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Prime Time

Prime Time. Whole Number Arithmetic. A little vocabulary. Factors are one of two or more whole numbers that are multiplied together to get a Product 5 x 7 = 35 Five and seven are factors, and thirty-five is a product. There are 2 kinds of numbers…. Prime Numbers

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Prime Time

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  1. Prime Time Whole Number Arithmetic

  2. A little vocabulary • Factors are one of two or more whole numbers that are multiplied together to get a Product • 5 x 7 = 35 • Five and seven are factors, and thirty-five is a product

  3. There are 2 kinds of numbers… • Prime Numbers • These numbers have only two factors… One and themselves • 7 = 7 x 1 • 7: 1,7 • Composite Numbers • Have at least three factors • 10 = 5 x 2 and 10 x 1 • 10: 1,2,5,10 • 12 = 6 x 2 and 4 x 3 and 12 x 1 • 12: 1,2,3,4,6,12

  4. Factor Pairs 24: 1, 2, 3, 4, 6, 8, 12, 24 All factors have pairs

  5. Finding All of the Factors • You have to be systematic • Start with one and count up • Stop when you find two consecutive factors whose product is the number you are working on. 60: 1, 2, 3, 4, 5, 6, 10 6 x 10 = 60 so this is the central factor pair Now just match your other factors!

  6. Square Numbers • 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 • 3 x 3 = 9. This statement could be modeled like this:

  7. Square Numbers • 16: 1, 2, 4, 8, 16 • Because we don’t repeat factors in our lists, square numbers have an odd number of factors • 4 x 4 is the middle pair • The square root of a square number is a whole number! • Numbers between square numbers have square roots between the squares…

  8. Square Roots • The numbers between 25 and 36 have square roots between 5 and 6 (non whole numbers) • Examine Factor Pair Mountain (website) • Knowing our square roots, tells us when to stop looking for factors. • 27’s square root is between 5 and 6 so I only need to count to 5 to find all of the numbers up FPM • 27: 1, 3 – and we’re ready to climb down!

  9. Divisibility • 1: All numbers have 1 as a factor • 2: All even numbers (ending in 0, 2, 4, 6, 8) have 2 as a factor • 3: If the sum of the digits is divisible by 3 so is the number itself • 81 -> 8 + 1 = 9, which is divisible by 3, so 81 is too! • 372 -> 3 + 7 + 2 = 12, so 3 is a factor • 109 -> 1 + 0 + 9 = 10, so 3 is not a factor

  10. Divisibility • 5: Numbers that end in 5 or 0 have five as a factor • 6: Numbers that have 2 & 3 as factors have 6 • 9: Same trick as for 3! • 162 -> 1 + 6 + 2 = 9 • 162 = 9 x 18 So what about the rest of the numbers? This is where it gets interesting…

  11. Stretching Numbers How do we know if 8 goes into 140? Our math facts don’t go up that high! Long division? Pick a number close to 140 that you know 8 goes into… how about 80. Look at the difference (60) If 8 goes into the difference, it goes into the number

  12. Stretching Numbers See if 8 goes into 224 You might use 8 x 3 to come up with 8 x 30 Since we know 8 goes into 240 all we have to do is check the difference – in this case 16. This means that 8 is a factor of 224. (8 x 28 = 224) Let’s practice!

  13. Primes • The Fundamental Theorem of Arithmetic • All numbers are the unique product of prime numbers • I think of this as a unique fingerprint for each number, or perhaps a number’s true name (Eragon)

  14. Factor Trees 36: 2 x 2 x 3 x 3

  15. Factor Trees 42 = 2 x 3 x 7

  16. Factor Trees • You can start any way you want, but the end result will always be the same – the order of the factors does not matter • Examine different factor trees for 100 • Practice!

  17. LCM & GCF • Knowing the prime factorization of numbers allows us to see what they have in common. • Greatest Common Factor (GCF) • ALL of the prime factors that any two (or more) numbers have in common. • INTERSECTION • Least Common Multiple (LCM) • ALL of the prime factors (without duplications) • UNION

  18. LCM & GCF 2 2 3

  19. Going Forward • Computation is probably always going to be a part of your child’s math education – have them master the facts now. • Do mental math at every opportunity with them • Not just the facts • Estimation • Divisibility tests • Large numbers

  20. Next Units • Bits and Pieces (CMP) • Fractions – representations and computation • Potential parent night? • Division (Additional Materials) • Long Division – standard algorithm • Estimation • Mental Math

  21. Thanks • There will be more group practice if time permits

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