Creating Math Moments: How we can transform typical textbook problems into moments that matter.

In an ongoing effort to demonstrate that we can apply the 3 part framework of Spark Curiosity, Fuel Sense Making, Igniting Teacher Moves to any lesson in math class we’ll tackle the common problem of finding the equation of a line between two points. Like this basic problem:

Sometimes textbooks may even jazz it up a bit to give it some context like this one.

Let’s re make this lesson to fall under our 3-part framework.

Even though this is a grade 9 and 10 expectation here in Ontario you’ll find that this problem is quite accessible for many grades.

In particular you could use it to uncover:

  • Find the slope (rate of change) of a line between two points;
  • Find the equation (rate of change) of a line between two points;
  • Model real-life relationships involving constant rates;
  • Model linear relationships using tables of values, graphs, and equations.

Spark Curiosity

We’ve been arguing that instead of finding a context that will make students interested we should follow the curiosity path which we’ve described in Lessons One and Two from our 4 part video series instead.

Let’s consider the big idea here: We want students to build an algebraic representation of a linear relation using only two values.

To withhold information and build anticipation we will strip all the numbers and questions and ease into the lesson. To help create the classroom culture that values student voice, student thinking, and growth we’ll ask students to fill in the two blanks here:

Setting the floor low will help our students feel attached to the math problem that is coming. The more attached and invested they will feel the more internal motivation they will have to pursue the problem to the end.

In behavioural economics there is a theory known as The Sunk Cost Fallacy.

Or also known as Escalation of commitment.

From Wikipedia,

Escalation of commitment is a human behavior pattern in which an individual or group facing increasingly negative outcomes from some decision, action, or investment nevertheless continues the same behavior rather than alter course. The actor maintains behaviors that are irrational, but align with previous decisions and actions. 

You see, we humans are inclined to avoid loss. We will continue with a project or line of thinking if we feel that if we abandoned it we would incur loss. Even if that abandonment was better for us.

For example, my first car, a 1993 Ford Escort – you know, this is the car that had the automatic seatbelts. When you sit down and turn the car on the seat belt came up and automatically moved over your shoulder. One winter the heater in the car stopped working and I paid over $1000 to have it fixed. Then not long after something else broke on the car and instead of saying enough with this car I said, “Well, I just paid $1000 to fix it if I don’t fix it now then it’s like my $1000 was wasted.” This bias I just exhibited is an example of the sunk cost fallacy. I wanted to throw bad money after good. The $1000 I previously spent was a sunk cost and there’s no way I could get that back so the $1000 shouldn’t play a roll in my new decision to fix the car. I should decide to fix the car or not fix the car without letting that $1000 affect this decision.

The sunk cost fallacy makes us feel that if we invest time, money, resources into a project or decision that we should keep going with that project or decision so we avoid loss. The escalation of our commitment keeps us in the game.

In math education we can use our students own tendencies of avoiding loss for their own good. By setting a low floor in activities, we are easing our students into those activities and lessons so that it will be harder for them to just quit and give up once they are deeply invested in the activity . They won’t want to feel that what they’ve done so far in the activity was a waste of time and resources. They’ve sunk a cost into the activity and will continue with it to avoid loss. You can read another application in education of the Sunk Cost Fallacy from Robert Kaplinsky.

Your students will fill in various items and values for this problem. In my class this was a fun moment as we shared out what they wanted to buy and for how much.

Fuel Sense Making by Revealing Information

So now we’ll move down the Curiosity Path and narrow the focus to give a little more information.

How much would 12 shirts cost?

Students can make quick predictions before revealing the information slowly…

We don’t want to waste all the work we’ve done on escalating our students commitment so we’ll move down the curiosity path a little bit more and avoid the rushing to the algorithm. Students will use the given information and their prior knowledge to build a strategy to solving this problem.

Fuelling Sense Making by Anticipating

We are strong believers and practitioners in the PDF or the book 5 Practices For Orchestrating Productive Mathematics Discussions. So in preparation for this lesson we used our Anticipation, Selection, and Sequencing template to brainstorm possible solutions and strategies our students will try.

You can grab a blank copy of this template here.

To maximize your mathematical discussions you may want to sequence the strategies from most common to least common.

For example;

You can expect many students to try to find a unit rate to solve this problem. This is quite natural! It makes sense to find the price per shirt. However, not all situations are directly proportional. We can ask our students: How do we know this is a direct proportional relationship?

When students find the unit rate for 12 shirts at $122 and then again for 24 shirts at $209 they will see that it doesn’t cost the same per shirt! WHAT!?

Something else is going on here. You may want to give a small hint here asking, “hmmm, If 12 shirts cost $122 does 24 shirts – which is double the amount of shirts cost $122 x 2? How much more does 24 shirts cost? What would 36 shirts cost?

Students who noticed this right away may draw a double number line to show the changing prices and eventually determine the cost per shirt.

Students who have found the cost per shirt will still notice that simply multiplying the cost per shirt by the number of shirts STILL doesn’t get the cost — there is some other value that consistently needs to be added – The initial value or fixed cost.

Have a discussion at what this fixed cost could be — shipping charges? Overhead costs? Printing rental fee? ect. With this new calculation rule students can move on to verify that it does indeed work with 200 shirts, and then finally find the cost of 1100 shirts. You may even want to steer your discussion towards finding an algebraic representation of this relation.

Some students may represent this pattern as a table instead of a double number line. Depending on your grade level you may also want to use the word slope to represent the cost per shirt. If you see this solution from your student you’ll want to push for an algebraic representation

You may see some students turning toward Desmos and graphing the points to find an algebraic representation. We definitely anticipated this having taught this lesson in a grade 10 applied class.

The order you present these strategies/solutions will depend on your lesson goal. If you are trying to achieve the goal from the top of this post (Finding an equation of line between two points) then you most likely will want to end with finding the algebraic representation and then showing how you can use Desmos to verify that representation.

Finally we can show students that if the relation is linear, we really only need two points.

We feel that if we can take this particular learning goal and modify the delivery and teacher moves to create a math moment that matters we can do this with any textbook problem. What lesson should we make over next?

If you haven’t checked out our 4-part video series yet get over there now!

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. 

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.

Promote Struggle – A Hero’s Journey in Math Class

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….

Screen Shot 2015-11-17 at 9.54.28 PM

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. 

Screen Shot 2015-11-17 at 9.55.06 PM

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. 

Screen Shot 2015-11-17 at 10.15.32 PM

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 change 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 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. 

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.

 

 

How We Can Avoid a Major Lesson Planning Misconception.

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, 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 stepJoin 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!

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!