Hour Glass Multiples

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.

DOWNLOAD THE LESSON FILES:
VIDEOS & IMAGES

Download the lesson files so you can run bring out great moment around least common multiples. 

 

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.

New to Using 3 Act Math Tasks?

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!

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

Mike said,

Bryan also was thinking it was screaming LCM.

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.

Pumpkin Time-Bomb Activity

For the last few years  I’ve shared out a Google Form for classes to record measurements around their pumpkins and make them explode! I shared that form on Twitter so that we could crowd source as many pumpkins as we could to make the sample size large enough. I was pretty shocked at how many schools from North America took on Pumpkin Time-bomb. By the time Halloween was over the spreadsheet had over 90 entries. That’s over 90 pumpkins exploded in the name of math and data collection.

[Update] – October 2018 – The form now has over 500 entries!!

This coming week let’s add to the data and use the it in our classroom to discuss: Scatterplots, Trends, Correlation strong, weak, no-correlation, lines of best fit, correlation coefficient, etc.

Here’s a sample lesson you could use on the day you make your pumpkin explode.

SPARK Curiosity

Play this video which shows Jimmy placing rubber bands around a pumpkin.

NOTICE & WONDER

Using a notice & wonder strategy, have your students record anything they notice and anything they wonder from the video.

ESTIMATION:

Steer you class’ wonders toward the questions: How many rubber bands will make the pumpkin explode?
Have students write down a guess that is too low. Too high. Then estimate their best guess.

 

If you’re looking for your lesson goal to be around estimation then show the act 3 video next, but if you’re looking to go further and tackle a learning goal around Using scatterplots, lines of best fit, or linear regression jump down the post.

Show the Act 3 Video

 

Using Scatterplots & Trends to Improve Your Prediction. 

Alternatively, to Spark Curiosity you could use this pre-made Desmos Activity! which allow you and your class to follow a Curiosity Path.

 

WITHHOLDING INFORMATION to create ANTICIPATION: 

Use the PAUSE tool on the activity to lock their screens while you show your students the video on your main screen. Encourage your kids to discuss what they notice and wonder from the video! In pairs, I have my students TALK first and then TYPE second when collaboratively working on a Desmos activity.

ESTIMATION:

Consider pausing the screen again while you use the snapshot tool to grab student responses! This will lead into predicting how many bands will make Jimmy’s pumpkin explode. Have your students TALK first and TYPE second on screen 2 to make a prediction. Again, share students predictions using the conversations tools Desmos provides.

FUEL SENSE MAKING – IMPROVE YOUR PREDICTION: 

Bring your students down the curiosity path a little more. Ask them about how we can improve our predictions? What other information would you like to know about the pumpkin or the bands?

Have a discussion on variables & relationships. Write all the variables on the board they come up with. Narrow down the list to items that are measurable with the pumpkin we have in the class. What affects the explosion the most? Height, diameter, circumference, thickness of the wall?

Using the PACING tool in Desmos move your students few the next few screens to make a scatterplot prediction of the relationship between the diameter of a pumpkin and how many bands will make it explode.

 

Screen 5 shows a scatterplot of pumpkins that have already been blown up and the relationship between diameter and bands (or non relationship). Have your students move the orange point to a place that helps them predict the number of bands. What placement would be wrong?

The next few screens ask your students to do that all over again while looking at the relationship between the height of the pumpkin and the number of bands.

Finally, reveal the answer after students have improved upon their predictions.

Now Bring out your pumpkin for the class to see! Have them predict how many rubber bands it will take before it will explode. Repeat the estimation process. Have them save their guess till the end of class. Where will YOUR pumpkin fit on the scatterplots shown in the Desmos activity?

 

If you are not planning on using the Desmos activity then you can use the original activity post from October 2015.

FUEL SENSE MAKING – Making A Model

Throw out the question: “What about the pumpkin do you think affects how many rubber bands are used to make it explode?” Let your students brainstorm a list of variables. Have a discussion on variables & relationships. Write all the variables on the board they come up with. Narrow down the list to items that are measurable with the pumpkin we have in the class. What affects the explosion the most? Height, diameter, circumference, thickness of the wall?

Screen Shot 2015-10-24 at 6.34.42 PM

Have them choose a variable that they feel should have a relationship with the number of rubber bands. Fill out the prediction part of the handout.

Screen Shot 2015-10-24 at 4.54.15 PM

Click here to grab a copy of the prediction handout

As a class measure all variables needed. Write them on the board for all to see.

FUEL SENSE MAKING – Analyzing Data

Give students the link to the spreadsheet of all the pumpkins to date (You should copy and paste the data to your own sheet so you can filter/sort the results and share that sheet out to your students.)

Discuss with your students the lack of consistency in the selection of rubber bands from all over the country. How can we minimize this variable skewing our results? Filter the data with your students(or before hand) showing one type of rubber band (Most common is a rubber band of length 8.65 cm). This will only show all the pumpkins that have been destroyed using that type of band.
Screen Shot 2015-10-24 at 2.48.38 PM
Get your students to grab the data that relates to their relationship.

For example:
If Kristen chose the relationship Circumference vs. Rubber bands she should copy and paste the circumference column and the rubber bands column into a new sheet side by side. Then copy and paste all that data into the pre-made Desmos File.
Screen Shot 2015-10-24 at 5.14.57 PM
She can adjust the scale in Desmos as needed. Have her move the movable point and drop it where she thinks your class’ pumpkin will lie. Or you can have her find the line of best fit to help predict how many rubber bands it will take. Either way we want her to predict with more accuracy.

