Puzzling Dimensions

I wanted to grab some thoughts from you on a possible lesson idea.

On the weekend Jules and I worked on some puzzles. After we dumped the puzzles out I would ask her how big will the puzzle be? She would look up, and with that wondering look say …”pretty big” or “not too big,” but nothing exact….she’s only 6. I narrowed the question down. How many pieces would be along the bottom? Along the side? And she would make a guess. We would do the puzzle and then find out.

This got me thinking.

I was thinking about factors of numbers and how that relates to the dimensions. I also thought about optimal dimensions of rectangles given a set area.

If a puzzle had 60 pieces what could the dimensions be? 100 pieces? 1000 pieces? 

Take Elsa for example. With 48 pieces do you know what the dimensions will be? Think of some possible combinations. Got them? 


And….. bam! Did you think of 8 pieces by 6 pieces? 


The puzzle we worked on had 100 pieces and it was a 10 piece by 10 piece puzzle.
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I feel like there is a lesson here but I’m not 100% sure where it fits. It may fit in many places.

If Act 1 is a short clip of us putting a puzzle together like below, then how does the rest of the lesson go? How do you see the rest of the lesson play out if you teach Kindergarten? Primary grades? Middle school? High school?

I would love to hear your ideas on the lesson goals as well as the lesson format! Together we can do it!

Favourite & Fix: Nov. 11

For the Favourite & Fix series each week I’m posting one idea from my lessons that was my favourite and one topic that I need help on. A topic I hope to fix. I’m hoping that in the comments or on Twitter #Fav&Fix you amazing readers can help me out with some hints, tips, and suggestions.

Favourite:

This week I introduced the unit circle to my MHF4U class. I wasn’t happy with the way I introduced the circle in years past so I made a change. I want students to see that our special triangles are just reflected around the circle. Instead of drawing them, or imagining that they are reflected….I wanted them to physically pick them up and flip them and move them. I wanted them to see that the lengths are the same. So I cut out 30-60-90 triangles and 45-45-90 triangle each having a hypotenuse of 10 cm. I created a circle with radius 10. Now each time you place the triangle on the circle we can easily see the principal angle it creates and the coordinates of the point on the circle….It’s the lengths of the triangle….and since the hypotenuse is one the lengths correspond to the Cosine and sine value of that angle. The physicality of this I believe helped allow the students to grasp what the circle shows.

Fix

This week in our MEL3E class used Fry’s Bank from Dan Meyer.

This problem, like many 3-Act Math problems, allowed my class to discuss, question assumptions, and uncover math. The problem helped restore some of our great classroom atmosphere that we’ve been missing lately. I want more! This coming week we’ll keep working through compound interest problems. I’m planning on doing Robert’s Not Cashing the Cheque problem.  After that my resources for compound interest problems are pretty thin. I want to continue posing interesting problems to my students. Can you suggest some? Do you have great compound interest problem that keep students curious and questioning? I’m looking for some!! Share those great problems here or on #fave&Fix on Twitter. I’m looking forward to what you come up with.

[Update]

Thank to all of you who commented through Twitter on great compound interest problems. Here is one from Diana,

Here’s where our class went on Monday:
We started off with Robert Kaplinksy’s How Much Did Patrick Peterson Lose By Not Cashing His Check problem. Go ahead and read his lesson plan.

What made this problem great for our class was the discussion that occurred before any math happened. An amazing argument bubbled up with one side saying “Who cares….what’s the big deal” and the other side saying “That’s just super insensitive…..I could use the interest off that account”. My class from the beginning of the year was back! They had put away the drama that had happened between them and focused on the problem. We guessed at the interest he was losing daily. And then using the info from Robert’s site calculated the interest in the first couple of days. Then broke out the Finance Solver to determine how much was lost for the 27 days.

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The class wanted to know more! They wanted to know what he would lose if he didn’t cash it for the year, 2 years, 5 years!! So we did that too.

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Next, we investigated Robert’s How Much Should Dr. Evil Demand?


Read Robert’s post to see the plan.

Again, with this group, we didn’t use exponential functions…but the Finance Solver to determine what $1000000 would be worth 3o years later with average inflation of 5.33% per year. We also extended to find how long we would have to wait for $1 million to be worth $100 billion.

Thanks for the help!

 

One Favourite, Two Fixes: Nov 5

I need help! Every time I go through a lesson there is aways something I want to fix. Something that can be improved upon. Something that can get my students to think a little deeper about the math or something that can make my organization better, or class management better. Sometimes I can think of how to fix it, sometimes I know there is a fix out there and I can’t see it. That’s where you come in.

Each week I’m going to post one idea from my lessons that was my favourite and two topics that I need help on. Two topics I hope to fix. I’m hoping that in the comments or on Twitter you amazing readers can help me out with some hints, tips, and suggestions.

Favourite

For the last two weeks one of my class’s atmosphere seemed poisoned (see below).  My favourite this week though helped restore (for a short time) that atmosphere back to where we have spent most of the year. Warm-ups to the rescue! Our warm-ups dive into our Everyday Math curriculum like nobody’s business. Sometimes they take 10 minutes, sometimes 20….and I’m ok with that. Each one has my class engaged for that time. This week my favourite was Dan Meyer’s Dueling Discounts.

The kids each had a copy of a $20 off coupon and a 20% off coupon. For each item I showed, I had them hold up their choice of coupon to use. I was loving that all students were actively engaged and WANTED to know which was the better deal. It was so nice to see this with my kids again.

First Fix

Like I said above, my class’ atmosphere has seemed poisoned for the last two weeks. Before that we had an amazing atmosphere…..all kids worked well together. They sat in random pairs everyday, they were engaged! Then a few things happened.

  • Outside of class somebody was texting things they shouldn’t to somebody else….VP said they are not to work together.
  • A new student was added and talks a lot
  • Two others can’t work together because of a fight they were in last week.

So now we don’t have the awesome —we all can work together and build off each other atmosphere I love and had. I’m looking for tips to try so we can re-create our atmosphere we started with. Any ideas?

Second Fix

In my grade 9 mfm1p class this week we worked on solving proportions through the Smart Car Smash activity. screen-shot-2016-11-06-at-9-22-20-am

After going through super gross lesson and seeing the kids smile and cringe at the same time it was time to practice our strategies. I gave out this sheet below.

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This is where all the excitement for math was lost….”ah man, a worksheet”. Most students solved the problems. Some kids who were actively engaged with the first problem now were shut down. How can I keep the practice portion that I need but keep engagement up?

Thanks for reading. I would love your help. Share your suggestions on Twitter or below in the comments.

Helpful Fixes from Readers:

Updating the MFM1P Spiral

“Have you taught for 25 years? Or have you taught one year 25 times?”

I don’t think I’ve taught the same course the same way ever. Why would we? We don’t have the same kids in front of us. And especially with the resources at our finger tips from our colleagues inside and outside of our schools. I’ve wrote before about the power of #mtbos and it changes the way you teach.

I started spiralling the MFM1P course a few years ago with Kyle Pearce. Since then I’ve taught that course 3 or 4 semesters in row…..and never the same way. New amazing lessons and tools are springing up. For past lessons I wasn’t completely happy with I’ve got to see if this new lesson or that lesson will help my students understand the concepts more deeply.

One change I wanted to make was to include solving equations earlier in the course. In my old plan I waited to introduce it after introducing linear relations. But, after teaching solving equations using the Double Clothesline and the puzzle nature of learning it that way….I can introduce it now and continually practice our skills through warm ups.

If you want to follow along as my day-to-day plan unfolds follow this link! If any of you have been spiralling MFM1P I would love compare notes, or see your plans.