Sum It Up! 

Here is a small strategy I used for our review day today.

In Advanced Functions we were reviewing average rates of change and instantaneous rates of change.

Instead of just doing problems from the book I organized the rate problems in 4 sets, polynomial, rational, trigonometric, and exponential/logarithmic functions. Each set had 3 problems.

I organized the class in groups of 3 and gave each group a set. They were to complete the problems and check the answers.

The kicker here was that the sum of the 3 answers in a set was written on the board.


If the group got all questions correct their sum of their answers would match correctly. But…… If one was wrong they had to check all solutions and decide which needed fixing.

First, this small change made all groups work more carefully than usual. And second, when they had to double back, you could see them rationalizing which answers “had” to be correct and which answers were “toss ups”.

It was a great tool for students to self assess and evaluate their own readiness.

double desks

After each group completed a set they would swap with another group… Eventually completing all sets.

Promote Struggle – A Hero’s Journey in Math Class

While in Miami for the Apple Distinguished Educators Institute we saw a speaker from Pixar (I can’t recall his name) 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 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. 

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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.

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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…

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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.

Final Assignment – Estimation Challenge

For our final activity I started the off with this……

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we filled in too high, too low, and best guesses! Then we checked the answer…..

Completing this challenge got the students pumped and hooked into doing some math on our very last days of class (especially with some students exempted from the final exam). Our final assignment is to …

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We got out the iPads and I let the kids work….. here is what a few came up with:

A lot of kids did water filling or post it note covering estimates. Some kids ended up making an all-out 3 Act math problem.


How many caps will fill the marker?

Estimate & Answer



How many cups to fill the shape?
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A 3- act task in Explain Everything:

How many post-its will cover this triangular wall:

Act 1: She put a photo and a small video in Explain Everything to start us off.
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Act 2: She provided us with a little more info after we made some guesses.
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Act 3: Made a time lapse video and provided a screen shot with the answer

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Grab her Explain Everything File here




How many stickies to fill the door window?

FullSizeRender 17

And hit us up with a time lapse video for the answer:




How many water bottles will fill the hexagonal prism?


and the answer


Watch the water fill up by grabbing her Explain Everything File

A fun last few days….and I’ve got some new estimates for class next year.

3 Act Math – Ground Cover

With inspiration from Dan Meyer through Kyle Pearce, I created a Three-Act Problem. My grade 11 class just finished finding side lengths and angles in right triangles using the primary trigonometric ratios, and we’re moving on to solving problems! In the spirit of solving problems… is a Three Act Problem.

Ground Cover

Act 1: Video

Act 2: Pictures

Sod Dimensions

Garden Measurments

Act 3: Video