Students will learn how to do basic calculations using binary numbers on cards
| Creator | Mikko Muilu |
| Subject | Math, ICT |
| Length | 45 minutes |
| Pedagogical Approach | Phenomenon-based learning |
| Competences | Students learn to understand how binary numbers work on basic math solutions |
| Grades | Students aged 9-12. |
| Technologies | Pen and paper |
Binary numbers with cards
Description:
Computers are made of transistors and they can’t calculate or understand numbers in the very basic level like we do. Computers work with ones and zeros, which can be marked with electrical voltage on or off. No voltage means 0 and voltage on means 1. This is easy enough to understand, but what if we need other numbers or other symbols than just 0’s and 1’s.
We are familiar with numbers from 0 to 9. When we start calculating up from zero, what do we do, when we come to the last number, nine?
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 … 10.
We started again with the same numbers, but we put one in the front. Now we are using two symbols (1 and 0) to present the number 10.
This is the same series, but with zeros marked also in front of the first numbers
00, 01, 02, 03, 04, 05, 06, 07, 08, 09 … 10.
We could put as many zeros in front of the number as we wanted and we could still understand what the number is. 00005 is still just a 5.
Computer scientists are using this same trick, but with only numbers 0 and 1 in use. Latin word bi means a pair or “two”. In binary number system we are only using two symbols (0 and 1) and this is why it is called binary.
00 = 0
01 = 1
10 = 2
11 = 3
This can be presented also with cards
Exercise 1:
Teacher paires the students and gives three cards to each pair. If the exercise is done for the first time, students can start up by cutting out the cards from the papers. Only the first two are used in the beginning.
Card with one dot is on the right, the card with two dots is on the left.
Students have to turn as many dots up as teacher asks them to. With two cards numbers can be between 0-3. It is important to keep the cards in the same order on the table in front of the students.
For example Teacher asks students to turn the cards so that only one dot is visible. This should look like this
Students can now write the number in binary code. If the leftmost card is turned facing down, it’s a 0 and if the dot’s are visible, it’s 1. Above number 1 is 01 in binary.
Number 2 is 10 in binary
Number 3 is 11 in binary.
Discussion:
Everytime a new card is added, the possible numbers double. Let’s continue with three cards. Now the possible binary numbers are as follows.
000 = 0
001 = 1
010 = 2
011 = 3
100 = 4
101 = 5
110 = 6
111 = 7
Again, the number of dots visible state the binary code, when the cards are in this order.
Exercise 2:
Teacher gives numbers 0-7 and students try to figure out how they are presented in binary.
For example number 5 is 101 in binary.
Discussion:
The number of cards used would be called how many bits the numerical system has. The two cards in exercise 1 are a 2-bit system. The three cards in exercise 2 are a 3-bit system.
Exercise 3:
Now lets’s practice with a 5-bit system. With 5 cards we can present numbers from 0-31. This is going to be slower than with previous exercises, so take your time.
When all of the cards are facing up, we can see 31 dots and this is presented as 11111 in binary.
Above we can see 21 dots and it’s 10101 in binary.
Students realize pretty quickly how the numbers should be formed. If the asked number is 19, students should look first the leftmost card and compare it to the number asked. In this case the leftmost card presents 16, so it is taken in to account. 19-16 = 3 and now we’re moving right to find out how to present 3 with rest of the cards. The next card is 8, which is too big. The next one is four, which is also too big. The next one is 2, which is smaller than 3, so we’ll need it. 3-2 = 1, which means we will need also the last card with one dot. From this we can deduce that 19 is 10011 in binary.
Discussion:
Modern computers use 64-bit architecture, which allows a mindblowing number of different numbers. The numbers can be calculated up to 2^64. This means computer has 64 parallel wires in the processor and the numbers are presented with voltage either on or off in each wire.
30 years ago 8-bit systems were still used and the individual cables could be seen (2^8 = 256 different values). They worked with the principle presented above.
