# How Can We Anticipate to Fuel Sense Making? Stretching Trees

Are you looking to avoid “Lesson Flops” and bring on a “Lesson Successes?” I sure do. That’s why I plan with anticipation of student thinking in mind.

I want to share a lesson I co-created with Brian McBain and teachers at Wallaceburg Secondary School and how anticipating student thinking helped avoid those flops!

Let’s run through the lesson first, then I’ll give you a window into how we planned it.

In random groupings students went to their wall space and were presented this first task.

I have to admit when we planned it we anticipated everyone to draw Christmas trees but after showing the image they all drew a variation of the one above.

Drawing the trees was no biggie since our new amazing whiteboards from Wipebook.ca (wipebook.com) has grids on them. Students counted up 20 units and drew their very best tree! Onto the next part of the lesson.

Draw another tree that has a height that is less than 50% of the first tree’s height.

Here is a typical drawing from my students.

In small groupings and also as a whole class I asked and discussed “How do you know the height is less than 50% of the original tree?”

And then we moved onto this…

Draw a tree with a height that is more than 50% of the original tree’s height.

With this prompt we wanted to dive into the answers a bit more. “How do you know it’s more than 50%?” “How can we verify that 16 units high is more than 50%?” Also with this we had students drawing trees higher than 100%. We paused the class and verified and shared out the different tree heights around the room.

I prompted them to draw a tree that was exactly 30% of the original tree’s height.

This is where I was super interested to see how they would solve this. Their solutions were going to fuel the discussion going forward (Check below to see how we anticipated what they would do).

Most groups of students used the grid and found a unit rate. Can figure out this strategy?

This group knew that 10 units would be 50% so they took the 50% and divided it up into 10 units giving 5% per unit. Then they counted up by 5s until they reached 30% and got 6 units high. Other groups took the whole 100% and divided it by 20 to get 5%/unit. As a class we gathered around these solutions and explained the strategy. Any group that was stuck went back to their boards to use this newly presented strategy and the other groups pushed forward with this new prompt.

Groups progressed through this prompt at different times, but when they were ready I gave them this one: “Draw a tree with a height that is exactly 62% of the original tree.”

This is where the struggles happened. Again we were interested in HOW students solved this problem. Most new that 60 was going to be 12 units high….and then just estimated from there how high the tree would be. Some did guess and check to narrow down how high exactly 62% was. This was exactly what we had planned. We had wanted and led the students here to create this struggle so that we could step in and teach them a strategy!!

We used a double number line: One side showing percent from 0 to 100 and the other side showing the heights of the tree. But instead of a horizontal number line we tipped it up and made it vertical!

We had a discussion on proportions: “Is this a proportional relationship?” “How do we know?” Yes….so we can apply a proportional strategy to solve this. After that the students had a new and improved strategy to try the next few prompts:

Draw a new tree that has a height that is 17% of the original tree’s height.

and then,

And then we switched to a new “starting” tree.

And kids drew this.

After I felt that groups were comfortable, their next task was given out (which stretched into day 2). We changed the scenario from trees to colouring.

Want to get the PDF with all the image prompts and handouts? Click Here.

When reflecting back this lesson was not one of the “flops” it was a “success” and most of the credit has to go to planning with anticipation in mind.

## Anticipating to Fuel Sense Making

When Brian, the team, and I set out to design this lesson we were looking for a way for students to feel like they weren’t learning something new. That they could take the idea of percent and just use it like they have already solved proportion problems. We also wanted students to follow the Hero’s Journey and feel that there was a definite need to use a proportion strategy.

Here is what the early stages of the planning process looked like. Yep, scribbles in a journal. We spent a lot time thinking about the right progression of prompts so that we could maximize student work and use their strategies to push learning forward.

