Factor Trees

I can guess that just like myself, a lot of parents do not understand factor trees; what they are, how to make one, what one looks like, what they are used for, etc... I didn't fully understand them, and their importance in math until I became a college student. 

To start understanding a factor tree, you must first learn the official definition of one. A factor tree, as defined by factortree.com, is a diagram used to break down a number by dividing it by its factors until all the numbers left are prime. Factor trees are most commonly used in elementary math to find out the Greatest Common Factor (GCF) between two numbers. 

Once you understand the definition, it you now need to learn how to construct one. Below is a video showing how to factor tree the number 72:

You can also visit coolmath.com to get a visual representation of how to make factor trees, if you prefer reading.

I was quite amazed upon my researching of factor trees to find factor rainbows. I think that by showing my future students factor trees or factor rainbows will give them a sense of control because they can chose for themselves which method they like. 

I have also found a couple of great Websites that have factor tree games for students to practice prime factoring numbers:

• Factorization Forest 
• Factor Trees

I believe it is extremely important for us as not only teachers, but parents of children to know and understand what our children are doing in math class. The Internet, and your child's teachers are full of information, games, videos, songs, etc... to help us, help our children. When our children, I know my step-son for sure, see that we understand what they are doing and can do it too, it makes them more confident and hungry to learn more.

The Rule of Signs

Probably one of the most confusing things for kids in math class is positives and negatives. I know that when I was in grade school, and still to the day, I struggle with this. I have found printable charts of the rules of signs by searching the World Wide Web. I have found that the only way for me to remember and apply the rules is by having a visual representation of them at all times. I can see this being a great thing for my future students. I plan on making a HUGE printable of the following pictures:

I am a strong believer in charts, but they can't be too busy, I think it makes it harder for a child's mind to remember for later work. I have actually made both of the above charts into jumbo size and glued them to red and blue construction paper, and hung them in my own classroom for the school-aged children that I help with homework. The rules of signs is another big thing I have noticed students that I help struggle with. I have found that by having huge charts in my room has really helped them to become quicker, and more confident in solving math problems that involve positives and negatives.

I did some further searching on the Internet and found a couple of great songs to help students in remembering the rule of signs, please check them out below:

I have actually introduced these two, and some more, songs to the school aged children at my work, and now they can't stop singing them. Okay I can't stop singing them either. I have got to admit, using songs to help children learn is a great idea. I use songs a lot in my own classroom. We have songs for skip counting, the vowels and consonants, the seasons, the months of the year, and much much more. I have found that using songs to teach somewhat complicated subjects acts like a superglue; sticking the ideas and concepts in the children's minds and never letting go. I have had students from 5+ years ago still come up and sing some of the songs I taught them. If teachers could only be more open to newer and creative ways for teaching...

Tests for Divisibilty

Maybe I am getting old, and don't remember how to test numbers for divisibility, but I have learned some really cool tricks that I would like to now share. I will be explaining how to test for divisibility by 2,3, 4,5,6,7,8,and 10. You can find a convenient Divisibility Rules Table here, which I have a printed copy of and have placed on my refrigerator for my step-son that we use quite frequently!

To test a number to see if it is divisible by 2:

You simply look at the last digit in your number, if the last number is any multiple of 2 (0,2,4,6,8) than your number can be divided by 2.

To test a number to see if it is divisible by 3:

You take your given number and add up all the digits, if the sum of digits is a multiple of 3, you have a number that is divisible by 3.

To test a number to see if it is divisible by 4:

You take the last two digits of a number and add them together, if the sum is a multiple of 4, you have a number that is divisible by 4.

To test a number to see if it is divisible by 5:

Check to see if the last digit of given number is a 5 or a 0.

To test a number to see if it is divisible by 6:

Look at the last digit of your given number, decide if it can be a multiple of 2 or 3. For which ever one it is  multiple of, you then add up all the digits, if the sum of digits is multiple of 2 or 3, your number is divisible by 6.

To test a number to see if it is divisible by 7:

You will want to subtract 2 times the last digit from the other digits. 

224: 22-8=14

To test a number to see if it is divisible by 8:

Check to see if the last 3 digits of your number add up to a sum that is a multiple of 8.

587320 is divisible by 8 because 320 is divisible by 8.

To test a number to see if it is divisible by 9:

Add all the digits of the number in question, check to see if it is a multiple of 9.

To test a number to see if it is divisible by 10:

Look at the last digit in the number you are given, if it ends in 0 than it is divisible by 10.

