1150 day 4

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1150 day 4

  1. 1. Whole-number operations
  2. 2. Why do we use a base-ten number system?
  3. 3. 1
  4. 4. 2
  5. 5. 3
  6. 6. 4
  7. 7. 5
  8. 8. 6
  9. 9. 7
  10. 10. 8
  11. 11. 9
  12. 12. 10
  13. 13. Grouping in base ten 14
  14. 14. Base-ten number system 24123 Allowable digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base-ten place value 2 4 1 2 3 ______ ______ ______ ______ ______ 104 103 102 10 1 10000 1000 10024123 = 2·10000 + 4·1000 + 1·100 + 2·10 + 3·1 expanded form
  15. 15. How would we count if we only had one hand?
  16. 16. 1
  17. 17. 2
  18. 18. 3
  19. 19. 4
  20. 20. 10Base-five number system
  21. 21. Grouping in base five 24five
  22. 22. Base-five number system Allowable digits: 24123five 0, 1, 2, 3, 4 Base-five place value 2 4 1 2 3 ______ ______ ______ ______ ______ 54 53 52 5 1 625 125 2524123five = 2·625 + 4·125 + 1·25 + 2·5 + 3·1 = 1250 + 500 + 25 + 10 + 3 = 1788ten
  23. 23. 203five = _______ ten Base-five place value 2 0 3 ______ ______ ______ 52 5 1 25203five = 2·25 + 0·5 + 3·1 = 50 + 0 + 3 = 53ten
  24. 24. Base-four number system Allowable digits: 20213four 0, 1, 2, 3 Base-four place value 2 0 2 1 3 ______ ______ ______ ______ ______ 44 43 42 4 1 256 64 1620213four = 2·256 + 0·64 + 2·16 + 1·4 + 3·1 = 512 + 0 + 32 + 4 + 3 = 551ten
  25. 25. Base-seven number system Allowable digits: 261seven 0, 1, 2, 3, 4, 5, 6 Base-seven place value 2 6 1 ______ ______ ______ ______ ______ 74 73 72 7 1 2401 343 49261seven = 2·49 + 6·7 + 1·1 = 98 + 42 + 1 = 141ten
  26. 26. Base-twelve number system Allowable digits: 2TEtwelve 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E Base-twelve place value 2 T E ______ ______ ______ ______ ______ 124 123 122 12 1 20736 1728 1442TEtwelve = 2·144 + 10·12 + 11·1 = 288 + 120 + 11 = 419ten
  27. 27. 2111281ten = ________fiveBase-five place value 2 1 1 1 ______ ______ ______ ______ ______ 54 53 52 5 1 625 125 25 281 31 6 250 25 5 31 6 1
  28. 28. 1012197ten = ________threeBase-three place value 1 0 1 2 1 ______ ______ ______ ______ ______ 34 33 32 3 1 81 27 9 97 16 7 81 9 6 16 7 1
  29. 29. 1E5281ten = ________twelveBase-twelve place value 1 E 5 ______ ______ ______ ______ 123 122 12 1 1728 144 281 137 144 132 137 5
  30. 30. Addition Algorithms 15 + 16Using Base-ten Blocks 31
  31. 31. Expanded Algorithm Traditional Algorithm 1 15 15 + 16 + 16 11 (add ones) 31 20 (add tens) 31
  32. 32. Scratch Addition 3 2 2 1 3 4 7 8 2 6 52 98 75 61 4 3 91 86 3 + 3 92 + 6 73 9 7 14 2 27 3 7
  33. 33. Base 4 Addition Allowable digits: 0, 1, 2, 3 + 0 1 2 3 1 0 4ten = __ __ four 0 0 1 2 3 4 1 1 1 2 3 10 1 1 5ten = __ __ four 2 2 3 10 11 4 1 3 3 10 11 12 1 2 6ten = __ __ four 4 1
  34. 34. 1 + 0 1 2 3 23four 0 0 1 2 3+ 32four 1 1 2 3 10 12 1four 2 2 3 10 11 3 3 10 11 12 11 203four+ 133four10 0 2four
  35. 35. 1 1 2 7ten = __ __ five 23five 5 1+ 14five 4 2five 11 1 4 12ten = __ __ eight 275eight 8 1+ 327eight 1 2 10ten = __ __ eight 62 4eight 8 1
  36. 36. 1 1 2 14ten = __ __ twelve T9twelve 12 1+ 75twelve 16 2twelve 1 6 18ten = __ __ twelve 12 1
  37. 37. Subtraction Algorithms “Take away” 32 – 15Using Base-ten Blocks Start with 32 Take away 15 17 left
  38. 38. Traditional Algorithm 2 1 32 - 15 17
  39. 39. Subtraction 2 6 4 9 4 15 32four 526seven T53twelve - 13four - 461seven - 528twelve 1 3four 3 5seven 5 2 7twelve
