The Human Binary Calculator: Investigating Positional Notation Subject: Mathematics/Computer Science Topic: Numbers Grade Level: 8-12 Time: 40-60 min Pre/Post Show Math Activity Introduction: In Show Math, the decimal expansion of Pi is converted into binary. This activity strengthens students’ understanding of positional notation by exploring the representation of numbers in a base other than base 10. By expressing numbers in binary, including values that are not whole numbers, students will gain an under- standing of how the value of a number depends on the placement of its digits. Learning Objectives: By the end of this activity students should: • Understand the principle of positional notation • Expand their knowledge of exponents and addition/subtraction of fractions • Know how to convert from binary to decimal and from decimal to binary Skills: Students should develop: • Team work skills • Communication and collaboration skills • Number sense skills Material & Resources: • Printed cards (see templates below) per group of five students • Worksheets for the class • Overhead of Powers of 2 WNCP Curriculum Links: Mathematics 9 Demonstrate an understanding of powers with integral bases (ex- cluding base 0) and whole number exponents by representing repeat- ed multiplication using powers. [C, CN, PS, R] Foundations of Mathematics 10 Demonstrate an understanding of powers with integral and rational exponents. [C, CN, PS, R] Apprenticeship and Workplace Analyze puzzles and games that involve numerical reasoning, using Mathematics 11 problem-solving strategies. [C, CN, PS, R] Apprenticeship and Workplace Analyze puzzles and games that involve numerical reasoning, using Mathematics 12 problem-solving strategies. [C, CN, PS, R] Foundations of Mathematics 12 Analyze puzzles and games that involve numerical reasoning and logical reasoning, using problem-solving strategies. [C, CN, PS, R] Computer Information Systems Differentiate between binary, decimal, and hexadecimal number 11 systems. Background: Throughout history different societies and cultures have used different types of number systems. There are records of ancient civilizations marking tallies, using lines to represent numbers. For example \\\\, represents 4. This is called a unary numeral system. It is a number system with base one and is the simplest numeral system to represent the natural numbers. The standard method of counting on one’s fingers is effectively a unary system. By introducing new symbols to the unary system, we can make it simpler to count higher. It is common for these new symbols to be a power of 10, for example if / is one, + is ten and – is 100, then 304 is ---////. Notice that the position of the symbols is irrelevant; 304 could be written ---//// or ////--- or any variation of that and still have the same meaning. These numeral systems are called additive systems. The type of numeral system used today is the Hindu-Arabic numeral system. In this number system the base is 10 and is therefore referred to as the decimal numeral system. There are ten different digits (0 to 9) used in this number system. The position of each digit within a number denotes the multiplier (power of ten) multiplied with that digit - each position has a value ten times that of the position to its right. For example, we can expand 304 to 3×100+0×10+4×1. This system of numbers originated in India and is now used throughout the world. Positional systems make it easier to perform addition, subtraction and multiplication than with the additive number system. Imagine how much more difficult adding large numbers if you were counting up tally lines (and that is not to mention multiplication and division). As well the positional number systems require a finite num- ber of digits to write high number where as additive will require different symbols for higher powers of ten. The binary number system is a positional number system with base of 2. It uses the symbols 0 and 1 to repre- sent numeric values. In decimal, after a digit reaches 9, an increment resets it to 0 but also cause an increment of the next digit to the left. In the binary, counting is the same except that only the two symbols 0 and 1 are used. Since binary is a base-2 system, each digit placement represents an increasing power of 2, with the rightmost digit representing 20, the next digit to the right representing 21 and so forth. So 304 in binary is (1×28) +(0×27) +(0×26) +(1×25) +(1×24) +(0×23) +(0×22) +(0×21) +(0×20) =100110000. Fractions in binary only