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Transcript
Number patterns Introduction to number patterns Special groups of whole numbers 7 Fibonacci sequence 9 Factors and multiples 10 The question of divisibility 12 Suggested answers to activities 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 3 P0025398 17 1 2 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 Introduction to number patterns Everyday tiling patterns are made by interesting repetitions of basic shapes. Draw what comes next in this tiling pattern. 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 3 Numbers can make some interesting patterns too: 1 Try these, using your calculator: 37 3 = ___ 37 6 = ___ 37 9 = ___ 37 12 = ___ Keep the pattern going: 37 ___ = ___ ___ 18 = ___ ___ ___ = ___ 37 24 2 = ___ How about this one? Using your calculator, write down the answer to these: 1 9 = ___ 2 9 = ___ 3 9 = ___ 4 9 = ___ Keep it going: ________________ ________________ ________________ ________________ 4 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 3 Now for something a little more adventurous: 1 9 + 2 = 11 12 9 + 3 = ___ 123 9 + 4 = ___ 1234 9 + 5 = ___ Keep going: ________________ ________________ 1 37 3 = 111 37 6 = 222 37 9 = 333 37 12 = 444 Keep the pattern going: 37 15 = 555 37 18 = 666 37 21 = 777 37 24 = 888 2 1 9 = 0.1111… 2 9 = 0.2222… 3 9 = 0.3333… 4 9 = 0.4444… Keep it going: 5 9 = 0.5555… 6 9 = 0.6666… 7 9 = 0.7777… 8 9 = 0.8888… 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 5 1 9 + 2 = 11 3 12 9 + 3 = 111 123 9 + 4 = 1111 1234 9 + 5 = 11 111 12 345 9 + 6 = 111 111 123 456 9 + 7 = 1 111 111 Completing a pattern is as easy as identifying similarities and differences. For example, write the next two numbers for the number pattern. 1, 3, 9, 27, ____ , ____ Each term is three times the term before it. That is, Thus the pattern is: 1, 3, 9, 27, 81, 243. Activity 3A Write down the next two numbers in each of the following number patterns. 1 5, 15, 25, 35, 2 3, 6, 12, 24, 3 1, 3, 5, 7, 4 10, 7, 4, 1, 5 101, 201, 301, 6 96, 48, 24, 7 80, 70, 61, 53, 46, 8 4, 11, 18, 25, 6 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 9 1, 4, 9, 16, 25, 10 1, 3, 6, 10. Check your answers against those at the end of the topic. Special groups of whole numbers You might have recognised some of the patterns in Activity 3A as belonging to special groups of whole numbers. Odd numbers The pattern 1, 3, 5, 7, … for example, is the group known as odd numbers. If we were to draw a diagram representing the group of odd numbers, it would look like this: there is always one ‘unpaired’ dot Even numbers The other half of the group of counting numbers is called the group of even numbers. A diagram of the first few even numbers looks like this: all dots are ‘paired’ Square numbers Did you recognize the pattern 1, 4, 9, 16, 25, …? This is the group of square numbers, found by multiplying each of the counting numbers by itself. 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 7 1 What is the 7th square number? ________________________ ________________________ 2 What is the 12th square number? ________________________ ________________________ 1 72 = 7 7 = 49 2 122 = 12 12 = 144 It is easy to see where the name ‘square’ comes from, if you look at the pattern square numbers make. Triangular numbers Did you recognize the pattern 1, 3, 6, 10, …? You should have found that the pattern is formed by adding one more each time: That is, add 2, then add 3, then add 4, then add 5, and so on. Mathematicians love their shapes so much that they call this group triangular numbers, but they should be called ‘ten-pin bowling numbers’, don’t you think? 8 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 Fibonacci sequence This pattern is more challenging. See if you can write down the next two numbers: 1, 1, 2, 3, 5, 8, 13, 21, ____ , ____, … Can you write the rule, in words? