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LCM – with Algebra

LCM – with Algebra. Stepping Up Our Level of Thinking. Number Talk. What does Y have to be?. First, let’s look at Least Common Multiples from a 6 th grade perspective. Find the LCM of 4 and 6 List of Multiples for both 4 and 6. 4: 4, 8, 12, 16, 20 . . . 6: 6, 12, 18, 24, 30, . . .

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LCM – with Algebra

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  1. LCM – with Algebra Stepping Up Our Level of Thinking

  2. Number Talk • What does Y have to be?

  3. First, let’s look at Least Common Multiples from a 6th grade perspective • Find the LCM of 4 and 6 • List of Multiples for both 4 and 6. • 4: 4, 8, 12, 16, 20 . . . • 6: 6, 12, 18, 24, 30, . . . So the Least Common Multiple of 4 and 6 is 12 Find the Least Common Multiple of 8 and 12 List of Multiples 8: 8, 16, 24, 32, 40, 48 12: 12, 24, 36, 48, 60

  4. Now that you’re in 8th grade • You don’t need to make a whole list. The list can be done in your head. • Try finding the LCM of these • 5 and 12 • 12 and 18 • 10 and 15 60 36 30

  5. Here’s a different perspective on finding the LCM • Prime Factorization • Example: Find the LCM of 4 and 6 • First, we could find the prime factorizations • 4 = 2*2 • 6 = 2*3 • First you find the common factors, then you multiply that by the unshared factors

  6. Let’s look at another example • Find the LCM of 12 and 15. • Prime Factorizations • 12 = 2*2*3 • 15 = 3*5 • Again, bring down the common factor(s), then multiply by the unshared factors.

  7. Try these for yourself • Using Prime Factorization, find the LCM of these • 20 and 30 • 8 and18 • 15 and 21 20: 2*2*5 30: 2*3*5 Shared Factors are 2 and 5 Unshared Factors are 2 and 3 2*5 * (2*3) = 60 8: 2*2*2 18: 2*3*3 Shared Factor is 2 Unshared Factors are 2*2 and 3*3 So 2*(2*2*3*3) = 72 15: 3*5 21: 3*7 Shared Factor is 3 Unshared Factors are 5 and 7 So 3*(5*7) = 105

  8. However, there was another way to find the LCM • Look at the following prime factorizations written using exponents and the LCMs. • LCM of 20 and 30 was 60. • 20 = 22 * 5 • 30 = 2*3*5 • 60 = 22*3*5 • Using just the exponents, can you see how to find the LCM?

  9. Let’s look at another one • LCM of 8 and 18 was 72 • 8 = 23 • 18 = 2*32 • What is the largest exponent of the 2’s? • 23 • What is the largest exponent of the 3’s? • 32 • What does 23*32 =? • 72

  10. Let’s look at just one more. • Find the LCM of 72 and 96 • 72 = 23 * 32 • 96 = 25 * 3 • What is the largest exponent of each prime number? • 25 and 32 • LCM = 25 * 32 =288

  11. Summary • What are the three ways we can find the LCM? • (6th grade) Make a small list of multiples and find the least common multiple • Good for small numbers • (7th /8th grade) Use the prime factorization to locate shared factors and unshared factors. Multiply the single shared factors and multiply all the unshared factors. • Works better for larger numbers • (7th / 8th grade) Use the prime factorization and write them using exponents. Then multiply the values with the largest exponent for each prime number. • Works better for larger numbers

  12. Use any method you’d like. Some choices are wiser than others Find the LCM of the following • 15 and 35 • 12 and 21 • 96 and 50 • 200 and 49

  13. Warm-Up • Find the LCM of the following numbers • 1) 15 and 55 • 2) 18 and 24 • 3) 25 and 12 • 4) 9 and 24

  14. Let’s use the perspectives we learned earlier • Suppose I have xy and xz • The factorization of xy and xz are • xy: x*y • xz : x*z xy: x*y xz: x*z Shared Factor is x Unshared Factor are y and z So the LCM is x*(y*z) = xyz

  15. Alternatively • xy: x*y • xz : x*z • What’s the largest exponent for x? • 1 • What’s the largest exponent for y? • 1 • What’s the largest exponent for z? • 1 • LCM equals x*y*z = xyz

  16. Suppose I need the LCM of x2y3 and x2yz • Factorization of both terms x2y3: x*x*y*y*y x2yz: x*y*z Shared Factors are x2 and y Unshared Factor are y2 and z So the LCM is x2*y*(y2*z) = x2y3z Using the alternative method x2y3 X2yz What largest exponent for each variable? x2 andy3 and z = x2y3z

  17. Try these on your own • Find the LCM of • a3b and ab2 • g5h2 and g2h • O11f14 and O4f6 a3b2 g5h2 o11f14

  18. What about when we put coefficients in? • 2x and y • 2x: 2*x • Y:y • Since they have no common factors, the LCM is the product of the two. • 2x*y = 2xy • 12x and 8x2 • 12x: 2*2*3*x = 22*3*x • 8x2: 2*2*2*x*x = 23*x2 • LCM of 12 and 8 is 24 • LCM of x and x2 is x2 • LCM = 24x2

  19. Try these on your own • Find the LCM of • 3x3 and 6x • 10x2y2 and 15xyz2 • 3(x-4) and 9(x-3) 6x3 30x2y2z2 9(x-4)(x-3)

  20. Warm-Up • Find the LCM of the following • 1) xyz and z • 2) 2x2y4 and 3x3y • 3) (x-2) and (x-2)2 • 4) 9y and yz2 • 5) 5b2 and 15ab2 • 6) 2(m-1) and (m-3)

  21. Why do we do this? • It’s so we can add or subtract algebraic fractions • To add fractions that have algebraic expressions in the denominator, we need a common denominator • Example • The denominators need to be the same in order to add these. So we need the LCM of x and 3 first. • LCM=3x

  22. Some more fractions • First, find the LCM of the denominators • LCM of 4y and x is 4xy. • What do we need to multiply 4y by to make it 4xy? • What do we need to multiply x by to get 4xy? • Put together • Then get your final answer

  23. Try these on your own

  24. Since subtraction is ALMOST identical to addition. Try these next

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