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1150 day 6

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1150 day 6

  1. 1. Integers andDivisibility
  2. 2. Counting (natural) numbers 1, 2, 3, …Whole numbers 0, 1, 2, 3, …Integers … -3, -2, -1, 0, 1, 2, 3, …
  3. 3. Absolute value - the distance between anumber and zero |4| = 4 | 2| = 2
  4. 4. Models for integer additionNumber Line Model2 + –3 = 1 1+ 2= 3
  5. 5. Charged field (chip) model for Addition 2+ 3= 5 3+3= 0 - - - + - - - + - - +1+ 2= 1 Two numbers are + - additive inverses if - their sum is zero.
  6. 6. Charged field (chip) model for Subtraction “Take away” 2 ( 1) = 1 Start with 2 negatives - Take away 1 negative 1 negative left - 1 ( 2) = 3 Start with 1 positive + Take away 2 negatives 3 positives left
  7. 7. 2 1= 3 - Start with 2 negatives Take away 1 positive - 3 negatives left
  8. 8. Number Line model for subtraction “Walk direction to face the line” Negative numbers “face left” Positive numbers “face right” how to walk Subtraction “walk backward” Addition “walk forward” 2 1= 3 2 + ( 1) = 3 -1 -2Relating subtractionto addition:a – b = a + (– b)
  9. 9. Number Line model for subtraction direction to face how to walk Negative numbers Subtraction (backward) Positive numbers Addition (forward) 2 ( 3) = 1 -(-3) -2 2 + (3) = 1
  10. 10. Integer MultiplicationCharged Field (Chip) Model -- 3( 2) = 6 --Add 3groups of two negatives -- 3(2) = 6 -- -- ++Take away of two ++ -- 3 groups positives ++
  11. 11. Number Line model for multiplication arrows Negative numbers Positive numbers -1 -1 -1 3( 1) = 33 groups Of -1 arrows -1 -(-1)-(-1) (-1) - 3( 1) = 3Reverse Of -1 arrows3 groups
  12. 12. Integer Multiplication – An investigation of patterns 3·3= 9 3 3· 3= 9 +3 3·2= 6 2· 3= 6 3 +3 3·1= 3 1· 3= 3 3 +3 3·0= 0 0· 3= 0 3 +3 3· 1= 3 1· 3= 3 3 +3 3· 2= 6 2· 3= 6 3· 3= 9 3 3· 3= 9 +3Positive · Positive = Positive Positive · Negative = NegativePositive · Negative = Negative Negative · Negative = Positive Same Signs – Positive answer Different Signs – Negative answer
  13. 13. Multiplication and Division a · b = c means c b=aExample: 3 · 4 = 12 means 12 4 = 3Integer Division pos · pos = pos so pos pos = pos pos · neg = neg so neg neg = pos neg · pos = neg so neg pos = neg neg · neg = pos so pos neg = negSign rules for division are identical to multiplication
  14. 14. Using the Difference of Squares formula to multiply (a + b)(a – b) = a2 – b2Multiply 42 · 38 Multiply 107 · 93 =(40 + 2)(40 – 2) =(100 + 7)(100 – 7) = 40 2 – 22 = 1002 – 72 = 1600 – 4 = 10000 – 49 = 1596 = 9951
  15. 15. DivisibilityIf a and b are integers, then b divides a ifthere is an integer c such that a = b · c Why?Does 3 | 12 Yes Because 12 = 3 · 4Does 6 | 12 Yes Because 12 = 6 · 2Does 24 | 12 No Because 12 = 24 · integer
  16. 16. Divisibility tests2 Even number (ends in 0, 2, 4, 6, 8)3 Sum of digits is divisible by 34 Last two digits divisible by 45 Ends in 0 or 56 Divisible by both 2 and 37 Cross out, double, subtract8 Last three digits divisible by 89 Sum of digits divisible by 910 Ends in 011 Difference of alternate digits (ocean waves)
  17. 17. Test 5182 for divisibility by Why?2 Yes 5182 is even3 No 5 + 1 + 8 + 2 = 16, and 3 | 184 No 4 | 825 No 5182 does not end in 0 or 56 No Not divisible by both 2 and 3
  18. 18. Test 5182 for divisibility by 5182 Why? -47 No 7 | 43 514 -88 No 8 | 182 439 No 5 +1 + 8 + 2 = 16 and 9 | 1610 No 5182 does not end in 0 1311 No 11 | 10 5182 13 – 3 = 10 3
  19. 19. Test 3,885,840 for divisibility by Why?2 Yes 3,885,840 is even3 Yes 3+8+8+5+8+4+0=36, and 3 | 364 Yes 4 | 405 Yes 3,885,840 ends in 06 Yes Divisible by both 2 and 3
  20. 20. Test 3,885,840 for divisibility by 3,885,840 Why?7 Yes 7 | 21 -0 388,5848 Yes 8 | 840 -8 38,8509 Yes 9 | 36 -0 3,88510 Yes Ends in 0 -10 37811 No 11 | 2 19 -16 21 3885840 17 19 – 17 = 2

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