Sparking Curiosity & Fuelling Sense-Making with the Least Common Multiple.
In this 3-Act Task students will be presented with a puzzling video of 3 “hour glass” sand timers. They’ll solve a brain-teaser like problem while ultimately learning about common multiples and the least common multiple (LCM).
In this particular, this task can be used for
- estimation;
- spacial sense;
- volume;
- counting in multiples;
- least common multiple;
Act 1: Sparking Curiosity
Ask students to create a notice/wonder table or you can use one that Kyle Pearce and I built for our online workshop Making Math Moments That Matter.
Ask your students to write down anything they notice and anything they wonder while viewing this video:
Then have them share with elbow partners and then finally with the entire class.
Some possible notices and wonders:
- I see three different colour timers.
- Is that sand?
- Whose house is that?
- Are they timing the same amount?
- What times will they time?
- Will all three timers ever end at the same time? If so, when?
- Is the timer in minutes?
- I think the yellow timer times for 3 minutes.
After capturing all the notice and wonders on the front board steer the class to working on the problem
“Will all three timers ever run out of sand at the same time? If so, when? If never, why not?”
Assume that we will keep turning over a timer after the sand runs out.
Take a few minutes to have your students estimate when the timers will all run out at the same time –> “Predict with reasoning”.
Act 2: Reveal Information to Fuel Sense-Making.
To avoid rushing to the algorithm push down the curiosity path some more. Instead of just handing over all the necessary information to solve a problem ask the students what they want to know more about. For example student 1 might say “I’d like to know the times of all the timers”. As a teacher your next question should be: “I see, and if I gave you that information what would you do with it?” We can learn what our students understand and are thinking with their response to one prompt. By asking them to anticipate what they need forces them to develop a problem solving strategy.
After hearing a few students out, give them this information: But make them guess first. What time does each timer time?
Reveal the timers:
After this reveal send students to their vertical spaces to explore the strategies they began in the anticipation stage to determine when the timers will run out of sand at the exact same time.
Strategies you may see:
- Drawings that show how much time is left every time one timer runs out.
- lists of the multiples of 2, 3, and 5.
- tables that track minute by minute.
Fuel Sense-Making to Consolidate Learning.
Depending on your grade range and student ability you’ll want to frame your consolidation so showcase your target learning goal.
I’m sure most learning goals will include a triple number line showing how multiples of 2,3, and 5 overlap.
Clearly show using the lines how the 2 and 3 minute timer will be turned over at the same time at the 6 minute mark. Then show them all the common multiples between 2 and 3.
Finally bring in the multiples of 5 to the mix.
As part of your consolidation show this video which overlays the common multiples as they occur in the reveal video. Students can clearly see that when the timers are turned over at the same time we have a common multiple.
Here is a reveal video without the number line overlay.
Try this lesson out in your class and report back here in the comments to tell us how it went.
Are you new to 3-Act Math problems? Grab our guide to running these problems in your classroom. Learn tips, suggestions, and avoid common mistakes of using these types of tasks.
Acknowledgements.
I want to thank Michael Jacobs for turning my thinking towards thinking about the least common multiple. The creation story of the above task comes from this hour glass timer I bought from David’s Tea
I just bought this triple “hour” glass timer! I think we can pull some math from it. What do you wonder? What do you want to know more about? Let’s build some math lessons together. What can we do with this? #iteachmath #mtbos #mathchat pic.twitter.com/6FCz8ivpsO
— Jon Orr (@MrOrr_geek) November 6, 2018
Mike said,
if you flip it over every time the sand runs out from one timer, how many times will you flip it over before all the timers are empty at the same time?
— Michael Jacobs (@msbjacobs) November 6, 2018
Bryan also was thinking it was screaming LCM.
Doesn’t this scream LCM to you??🤩
— Bryan Penfound 🌐 (@BryanPenfound) November 7, 2018
Which made me start thinking about how that couldn’t work with all three timers attached. So I set off to buy some new timers. I found the ones you see in the problem above.
4 thoughts on “Hour Glass Multiples”
Great Lesson! I really like this use of hour glasses as a way to talk about least common multiples. I think this activity aligns perfectly with the 3-Act Task Format. The videos you included were great as well, I liked the last one how you put the number line over the video and updated it along with the video. I am thinking of all the possible extensions/adaptations of this lesson, you could do things like balls bouncing at different intervals, two people running back and forth from different places at different speeds, etc. But I think this is one of the most straightforward ways of thinking about least common factors in the real world. Thanks for the great idea for a lesson that I will definitely try out in the future!
I tried this out with my fifth graders. They were very intrigued by Act 1, but no one came up with the question this lesson focused on. It’s a great application of least common multiples, but if this were on http://www.101qs.com, I wonder what the most common question would be.
Adina,
Good question? What we’re those first wonders your students had?
The most mathy questions I got were, “Why is one faster than the other?” , “Which one is the fastest?”, “I wonder if one has more sand in it than the others”, and “Maybe one has a bigger hole to let the sand through it so it’s going faster.”
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