Minnesota Math Standards

Who, What, Where, When, Why

Does anyone really truly understand the “Math Standards?” I know myself, like many other parents do not fully understand these standards. Straight from the Minnesota Department of Educations handbook on math standards I would like to share the opening paragraph:

“The Minnesota Academic Standards in Mathematics set the expectations for achievement in mathematics for K-12 students in Minnesota. This document is grounded in the belief that all students can and should be mathematically proficient. All students should learn important mathematical concepts, skills, and relationships with understanding. The standards and benchmarks presented here describe a connected body of mathematical knowledge that is acquired through the processes of problem solving, reasoning and proof, communication,

connections, and representation. The standards are placed at the grade level where mastery is expected with the recognition that intentional experiences at earlier grades are required to facilitate learning and mastery for other grade levels.”

For the next 44 pages one can see all four categories that students are assessed on, by grade level: Numbers and Operations, Algebra, Geometry and Measurement, and Data Analysis and Probability. To help parents better understand the handbook, you can visit the Minnesota Association of School Administrators (MASA) website, or click this link, http://www.mnasa.org/cms/lib6/MN07001305/Centricity/Domain/76/acceleration_delaney.pdf, to get all your questions answered. There is also a website for teachers to help them translate the standards into their own classrooms, http://scimathmn.org/stemtc/

After looking around the net, I discovered many websites out there designed to help students reach important benchmarks associated with the standards, and help them prepare. Here are the two best website I found:

·       http://www.ixl.com/standards/minnesota/math

·       http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?keyword=mm1+test+prep+minnesota

Division as Repeated Subtraction

Division has got to be my worst fear when it comes to math. When I was younger, I learned how to divide by doing the classic long hand method. This was incredibly time consuming, and confusing; if you misplaced just one number, the entire problem is wrong and you end up having to start all over. Most of the time I would get so frustrated that I would just whip out my calculator and solve the division problems without really understanding the method. But, as an adult I have learned about a new way math teachers are teaching division, through repeated subtraction. 

One way to use the repeated subtraction method is by subtracting off as many hundreds as you can, then tens, and then ones. This method would look like my example below:

Another way to use the repeated subtraction method is my subtracting the divisor from the dividend until you reach zero, or close to zero. The video below does a wonderful job of illustrating this:

 

The last method I have discovered for using the repeated subtraction is by using a simple number line. The picture example below shows how this would be done:

Looking at division problems this way makes it much less scary, confusing, and time consuming. The fact that the repeated subtraction to solve division problems has three different ways makes it even more important for us as teachers to be utilizing in the classroom. As an educator myself, I know that children all learn differently, and I believe that teaching division as repeated subtraction and having three different ways to accomplish is great because some students may find using the number line way more suited to them, or some may prefer the subtracting of the divisor from dividend, and if all else fails, some may be more comfortable using the subtracting of hundreds, tens, and ones to their liking. In my future classroom I fully intend on using this method in teaching my students division!

Expanded Notation for Multiplying Whole Numbers

Using Expanded Notation has got to be the best way I have seen for solving multiplication problems! It is so easy and simple to use, and you can solve your problems faster and with accuracy. Below are some pictures of what large numbers look like in  Expanded Notation. (I like the idea of using expanded notation for large number multiplication; makes it much less confusing and time consuming)

I find this to be a very clever way to be teaching students place value, by breaking down the numbers into their respective parts (hundreds, tens, and ones). 

Now when multiplying, lets say with a problem like 8x469, you would first start with converting 469 into Expanded Notation which would look like so:

400+60+9=469

Now that we know what 469 looks like in Expanded Notation we can move onto our next step of the problem. We need to now multiply our 469 by 8, and to do this we would put our Expanded 469 into parenthesis and write 8 before it like so:

8x(400+60+9)

From here the next step is to distribute the 8 through the parenthesis, and would look like this:

8x400+8x60+8x9

From here you would solve each multiplication problem, looking like this:

3,200+480+72

To finish you just go through and add up everything giving you your answer of:  3,752

I personally have used this method while helping the school-aged children at my place of employment, and with much success. At first, it is overwhelming, but after a couple of days they no longer need my help (which makes me kind of sad). I wish that when I was younger I could have been taught this way to do multiplication problems. I am stuck in the long-hand way of doing multiplication, so doing this at first was very hard for me to get used to. But, through using this method with the students I tutor at work, I have come to always use this method for multiplying. Study Jams is a wonderful website that offers students training, and testing on expanded notation I myself have used this website many times in helping my step-son practice writing numbers in expanded notation for many math assignments.