octal system will be 1’s, (8 ⁰), 8’s, (8 ¹), 64’s (8 ²), 512’s (8 ³) and so forth.
• Demonstration 20min - Demo / Modeling

  On the board, begin counting in octal. Have the students count with you. 0,1,2,3,4,5,6,7,10,11,12,13— Stop and ask them what the number 13 in octal is in decimal. It would be 11 in decimal! Write the two number side by side, 13 and 11. How would a computer programmer know just by looking at one of these numbers what the actual amount was?

  One way would be to add the notation mathematicians use- a subscript number after the number to indicate the base. For example, 13 ₈ would mean the number 13 counting in base 8 or octal. However, this is difficult to do in computer work since there is no easy way to create a subscript in a simple line of text like a computer would see, so computer programmers have come up with another way. They put a zero (which you would not normally begin a number with) and then a letter to indicate the most common types of numbering systems computers use. So if a number begins with 0b, like 0b1101, it means the number is binary. If it begins with a zero, like 0356, it probably is octal. To indicate hexadecimal, they begin the number with 0x, like 0x856a. Normal decimal numbers do not have the leading zero.

  Add a zero before the number 13 to make it 013. Now can they tell which number is octal?
• Check for Understanding 5min - Check Understanding

  Write several numbers on the board-

  015, 16, 31, 025, 25, 016.

  Ask the students which of these numbers indicate the same quantity. If they are understanding correctly, they should answer 025 and 31.
• Guided Practice 20min - Guided Practice

  Have the students take out a piece of paper and write their names in the top right corner. Have them label it “Octal Numbers Lab”.

  Have them fold the page lengthwise into quarters. Then have them number in the first column to 25, move to the next column and number from 26-50 and so on until they reach 100.

  Now tell them to begin counting in octal beside the number- 0,1,2,3,4,5,6,7,10,11,12- all the way to 100. Tell them to be sure to leave space after the octal number, as there will be one more number added later. When they reach 100 decimal, what number do they get octal? They should get 0144.

  Write this question on the board—

  “Why do computer programmers always mix up Halloween and Christmas?”

  If they are having difficulty, have them circle the number pair 25-031 on their papers. If they still don’t get it, the joke is that decimal 25 and octal 31 are the same number- Dec.25 and Oct. 31, the dates of Christmas and Halloween!

  For a little added humor, play the attached video from “Nightmare Before Christmas” where Jack Skellington from Halloween is trying to understand Christmas.

  Now have the students write the same number in binary beside the octal number. So, for example, the number 25 would look like this: 25. 031 0b11001. Have them draw lines dividing the binary into groups of three bits each. So the binary equivalent of decimal 100 would be 1|100|100. Have them compare the groups of three bits with the octal number. Point out that every three binary bits can be represented as one octal number. In this case, 1=1, 100=4 and 100=4 for binary 0144. This is one of the reasons computer programmers find it easier to work in octal. Octal can be more easily translated back and forth to binary.

  • Jack Skellington tries to understand Christmas [ Watch Video ] [ Download Original Video ] Jack Skellington tries to understand Christmas
• Activity #6 30min - Lab / Shop

  Now have the students turn their pages over. On the back, have them title the first three columns “decimal” “octal” and “binary”. Have them think up 20 numbers that can be either decimal or octal or binary and put them in the correct column, using correct notation. Remind them to be reasonable and keep the numbers below 1000 decimal. Now have them trade their papers with another student and have them fill out each other’s papers solving the numbers for the missing decimal, octal and binary. When they are done, have them compare their results with each other and check for accuracy. Collect the papers and check them.
• Closure 45min - Closure

  Tell the students that there also is a way to count on your fingers in binary. Show the video clips below and challenge them to learn to count in binary through 31 on their fingers. 31 is how high you can count on one hand. Have them work together in small groups until each student can count to 31. Have them demonstrate their new skill to you.

  • How to count on your fingers in binary [ Watch Video ] [ Download Original Video ] How to count on your fingers in binary
  • Count to 31 on one hand using binary [ Watch Video ] [ Download Original Video ] Count to 31 on one hand using binary
• Assessment - Assessment

  Hand out the “Decimal Octal Binary Lab” below. Do not let the students work together so you can monitor whether all of them are understanding the concept, not just copying the others. Collect and check.