We also spent a great deal of time planning out the different strategies we thought students would use to solve the original prompt “Draw a tree that is 30% of the original tree’s height”

We outlined the strategy of finding the unit rate of 5% per unit, we thought many students would already know the “rule”: Turn the % to a decimal and multiply (But no one did do that in my class). We thought it was possible for them to create a proportion. We thought some groups would try a guess and check strategy. Like: “I think the height is 7. Let’s see if 7 out of 20 is 30%.” Only a few groups did this.  We ranked each strategy in order of most likely to least likely.

Anticipating their solutions and strategies puts me in a better position to understand their thinking and help shape that thinking. For each possible attempt I need to be ready to provide feedback to help them achieve our goals.

We take for granted how much time is needed to prepare and anticipate adequately. It takes time to make this happen, but that time is worth every minute. Especially if it puts me in a better place to understand what my students are thinking.

This has been my assessment goal: Understand their thinking in order to push them further. That’s it! That’s the main idea.

Anticipating their thinking will always put me in a better position to fuel their sense making.

Read More: Fuel Sense Making & Black Box Defrost

# Why Consistency Is More Important Than Intensity: Culture in the Math Classroom

Ok so you tried using a problem based lesson like a 3-Act Math Task or maybe you had students solve a task in groups with your brand new whiteboards and it …….flopped! Yep you’re worst nightmare was there with you in that classroom! A failed lesson! You were sure it was going to be a success. You heard that that activity was a great one but for you it just didn’t work.

Don’t worry. It wasn’t you. It’s normal. This comes down to an example of Intensity versus Consistency. I first heard Simon Sinek (an inspiring speaker and author on business and leadership) talk about this on the Tony Robbins Podcast.

Wait, what’s Intensity versus Consistency???

Ok, say I’m looking to get into shape. I want to be physically healthy and fit. Everybody knows that I can’t just go to the gym for 6 hours today and expect to instantly be fit and ripped! No one expects that. That is an example of intensity. Or consider brushing teeth. I can’t just brush my teeth once and hard and then expect my teeth to stay healthy! They’d all fall out after a while! Or let’s say reading books with my kids at bedtime. If I read for two hours before bed tonight with my daughter Lucie it’s not going to make her a better reader. Those are all acts of intensity. Brief intense moments of an event.

We know that we need to brush our teeth twice a day everyday to see results. We know that to go to get in physical shape we need to routinely work out 20 minutes a day and we will see results. We will absolutely get into shape. If we read with our kids every night then they will become better readers and better writers! We know this. These are acts of consistency.

It’s little consistent events that will make the difference not a big one-time or two-time event. But the problem is that we can’t see the benefits of the small acts in real time. I don’t see that my teeth are healthier after the one time brush or that my daughter reads any better…..I have to wait to see those benefits. And it’s hard because those benefits come at different times for different people.

When Simon talks about great leaders he says that great leaders have to build consistency and not intensity. It’s all the little things they do to create a great work culture and not the big hoopla one time event. A great leader can’t just throw an amazing holiday party and expect that to be the solution to a great work environment. They have to show acts everyday that they value their employees. That consistency will create a great work culture!

## Intensity versus Consistency for Math Education:

That lesson that flopped was a one-time event! It was because we viewed it as an act of intensity. In order for those lessons to be successful and to bring out deep meaning and learning for our students we need acts of consistency. We need to do this as part of our routine.

That consistency will help create the amazing classroom culture you are dreaming of. That positive, safe, fun learning environment where kids learn with each other and with the teacher! But we have to be willing to put in the work to build consistency.

And the students are not going to be the ones to magically make this happen. We have to do it. We’re talking about middle school or high school here. Students would rather NOT talk to other peers they don’t know. We need to teach them how to help create this environment.

And It’s all the little things we do everyday that will make this happen.

• It’s the Daily Warm ups where you have kids discussing arguing defending.
• It’s routinely asking kids to struggle that teach them resilience.
• It’s the Random Grouping everyday and using Whiteboards that show them that you value their voice.
• It’s the problems you use to teach with.
• It’s how you demonstrate to your students what you value in learning.

