Y8 SUPPORT AUTUMN TERM UNIT: Number and Algebra 1 – Numbers and Sequences TIME ALLOCATION: 6 Hours PRIOR KNOWLEDGE integer, negative, positive, sign, square, squared, square root, cube, factors, prime factors, multiples Recognise negative numbers sequence, term, nth term, consecutive, predict, rule, in contexts (e.g. on a generate, continue, number line, on a ascending, descending, temperature scale) symbol, algebra, index recognise and extend a number sequence by counting on or back by a constant number. Recognise multiples of 2, 3, 4, 5, 6, 7, 8, 9 and 10 Describe what a prime number is. LEARNING OBJECTIVES LEVEL 4 test to see if a number can be divided by 2,3,4,5,10 or 100 KEY WORDS find the pairs of factors of a 2digit number. know all the prime numbers below 20. order negative numbers. find the differences in temperatures STARTER Sequences in words 1, 2, 3, 4 Find factor and multiples (choose easy) Sequences Square roots – using the Chinese method LEARNING OUTCOMES simple tests of divisibility, such as: 2 the last digit is 0, 2, 4, 6 or 8; 3 the sum of the digits is divisible by 3; 4 the last two digits are divisible by 4; 5 the last digit is 0 or 5; 6 it is divisible by both 2 and 3; 8 half of it is divisible by 4; 9 the sum of the digits is divisible by 9. the pairs of factors of 51 are 1 × 51 and 3 × 17; the pairs of factors of 56 are 1 × 56, 2 × 28, 4 × 14, 7 × 8. the final position of an object after moves forwards and backwards along a line; a total bank balance after money is paid in and taken out; the total marks in a test of 10 questions, with +2 marks for a correct answer and –1 mark for an incorrect answer; Know that a sequence can have a finite or an infinite number of terms, and give simple examples. For example: the sequence of describe what is meant by the position and term of a sequence. To be able to generate and describe integer sequences given a simple term-to-term rule, or sequence To be able to generate sequences from simple practical contexts. LEVEL 5 To be able to find squares, square roots and cubes, and use appropriate notation. recognise triangular numbers find highest common factors and lowest common multiples in simple cases To be able to add and subtract negative numbers. To be able to generate a sequence given a rule for finding each term from its position in the sequence: counting numbers, 1, 2, 3, 4, … is infinite; dots indicate that counting continues indefinitely. The sequence of two-digit even numbers, 10, 12, 14, …, 98, is finite; dots indicate that the sequence continues in the same way until the final value 98 is reached. Generate sequences by counting forwards or backwards in increasing or decreasing steps. Find the first few terms of the sequence. Describe how it continues by reference to the context. Begin to describe the general term, first using words, then symbols; justify the generalisation by referring to the context. Eg. Growing matchstick squares or Squares in a cross know the square numbers up to 12x12 and can find their square roots know that square numbers make square patterns of dots and triangular numbers makes triangles Find the lowest common multiple (LCM) of two numbers, such as 6 and 8; 6 times table: 6 12 18 24 30… 8 times table: 8 16 24 32… The lowest common multiple of 6 and 8 is 24. Find the highest common factor (HCF) of two numbers, such as 18 and 24; The factors of 18 are 1 2 3 6 9 18 The factors of 24 are 1 2 3 4 6 8 12 24 1, 2, 3 and 6 are common factors of 18 and 24, so 6 is the highest common factor of 18 and 24. Extend patterns such as: 2+1=3 –3 – 1 = –4 2+0=2 –3 – 0 = –3 2 + –1 = 1 –3 – –1 = –2 2 + –2 = 0 –3 – –2 = –1 2 + –3 = –1 –3 – –3 = 0 Start at 0 and count on in steps of 0.5. 0.5, 1, 1.5, 2, … Each term is a multiple of 0.5. Start at 41 and count back in steps of 5. 41, 36, 31, 26, … Each term is a multiple of 5 plus 1. ACTIVITIES ICT Sieve of Eratosthenes to find primes On a hundred square, colour in 1, then the multiples of 2 that are greater than 2 in one colour, the multiples of 3 that are greater than 3 in another colour… so that the remaining uncoloured numbers are the primes. www.mymaths.co.uk negative numbers 1 and 2 odds, evens, multiples multiples factors and primes sequences Using matchsticks in pairs, one person starts a pattern can the other person continue it Factor generals – pupils organize themselves into groups – discuss factors and primes. RESOURCES Multiple and Factors resources Divisibility tests Negative number ladders Fast factors Hot numbers Finding squares, multiples etc Factor trees Guess the number Number sequence game Functions and sequences investigation Counting stick - can hide numbers in sequences Term to term sequence Next term and first term Negative term to term FUNCTIONAL SKILLS and MPA OPPORTUNITIES Handshakes investigation How many handshakes would there be if 2, 3, 4 etc, people shook hands with other? What is special about the numbers of handshakes? Negative Numbers First Connect Three Factors and Multiples Game, Factors and Multiples Puzzle Picturing Square Numbers PLENARIES AND KEY QUESTIONS Which numbers less than 100 have exactly three factors? What number up to 100 has the most factors? Find some prime numbers which, when their digits are reversed, are also prime. There are 10 two-digit prime numbers that can be written as the sum of two square numbers. What are they? The answer to a question was –8. What was the question? The result of subtracting one integer from another is –2. What could the two integers be? The temperature is below freezing point. It falls by 10 degrees, then rises by 7 degrees. What could the temperature be now? Can you give me an example of a number greater than 500 that is divisible by 3? How do you know whether a number is divisible by 6? etc Can you give me an example of a number greater that 100 that is divisible by 5 and also by 3? Is there a quick way to check whether a number is divisible by 25? How do you go about finding the rule for a sequence of numbers? Where do you start? What do you look for? How many terms do you check before you are convinced you know the rule?