Occam’s (or Ockham’s) razor is a principle named after the 14th century mathematician and friar, William of Occam. Ockham was the village in this English County where he was born. There are many resources to investigate this man and his theories. This is not about him but his thinking. Thinkers are important to the world. Over thinking something can be the death of it.

Most people have never heard of this and yet with the logical thinkers of today it is almost built into our genetic code. We know things without realizing how or why we do. The universe as a whole is almost emanating this into our very souls. Our brains absorbing codes that alter our thinking giving the same idea to the masses at the same time. I don’t completely understand everything. When I hear something my brain lets me know that logically the information is even viable. The brain will calculate out many different scenarios. You will start to evaluate your own opinions, theories and reason as to why one thing sounds right vs. the other. The Occam’s razor is a logical way of thinking.

Short excerpts from the 14th century theory:

“If you have two theories which both explain the observed facts then you should use the simplest until more evidence comes along”

“The simplest explanation for some phenomenon is more likely to be accurate than more complicated explanations.”

“If you have two equally likely solutions to a problem, pick the simplest.”

“The explanation requiring the fewest assumptions is most likely to be correct.”

“Keep things simple!”

You have heard many of these concepts built in to many popular slogans and methods of achieving a goal. The Occam’s Razor does not only have to applied to only scientific experiments but it can be applied to every day life.

This is a great scholarly way of looking at things. The way you look at things dictates how you decipher, translate and learn things. Then if you can learn things you can implement them into discovering the worlds secrets.

Base ten blocks are an excellent tool for teaching children the concept of addition because they allow children to touch and manipulate something real while learning important skills that translate well into paper and pencil addition. In this article, I will describe base ten blocks and how to use them to represent and add numbers.

The numbering system that children learn and the one most of us are familiar with is the base ten system. This essentially means that you can only use ten unique digits (0 to 9) in each place of a base ten number. For instance, in the number 345, there is a hundreds place, a tens place and a ones place. The only possible digits that could go in each place are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. In this example, the place value of the ones place is 5.

Base ten blocks turn the base ten concept into something children can see and touch.

Base ten blocks consist of cubes, rods, flats, and blocks. Cubes represent the ones place and look exactly like their name suggests - a small cube usually one centimeter by one centimeter by one centimeter. Rods represent the tens place and look like ten cubes placed in a row and fused together. Flats, as you might have guessed, represent hundreds, and blocks represent thousands. A flat looks like one hundred cubes place in a 10 x 10 square and attached together. A block looks like ten flats piled one on top of the other and bonded together.

In order to use base ten blocks to add numbers, students should be familiar with how to represent numbers using base ten blocks. To see what base ten blocks look like, and to try them out, go to the National Library of Virtual Manipulatives:

nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html

To represent a number using base ten blocks, make piles of base ten blocks to represent each place value. If your number was 2,784, you would make a pile of 2 blocks, a pile of 7 flats, a pile of 8 rods, and a pile of 4 cubes. It is useful to arrange the piles in a row in the same order that they appear in the number as that will be useful later on when children learn the paper and pencil algorithm.

Another useful skill to practice is trading base ten blocks. Each block can be traded for 10 flats, each flat for 10 rods, and each rod for 10 cubes. Going the other way, 10 cubes can be traded for one rod, 10 rods for one flat, and 10 flats for one block.

One simple use of base ten blocks that translates well to a paper and pencil method of addition is to add by regrouping. To add two or more numbers, start by representing each number with base ten blocks. Put all of the cubes from both numbers in the same pile; do this with the rods, flats, and blocks as well. Next, trade any groups of 10 cubes for a rod. Trade any groups of 10 rods for a flat; then trade any groups of 10 flats for a block. To read the resulting number, count the number of base ten blocks left in each pile and read the number.