  40. 40. Whole Number multiplication 3 x 5 = 15factors product
  41. 41. Models for whole-number multiplication 3 x 5 = 15Repeated addition model “three fives” 5 + 5 + 5 = 15
  42. 42. DESE MAP 4th grade Released item, 2004
  43. 43. Models for whole-number multiplication 3 x 5 = 15Array model (grid model) Count intersections 3 5
  44. 44. Models for whole-number multiplication 3 x 5 = 15Area model Count rectangles 3 5
  45. 45. A set is closed under multiplication if theproduct of any two numbers in the set is still inthe set.Which sets are closed under multiplication: a) {1, 2, 3, 4, …} Closed b) {0, 1} Closed c) {0, 1, 2} Not closed (2 x 2 = 4)
  46. 46. Properties of whole-number multiplicationCommutative property Identity Property of of multiplication multiplication axb=bxa ax1=a 3x5=5x3 7x1=7 1 is the multiplicativeAssociative Property Identity of multiplication a x (b x c) = (a x b) x c Zero multiplication 5 x (2 x 7) = (5 x 2) x 7 property ax0=0 7x0=0
  47. 47. Distributive Property of Multiplication over Addition a(b + c) = ab + acRectangle (area) model for the Distributive Property 3(4 + 1) = 3 · 4 + 3 · 1 4 + 1 4 1 3 = 3 + 3
  48. 48. Multiplication Algorithms Connection to Algebra: (36)(52) 36 x 52 = (30 + 6)(50 + 2) = 1500 + 60 + 300 + 12 = 1872Partial Sums 36 x 52 12 2x6 60 2 x 30 300 50 x 6 1500 50 x 30 1872
  49. 49. Traditional Algorithm 1 3 36 x 52 72 18 0 0 1872
  50. 50. Lattice Multiplication 36 x 52 3 6 1 3 1 5 0 5 0 1 8 6 2 2 7 2 1872
  51. 51. Lattice Multiplication 93 x 83 9 31 7 2 7 2 4 8 2 0 7 7 9 3 1 9 7719
  52. 52. 1 2 23four 1 2 6ten = __ __ four x 32four 4 1 11 2 1 1 5ten = __ __ four2010 4 1212 2four 2 1 9ten = __ __ four 4 1 2 0 8ten = __ __ four 4 1
  53. 53. Lattice Multiplication 1 2 6ten = __ __ four 23four x 32four 4 1 2 1 9ten = __ __ four 2 3 4 1 1 1 2 2 2 1 3 1 0 4ten = __ __ four 1 1 4 1 1 0 2 2 1 1 5ten = __ __ four 4 1 2 22122four
  54. 54. 1 9 4Ttwelve 1 8 20ten = __ __ twelve x E2twelve 12 1 98 9 2 110ten = __ __ twelve4520 12 145E 8twelve 4 5 53ten = __ __ twelve 12 1
  55. 55. Whole Number Division 6 3=2dividend divisor quotient Division is related to multiplication: 6 3 = 2 if 6 = 3 · 2
  56. 56. Models for whole-number divisionSet (partition) model 6 3=2 6 objects 2 objects in each set Divide into 3 equal sets
  57. 57. Set (partition) model 6 2=3 6 objects 3 objects in each set Divide into 2 equal sets
  58. 58. Models for whole-number divisionMissing Factor Model 6 3= 6=3· =2
  59. 59. Models for whole-number divisionRepeated Subtraction Model 6 3=? 6 3 3 =0 2 subtractions
  60. 60. Remainders (using a set model) 7 3 7 objects 2 objects in each set Divide into 3 equal sets 1 object left over 7 3=2R1
  61. 61. Division AlgorithmsRepeated Subtraction 678 6 = 113 6 678 - 600 100 sixes 78 - 60 10 sixes 18 - 18 3 sixes 0 113 sixes
  62. 62. 32four 2four = 13fourSet model
  63. 63. Traditional Algorithm 13 2four 32four 2four x 2four = 10four - 2 12 2four x 3four = 12four - 12 0
  64. 64. 12 2101five 12322five 101five x2 - 101 202five 222 - 202 20 2 -202 0
  65. 65. Order of Operations Parentheses Exponents Multiplication / Division (Left to Right) Addition / Subtraction (Left to Right)Simplify 15 + 6 · 4 – 10 = 15 + 24 – 10 = 39 – 10 = 29

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