In this lesson, you will learn more about binary numbers. You will also move from the shape-based number system to one that uses numbers. The system you create in this lesson will be infinite.
Students will be able to:
Number systems help us express and reason about quantities.
Early number systems were merely a system of tallies that allowed humans
to record and perform simple arithmetic with values.
The number system we use today uses the concept of place value
to allow us to express any value we wish by combining only 10 symbols (0, 1, 2, ...).
We therefore call it a "base 10" number system.
When developing a number system for a computer, we only have two symbols available
to us, corresponding with the two states of a single bit.
However, the power of place value allows our binary or "base 2" number system to express any value we wish.
When using this binary representation of numbers, certain values (1, 2, 4, 8, 16, etc.) are seen repeatedly. When written in binary, these values are 1, 10, 100, 1000, 10000, and so on, and so are the incremental place values in this binary number system.
Remember the last lesson when you used shapes and items to create a number system? What if you only had two items, such as a circle and square? How many different three place patterns can you make? Discuss your answer with a partner.
Why would you want a number system with only 2 symbols? That would allow us to use bits as the building blocks of the number system.
How large of a number can this system represent? Can you make it larger? Add more bits, and more permutations, to make the number go higher.
Print and cut the Flippy Do worksheet along the dotted lines at the bottom. To fill out the Flippy Do, the top row from right to left is powers of 2, starting with 2 to the 0 power. The next row is the value of that exponent. The third row is all 0s. The last row is all 1s, written on the back of the paper.
Each column is a place value, just like in base 10. The value in the second row is how much that place is worth in base 10. For example, the far right column is worth 2 to the 0 power, or 1 in base 10. The second column from the right is worth 2 to the 1st power, or 2. The third column is 2 to the 2nd power, or 4. To show the value 3 in binary, it would be 1 1 because the the right place is 1, and the left place is 2. One plus 2 is 3. How would you show 5?
Use the Flippy Do to convert between base 10 and binary numbers. Below is a visual representation of the Flippy Do using the instructions above.
Use the "Binary Calculator Widget" in the Resources section to observe counting across different number systems. You will be asked to enter a number in base 10, or decimal, and the calculator will convert that value to other number systems. Do you notice any connections between different bases?
You have learned to convert base 10 to binary, and binary to base 10. If possible, discuss your answers from the worksheet with others.