Binary Numbers	Reformatted August 2009
	Objectives: Students will use patterns to reason about place value as it is used in binary notation. Students will use the numeration system used by computer engineers. Background: These lesson were used successfully in a 1st-2nd grade multi-age classroom setting. The students enjoyed being able to write their numbers in binary and show how that relates to light switches being on or off.
	Computer Basis for Binary: Binary, base 2, is the way that computers "know" numbers, much as we know our numbers in base 10, also called decimal notation or Arabic numerals. Computers use binary because it correlates well with electronic switching: Off = 0, and On = 1. For example, if you have two light switches in your room you can demonstrate four numbers: Switch A Switch B Binary No. Decimal No. Off Off 00 0 Off On 01 1 On Off 10 2 On On 11 3 Challenge questions: We can see that we can represent the numbers 0-3 (four numbers) with two switches. How many numbers do you think we can represent with three switches? What about four, or five? Here we see bulbs in the sequence: on-off-off-on, representing 1001, as a binary number. Let students represent numbers with lamps or switches set on or off.   1 0 0 1	Related pages: place value lessons binary decision tree lesson digital displays lesson
	Structure of Binary Binary is a place value system much like base ten, only it is based on 2 rather than 10. Although your students probably do not know exponents, notice there are simple patterns they can figure out. In fact, if they only know addition they can still figure it out: Binary Place Multiplication Pattern Addition Pattern 1 1 1 2 1 * 2 = 2 1 + 1 = 2 3 2 * 2 = 4 2 + 2 = 4 4 4 * 2 = 8 4 + 4 = 8 5 8 * 2 = 16 8 + 8 = 16 6 16*2 = ? 16+16 = ?   ? * 2 = ? + ? = That sequence of numbers gives us the place values for binary: 16s 8s 4s 2s 1s Decimal Equivalent Numbers 0 0 0 1 0 2 0 0 1 0 0 4 0 0 1 1 0 4+2 = 6 0 1 0 1 0 8+2 = 10 1 0 0 0 1 16+1 = 17 Questions: What will be meant by a 1 in the sixth place? What will be meant by a 1 in the seventh place? What is the largest number you can write with 3 bits (places)? what is the largest you can write with 5 bits? Patterns in Binary If we write a sequence of binary numbers we should see some patterns that might help us remember. Decimal Binary 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111 Patterns: [1] All odd numbers end in 1, all even numbers end in zero. What will the last bit be for 277? [2] The last bit (1s column) alternates: 0-1-0-1-0-1-0-1. [3] The next column (2s place) alternates: 0 0 1 1 0 0 1 1 [4] the next column (4s place) alternates: 0 0 0 0 1 1 1 1. What do you think will be the pattern for the 8s place? Converting From Decimal to Binary - an algorithm We do not encourage memorizing algorithms as the correct understanding of mathematics. However, here is an algorithm that may be discussed: Subtract out the highest power of 2. Put in a one to represent that place value. Take your answer, and repeat step 1 until reaching 0. Fill in all other place values with 0, counting the 1s place as the 0th bit. Examples: 37 37 - 32 = 5 5 - 4 = 1 1 - 1 = 0 37 in binary will be: starting number 32 = 25 --> 100000 4 = 22 ---> 100100 1 = 20 --> 100101 100101 13 14 - 8 = 5 5 - 4 = 1 1 - 1 = 0 13 in binary will be: starting number 14 = 23 --> 1000 4 = 22 ---> 1100 1 = 20 --> 1101 1101 Converting Binary to Decimal - an algorithm Simply determine the place value of each 1 and add all the place values together: 100101---> 26 + 22 + 20 = 32 + 4 + 1 = 37 1101 ----> 23 + 22 + 20 = 8 + 4 + 1 = 13 Comments: This set of lessons was first introduced using patterns so that the students could figure out for themselves. After that it was reinforced with daily number practice, e.g.: Three volunteers write the day's date in decimal, Roman numerals, and binary. This led to questions like how long will it be before we get to use the next bit, and the observation the the 32s place, bit #6, will never get used to write the day of the month.
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