NOTE: This article is a follow-up to the earlier one, titled The M.O.M. Game. It is recommended that the first one be mastered before using this one.
After students have become skilled in determining the LCM of two or more numbers via the calculator, they should be given the opportunity to use this ability for higher level thinking on more advanced situations. Therefore, this article will describe an activity that can be used to achieve this aim. Again teamwork and cooperation should be stressed.
First, instruct the students to set up a "t-chart", as shown here:
numbers | match ------------|-------- 3 and 4 | | | | |
They should then find the "match" (LCM) for 3 and 4 by the method of the first lesson. It is 12, of course; it should be entered into the chart in the second column.
Little by little, one pair at a time the remaining pairs are presented to the students. After a while, the following t-chart will result:
numbers | match
3 and 4 | 12
4 and 5 | 20
5 and 6 | 30
6 and 7 | 42
7 and 8 | 56
8 and 9 | 72
(If the teacher deems it desireable, a few more pairs may be offered. This is a flexible point.)
By now, the goal of the activity should begin to become apparent, at least to those individuals who have been trained and encouraged to look for patterns in their math work. Since so few have been so trained, especially at lower grade levels, say 4th grade or so, the teacher may have to direct the students' attention to the pattern. Here, in this chart the "big idea" is:
Or in other words: "the LCM of two consecutive counting numbers is their product." This means, in practical terms, all the long work was really not necessary after all. But one should not overlook the value of the longer method. It presents the true conceptual meaning behind the term "LCM", which is the object of our quest, namely to find the lowest (or least) of the common multiples of the respective numbers, by actually observing the multiples of those numbers.
Once the students have discussed this pattern, they should be directed to move on to Problem #2. It is shown in its final form below, but of course should produced in a manner similar to the first one.
numbers | match
3 and 5 | 15
5 and 7 | 35
7 and 9 | 63
9 and 11 | 99
11 and 13 | 143
To be observed or pointed out is that the numbers used this time are consecutive odds. And, perhaps surprisingly (to some students, at least) the match is again the product of the two numbers being considered. The conclusion should be stated in words such as these:
At this point, students should be getting the idea that it's pretty easy to find the LCM of two numbers: just multiply them and presto!, you have it. So it is time to present a situation where such is not the case. And you don't have to look far to do it. Problem #3 considers consecutive even numbers.
In order to motivate the search for the truth in this situation, a change to our t-chart is required -- a third column is necessary.
numbers | product | match
4 and 6 | 24 | 12
6 and 8 | 48 | 24
8 and 10 | 80 | 40
10 and 12 | 120 | 60
12 and 14 | 168 | 84
One way to proceed is to ask the students to enter the products of the numbers, and then return to the old way to find the LCM. It should produce a bit of a shock that "now the product is NOT the same as the LCM/match. But then, just what is going on here? Obviously, the LCM is now half the product. Similar and related though it may be, this problem is not the same as the first two. And therein lies the point: before one draws a conclusion about a pattern, one should test and probe the case thoroughly, double checking things as much as possible along the way.
[Of course, for those who know a little bit more about number theory, the reason for the different result in #3 was that consecutive even numbers, or any two even numbers, are not relatively prime, whereas the number pairs in the first two problems were.]
To evaluate the learning that the teacher hope is occuring in this work, the following problem could be given:
Here is the t-chart:
numbers | product | match
6 and 9 | |
9 and 12 | |
12 and 15 | |
15 and 18 | |
18 and 21 | |
What happens now?
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