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:

- Describe how to use bits to creat a functioning number system
- Understand te relationship between the poswers of 2 and the number of bits needed to express a number of a certain magnitude
- Determine, for a given number of bits, both the number of possible numbers that can be represented and also the range of those numbers

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.

- Flippy Do worksheet in print
- Flippy Do worksheet in braille, Duxbury file
- Flippy Do worksheet in braille, .brf
- Activity Guide in large print
- Activity Guide in large braille, Duxbury file
- Activity Guide in large braille, .brf
- Binary Calculator Widget

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.

- Demonstrate how values can be calculated by flipping up a 1 for the values.
- Remember place value in base 10, and think how it converts to place value in binary.

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.

- Turn in your worksheet.
**Written Prompt:**Explain why this joke is funny: "There are 10 kinds of people in the world, those who know binary, and those who don't."**Written Prompt:**In 100 words or less, describe how place value is used in the binary number system. How is it similar or different from the way place value is used in the base 10 number syster?**Reflection:**Reflect on learning in this lesson. Write 3 recollections (things you remember), 2 observations (things you noticed), and 1 insight (something of which you fully understand the significance).

- How much memory does your music player, phone, or computer have? Do you notice anything related to binary?
- Practice translating between bases using the Binary Calculator or Flippy Do.

**CSTA K-12 Computer Science Standards (2011):**CT.L2:7, CT.L2:8, CT.L2:9, CT.L2:14**Computer Science Principles:**2.1.1 (A, B, C, D, E, G)**Computer Science Principles:**2.1.2 (A, B, C, D, E, F)**Computer Science Principles:**2.3.1 (A, B, C, D)**Computer Science Principles:**2.3.2 (A, B, C, D, E)**Computer Science Principles:**3.1.3 (A, B)