To illustrate this procedure, picture the addition question, 568 + 693. After representing both numbers with base ten blocks and combining the piles of like base ten blocks, you should have a pile of 11 cubes, a pile of 15 rods, and a pile of 11 flats. Trading 10 of the cubes for 1 rod means you now have 1 cube, 16 rods and 11 flats. Trading 10 of the rods for one flat results in 1 cube, 6 rods, and 12 flats. Trading 10 of the flats for one block gives you your final piles of 1 cube, 6 rods, 2 flats, and 1 block. The answer to the addition question, therefore, is 1,261.

If you don’t have base ten blocks, you can use the virtual base ten blocks or make paper versions. If you need addition questions (with the answers included), you can access thousands of free math worksheets at .math-drills.com

In future articles, I will describe more uses for base ten blocks including subtraction and multiplication, and I will continue the series with other manipulatives that can help your child or student learn math.

More than likely, when you learned how to add, you started on the right and moved to the left. If you were adding whole numbers, you added the ones, “carried” if necessary, and repeated for the tens, hundreds and so on. This works well on paper, and it is the most efficient paper and pencil method; however, adding in the other direction has several desirable advantages: the left to right method promotes a better understanding of place value, it can be done mentally with much greater ease, and it does not require that numbers be lined up in a column. Students can learn left to right addition, so they have another method to choose from when presented with addition problems.

Left to right addition involves adding the largest place values first. As you move from left to right, you keep a cumulative total, so it is simply a number of smaller addition problems. To give you an idea of how it works and what it sounds like, consider the example, 677 + 938.

Begin by adding the left most place values. In the example this is 600 plus 900 equals 1500. Add the values in the next place, one at a time, to the previous sum, and keep track of the new sum each time. In the example, 1500 + 70 is 1570, 1570 + 30 is 1600. For students who are more proficient at this algorithm, they don’t necessarily think “plus 70″ or “add 30.” Their thought process, if said out loud might sound like, “600, 1500, 1570, 1600, . . .” Continue adding the values in each subsequent place until finished. The final steps in the example are 1600 + 7 is 1607, 1607 plus 8 is 1615. The sum is 1615.

As you can imagine, students need to be proficient at single digit addition and have an understanding of place value before attempting left to right addition. When they are first learning it, they might try repeating sums as they go along (e.g. 1500, 1570, 1570, 1570, 1600, . . .) to help them retain the newest sums. They might also cross out digits as they are adding. There is no rule about having to add in this way mentally. Students could write down the sums as they proceed.

Left to right addition promotes a better understanding of place value than right to left addition. In right to left addition, single digits are carried or regrouped with little emphasis placed on what the value of those carried digits are. In the example, 1246 + 586, students add 6 + 6 to get 12; they write down the 2 and carry the 1 when they should be carrying the ten. In the next step, they add 8 + 4 + 1 to get 13; they write down the 3 and carry the 1 when they should be adding 80 + 40 + 10, writing the 3 in the tens place (i.e. 30) and carrying the hundred. Essentially, right to left addition excludes vocabulary related to place value. Left to right addition, on the other hand, promotes an understanding of place value as each digit is given its correct value. In the example, the one in the thousands place is one thousand, the two in the hundreds place is two hundred, and so on.

Left to right addition is well-suited to mental addition since the sum is cumulative with no steps in between; in other words, there is nothing for the student to keep in mind except for the cumulative sum. In right to left addition, several numbers must be remembered as the student proceeds. To illustrate this, consider the simple example, 64 + 88. In left to right addition, the sum is simple to find: 60, 140, 144, 152. Only one number had to be remembered at any point. In right to left addition, 4 + 8 is 12, so there are already two numbers to remember: the two in the ones place and the regrouped ten. The next step is to add 60 + 80 + 10 to get 150. At this point, the two must be recalled and added to the 150 to get 152. Although this sounds simple, it becomes more complicated with more digits.