Screen Shot 2015-10-24 at 5.17.17 PM

So Kristen would predict that if her circumference was 90.5 cm then it will take 272 rubber bands to blow up the pumpkin!

Screen Shot 2015-10-24 at 6.28.55 PM

Now if Kristen chose a variable that it was clear there is no relationship then you get to have a discussion about correlation vs. no correlation. Have her choose new variables to predict on.

Once everyone in the class has a new prediction start wrapping bands around that pumpkin (You may want to start this as early as possible).

Watch your pumpkin explode and give congratulations to the student who predicted closest to the actual number of rubber bands.

Don’t forget to enter all your data to the sheet by filling out this form (you can also use the form to show the videos to the class).

 

[Updated] – You can use this Desmos Activity Builder Activity to facilitate the lesson. It includes only data for Diameter and Circumference.

[Updated] – You can grab a copy of the spreadsheet to save in your Google Drive. From here you can modify. 

Access the Form

Access the Data

From Oct 30. 2015

A few pumpkins from 2014 & 2015

Eye To Eye – A Similar Triangle Problem

Here’s a common similar triangles application problem that shows up in most middle and high school textbooks. A mirror is placed on the ground between two objects, showing two triangles with a bunch of measurements given and we’re supposed to find the height of one of the objects. 

A typical approach to showing how this problem is modelled with similar triangles is to walk students through a full solution. 

 

In lesson 1 of the video series that Kyle Pearce and myself have shared to make math moments that matter in your class we outline how why and how we can reshape our lessons to become more curious. If you haven’t yet watched the video series go ahead and watch video one now!

 Let’s take this similar triangle problem and remodel it so it follows a Curiosity Path so we can fuel student sense making with similar triangles. 

Recall that the first part of changing a problem to include more curiosity is to determine how you can withhold information to create anticipation. 

Here’s my attempt at doing this for our students. 

Have your students set up their page or whiteboards to record what they notice and what they wonder after watching this very short video clip. 

After discussing what students notice and wonder, bring out the wonder (if your students didn’t already) — Will they see eye to eye through the mirror?

Allow your students to analyze the video again and have them predict if they could see eye to eye. Then hit them with these three images one at a time. 

For each image, ask them to predict the answer to: Can Danielle and Dylan see eye to eye? Which image is it easy to see that the two can’t see eye to eye? Which image is harder?  Why is it easier in one image over another? Have your students draw a picture to show you why Danielle and Dylan can’t see eye to eye in the second image? To bring students down the curiosity path a little further and deepen their investment into this problem ask them to predict where Dylan SHOULD stand so that they can see eye to eye. 

What information is useful to know? Hearing your students insights at this moment is fuel for your formative assessment of their understanding and their problem solving toughness. When a student asks for the Danielle’s distance from the mirror ask “What would you do with that information if I gave it to you?”  Listen closely to the answer of that question. You will discover quite quickly who is anticipating possible strategies and the reasonableness of those strategies and who’s strategies will need some assistance. Consider giving Danielle’s distance from the mirror to help update their prediction. 

You can reveal the information as students request it. 

Now that we’ve build up student curiosity by bringing them down the curiosity path we reach the fork in the road we outlined in Video 2 and 3 of our series. We can either rush to an algorithm or we can keep following the path towards making a math moment that matters. 

In this activity we can fuel student sense making by having students experience what it’s like to see eye to eye. Students can mimic what they saw in the video to see how far a partner should stand away a mirror so the two partners will see eye to eye. 

Students will arrange themselves as shown in the activity handout, determine how far one partner must stand to see each other in the mirror, then they test that distance to see if they actually see each other! Students will collaborate, peer and self assess, be active, and engage in purposeful practice. 

Finally, students re visit the three scenarios presented at the top of the lesson to determine if Danielle and Dylan will see eye to eye. They essentially will prove if the triangles are similar or not. 

An alternate or extension problem students can work on is “Where should we place the mirror so that they do see each other eye to eye? 

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ALL LESSON FILES

Grab the handout, images, and video files for your classroom!

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Making Math Moments That Matter – Live

What makes students remember the math they are learning? Is it because you’re using a real world problem that they can relate to? Is it because maybe you used a 3-Act task? Is it because they practiced the content over and over? Is it because you used spaced practice versus massed practice? My good friend Kyle Pearce and I believe it is much more than that.

While at Oame 2018 Kyle and I took a chance and hit record on Facebook Live during our 75 minute workshop title Going Deeper with Math Moments That Matter. If you missed it or want to learn more you can watch the whole thing right here!

Session Description:

What makes a memorable math moment? Is it a real world task? Is it relevant to your students? Is it media-rich or delivered in 3 acts? While many professional development sessions focus on a specific component of an effective math lesson, Jon Orr and Kyle Pearce will model what they believe to be the three key components of an effective mathematics lesson: sparking student curiosity, fuelling their sense making and igniting your next steps. Join them as they lead a task to break each component down and then build it all back up to create a memorable math moment.

[UPDATE] – Facebook has removed our video — maybe we were too awesome?? So I’ve included three short snippets from other live workshops here:

and another,

and another,

What were your moments that you remember from math class?

What do you want your students to remember 5 years from now? Leave comments below. Or jump over to my Facebook Group and you can comment there.

Grab the Making Math Moments Matter Curious Task Template and our file with support resources over at makemathmoments.com

Thanks for being here with us!