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

Halfway numbers present some very good mental math opportunities for middle school students. Halfway numbers are numbers that are halfway between two other numbers. For example, students could be asked to find the number halfway between 48 and 64. When I first presented this type of question to my students, I had in mind two distinct methods that could be used. Obviously the halfway number would be the average of 48 and 64. (I call this "The Average Method".) So the student would have to add 48 and 64 in their head. A good mental math strategy for that could be to add 40 and 60 (100) and then add 8 and 4 (12) and then add 100 and 12 getting 112. That is called "Front End Addition". (There are other good mental math strategies for adding 48 and 64 but that is not my focus here.) 112 must then be divided by 2. A good mental math strategy is to "distribute" the division. 100/2 is 50. 12/2 is 6. 50 +6=56. Therefore the number halfway between 48 and 64 is 56. The other common strategy for halfway numbers I call "The Difference Method." The (positive) difference between 48 and 64 is found first. The student could mentally subtract 48 from 64 by first subtracting 40 from 64 getting 24 and then subtracting the remaining 8. 24-8 could be done by 24-10+2 (subtracting 8 is the same as subtracting 10 and adding 2.) 24-10+2=14+2=16. So the difference between 64 and 48 is 16. The halfway number is half of that difference (half of 16 is 8) away from 48 and from 64. So the student could add 8 to 48 or subtract 8 from 64. Either way you end up at the halfway number of 56. (By the way, although it is not my focus here, I did want to mention another method for finding the difference between 48 and 64. In general, one of the best ways to subtract in your head is to figure out what you have to add to the smaller number to get to the larger number. In this situation it would be natural to think that you would add 2 to 48 to get to 50 and then add 14 more to 50 to get 64. Thus in total you would add 2+14 or 16 to 48 to get 64 and therefore the difference between 48 and 64 is 16. The concept of “adding on” is a great way for people to do mental math subtraction.)

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.