Right to left addition does not require numbers to be lined up in a column, but it is often taught that way because the method tends to ignore place value and relies on a student’s ability to line up the place values to compensate. Many errors that students make in right to left addition occur because they don’t have a strong knowledge of place value, and they forget or don’t realize that like place values need to be lined up. They might, for instance, add a digit in the tens place to a digit in the hundreds place. Another scenario is a sloppy recording of numbers where a digit is mistakenly added to the wrong column. In left to right addition, the emphasis is on finding a certain place value in each number rather than relying on the place values being aligned. Students, of course, need to be able to recognize place value before they can be successful at this method. For instance, they should be able to recognize that the ones in the numbers: 514, 1499, and 321 are in the tens, thousands, and ones places respectively. If they can’t, further teaching on place value is required before addition can be taught effectively.

Although left to right addition has several advantages, it isn’t suggested that you scrap everything else. Learning a wide variety of addition methods allows you latitude in problem solving situations. By teaching students this method, you give them another option when they are tackling addition questions.

Ever wonder if you and your students could create your own TV news show? Would you like to have announcements and school/class information available to students all class long? Would you like to avoid those students who were absent constantly asking you, “What did we do in class yesterday?” It isn’t only possible to do, but with a few pieces of equipment, it’s easy to set up and run.

You, of course will need several pieces of hardware, including a TV or (digital projector) and a computer. You will also need the proper cables to connect the two. We’ve discovered that sometimes the resolution on some computers needs to be adjusted or changed, so check your monitors setting. You might even need a scan-converter if all else fails. Such a TV network can also be simply set up on a computer monitor which is turned to face the students.

Your computer will also need PowerPoint (or an equivalent presentation software). We’ve used such programs effectively on Macs, as well as Linux and Windows machines, and they all work well for this application.

PowerPoint has the feature of progressing through information or slides by either clicking your mouse, or by setting up timings between every action. Thus, you can have each word, line, paragraph, or even graphic animated automatically. You can change up the settings for different bits of info you have. Check the top menu for ’slide show’, and follow down the menu to ‘custom animation’ (or look for a similar command). Once there, you can select each element to animate, the type of transition to occur, any sound you want associated with it, and also the timing (automatic, not on a mouse click). You will want to practice a few times until your timing is good, and there are enough seconds to see or read each element before the next animation or transition.

Even your slides can be changed automatically. Go to the ’slide show’ menu and select ’slide transition’ or ’set up show’. From there, you can choose the type of transition, and even its speed of animation.

You may wish to check your computer’s settings so the machine doesn’t go to sleep on you, or change to a screen saver. That would definitely defeat your purpose!

Now that you know how to set up a show, you have to decide what material or information to put out on display. I put up basic information such as the lunch menu, school or class announcements, and homework assignments. I will also post a class schedule and switch times if the daily schedule is altered. For the students who were absent, we also display class notes from previous classes. Now there is no excuse for students missing assignments or class information! And this saves you from having to deal with every returning student asking what was missed and where to find it.

If you are brave and want to create a great class project, have your students run your daily announcements. You could partner them up and have your first class of the day create the announcements. Another project is to have your students create storyboards, where a short story is broken up among a number of slides, each slide including pictures, clip art, or graphics to illustrate the story. You can find many good images online or in the clip art of your program. If you have access to a digital camera, you can even have students take their own pictures and insert them.

Yet another project we’ve done is to create a PowerPoint to summarize one class or a week’s worth of class info. This becomes an animated newsletter or magazine. Again, assign a student to take photos on a digital camera during the class and combine these with articles on the various activities you’ve done. You might want to include students’ work as examples.

There are also advanced techniques you can experiment with as you get better with the program. Sound can be added, such as background music, songs, or voice recordings. There are also ways to include video. Become an expert with the basics, and you’ll be ready for these advanced techniques.

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Frank Holes, Jr. is the editor of the StarTeaching website and the bi-monthly newsletter, Features for Teachers. Check out our latest issue at:

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