Two Trains…

How many of you have seen a problem like this one?

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I’m a fan of taking a problem like this, one that you would assign for homework (in the “application” section of the exercises….and one that very few students even attempt….and someone will ask you to take it up next class) and bring it to the start of my lesson. I’ll teach our concept/idea through this problem. But we can’t just throw this problem up on the board and say “Let’s solve it”……because no will want to. There is no drive for any of us. Like Dan mentions here….who cares!

Who cares about the trains travelling…who cares that they are even trains….they could be bicycles, or cars playing chicken….but is changing the context really going to change how engaging the problem is to students? Dan argues no. I agree.  Before you read about this lesson check out this post on Real vs. Fake world….and the Circle Square lesson on 101qs.com which was an inspiration for changing the Two trains problem around.

Here’s my go at this one:

Show them this video:

ask What do you notice? What do you wonder?

Have students guess WHEN the two dots would meet?

Give some more info

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Have them guess on WHERE the dots will meet?

Have a discussion on what will be needed to determine the times and distances. Spend some time here on speed. Go over the relationship between distance, time, and speed.

Show them this image and have them makes some guesses on where the dots are now.

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then reveal

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Calculate the speeds of the dots. Have students go back to their original guess on time and find how far each dot would travel.  Who in the class is closest? Did anyone guess right?

Now help them generalize…

Create the equations

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If our lesson is on solving this using an algebraic technique we can teach them that here. Or maybe we want to show them the graphical solution. Either way we have taken the tougher question from homework that no one cares about and used it to set up and teach a skill.

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and finally,

I’m sharing this lesson now (before I teach it) with you hoping to get some feedback. Writing these lessons here also help me work out the details. This is week 4 of the #MTBos blogging initiative and its focus is lessons. I won’t get a chance to teach a lesson this week. Our school had final exams and then PD days in preparation for second semester. Good luck to all those starting up again!!

 

Catch the Spiral! 

Last May I shared my day-to-day planning spreadsheet for my grade 9 applied course. On that sheet I recorded the topic, tasks, and resources for each day of the semester. I used that as a resource for myself when teaching 1P through a spiral this semester. I found that having that sheet to go back too was super helpful and a time saver. This semester I followed that timeline except with a few tweaks here and there.

Since that sheet was so handy to have I made one similar for my MPM2D class. It was my first time spiralling that course and I wouldn’t go back to teaching through units again.

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I heavily relied on Mary Bourassa’s blog….she is amazing. She shares her day-to-day plan as posts on her blog and also shares all of her resources and handouts. Thanks so much Mary!!!

Spiralling in Academic vs. Spiralling in Applied

I struggled initially with deciding to spiral the MPM2D course because of my experience with MFM1P. I had previously taught the 1P course through activities and 3 act math problems so it was a no brainer to just mix up the order of the problems and tasks. It was an easy transition since I had all the resources. For the 2D course though, it had been a while and I had not taught it with a task/activity approach.

What I found to work best in the academic class was to learn all new ideas/topics through activities and productive struggle with some direct instruction thrown in as a consolidation. Unlike the 1P course where I switched tasks/topics daily, I stuck to a topic/idea for a few days or a week in the 2D course. Once, for example, the class was comfortable with transformations of quadratics we would switch to trigonometry for a week, then analytic geometry for a week, etc.

I felt that through spiralling and teaching through productive struggle my students were better problem solvers. They were not just waiting to be told how to solve a problem. They were always actively thinking about which ideas they had learned could apply to solve a particular problem. That confidence I saw allowed us to go more deeply into the content than ever before. We just didn’t skim the surface of the processes, algorithms, and algebra needed, we solved problems!!

If you wanted to spiral the 2D course or a similar course I thought I would share out my plan to help out. Here is my day-to-day plan with links, resources, Desmos activities, 3 Act tasks, assignments, homework, etc from my spiralled MPM2D course. (It’s not fully complete for every day but you’ll get a sense of how the class ran).

Most files are either Smart Notebook, Apple’s Keynote, or PDF.

Get Apple’s Keynote on your Mac or on iOS.

 

 

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.

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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.

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

Better Questions – Two Truths & One Lie

During these last few days of class we have been completing problem after problem to prepare for our exams. I wanted to start class off with a different question than “What problems should we take up?”

I’ve used and played the game Two Truths and One Lie as an ice breaker before and it popped into my head when I read this weeks blog post from ExploreMTBos.wordpress.com. Having students develop two truths and one lie about a math concept seemed super intriguing. The openness of this task made me eager to see what they would come up with. It would give my huge insight to what they know and what we would need to work on too.

This image was waiting for my 10th grade class when they came in.

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They worked in pairs as a team to develop the truths and lie. I overheard great discussions on what should be considered truth and how to choose a lie. I loved the variety of what they chose. No two groups had the same.

Here is one:

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After writing their statements on a sticky note they were to trade stickies with another group and identify which statements were true and which was the lie. This was the best part! They were competitive. I overheard groups saying “prove it” or “show why that’s true”. They were demanding each other to see work and evidence. They weren’t accepting guesses! The vocabulary was amazing too. Afterward I had them all write their truths and lies up on the board. We went through each statement discussing the strategies needed to verify.

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I also ran the same activity in my 12th grade Advanced Functions class. Here was their function: Screen Shot 2016-01-25 at 2.34.42 PM

and their stickies:

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Afterward I asked the class what we could do to make the activity better. Here are some suggestions I’ll try next time.

  • A point should be awarded for getting the two truths and a lie correct and not for “stumping” another group.
  • Each group should get their own function to create statements for. Then in each match the opposing team would have to verify everything. Lots of practice and lots more variety.
  • We ended up giving a point to groups who found errors in another groups statements.

Also thinking of putting a question like this on my next assessment/test. Have the student write and indicate which is the truth and which is the lie. Or give 3 statements and have them generate a graph or equation where two statements are true and the other is a lie.

If you have any other suggestions on how to change/modify this activity I’d love to hear about them.