## Sunday, January 13, 2008

### HALFWAY NUMBERS

Back on Nov. 27, Nov. 29, and Dec. 3rd,on this blog,I posted some information on Mental Math which included philosophy and numerous mental math questions. I would urge you to read that if you have time. The questions represented a wide variety of mental math situations and math content. If time does not permit you to do much mental math, with students or children, but you want to do something rather than nothing, then "halfway" numbers might be the solution. I thought of this by accident but it proved to be a very good concept and a source of good mental computation opportunities. It is something that can be tailored to almost any age group and/or ability. A teacher or parent that does just one or two of these a day can help their students develop some confidence and ability in mental math. Below is an explanation of the concept and techniques.

HALFWAY NUMBERS

One class period when I was doing halfway numbers with my students, one of them, named Gary, used a different method. He took half of each number and then added those together. He did get the halfway number. In my example above, that would mean taking half of 48 (24) and half of 64 (32). Then add 24 and 32. You get, of course, 56. I have to admit that at first my reaction was that it was a lucky coincidence. After school that day I looked at it algebraically, and it turns out that "The Gary Method" does work all of the time. You just have to show that a/2 + b/2 = (a+b)/2. As you can see in this example, "The Gary Method" might be the easiest way to find a halfway number between 48 and 64.

Finally, one day, when I was explaining "The Difference Method" I used a number line diagram. I drew a number line putting a tick mark on the left labeled 48 and a tick mark on the right, labeled 64. I put a tick mark halfway between 48 and 64 and said we needed to find what that number was. I drew a little arc from 48 toward 64. The arc represented a jump of two to 50. Then I drew a similar arc from 64 to 62. So now the halfway number is halfway between 50 and 62. Some students can see right then that the halfway number has to be 56. If others don't see that you can continue moving toward the halfway number equal amounts until the halfway number is obvious. I ended up calling this "The Number Line Method." They do need to be able to do it in their head, of course. Some students really like this method. They "see" what is going on a little better.

Give them some easy examples at first and let them try to find the answer their own way. Students will naturally gravitate to one or two of the above methods. You can gradually introduce the other methods and then show how each method could be the best way to do certain halfway number questions. Who knows? One of your students may discover an altogether different method that also works.

By the way, when students are first asked to do any type of mental math question, they will tend to try to do the paper and pencil method in their head. I do everything I can to discourage that. Pencil and paper methods are good when you use pencil and paper. When doing mental math we want to use strategies that are appropriate for mental math. Thinking this way and using the strategies of mental math can be very empowering to students who previously assumed that they were not good at math. They may realize that they are good at math when they are taught and allowed to use their brain in a natural way.

Anonymous said...

Anonymous said...

Picture!!!!!!!!!!!!!!

what numbrr is half way beetween 12 and 56

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