Traditional vs. Friendly Number Algorithm-A False Dichotomy

Let me start off this post by saying that I do not think that this is necessary for all students but it may be necessary for some. After seeing a couple of students who still cannot not carry out the traditional division algorithm (including writing down the acronym D.M.S.B.-divide, multiply, subtract, bring down), I feel it is necessary to present to these students a different way of looking at it. Now I believe that these students are beyond the repeated subtraction phase of division but have yet to grasp the underpinnings of the traditional algorithm (which is suppose to be faster).

So let us take a look at the traditional algorithm, using the problem 3672 divided by 8.

As you can see the first thing we would do is to see how many times 8 goes into 3 (which is really 3,000) and it doesn’t so you move to the next digit and figure out how many time 8 goes into 36. We determine 4 times, which is 32. You then subtract 32 from 36 and are left with 4. We are then taught to bring down the next digit (the B in “D. M. S. B.”) which is a 7. We then begin the process again-determine how many times 8 goes into 47, which is 5, 8 times 5 is 40, and 47-40 is 7. We bring down the last digit, 2, and repeat the process one more time, which leaves us with no remainder. This is fine as long as the student understands what is going on.

For example, I would want them to realize that when we are talking about 36 divided by 8 we are really talking about 3,600 and the choice of 4 is 400. When we bring down the next digit, that is what it is not a number. According to Merriam-Webster OnLine a digit is; “a: any of the Arabic numerals 1 to 9 and usually the symbol 0 b: one of the elements that combine to form numbers in a system other than the decimal system” and a number is; “ a sum of units : total c (1): a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication.” For more information on digits and numerals you can go to Wikipedia.

Let’s take a look at the friendly number method. As you can see this method helps build on the strengths students already posses (rounding, multiplying by powers of 10, basic facts). It also looks similar to the traditional algorithm which would make the transition from the friendly number method to the traditional algorithm easier.

As point of emphasis, when working with struggling students, or using this for the first time, I would begin with 10 instead of 100. Students are usually strong with the 10-facts and they could then see how cumbersome and time-consuming it would be to use 10 over and over again. You could move them from 10 to 100, using their knowledge of powers of 10, then to any basic fact like in the second step (300 x 8 because they know 300 is 3 x 100 and therefore they can multiply 3 x 8 and 100 x 8).

As they become more comfortable and use this method they will begin to see the connection between the rounding and the numbers they choose to solve. At first they will use the rounding to see if their answer is “in the ballpark.” You can get them to see that they can use the estimate to help them right away. By that I mean that instead of starting with 100, they could choose 400 because they already know that 500 x 8 is 4,000 (inverse relationship between multiplication and division). They would therefore be as efficient as they can be and only need three steps to solve the problem as opposed to the 4 (above) or more.

Once they have became efficient and effective with this method you could transition them the friendly number to the traditional, if needed. The students can also compare and contrast the two methods to discuss the similarities and differences.

As you can see when dividing 36 by 8 you are really dividing 8 into 3,600. The same applies to the other two steps. The students can see how the 0’s have been eliminated in the traditional algorithm to further the comparison.

I think that for students who do not understand the traditional algorithm, this strategy can be beneficial.

* Division pictures written on ArtRage 2.5 Starter Edition

Advertisements

5 thoughts on “Traditional vs. Friendly Number Algorithm-A False Dichotomy

  1. I LOVE your explanation, and the fact that you care enough about your students to offer them alternatives.
    You might want to check out another way to attack division, which is presented at Easy way to do Long DIvision at MathMojo.com.
    It’s not necessarily better than the way you explained, but it might help some students.
    Keep up the good work!
    – Brian (a.k.a. Professor Homunculus at MathMojo.com )

  2. All for methods like the one above that get kids thinking about numbers and operations. The post that applauds the variety of method employed by this teacher gets a second from me. Instead of asking “what do I do?” kids are encouraged to ask how a number is like another, how different it is. These are analytical, that is, higher order thinking than following a particular regimen, which is application, a foundation thinking task (in the bloom hierarchy). Note how the former is more in the spirit of inquiry than the latter. Certainly there is plenty of room in a kid’s mind for both, but we should admit that we (curricula and the standardized tests) tend to favor the foundation thinking skills over the HOTS. So let’s hear about more methods. One more thing: it is a sign of respect and care for children when we ask them to act from their strengths rather than drill at their uncertainty. The social development of our kids ( trust building, autonomy/risk taking, initiative/inventiveness) should be kept in mind as we teach and learn, offering alternative pathways to understanding and mastery.

  3. I love the how you explained it because you make it easier for the students to understand each of the stratergies like partial quotients is kind of like the same way you do it with the friendly numbers but I really love how you explained because you explained it well and you put a lot of discription on it

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s