Author: Jon Orr

Eye To Eye – A Similar Triangle Problem

by Jon Orr

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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A Squiggle-Line Dilemma: How Creating Bends Gives us Freedom in Planning

by Jon Orr

Have you read one of my all-time favourite books The Dot and the Line: A Romance in Lower Mathematics by Norton Juster? It’s not a new book it was originally written in 1963.  I not only read it to my classes on Saint Valentine’s Day but I gave it to my wife as a present way back on our first valentine’s day together.

I love that every time I read it it makes me reflect on who I want to be as a human and also as a math teacher!

If you haven’t read it you can watch the Academy Award winning Short Animation by Chuck Jones right here, now! Watch it before reading the rest.

Lately I’ve been thinking about this story as it relates to how we math teachers feel the need pursue the “perfect” math lesson or that shiny new tool/technique we hear we should try.

We seem to be after the perfectly engaged class (behaviourally and cognitively) learning the chosen standard at just the right pace for all students. And why shouldn’t we? It sounds great. But, what is the likelihood that we’ll ever achieve this “perfectness”. The reality is that teaching is messy; all classrooms are different.

We see so much positivity on the internet and from our peers. Looking at twitter or blog posts suggests that so many teachers are having these perfect classes or that the shiny new tool/technique solves all our problems. And it leaves us sometimes feeling inferior and overwhelmed.

I think some of us feel that we need to be using that iPad, or new tech tool, or shine new learning model everyday to create this perfect happy class.

Let’s relate this situation to The Dot and the Line story.

Imagine for a moment that you are the main character from the book; the line. The dot is ….. well, the dot is that “perfect” class lesson where all students are using that new shiny tool or technique that we’re not quite sure about.

When the line first meets the Dot and sees that “she” only has eyes for the whimsical squiggle, the line feels that “he” needs be more like the squiggle.

Many of us teachers also feel or have felt that we have to become the whimsical squiggle to win the dot to our side. We feel that we have to become not just entertainers, but we have to become someone we are not. Many teachers also feel that we have to give up core beliefs on what creates good a good learning moment so we can have this other, supposedly great learning tool or technique. 

But that’s not true.  We don’t need to change our core beliefs of what creates great learners. We don’t need to give up on teaching students dedication, determination, and rigour to bring in curiosity, creativity and openness into our lessons.

For example, some math teachers believe that by teaching through problem solving with tasks like Popcorn Pandemonium, or Kyle Pearce’s Candle Burning problem you HAVE to sacrifice procedural fluency.  They believe that you can’t have both mathematical rigour and learning through problem solving. You either have to be a squiggle or a straight line. They believe it’s one way or the other.

What I believe is that we may have to BEND, just like our pal the Line to truly create math moments that matter for our students.

Like the line, Bending gives us permission that it’s not an all or nothing transformation. We don’t just have to choose between a squiggle and a perfect line.

Like the line, bending means though that we may have to work harder and smarter.

Like the line, bending means that we can teach through problem solving as well as getting students the practice they need to become fluent without sacrificing time.

Like the line, bending means that we can recreate ourselves — in a stronger way that supports learning.

Bending means that we need to actively think about how we can incorporate our core beliefs of good learning in our lessons while meeting the needs of ALL our students.

To address one common Line vs. Squiggle comparison:

How do we incorporate practice and procedural fluency in lessons while building resiliency in problem solving — without sacrificing time?

I use purposeful practice routines that encourages student discourse, self assessment, peer assessment, movement, and error checking that bring my students closer to procedural fluency after we’ve used productive struggle to learn a topic.

Download and learn more about 5 practice structures I highly recommend you add to your practice routines. 

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5 Practice Structures in Math Class

Learn about 5 of my go-to practice structures for self assessment, peer assessment, movement, and error checking!
Yes, I want the Guide

Further reading:

Jimmy Pai’s Be You. 

Kyle Pearce’s Automaticity vs. Memorization.

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My #1 Go-To Tool In Math Class

by Jon Orr

Let’s start with this one question:

For me, I use a set of 4 criteria to evaluate all resources, tools, and lesson ideas. It helps me quickly narrow down whether a tool will help me achieve the desired results I look for in my classroom.

Here are the four criteria.

  1. I want ALL my students to show me their thinking and understanding in interesting ways. I want them to show me what they think first instead of just telling them what to think! I want to open up the questioning that goes on in my room. So I look and create lessons that allow for this.
  2. I want my students to discuss, collaborate, argue, defend, and justify with each other. I believe this helps clarify their learning and understanding so I must make sure that discussion and collaboration happen in my best lessons.
  3. I am always assessing for growth! I’m constantly looking to see who gets what we are doing and who needs help. I need to be able to assess quickly the abilities in my room so I can use that on the fly to decide where to go next. Assessing easily must be apart of my lessons.
  4. Every lesson or activity must have a ratio between the cost of set up and the payoff where the payoff heavily out weighs the set up. Nothing is worse than spending a huge chunk of time, making, cutting, designing and then when you run it the learning outcome wasn’t worth it. The payoff must out weigh the set up.

My #1 Go-To tool/technique is WHITEBOARDS!! Having my students work in random pairs daily at vertical whiteboards is the tool that easily allows my students to get to work faster and hit all four of my criteria.

How do whiteboards stack up against my criteria?

