Prime TimePrime TimeWhole Number ArithmeticWhole Number Arithmetic

A little vocabularyA little vocabularyFactors Factors are one of two or are one of two or more whole numbers that more whole numbers that are multiplied together to are multiplied together to get a get a ProductProduct

5 x 7 = 355 x 7 = 35Five and seven are factors, Five and seven are factors, and thirty-five is a productand thirty-five is a product

There are 2 kinds There are 2 kinds of numbers…of numbers…

Prime NumbersPrime NumbersThese numbers have only two factors… These numbers have only two factors… One and themselvesOne and themselves

7 = 7 x 17 = 7 x 17: 1,77: 1,7

Composite NumbersComposite NumbersHave at least three factorsHave at least three factors

10 = 5 x 2 and 10 x 110 = 5 x 2 and 10 x 110: 1,2,5,1010: 1,2,5,1012 = 6 x 2 and 4 x 3 and 12 x 112 = 6 x 2 and 4 x 3 and 12 x 112: 1,2,3,4,6,1212: 1,2,3,4,6,12

Factor PairsFactor Pairs

24: 1, 2, 3, 4, 6, 8, 24: 1, 2, 3, 4, 6, 8, 12, 2412, 24

All factors have pairs All factors have pairs

Finding Finding AllAll of the of the FactorsFactors

You have to be systematicYou have to be systematicStart with one and count upStart with one and count upStop when you find two Stop when you find two consecutive factors whose product consecutive factors whose product is the number you are working on.is the number you are working on.

60: 1, 2, 3, 4, 5, 6, 1060: 1, 2, 3, 4, 5, 6, 106 x 10 = 60 so this is the central 6 x 10 = 60 so this is the central factor pairfactor pairNow just match your other factors! Now just match your other factors!

Square NumbersSquare Numbers1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144100, 121, 1443 x 3 = 9. This statement could be 3 x 3 = 9. This statement could be modeled like this:modeled like this:

Square NumbersSquare Numbers16: 1, 2, 4, 8, 1616: 1, 2, 4, 8, 16Because we don’t repeat factors in our Because we don’t repeat factors in our lists, square numbers have an odd lists, square numbers have an odd number of factorsnumber of factors

4 x 4 is the middle pair4 x 4 is the middle pair

The square root of a square number is a The square root of a square number is a whole number!whole number!

Numbers between square numbers have Numbers between square numbers have square roots between the squares…square roots between the squares…

Square RootsSquare RootsThe numbers between 25 and 36 have The numbers between 25 and 36 have square roots between 5 and 6 (non square roots between 5 and 6 (non whole numbers)whole numbers)

Examine Factor Pair Mountain (website)Examine Factor Pair Mountain (website)

Knowing our square roots, tells us when Knowing our square roots, tells us when to stop looking for factors.to stop looking for factors.

27’s square root is between 5 and 6 so I 27’s square root is between 5 and 6 so I only need to count to 5 to find all of the only need to count to 5 to find all of the numbers up FPMnumbers up FPM

27: 1, 3 – and we’re ready to climb down!27: 1, 3 – and we’re ready to climb down!

DivisibilityDivisibility1: All numbers have 1 as a factor1: All numbers have 1 as a factor2: All even numbers (ending in 0, 2, 4, 2: All even numbers (ending in 0, 2, 4, 6, 8) have 2 as a factor6, 8) have 2 as a factor3: If the sum of the digits is divisible 3: If the sum of the digits is divisible by 3 so is the number itselfby 3 so is the number itself

81 -> 8 + 1 = 9, which is divisible by 81 -> 8 + 1 = 9, which is divisible by 3, so 81 is too!3, so 81 is too!372 -> 3 + 7 + 2 = 12, so 3 is a factor372 -> 3 + 7 + 2 = 12, so 3 is a factor109 -> 1 + 0 + 9 = 10, so 3 is not a 109 -> 1 + 0 + 9 = 10, so 3 is not a factorfactor

DivisibilityDivisibility5: Numbers that end in 5 or 0 have five as 5: Numbers that end in 5 or 0 have five as a factora factor

6: Numbers that have 2 & 3 as factors 6: Numbers that have 2 & 3 as factors have 6have 6

9: Same trick as for 3!9: Same trick as for 3!162 -> 1 + 6 + 2 = 9162 -> 1 + 6 + 2 = 9162 = 9 x 18162 = 9 x 18

So what about the rest of the numbers?So what about the rest of the numbers?

