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by Jon Orr

Sparking Curiosity

Students will extend their use of patterning to connect different representations of linear relations. More specifically, students will be exposed to rates of change, calculating the slope of a linear relation, building linear equations, and solving linear equations.

The purpose of this Task is:

  • to help students develop an understanding of how various representations of linear relations are connected;
  • create a linear equation given two points; and,
  • solve linear equations.

Intentionality

As is true for any task, the intentionality or learning objective can vary depending on what mathematical thinking you are hoping to elicit.

The mathematical ideas we are trying to elicit when we use this task are connect various representations of linear relations, build linear equations from two or more points, and solve linear equations.

This will be achieved by first having students watch how the cost of renting a scooter is related to the time rented. You can learn more about the actual scooters and the pricing structure at lyft.com.

Spark Curiosity

WARNING!!! Be sure to share with your students the importance of wearing Bike Helmets while riding; we certainly should have. We realize that you may not want to do this activity with your students as a result of our poor judgement.

Show students this video:

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

Here’s some of the “everything and anything” students noticed and wondered on chart paper:

  • I noticed that they were riding scooters;
  • I noticed that they weren’t wearing helmets;
  • I noticed the map;
  • I noticed that the cost changes;
  • I wonder where that was?
  • I wonder how much the scooters cost?
  • I wonder what the range means?

Now you can focus in on the big question of the task.

How much does it cost to ride the scooter the entire length of the outlined route?

We can now ask students to make a prediction using their estimation skills. Ensure you use the Too high and too low strategy. Ask them what is a wrong answer? How much would be too high? How much would be too low?

Students will also be uncomfortable here because the length of time of the trip or how the scooter charges customers has not been revealed yet. We encourage you to hold off on revealing these answers because it will build anticipation. Anticipation is what students need so they can start formulating a plan.

At this point, we want to give students the opportunity to improve their predictions by engaging in developing a problem solving strategy.

Ask them: If we are going to improve our predictions what information will we need? Have students share with an elbow partner before sharing this information with the entire class.

As students voice the information they wish to see.

Ask them: “And what would you do with that information if I gave it to you?”

Listen in very closely here. Their responses will give you allow you to assess their prior knowledge and also their thinking into solving this problem.

Slowly Reveal More Information

Once students have asked for information reveal the information that you do have.

Reveal a key fact:

The cost of renting a scooter depends on the time the scooter has been rented for (You may want to reveal depending on your students’ prior knowledge that there is a flat fee to get on the scooter).

Reveal three snapshots showing the cost at different points in the trip. We include three here so that we can verify that the relationship between the cost and the time is linear.

Reveal the total cost of the entire trip.

With this information students can start to develop strategies to determine how much it will cost for a 15 minute and 9 second trip.

Fuel Sense Making:

LEARN HOW TO FUEL SENSE MAKING

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After consolidating the learning using student generated solution strategies and by extending their thinking intentionally, we can share what the actual cost of the trip was:

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….

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 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.

Gaining Insight

by Jon Orr

As the year closes down I think back on 2017.  I was curious about some of the stats on this site and was blown away at some of the numbers. I never thought that when I started sharing what goes on in my classroom that I would have over 150000 views in a single year! Amazing….and thats all because of you! I dug a bit deeper and found the three most popular posts from this year.

  1. Angry birds Parabolas
  2. Flippity Flip Bottle Flip
  3. Spiralling Grade 9 Math

At first glance I thought, “Yeah, those top 2 posts make sense. Their kinda gimmicky and fads. We search for those relevant topics our students are into; games and bottle flips! I’m sure if I wrote a post on fidget spinners it would be up there too.”But after thinking back on those activities and comparing them I think both their value come from being able to gain great insight into student thinking. And it’s that ability to assess our students deeper thinking here that teachers are drawn to.

Take the Angry Birds lesson for example, the creativity that is embedded  throughout the lesson is everything. Students get to choose how their flight paths look and act. There’s a story behind every arc they put into their activity. Their thinking can’t help but spill out all over, and I get to use that knowledge I gain to help push them along. Take away the angry birds and you still have a great creative lesson.

Replace it with a drawing, or trace of a picture or even a marble run and students experience the exact same creativity and learning goal expectations. The activity still allows me to have those insightful conversations.

The bottle flipping activity is a formative assessment gold mine. Again take away the bottles and replace with paper balls or card tossing and this lesson is identical, and I have just as much success at seeing into my students thinking.

It’s this insight that we all want. It’s this insight we need. Insight allows us to what Kyle Pearce and I have been calling ignite our moves. Seeing how a students thinks in live time allows us to act. We may act to address a misconception. We may act to push learning further. We may act to plan our next lesson. We may act to change our planned lesson into something that the students need at that moment. Lessons that allow insight into student thinking must be our norm.

This fits with the 3rd top resource. Spiralling Grade 9 Math. The file found on this post give us a day-by-day to teach with lessons just like the ones above. Not gimmicky lessons —  Lessons that spark curiosity! They are lessons that provide great insight so I can ignite my moves and fuel my students sense making. And fuelling sense making has to be our main purpose.

Have you used any of these resources? Comment below to share how?

3 New Desmos Activities: Talkers & Drawers

by Jon Orr

Goals of the activity:

Students will:
  • Begin to recognize characteristics of linear, quadratic, or periodic functions.
  • Generate a need to use proper vocabulary around linear, quadratic, or periodic functions.

Specific recommendations:

  • The “talker” cannot use their hands and should keep them behind his/her back. This will help the student be careful and direct the language they choose to describe the graph.
  • The “drawer” cannot talk.
  • Set a time limit. Possibly 3-4 minutes for the “talker” to describe the graph to the “drawer” with the goal to reproduce the graph.
  • Consider having all the “drawers” reveal the graphs at the same time for dramatic effect.
There are three different versions of the activity based on topic
Links to the three activities:

What the student experiences:

Once students choose a role tell them “Talkers, your goal is describe the graph perfectly to the drawer. Drawers, your goal is to listen carefully and without talking try to match the talkers graph. You will have 3 to 4 minutes for each graph.”
When the time is up, tell all the drawers to click the REVEAL button at the same time to see how close your sketch was.
 

What the teacher experiences:

While students are describing and sketching take time to listen to the words they use. Store these words for later in the class so you can link them to the proper names.
Example: 
You heard Jose Adem Chain say, “The pattern starts at 2 and goes up…” If most students are using the phrase “starts at..” We can introduce the term y-intercept.
Or on the periodic function version:
A student might say, “…it does that and then repeats 4 units later” You now have a gateway into introducing the period of the function.
After each round use the Teacher View to showcase some student graphs to the class.
Consider restricting the students to the current sketch and move from sketch to sketch as a class.
Last question.

The words generated on this slide will most likely be informal. As a class discuss the informal use of the word and then introduce the more formal words relating to the topic.
Inspired by Brian McBain and also the team at Desmos