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ The next two numbers are 34, 55. Except for the first two in the pattern, each term is the sum of the two terms before it. That is 1+1=2 1+2=3 2+3=5 3+5=8 5 = 8 = 13 8 + 13 = 21 and so on. This pattern of numbers is called the Fibonacci sequence. 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 9 Activity 3B 1 Complete this table for the special groups of numbers above: Number group Pattern (a) even numbers 2, 4, 6, 8, …. (b) odd numbers (c) square numbers (d) triangular numbers 2 (a) What is the 20th square number? (b) What is the 8th triangular number? 3 34, 25, 43, 100, 81, 27, 111, 2, 70, 9, 64, 49 Which of the numbers in the box above are: (a) even numbers? (b) odd numbers? (c) square numbers? (d) both square and even? Check your answers against those at the end of the topic. Factors and multiples A factor of a counting number divides into it exactly, that is, with no remainder. For example, 3 is a factor of 15 because it divides into 15 exactly 5 times while 4 is not a factor of 15 because 15 4 = 3 and something left over. Some numbers have several factors, others do not. 10 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 Example 1 List the factors of 15. Answer: 1 15 = 15 and 3 5 = 15. So the factors are 1, 3, 5, 15 Example 2 List the factors of 36. Answer: 1 36 =36 2 18 = 36 3 12 = 36 4 9 = 36 6 6 = 36. So the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. Example 3 List the factors of 17. Answer: 1 17 = 17. So there are only 2 factors of 17: 1 and 17. Some comments A number like 17 with only 2 factors (itself and 1) is called a prime number. Numbers like 15 and 36, which have more than 2 factors, are called composite numbers. List all the prime numbers less than 20. _______________________________ 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 11 2, 3, 5, 7, 11, 13, 17, 19. Did you wonder what to do with the number 1? 1 is neither prime nor composite. A multiple of a counting number is found when you multiply that counting number by any other counting number. For example, 10, 20, 30, 40, 50, …………… are all multiples of 10. 3, 6, 9, 12, 15, …………… are all multiples of 3. List the multiples of 7 between 20 and 29. __________________________________ 21, 28 The question of divisibility Determining whether or not one number is a factor of another number is really just a matter of knowing your times tables! But there are some other helpful tricks, too. Here are some of them. Do you remember the group of even numbers we looked at earlier? It was the group that looked like this. 2, 4, 6, 8, 10, … All even numbers are divisible by 2. 2 is a factor of any number that ends in 2, 4, 6, 8, or 0. 12 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 Here is an interesting fact. A number is divisible by 3 if its digits add up to a number divisible by 3. For example, is 141 divisible by 3? Test: 1 + 4 + 1 = 6, and 6 is divisible by 3, so yes, 141 is divisible by 3. 3 is a factor of any number whose digits add to be a multiple of 3. No matter how large a number you are testing, the number will be divisible by 4 if its last 2 digits are divisible by 4. What a time saver! For example, does 4 divide evenly into 13 580? Test the last 2 digits. 4 does divide evenly into 80, so 4 must also divide evenly into 13 580. 4 is a factor of any number whose last 2 digits make a multiple of 4. The group of multiples of 5 is always easy to identify. 5, 10, 15, 20, 25, 30, 35, … They all end in a ‘5’ or a ‘0’. 5 is a factor of any number which ends in 0 or 5. Multiples of 10 are even easier to see. 10, 20, 30, 40, 50, … They all end in ‘0’. 