Pondering the GCF & LCM

Some ideas are a little bit like raindrops that have splashed onto my puppy in a summer rainstorm.    They land lightly on her but are quickly shed as she shakes her body in a happy dance.   They touch her for only a moment and are gone.   Some ideas, though are more like a piece of sticky gum on the sidewalk on a hot summer day.    They seem to be permanently attached.    For some reason, that is how Mark’s post on exit cards has been for me these last few weeks.   I keep pondering what exactly I should want to know about my students.

This question has been dancing around in my head as I have been planning the first unit for my sixth graders.   This particular unit incorporates the ideas of the Greatest Common Factor and the Least Common Multiple (among other things).   For some reason, a lot of students seem to lump the two ideas into a single muddy mess.   I think part of the problem is the language – the words are so similar and yet so different (greatest/least, factor/multiple).    Even parsing out just the factor/multiple piece is tricky.   So many students confuse the two words.    They have trouble making them their own.  I also think part of the problem is that students have some measure of difficulty determining just what a Greatest Common Factor or a Least Common Multiple is.   I don’t mean the question of how you find them, I mean the question of exactly what are they and why should anyone care.

As we explore this larger question in the unit, we will do so in a problem context using Connected Math.

  • We’ll start with a scenario in which two siblings are going to ride two different Ferris Wheels, one larger and one smaller, at a carnival.   Given the amount of time for a single revolution for each Ferris Wheel, students will consider how long it will take for the two siblings to both be at the bottom of the revolution simultaneously and how many rotations each wheel will have made.    I usually see a fairly wide range of strategies as students tackle the question.    Some students will start making a list of the times for each sibling to return to the bottom and then find the common time in each list.   Some students will use this strategy but will organize the information in a table (which makes it so much easier to make sense of the information).   Some students will really struggle to make sense of it and we’ll need to talk about what kind of strategies they might use to help make sense of the situation.   Eventually, they try something like drawing a picture or modeling the scenario using some different colored cubes.    A few will immediately see that they need the LCM.
  • Next, we’ll move onto an exploration of the life of two different cicada species, one of which emerges every 13 years and one of which emerges every 17 years.   Students will explore the question of how often both species emerge simultaneously.    Generally, the thinking will be similar to that in the Ferris Wheel problem but it will come together a little more readily because of the discussion that ensued after the Ferris Wheel problem.
  • Our next exploration will consider a scenario in which a girl is planning snacks for an upcoming hike.    She has a specified number of apples and a specified number of bags of trail mix and needs to determine how many snack bags she can make if each snack has the same amount of food and there is no food left over.    Initially, some students want to apply that same hammer – they want to find the LCM because that worked on the last two problems so it must be the way to go on this one as well.   Then, they get an answer that doesn’t make sense – how can you have more snacks than you do apples?   I like when this happens because it makes them step back and try to actually make sense of the problem – to realize that sense-making is what math is really all about.   Some students will reach for some blocks, using one color for apples and another for trail mix.   After they play around with it for a few minutes, they usually realize that they are dividing into groups and move forward without actually modeling the whole problem.   Some students will do the same kind of modeling just sketching or tallying and reach the same end result.   A few students will just seem to see it and will find the GCF.    I always try to bring out the different strategies as we debrief the solution and ask a lot of questions about why they chose that strategy and what that strategy showed them.   We follow that up with a scenario in which the canary ate some of the trail mix and the numbers suddenly get messier.   The idea remains the same, though.
  • Finally, we consider a scenario in which someone has donated some cans of juice and some crackers to the school for snacks on a field trip.    Students need to find out how many kids can be served by the donation if each kid is going to get the same amount.   This problem seems to come together pretty quickly since it is the second GCF problem and follows a pretty solid debrief of the first problem.

Following this sequence, I expect some of the students will have made sense of the concepts of GCF and LCM, but some of them will not have.    Some of them will still struggle on the next novel scenario to figure out how to solve it.   I am wondering if my question (exit ticket) needs to be something like this.

What kind of problem requires you to find the LCM?   What kind of problem requires you to find the GCF?

I want to know if they have been able to generalize the concept.    Do they realize that LCM problems are really problems of alignment and that GCF problems are really problems of grouping (dividing)?   This is a shift for me.   In years past, my exit card would have been another problem and I would have evaluated their mastery based on their ability to solve that problem.   With this question, I am asking them to think on a different level.   I want them to consider what is the same about LCM problems, what is the same about GCF problems and how are the two different.

I’m not sure if this will be my exit card question, yet.    I have a little more time to ponder.   Am I asking the right question?

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