It’s the things you do everyday that matter. Those are the things that will build the culture you are looking for. Routine and Consistency are what will drive change in your classroom and student learning.

This takes dedication. I know it’s hard so to help you out I’ve put together a handout that you can download, read and share with other teachers on 5 tips to to build amazing classroom culture. And you guessed it all of them involve being consistent instead of intense.

What are your tips to build amazing classroom culture in your math classes? What are we missing? Just add those in the comments below.

# Spiralling in Advanced Functions (MHF4U)

I’ve been spiralling my courses for the last few years, but this last semester was the first time I spiralled the Advanced Functions MHF4U course. If you’re new to the spiralling idea check out the blog post from Mary Bourassa and the MHF4U website from Al Overwijk and Janice Bernstein. They’re great resources to get you going.

This post is really to remind my future self on what I did this semester and for anyone else asking spiralling questions.

### On Planning

Occasionally I will get an email from a teacher who is interested in trying spiralling and the question they usually ask is, — Where do I start? I think most of us need someone to shine the flash light down the path for us to see where to head. I usually start with a table that shows the strands of the course and where the major skills (overall expectations) fit in. I try to group them by themes. This year I my cycle one was about introducing the functions and focusing on graphing characteristics. Cycle two focused on linking algebraic representations with graphical. See below.

From there I keep an ongoing day-to-day plan.

Click to see the live version

### On Homework:

In the past I’ve given out homework in a very traditional way, “Tonight, complete page ___ Questions #__ to ___. Tomorrow we’ll take them up.” And what did homework take-up look like in a grade 12 course? Well, for me, it was always “What problems did you have trouble with? Number 8b? Ok, does anyone have that one completed? Kearra can you put that solution up on the board?” If no one had that question right, then I would put up a solution. And everyone watched, twiddling their thumbs (or more realistically — texted) while I put that solution up….or we all watched Kearra put the solution up. Not a great use our of time.

I’ve changed that process over the last year or so. For me, giving out homework comes in a homework set. I got the idea from Al Overwijk and Mary Bourassa. The sets not only have practice problems from the ideas from that day, but also practice problems from other areas of the course. Each night of homework they are practicing most strands of the course. It keeps concepts fresh in their minds and keeps practice going all semester.

a typical homework set

When students come to class they get a playing card that randomly assigns them a partner. Instead of asking which question we should put up, I choose two or three from the set and the pair has to put them up on the vertical whiteboards/blackboards around the room. They are only allowed one piece of chalk or marker between them. I circulate around the room to give feedback and check for understanding/thinking. I’ll routinely yell out to “switch the marker” which forces students to communicate, error check, and defend their work. A better use of our 10 minute homework take-up time. After, students hand in their homework which allows me to check their understanding and gives me insight on what skills we need to improve on (I choose one or two questions to focus on). Gone are the days where I give out homework and I don’t find out what they really know until test time. Now, I know daily. Is it more work for me? Yes it is. But it’s worth it.

Can’t see the video? Click through to the post

After homework take up.

### Whiteboards & Note-taking:

Most of our problem solving and practice work in class this year was done on non-permanent surfaces. For some students, parents, and teachers this is a concern since they are wiped away and there is not a record of that work. Here is an email response I sent a fellow teacher this year to address the concern:

“Do your students need the note? Are they asking to take notes? If so, have a conversation with them about what they need and teach them to take pictures of what they need or make notes for themselves. Or have them summarize what they’ve learned after doing the problems as an exit slip.
I sometimes do “important” solutions on chart paper and then they stay up in the room so we can refer back to them.”

### Changes:

As always I’ll be making changes for the next time I teach the course. I want to include solving equations earlier in the course. This year I didn’t bring it in until cycle 3 and I feel like we could have benefited from more exposure. Also, radians need to be introduced in cycle 1 so that it can fuel all of trig for the rest of the year. I feel like it was crammed into the last cycle.

### Day-to-Day Outline and resources for MHF4U

See the outline as a webpage