10 is a factor of any number which ends in 0. Challenge You probably know how to spot multiples of 11 that are under 100: 11, 22, 33, 44, 55, …They are all double-digit numbers. Now, with this nifty party trick, you might be able to identify some in the hundreds as well. Try this short-cut way to multiply a two-digit number by 11. Take as an example, the number 23. Write 2 on the left, leave a space, then write 3 on the right: 2_3 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 13 Now add the 2 and the 3 to get 5, and write the 5 in the gap you have left in the middle: 253 Result: 23 11 = 253 Check on your calculator. Try another: 52 11 = ___________ And another: 81 11 = __________ Were your answers 572 and 891? Another challenge What happens to our rule when the sum of the two digits is 10 or more? This means you will have to carry. Example 47 11 = ? Use your calculator to find the answer, then see if you can find the rule. Test your theory on several more examples. Write your test examples and your rule in the space below: _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ 14 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 Activity 3C 1 Complete this table by giving the meaning of each word. Word Meaning factor multiple prime composite divisible 2 Use your definitions in Question 1 to help answer these questions. (a) List the factors of 24. (b) Write the first 5 multiples of 4. (c) Write the multiples of 3 between 20 and 30. (d) Write down the prime numbers between 20 and 30. (e) From this list, circle the composite numbers: 20 23 25 27 30 31 33 35 37. (f) Circle true or false for each of the following statements. (i) 4 is a factor of 348. T F (ii) 1255 is divisible by 3. T F (iii) 1 000 000 is a multiple of 5. T F (iv) 27 457 is an even number. T F (g) List the factors of 49. 3 There is one even number that is not composite. What is it? 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 15 4 (a) Complete the table: Factors of 20 Factors of 32 (b) For the numbers 20 and 32, common factors are factors common to both lists. (i) List the common factors of 20 and 32. The highest common factor is the largest number found on both lists. (ii) What is the highest common factor of 20 and 32? 5 (a) Complete the table for multiples less than 80: Multiples of 10 Multiples of 12 (b) The lowest common multiple is the smallest number common to both lists. What is the lowest common multiple of 10 and 12? 6 Lengths of framing timber are sold in multiples of 30 cm. Emma needs 80 cm. What is the shortest length she can buy for the job? 7 The Olympic Games are held in leap years, which occur every 4 years. Sydney held the Games in 2000. (a) Will 2050 be a leap year? (b) What is the first leap year after 2025? (c) What will be the year of the first Olympic Games held after 2110? Check your answers against those at the end of the topic. 16 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 Suggested answers to activities Activity 3A 1 45, 55 2 48, 96 3 9, 11 4 –2, –5 5 401, 501 6 12, 6 7 40, 35 8 32, 39 9 36, 49 10 15, 21 Activity 3B 1 (b) 1, 3, 5, 7, 9, 11, (c) 1, 4, 9, 16, 25, 36, (d) 1, 3, 6, 10, 15, 21, 2 (a) 20 × 20 = 400 (b) 36 3 (a) 34, 100, 2, 70, 64 (b) 25, 43, 81, 27, 111, 9, 49 (c) 25, 100, 81, 64, 49, 9 (d) 100, 64 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398 17 Activity 3C 1 A factor of a counting number divides into it exactly. A multiple of a counting number is found when you multiply that number by another counting number. A prime number has only two factors (itself and 1). A composite number has more than two factors. Divisible: A number that can be divided exactly by another number is said to be divisible by that number. 2 (a) 1, 2, 3, 4, 6, 8, 12, 24 (b) 4, 8, 12, 16, 20 (c) 21, 24, 27 (Note: Although 30 is a multiple of 3, it is not included because the word ‘between’ excludes the end points.) (d) 23, 29 (e) 20, 25, 27, 30, 33, 35 (f) (i) True (ii) False (iii) True (iv) False (g) 1, 7, 49 3 2 4 (a) (b) Factors of 20 1, 2, 4, 5, 10, 20 Factors of 32 1, 2, 4, 8, 16, 32 (i) 1, 2, 4 (ii) Highest common factor is 4. 5 (a) Multiples of 10 10, 20, 30, 40, 50, 60, 70 Multiples of 12 12, 24, 36, 48, 60, 72 (b) Lowest common multiple is 60. 6 90 cm 7 (a) No (2050 is not divisible by 4.) (b) 2028 (Make the last two digits a multiple of four.) (c) 2112 (Make the last two digits a multiple of four.) 18 4930CD: 3 Recognising Everyday Mathematics DET NSW, 2005/008/012/06/2006 P0025398