Number Concepts 2 Venn diagrams & binary numbers
Objectives
· Understand natural numbers, integers, rational and irrational numbers
· Use a Venn diagram to sort numbers
· Understand binary number system
· Understand the effect of multiplying and dividing binary numbers by 2
For this unit you will need:
calculator which shows at least 20 digits
Watch out for pupils who:
· find it difficult to grasp that decimals can continue forever;
· do not understand powers of 2.
Number Concepts 2 Session 1
Objectives: Understand natural numbers, integers, rational and irrational numbers; Use a Venn diagram to sort numbers
Teacher input with whole class
· Write headings on the board: natural numbers, integers, rational numbers, irrational numbers. Describe each and give an example, e.g.
· Natural numbers are counting numbers – i.e. whole and don’t include negative numbers, e.g. 5, 297.
· Integers – whole numbers including negative numbers, e.g. -5, 5, 297.
· Rational numbers – numbers which can be expressed as a/b where a and b are whole numbers, e.g. 5, (5/1), 1.5 (3/2), 0.125 (1/8), 0.333 recurring (1/3). These include decimals which repeat (recur) for ever, e.g. 1.1818181… (13/11). If you have a calculator which shows lots of digits, divide 1 by 1089 to give 1/1089 = 0.00 09 18 27 36 45 54 63 72 81… and they will see the 9 times table! Say that decimal equivalents to fractions are always either terminating decimals or recurring decimals.
· Irrational numbers - cannot be expressed as a/b, are decimals which go on for ever without repeating, even if the cycle was a million digits! Includes π and √2. (There is a proof that √2 is not rational, e.g. http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php.)
Paired pupil task
· Challenge pupils to work in pairs to draw a Venn diagram to show how these sets of numbers relate to each other.
Teacher input with whole class
· Ask different pairs to sketch their Venn diagrams on the board and others to agree/challenge them, e.g.
Number Concepts 2 Session 2
Objective: Understand the binary number system
Teacher input with whole class
· Write the headings 103, 102, 101 and 100. Ask pupils to work each out, reminding them that anything to the power of zero is 1, and write their answers underneath. Write a digit under each heading, e.g. to show 4326, explaining that this means we have four lots of 1000, three lots of 100, two lots of 10 and six lots of one. This our decimal number system based on powers of 10.
· Write the headings 25, 24, 23, 22, 21 and 20. Ask pupils to work out each and write underneath the powers. Write 1 under 16, 0 under 8, 1 under 4, 1 under 2 and 1 under 1. Say that this means one lots of 16, one lot of 4, one two and one 1, giving a total of 23. Explain that this is the way we write 23 using the binary system, which instead of having ten digits (0 to 9) like our decimal number system which uses powers of 10, but has only two digits 0 and 1 and uses powers of 2. Say that we have records of Indian mathematicians using the binary number system over 2000 years ago, but today they are used in computers.
· Ask pupils to discuss in pairs how they might write their age using the binary system. Take feedback.
Paired pupil work
· Ask pupils to work out how to write numbers to at least 20 using the binary number system. They also work out how to write three numbers between 50 and 100.
Teacher input with whole class
· Pupils share their larger numbers. Other work out what they are.
· Ask pupils to hold up one hand all fingers folded down. Explain that they are going to show binary numbers using their fingers. Fingers held up represent 1 and fingers folded down represent 0. So if they hold up the finger on the right of their hand as it faces them, this represents 1. If they hold up the finger next to this as well, that would show 3.
· Slowly as a class use fingers to count on in ones using binary numbers. They may need to their other hand at times to fold down their ring finger for example
Number Concepts 2 Session 3
Objectives: Understand binary number system; Understand the effect of multiplying and dividing binary numbers by 2
Paired pupil work
· Ask pupils to work in pairs to write two numbers less than 10, and then two between 10 and 50 using both the decimal and binary systems. They multiply each of their decimal numbers by 2 and write the answers using the binary system.
· They discuss what they notice, and then try other numbers to see if the same thing happens.
Teacher input with whole class
· Ask pairs to feedback and share their findings. Record some of their multiplications for others to see the pattern, e.g.
11 (3) × 10(2) = 110 (6)
100 (4) × 10(2) = 1000 (8)
111 (7) × 10(2) = 1110 (14)
· E.g. 3 written using binary notation is 11 and 6 is 110.
· Write 3 using binary notation, i.e. 11. Multiply 3 by 2 to give 6 and write binary notation, i.e. 110. What happens? Draw out that the digits shift one place to the left, and we fill the space on the right with a zero.
· When else have you seen this happening? Discuss how when using our decimal number system, digits move to the left when we multipy by 10.
· What do you think might happen when we divide binary numbers by 2?
Paired pupil work
· Ask pupils to work in pairs to think of even numbers, write them as binary numbers. They then divide them by 2 and write the answer using binary notation.
Teacher input with whole class
· Ask pairs to feedback and share their findings.
· How could we get digits to move TWO places to the left? Take and test pupil’s suggestions.