 

On a whiteboard students can easily show off their learning. They are quicker to get to writing on a whiteboard than on paper. Especially when the boards on the wall. Students get to defend, argue, justify their thinking with each other. I can easily see if students are understanding and the set up ratio is a no brainer. Here’s a whiteboard, marker….Go!

I’ve had students use small personal whiteboards at their desks before, but I couldn’t believe the change in active engagement and cooperation once they were standing. The discussions they were having about the math was much more insightful and meaningful.

Our whiteboard work usually starts as soon as the bell rings. In random pairs students would put up a few homework questions from the previous day. I could see students looking around verifying their work with their peers. They were self assessing.

We continue to make use of the boards while we work through our new challenges. Students have no problem leaving their space to go and talk to another group to gain some insight on new strategies….This is how we create great classroom culture. I can easily circulate the room to engage students in conversations and challenge their reasoning.

 

WHITEBOARDS IN YOUR CLASSROOM

Recently, Kyle Pearce and I took Wipebook Chart Packs on the road with us to some conferences and district presentations for our Making Math Moments That Matter workshop series and they worked great! In a breakout session we held at The OAME Leadership Conference in 2017, we had a room of over 100 teachers, consultants, and coaches up at the walls doing a math problem with us. It was fantastic!

We had so many participants asking for more details on how to get their own Wipebook Chart Packs that they are now offering an exclusive Educator Starter Pack you can only get through us!!!

 

EDUCATOR STARTER PACK

The education starter pack includes:

  • A Wipebook Flipchart (pack of 10);
  • A Wipebook Notebook Plain;
  • A Wipebook Workbook Graph;
  • A single correctable marker; and
  • A single tri-plus marker.
Your exclusive Making Math Moments That Matter offer is 25% off the list price value!!! 
 

Full disclosure: Kyle and I love this product so much that we became affiliates for Wipebook. Which means if you purchase through our link here YOU get a discount and WE get a small percentage with no extra cost to you! Win/Win! 

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***Any other country use the USA button above. 

Are you using whiteboards in your classroom? 

What is your #1 Go-To Tool?? 

Interested in learning how to assess for growth? We’ve got module 1 from our online assessment course for math teachers open for you to dive into right now!

Hurry Up/Kill Time Math Classes – How Desmos Can Help

by Jon Orr

Seth Godin brought up an interesting idea: If you think about it, everyone at the airport is in one of two modes. In a hurry, or killing time. You can imagine it right now! That impatient person in the TSA line just waiting to speed walk to the gate, or the group of people jockeying for position to board the plane first. On the other hand the only other people are just waiting around to speed up!

This is also happening in math classrooms. Both teachers and students.
Students want to hurray through lessons, get the homework done, move onto the next thing. Or they are just killing time. Teachers are hurrying up to start lessons, give examples, get the ideas out, give the homework. Some teachers are just waiting around until the day ends.

But this is not all students. Some are focused on learning to learn. And this is not all teachers. Many like you and me are actively making moments that matter in our students lives. Let’s help others slow down the “Hurry up to wait” classrooms.

One way I do this is to create and share those moments that matter through enhancing classroom discussions through a tech tool that creates discussions, not limits them like so many others do.

Watch,

 

Let’s Slow down the hurryers and energize the waiters. Let’s enhance our classrooms together. Check out a list of over 25 of my custom Desmos activities plus check out the hundreds of activities on  teacher.desmos.com  like the Pentomino Puzzles activity.

Here is a transcript of the audio in the video:

Hey, I’m Jon Orr, a math teacher from Chatham Ontario, Canada.

I’ve been on a mission lately to make moments in my math class memorable. Like you remember specific moments in your life because they were meaningful. Something that sticks with you. When I think back to my experiences as a math student I remember grade 4. I thought I was a master multiplier. My teacher even gave me stickers for doing extra work….and these stickers weren’t just normal stickers they were the ones that stand off the page like puffy stickers. You know, the ones that make the book not close all the way.

That sticks with me because of the feelings that go with the moments. I want to create those for my students; not with stickers, but memorable math moments. Like moments that students will remember years later. Like I have students years after my course still remember the toy car lesson we did or the pentomino puzzle solving lesson! I want this for every student in my class.

One tool that I think does this amazingly well is Desmos. And I’m not talking about just the online calculator. I’m talking about the earth shattering online activities that they create for us to teach with for free!

What I love is that each activity they build helps me make those moments. And they do that by allowing my students to show their thinking in interesting ways, they allow me quickly assess on the fly the abilities in my room…and they allow my students to have discussions! The tech creates discussions!

Here is one task that is great. Pentomino puzzles.

The activity is super easy to get into, just move this tile around until you cover a sum of 65. You can see students can easily share their thinking and strategies. I have kids use 1 device for 2 people so they can talk about their strategies. It keeps that collaboration I’m looking for. But then each new task builds towards solving the problems using an algebraic approach! I get kids to learn how to solve equations through this puzzling type of game!

As a teacher I get to see what student is on what screen, allowing me to help kids that need help and allow kids to move forward that are ready for it.

I can pause the screen on everyone’s devices so we can discuss strategies. The software is built to enhance classroom culture and discussions, not limit them like other tech does.

So one recommendation for you to try to make math moments matter for all your students is to explore the activities on teacher.desmos.com.

Thanks,

Find out more on desmos over on my website mrorr-isageek.com where I share all my custom made desmos activities and many other resources and ideas for your math classroom.

Take care.