terminate if the denominator has 2 as the only prime factor otherwise they will continue forever. As a result, 1/10 does Counting in Binary on Your Hands not have a finite binary representation. The digit directly to the left of the decimal place represents 2-1, the next left digit is -22 and so forth. The Using your fingers and the binary table below indicates the conversion from decimal to binary. number system you can count to 31 on one hand and 1023 on both hands. If you oriented your palms towards Decimal Form Expansion Binary your face, the values for one hand are: 2 1 0 4 (1×2 ) +(0×2 )+ (0×2 ) 100 0 3 (1×22) + (1×2 ) 11 Pinky Ring Middle Index Thumb 0 2 (1×22) + (0×2 ) 10 Power 4 3 2 1 0 of Two 2 2 2 2 2 1 (1×20) 1 Value 16 8 4 2 1 0 (0×20) 0 -1 1/2 = 0.5 (1×2 ) 0.1 This would continue on your -1 -2 -3 1/3 = 0.3333.... (0×2 ) +(1×2 )+ (0×2 ) other hand. The values of each -4 +(1×2 )+...... 0.010101... raised finger are added together 1/4 = 0.25 (0×2-1) +(1×2-2) 0.01 to arrive at a total number. In the 1/5 = 0.2 (0×2-1) +(0×2-2)+ (1×2-3) 0.0011.... one-handed version, all fingers +(1×2-4)+...... raised represents 31 because 16+8+4+2+1=31 and all your fin- gers are lowered if represents a 0. Resources: The figure to the • http://en.wikipedia.org/wiki/Positional_notation right represents Explains positional notation. 3 because the fingers representing • http://en.wikipedia.org/wiki/Numeral_system 2 and 1 are raised. Explains the history of different numeral systems. For more on counting in • http://en.wikipedia.org/wiki/Binary_numeral_system binary on your fingers visit: Explains the binary system in more detail. http://en.wikipedia.org/wiki/ Finger_binary • http://www.youtube.com/watch?v=qdFmSlFojIw Binary numbers explained in 60 seconds Activity Instructions: 1. Pose the question to the class: What is the highest number you can count with on your hands? Allow the class to come up with answers. If no-one is familiar with the binary system tell the class that you are going to teach them how to count to 31 using only one hand and 1023 using both hands. 2. Write at the top of the board 16, 8, 4, 2, 1. Ask the class, if they see a pattern in these numbers. Can these numbers be written in a different way? Once the class sees that they can all be written as 2n, explain to them how they will learn to count in binary. How is the word binary and the idea of “base-2” related? Normally we count in what is known as base-10 or decimal. 3. Have the students practice finger binary notation by calling out a few values to represent. Once they’re comfortable with the distinction between a value in base-10 and its finger representation, remind them that each finger up equals “1” of the appropriate power of two. Now have them write the value in binary using ones and zeros (i.e. “7=00111”). 4. Divide the class into groups of five students and give one set of cards to each group—one per member. 5. Each group will take a turn to create the base-2 representation of a number with their group, in front of the class. Group members should stand in order and when you call a base-10 number (eg “Twenty two”), each appropriate member should step forward and hold up their card if it’s part of the base-2 representation. 6. If you would like to make the activity a competition, have more than one group stand up and award the quicker group a point for every correct representation. 7. Call out numbers and have the rest of the class participate by writing the binary representation (i.e. 10110 for “Twenty two”) on a piece of scrap paper. Let the audience be the judges of correctness, and deciding which group got the correct representation first. 8. Each group can go up in turn and try to get the correct number. Let each group have a couple tries. 9. For advanced classes, hand out the decimal cards and assemble groups of 6—8 students and have them create mixed numbers. Write a decimal point on the board, and then write 1/2, 1/4, and 1/8 and their decimal expansions. Start with numbers that have finite representations in binary such as 1.25, 5.5, 4.625, 12.375, 31.875 etc. You can challenge them by asking them to represent seemingly simple numbers such as 0.7 (which cannot be represented exactly Value Line Activity in binary). Have them determine the closest representation to such values. This brings the discussion in the direction of computer Have one side of the room be representation of values —which is done in binary—and the strongly agree and the other side imprecision of different representations.