Author: Jon Orr

Hour Glass Multiples

by Jon Orr

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. 

DOWNLOAD THE LESSON FILES

 

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!

DOWNLOAD GUIDE

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.

Promote Struggle – A Hero’s Journey in Math Class

by Jon Orr

How many times have I seen a student give up before they even start an unfamiliar problem in my class? A lot! It happens way too much. How can we build resilience and determination in our students? One thing we can do is to let them experience unfamiliar problems regularly and help them struggle through the process of working on a solution.

Let me share with you how the Hero’s Journey story arc can help with learning productive struggle in math class.

While in Miami for the Apple Distinguished Educators Institute we saw a speaker from Pixar Randy Nelson discuss the aspects of Story. More specifically he spoke about the Hero’s Journey. That talk really hit home for me. Below is how I interpreted his message and how it relates to my classroom.

A Hero’s Journey

All of these characters take a hero’s journey….

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Since I’m a math teacher describing the Hero’s Journey is best done with……a graph (English teachers will know it’s shown as a cycle).

On a time vs. Tension graph the Hero’s Journey looks like this: Time is the length of the journey….or story. The tension is felt by the audience. 

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In the beginning the hero is introduced, the main conflict is introduced, his/her world starts to change. As the story continues the hero must battle the forces of evil & go through struggle. They must experience conflict. It’s the conflict that the hero learns about themselves. They learn their strengths and weaknesses. It’s the struggle that makes the ending awesome. Its the struggle that make the hero see the solution. It’s the lessons they’ve learned in the struggle that let’s them go aha! I know what I need to do! The story would mean nothing to the hero and the audience if the climax was much earlier in the timeline. As the story ends the character returns to a NEW normal. They take their learning and come out stronger on the other side. 

This curve we see above is nothing new to us. This curve is what learners go through. It’s a Learner’s Journey too.

Now, if we take a look at our traditional math classrooms we have a format much like this:

Photo credit: Kyle Pearce

Photo credit: Kyle Pearce

Let’s look at that structure on the Time Tension graph. 

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After we take up homework, we introduce the new lesson or topic or problem to work on. It’s unfamiliar so tension in our students starts to increase.  But what happens is that as the tension rises it immediately falls back down. And my good buddy Kyle Pearce mentioned to me that the tension doesn’t fall all the way back to the axis….a good number of our students feel that tension permanently. 

Why does the tension fall immediately?

We make that happen. We relieve students of their pain by immediately telling them HOW to solve the problem.

Screen Shot 2015-11-17 at 9.59.24 PMIt’s Our examples & solutions. Students don’t get a chance to struggle & discover, Therefore the math formula, strategy or algorithm means nothing to them! The memorizers will memorize and do ok, and the non-memorizers lose again. The ideas and strategies have no real value to them. 

I think students should feel the need for the math they learn. They should experience struggle ….just like the hero.Screen Shot 2015-11-17 at 10.20.30 PM

Let’s take the old model of our lessons and transform it to match the Hero’s Journey. It’s the struggle that adds value to their learning. Let’s move the reveal of math rules etc farther in the timeline. Let’s let the students productively struggle through math problems. The reveal of the “math” will mean so much more after students see and/or feel the need for it. 

Screen Shot 2015-11-17 at 9.59.55 PM

DOWNLOAD THE BUILDING RESILIENT PROBLEMS SOLVERS GUIDE

Download the 3-page printable guide that will give you 3 actionable tips to build resilient problem solvers in your math classroom. 

DOWNLOAD THE GUIDE

An example in my class this week came when I wanted to teach students how to determine an equation of a quadratic function when given some key points.

I gave them this simple Desmos Activity Builder slide from Match My Parabola

Screen Shot 2015-11-17 at 10.27.49 PM

Students already knew about vertex form of a quadratic function so I knew they could put in most of this equation. It’s the “a” value that they really didn’t know how to get efficiently. So I saw a lot of this…

Screen Shot 2015-11-17 at 10.28.41 PM

Students used trial and error to find -1/4 as the right “a” value. But we then asked “How do we know that’s the right one?” We then discussed plugging in a point to check to see if the right side equals the left side. They had a few more slides just like this but with different points. By the end of the last slide you could see that they really wanted a more efficient way of determining the “a” value than guessing and checking. This is where I stepped in and we discussed the idea of using one of the points and the equation to solve for the “a” value. Everyone was on board! They all had struggled before we discovered an efficient strategy. They all wanted it. If I had started class by showing them the first slide and then just telling them how to do it, I would see lack of understanding of why and bored faces.

It’s the struggle that makes the math worth it! Let’s let our students be Heroes. How are you promoting struggle in your classroom? I would love to hear of your ways. Leave a comment below.

Click here to grab the Desmos Activity Builder Activity I showed above.

The Hero’s Journey & Pentomino Puzzles

To help you wrap your mind around the Hero’s Journey as a lesson model I’ve created a Hero’s Journey Lesson Template. The exercise is to choose a lesson you have coming up in your class. How can you modify that lesson so that the flow follows a hero’s journey? Use the template below to help plan your lesson out.

Exemplar: I used the template to model how I use the Pentomino Puzzles activity to teach solving linear equations.

