GCF and LCM

Test on Wednesday already? You know what else is on Wednesday? The start of Spring Breakkk! I can't wait. Anyway, back to math...

My favorite way to find the GCF is using a factor tree. WE take a number a break it down until all the numbers are at the smallest possible and are prime. A good way to practice is to take a large number and break it down without using a calculator and finding all it's prime numbers.

Here is an Example:

Here we can obviously see all the prime numbers within 120 and 96. If we were to find the GCF between the two numbers we will line them up as so:

96: 2 2 2 2 2 3
120: 2 2 2 3 5

Take what numbers are lined up: 2 * 2 * 2 * 3 = 24; Therefore the GCF between 96 and 120 is 24. :) I love the way that this system shows how to find that so easy!

Now, to find the LCM between 96 and 120 is to take all numbers that represnted (only take them once!) and multiply together:

2 * 2 * 2 * 2 * 2 * 3 * 5 = 480! This is a way that we can use so much easier than actually going through every single multiple until we reach the same number. Getting to 480 would have taken so much longer if we did not use this method. :)

Fractions and Multiples

In our text (pg 216) it has a very interesting note on the relationship between a factor and a multiple. It states that if one number is a factor of a second number of divides the second then the second number is a multiple of the first.

For example,

4 is a factor of 20, as 20 is a multiple of 4.

Here is the rule for Factor and Multiple: If a and b are whole numbers and a does not equal 0, then a is a factor of b and only if there is a whole number c such that ac = b. We can say that a divides b or that b is a multiple of a.

This is a lot of words and there is always an easy, hands on way to to see how this works!

I will focus on two models to demonstrate factors and multiples much easier.

Linear Model:

This shows all the factors of 12 and the multiples of it as well. We can use this model to figure out the factors and multiples of any number.

Next, we have the retangular model. In this model, one number is shown by squares or tiles, and the two demensions of the rectangle will show the factors of the number. This is an easy effective way to teach ourselves and students how to know which numbers will evenly divided into eachother. Building longs of numbers and placing them together will determine the outcome. If there are left over tiles, then the number is NOT a factor.

Example: Try to make a rectangle out of the number 12. We can show this by having a 3 X 4 rectangle, 6 X 2 or 1 X 12. but, we cannot make a rectangle with any side using any other number. there would be left over units.

Multiplication

Thie blog will help with how to demonstrate multiplication using manipulatives. The rule for multipling whole numbers is: For any whole numbers r and s, the product of r and s is the sum occuring r times. Written out as such:
r * s = s + s + s .... + s -- r times
In our manipulative kits we can find the rectangular objects to help demonstrate this rule.
If we were to use a rectangle with 6 units on it, and place 5 of them in a row, it would then show 6 + 6 + 6 + 6 + 6 = 30. Then, take all 6 rows and place them into one large rectangle. This would give you a 5 X 6 shape indicated 30 units. As future teachers I find this to be a great example to use in the classroom to students who have a hard time understand why 6 * 5 = 30.

Using models for mulitplication algorithms can be used with larger numbers as well. Using base 10 pieces, we can see the relationship that numbers hold together when multipling.
Let's do the example for the text book (pg 166)
3 x 145

The number 145 is shown in model pieces and tripled because we multipling 145 by 3.
Grouping together all the flats, longs and units we will then see 4 flats, 3 longs and 5 units. This is because some longs became flats, units became longs and so on.

Using models is always a good example to share with people :)

Addition and Subtraction! Continued...

The first post regarding Addition and Subtraction properties focused on Addition.


This post will show you properties, examples and definitions to: Take Away, Comparison and Missing Addend.
Subtraction of Whole Numbers: For any while numbers, r and s, with r greater than or equal to s, the difference of r minus, written r - s, is the whole number c such that r = s + c. The number c is called the MISSING ADDEND. This dorky video helps with finding the missing addend in subtraction. Here is an example of the finding the missing addend:

12 - 6. To find the missing addend you must find the number that can be added to 6 to equal 12. Clearly, the answer is 6. Another fun example would to use money. I gave my waitress $2.00 and my change is 45 cents. The waitress will count up from 45 to 200 to determine my change.



