Sunday, September 29, 2013

More Fun (Thanks Mrs. Schwartz)

Friday, September 27, 2013

Just for Fun!

Why Prime Factorization Matters

The following video gives you a preview of some of the work we will be doing in the next unit. I briefly demonstrated how you could use prime factorization to reduce fractions to a couple of the classes. I will go into more detail about this in the future. But for now, this video is a great explanation and demonstrates the power of prime factorization.

Explanation of prime factorization

The following video is a good review of prime factorization.  Prime factorizations is possible because of the Fundamental Theorem of Arithmetic.  Think about what the word fundamental means.  It means the basis of something.  Therefore the Fundamental Theorem describes the basis of numbers which are products of their prime factors and therefore unique!

A little inspiration..

Friday, September 20, 2013

Finding LCM Using Prime Factorization

Like GCF, you can use prime factorization to find the LCM of two or more numbers.  First find the GCF since the LCM and both numbers will share this factor.

Then multiply the GCF to all the remaining prime factors.  You are doing this because the prime factor string for the LCM must account for all the prime factors in your original numbers.  If not then you do not have the LCM.

Notice every prime in the prime factor string for the LCM (2160) is used at least once.  It is important to understand that the prime factor string for the LCM (2160 in this case) is the shortest factor string possible that will account for the prime factor strings of 144 and 1080.  This is the very meaning of least common multiple.  If you multiply the factor sting for 2160 by 2, you have a common multiple of 144 and 1080 but it is not the least common multiple.

Finding GCF using Prime Factorization

You have lots of experience finding the greatest common factor with various methods (u-turn, listing, rainbow).  However, the most efficient way to find the GCF for any two or more numbers, especially large numbers, is to use prime factorization.  Remember, the prime factor string for any number is unique.  Breaking numbers down into their basic prime factors allows us to manipulate them in ways that help us do things like find GCF and LCM of numbers.  For those of us that stress over simplifying large fractions, prime factorization is invaluable.  See below.

Notice only the COMMON primes between the two numbers are highlight in red.  When you multiply these common primes together you get a product of 72.  This is the greatest factor that is shared among the two numbers.  You may also notice that 12 is a common factor because the prime factor string 2 x 2 x 2 x 3 is found within both of the factor strings of 144 and 1080.  Can you find more common factors of these two number just by looking at their prime factor strings?

Hopefully, you're thinking to yourself why would we need to know this.  Look at the image below.  You can use prime factorization to simply fractions with large numerators and denominators instead of painstakingly using divisibility rules and dividing!

For those of you who want to go deeper, you can find the GCF for any number of numbers.  You just need to make sure each number shares the common primes. Look!

If you need a refresher with prime factorization watch this video.

GCF & LCM Word Problems

Deciding whether to use common multiples or common factors to solve problems can be tricky.  Therefore to remember:     

GCF problems involve sharing/dividing evenly
 LCM problems involve things happening in  cycles/ same time

Here's a sample problem:  

There are 40 girls and 32 boys who want to participate on co-ed soccer intramural teams.  If each team must have the same number of girls and the same number of boys, what is that greatest number of teams that could participate?  How many boys will be on each team?  How many girls on each team?

Think about what you are being asked to do with the values you are given.  In this case you are being asked to divide/split your numbers into equal groups.  This is your hint to begin to find factors.  Best practice is to use the u-turn method to ensure that you find all the factors of a number in an organized manner.  When the problem asks for the "greatest" number it is asking for the greatest common factor (GCF).  Keep in mind greatest doesn't mean multiple even though multiples create larger and larger numbers.  Also be sure you are answering what is being asked.  Even though you may have found the GCF, you have to use it to answer the rest of the questions

Here's another sample:

Ms. Pearl is shopping at the supermarket.  She is getting things for the sandwiches she is going to prepare for next Sunday's picnic.  She sees that hamburger buns come in packs of 8 but the hamburgers themselves come in packs of 10.  What is the least number of packs of buns and burgers she should get so she has an equal number of buns and burgers?

For this problem you will need to find multiples because it only makes sense that you will be increasing the number of packs to be able to get equal numbers.  Because it is asking for the least number of packs you will be finding least common multiple (LCM).  However, the LCM only tells you the number of buns and hamburgers (which is 40 in this case) so you will have to use it to determine the number of packs of each type of item.

Tuesday, September 17, 2013

Prime Time Pretest

Prime Time Pretest

Things to think about for Thursday's Quiz...

Your first quiz of the year is this Thursday 9.18.13.  Here are some things to review:

  • Finding factors of a number
  • Finding Proper Factors of a number
  • Describing the best moves on the factor game and why they are the best moves
  • Describing a set of numbers in relation to their divisibility rules.  For example, list the multiples of 12 up to 100 and describe the divisibility rules for the multiples
  • Describe what makes a number prime
  • Describe the reasons why certain numbers do not appear on the Product Game board while others do
  • Know and use various divisibility rules
  • Identify primes and composites
  • Name at least two...
    • abundant numbers
    • deficient numbers
    • perfect numbers 
    • square numbers

Additional Important Vocabulary

Learning Vocabulary

The vocabulary assigned to various math ideas and concepts is not arbitrary (random). Often the words that name ideas and concepts actually describe the idea or concept.   For example, the term abundant number is used to describe a number in which the sum of the proper factors is greater than the number itself.  The word abundant means many, a lot, or sometimes excess.  If we understand the reason for the vocabulary we can better makes sense of the concepts.

Additionally, there are many strategies to understand and make sense of vocabulary.  The strategy of words maps is used below.  Understanding the vocabulary in math or science is half the battle in understanding the content of each subject.