You can see that we slowly build up the need for a helpful efficient strategy to solve the puzzles. When my students have struggled and persevered 3 or 4 times to solve a tough puzzle, the timing is now perfect for us to step in and help them develop that skill of solving equations.

Download your copy of the Hero’s Journey Lesson Template.

Want to dive deeper into learning how to teach through the Hero’s Journey? Dive into our self-paced online math educator pd course.

How We Can Avoid a Major Lesson Planning Misconception

by Jon Orr

One common misconception around how we should plan our lessons is that planning and creating lessons, course outlines, and assessments is all done in isolation.

There’s an iconic image of famous Fiction authors shutting themselves up in a cabin in the woods for months at a time and then emerge with this great manuscript.

This is actually a false image.

Most authors go through intense iterations of their books with many editors and audiences that provide feedback.

You many have this image that math lesson creators also lock themselves up in the teacher prep room to think up great lesson ideas only to miraculously emerge with perfect lessons. Or maybe you believe that we have magically created spiralled course outlines all by ourselves with little input from anyone else.

These things can’t be further from the truth. Every one of the lessons shared on this site and also any unit or course plans were all created in consultation with other teachers.

In fact, when Kyle Pearce and I first decided to change our course plans from the traditional textbook order to mixing up topics so we can maximize student retention through spiralling our math lessons, we created a joint outline with Google Sheets that we could each have input to. Planning lessons and courses should be collaborative effort.

In January 2018 I asked the twitter community “Your colleague is thinking of trying to teach through spiralling the curriculum. What are some SMALL changes they can make NOW so that’s it’s not overwhelming?”

Many teachers gave their suggestions but one comment really stuck with me, It was from Mary Bourassa,

She said,

“Lots of great replies but I would argue that most are not small changes. Switching to spiralling is a big change! My best advice is to plan a meeting with someone who has spiralled so that you can talk through your plan together. And make sure you know the curriculum really well.”

We need other people on our same teaching journey as we learn to create new lessons that meet our students need.

A book I highly recommend reading because it’s interesting with many great real-life stories and examples is The Creative Curve, How to develop the right idea at the right time. By Allan Gannett.

The main idea of this book is, and quoting from the publisher,

“We have been spoon-fed the notion that creativity is the province of genius — of those favoured, brilliant few whose moments of insight arrive in unpredictable flashes of divine inspiration.  And if we are not a genius, we might as well pack it in and give up. Either we have that gift, or we don’t. But Allen shows that simply isn’t true. Recent research has shown that there is a predictable science behind achieving commercial success in any creative endeavour, from writing a popular novel to starting up a successful company to creating an effective marketing campaign.”

One of Gannett’s Laws of creativity is the law of creative communities. He argues that creatives leaders like Paul McCartney, Steve Jobs, and  J.K. Rowling, didn’t create their great works in isolation, but were surrounded by a community of people. Gannett’s also argues that if you don’t have a community of supporting people around you then your chances for creating something is drastically reduced.

So, if you want to make math moments that matter for your students on a regular basis then you will need a community of supportive people!

Alex Overwijk is a high school math teacher in Ottawa Ontario Canada. What I admire so much about Al, is that after teaching math the “traditional way” for over 25 years he realized that he had been robbing his students of great thinking and made significant changes in his classroom routines with an emphasis on “Uncovering curriculum instead of covering curriculum”.

Al has written on his blog slamdunkmath.blogspot.com about Lesson study — a collaborative lesson design structure — that has led him to create many active great thinking lessons for his students.

Basically, lesson study in a nutshell is a group of educators, teachers, and administrators who will together plan a lesson for a teacher to deliver. They will all observe to witness how the students respond to the questioning and tasks included in the lesson, then they debrief to make changes. Then this process repeats. The group will plan, observe, and debrief for another teacher, and so on.

The group is planning lessons collaboratively, not in solitary isolation. The success/ or failure of the lesson is felt by the whole group and not just from the teacher delivery it.

When responding to teachers who say “I can’t afford to be out of my classroom that many times”…. Al says, “How can you not afford it? Your classroom will become a different place-a place you’re not familiar with. Your instructional practices will be challenged and will probably change as a result. Your belief in what students can do will change. You need to try this!”

Al and so many other teachers know that the success of great lessons and course plans can hinge on your access to a community.

What can you do? —- Find one or two teachers who also want to plan, talk ideas through, and collaborate on lessons or course designs. Please. Don’t do this alone. We need to avoid isolation. Sharing ideas, strategies and resources is how plans not only get created but how we stick to them.

Your next step to avoid Teacher Isolation → Join our closed Facebook group: Math Teaching & Learning K-12.  It’s closed so that you can feel comfortable asking math lesson related questions on Facebook without bothering your Aunt or your college friends with math related stuff. It’s a place just for us! It’s a place where if you’re feeling teacher isolation in your school come here and share your question or even just to vent. 

For example, a group member asked the following question….and other group members jumped in to help out.

Or here’s another example of a team effort

So, I’m hoping to see you in group! Remember, don’t do this alone! We can create better things together. 

Pumpkin Time-Bomb Activity

by Jon Orr

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.

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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