The Take-Away Concpet: Suppose that you have 15 apples and take away 5, how many apples to do you have left? 15 - 5 = 10. This will help as future teachers to teach to your students; giev them objects to "take away".

The Comparison Concept: Suppose you have 15 apples and someone else has 5 apples. How many more apples do you have than the other person? We then compare our collection to the other persons to see what the difference is! Which would then lead back to 15 - 5 = 10.

All these different concepts help us better understand subtraction and fun ways to teach our students someday! :)

Addition and Subtraction!

One thing that really helped me learn this lesson is being able to describe the Identity, Associative, Commutative, and Closure Properties for Addition.


As a reminder, I will write down the properties for those of you who want to use this to study from. (I started on page 143 of the text book.)

Additon of Whole Numbers: If set R has r elements and set S has s elements, and R and S are disjoint, then the sum of r plus s, written r + s, is the number of elements in the union of R and S. The numbers r and s are called addends.
- Remember that as a teacher, it children will learn how to add at an early age from adding objects together (and taking away). The same is with how we learn our addition today, just in different bases -- we put sets together and define their union together.

Closure Property: For every pair of numbers in a given set, if an operation is performed, and the result is also a number in the set, the set is said to be closed for the operation. If one example can be found where the operation does not produce an element of the given set, then the set is not closed for the operation.
- Another way to say this, or understand is say that this is the closed for the operation of addition. Here is a great video that simply shows the closure poperty clear and effective.

Identity Property for Addition: For any whole number b,
0 + b = b + 0 = b
and so on.
This is the easiest of them all! For both addition and multiplication, anything that equals itself (0 + x = x) and ( 1 * x = x) this will be the identity property.

Associative Property for Addition: For any whole numbers, a, b, and c,
a + (b + c) = (a + b) + c
It does not matter what order the numbers fall into, arranging numbers will not make a difference. Explination when using real numbers is show in This Video, and this really helped me understand.

Commutative Property of Addition: For any whole numbers, a and b,
a + b = b + a
When there are two whole numbers being added, they maybe changed (communted) without effected the outcome of the sum.

Numeration Systems

Since missing last week, it has been hard catching up today! But, at the same time it's been an interesting unit learning about base five, base ten, base twelve and so on. Also, 3.1 has more to offer than just the bases -- it shows a wide variety of different numerical systems.
Starting with Egyptian numerals, the text book (page 127) it gives numbers posted by Egyptian numerical system and ours. It is so fun to see what our numbers look like next to other historical number systems. This video is a good example and I found it helpful in understanding the Egyptian number system and how it cooresponds to the one we use in class, and in the United States. Some of the symbols are a bit humerous. I felt the man in the video was quite helpful in understanding what it is that they used to as base ten numbers with what we use.



Another interesting number system that is discussed in the book is the Babylonian numeration. The book (page 129) says that the Babylonian number system has a weakness. It says that the number system is missing a symbol for the number Zero. Here is a fun video to watch closely and see how the numbers aling, it's a confusing consept to unerstand -- my opnion at least!

Another number system mentioned in this chapter is the Mayan number system, it is really cool! Our book mentioned that thier number system was based on 20. One way to keep it line is to remember that they would says "one man for 20, two men for 40...". The book says that this system was used in New Guinea as well. Keep in mind that the Mayans from Yucatan and the Aztecs of Mexico used this number frequently.

Lastly, another great (and more common amoungst us Americans) is the Roman Numeral system. The Romans used based 10 in thier number system, like the Egyptians. We are used to see their most common symbols (for the numbers 1, 5, 10, 50, 100, 500 and 1000). The symbols used here are:
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000

It is interesting to note that the Romans used their number system so that one may read it decreasing order from left to right. One last video shows an easy way to follow the number system! :)