This is where it gets interesting…This is where it gets interesting…

Stretching Stretching NumbersNumbers

How do we know if 8 goes into How do we know if 8 goes into 140?140?

Our math facts don’t go up that high!Our math facts don’t go up that high!Long division?Long division?

Pick a number close to 140 that you Pick a number close to 140 that you know 8 goes into… how about 80.know 8 goes into… how about 80.

Look at the difference (60)Look at the difference (60)

If 8 goes into the difference, it goes If 8 goes into the difference, it goes into the numberinto the number

Stretching Stretching NumbersNumbers

See if 8 goes into See if 8 goes into 224224

You might use 8 x 3 to come up with 8 x 30You might use 8 x 3 to come up with 8 x 30

Since we know 8 goes into 240 all Since we know 8 goes into 240 all we have to do is check the we have to do is check the difference – in this case 16. This difference – in this case 16. This means that 8 is a factor of 224. (8 means that 8 is a factor of 224. (8 x 28 = 224)x 28 = 224)Let’s practice!Let’s practice!

PrimesPrimesThe Fundamental Theorem of ArithmeticThe Fundamental Theorem of Arithmetic

All numbers are the unique product of All numbers are the unique product of prime numbersprime numbers

I think of this as a unique fingerprint for I think of this as a unique fingerprint for each number, or perhaps a number’s each number, or perhaps a number’s true name (Eragon)true name (Eragon)

20: 2 x 2 x 5 23: 1 x 23 (prime)

21: 3 x 7 24: 2 x 2 x 2 x 322: 2 x 11 25: 5 x 5

Factor TreesFactor Trees

36: 2 x 2 x 3 x 3

Factor TreesFactor Trees

42 = 2 x 3 x 7

Factor TreesFactor TreesYou can start any way you You can start any way you want, but the end result will want, but the end result will always be the same – the order always be the same – the order of the factors does not matterof the factors does not matterExamine different factor trees Examine different factor trees for 100 for 100

Practice!Practice!

LCM & GCFLCM & GCFKnowing the prime factorization of Knowing the prime factorization of numbers allows us to see what they numbers allows us to see what they have in common.have in common.

Greatest Common Factor (GCF)Greatest Common Factor (GCF)ALL of the prime factors that any two (or ALL of the prime factors that any two (or more) numbers have in common.more) numbers have in common.

INTERSECTIONINTERSECTION

Least Common Multiple (LCM)Least Common Multiple (LCM)ALL of the prime factors (without ALL of the prime factors (without duplications)duplications)UNIONUNION

LCM & GCFLCM & GCF

Going ForwardGoing ForwardComputation is probably always going to Computation is probably always going to be a part of your child’s math education – be a part of your child’s math education – have them master the facts now.have them master the facts now.

Do mental math at every opportunity Do mental math at every opportunity with themwith them

Not just the factsNot just the factsEstimation Estimation Divisibility testsDivisibility testsLarge numbersLarge numbers

Next UnitsNext UnitsBits and Pieces (CMP)Bits and Pieces (CMP)

Fractions – representations and Fractions – representations and computationcomputationPotential parent night?Potential parent night?

Division (Additional Materials)Division (Additional Materials)Long Division – standard algorithmLong Division – standard algorithmEstimationEstimationMental MathMental Math

ThanksThanksThere will be more group practice if There will be more group practice if time permitstime permits