Engaging students: Finding least common multiples

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Theresa (Tress) Kringen. Her topic, from Pre-Algebra: finding least common multiples.

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What interesting word problems using this topic can your students do now?

While having students working on finding the least common multiples I could engage them by having them solve some word problems that would bring up real world problems in a way that they can relate what they learned to problems that deal more than with just numbers. One problem that could be presented ot the students is the following:

If you’re given packages of notebooks that contain 6 each and you are required to repackage them to send them to a school in need in groups of 22, what it the least amount of groups and original packages of notebooks that you can get without any notebooks left over?

In this problem, the students would be required to find the least common multiple of both 6 and 21. Since six doesn’t not go into 22 without a remainder, they would have to find lcm(6,22). Since the least common multiple of both 6 and 22 is 66, the students would have to apply what they know about least common multiples of numbers to figure out the word problem.

To continue with this, the students could then be asked to do the same thing for three numbers.

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How does this topic extend what your students should have learned in previous courses?

Students should have covered factors and multiples of numbers around fifth grade. Therefore finding the least common multiple of a number extends the topic from these previous topics. Since students can figure out the factors of a number, they should also know if one number is a factor of the second. If it is, then they will know that the second number is the least common multiple of the two given numbers. Say the students are given 3 and 9. The students should be able to tell right away that 3 goes into 9. Since 3×3=9 and 9×1=9 and since no number smaller than 9 can also be a multiple of nine, the least common multiple of 3 and 9 is 9.

When also looking at the least common multiples of a number, students know what multiples of a number are from previous courses. They will know that 18 is a multiple of nine as well as 27, 36, and 45. Students know that 3 times 9 is 27, but they will also know that since the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30, etc. they will also know that even though 3 times 9 is 27, that there is a number smaller than 27 that is also a common multiple of 3 and 9.

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How can technology be used to effectively engage students with this topic?

Students like games and it’s even better for the teacher if they are able to play while they learn or practice a given subject that they have learned. In order to engage each student, there a number of online games students can play to help them practice finding the least common multiples of given numbers. I have found a number of online games that students could go to for an activity. It pushes them, allows the students to go at their own pace, and allows students to be less worried about how fast or slow they are compared to other students.

One game is a timed game that gives the students two numbers to find the least common multiple of. They are given two minutes to see how many they can compute in that amount of time. They are still permitted to go at their own pace, but they are also pushing themselves to do better than the time before.


 A second game give the students two numbers and asks for the least common multiple. It is basically multiple choice since they are to select a number our of five or six different numbers. If they select the correct answer, they are permitted to “throw a snowball.” Each correct response helps them win the snowball fight.



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