Nrich – For What It’s Worth

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One of my favorite problems (and the one I presented at TMC this year) is What it’s Worth? from Nrich.  To say I “like” this problem would be like saying Sarah Hagan “likes” interactive notebooks.  Clearly an understatement.

Anyway, here’s the prompt:

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What I like most about this problem is that there are so many, OH so many, methods to solving it.  It is a FANTASTIC way to get students to focus on the pathways to the solution rather than the solution itself.  After the students figure out the value of the question mark, they go about discussing the numerous methods they used in order to arrive at their answer.  Furthermore, the problem includes 6 “beginnings” of solutions and learners then need to make sense of those as well as determine how a solution was reached along that path.

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Left: New format for discerning methods. Right: Old format for discerning methods.

To my surprise, along with Nrich’s site updates, this problem has improved as well.  Rather than showing a written start to the problem, provided are 6 visual introductions.

This allowed for an incredible amount of discussion involving each method.  And even those METHODS broke down into different methods.  It was method madness (awesome madness).  0917141216-1

Welding Math and Metal – Day 2

If I needed to choose the most productive portion of most students’ week, Monday morning, first hour would be pretty low on the hierarchy of engagement.  I was undeterred because making sure we had the correct solution was important.

We discussed that the radius of the spool would decrease every time a layer of wire was used.  They began calculating the resulting wire as layers were removed.  This served as an excellent opportunity to introduce summation notation and a great practical use for the mathematics behind it.  It seemed like a much better option than to add up dozens of calculations anyway.

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When we arrived at our correct answer (with the desired units) of 1.98 miles, the questions and estimating didn’t end.  They wanted to know how far they could stretch such a wire.  Would it go to the edge of our campus and back?  Would it go from here to the middle school?  Could you go all the way to the grocery store?

They settled on taking the wire, running it out to the edge of the soccer practice fields and then running it all the way to the middle school sign.  It ended up being, to the hundredth, the exact amount of wire we had, provided that someone would stand and hold the wire at the edge of the soccer field.  I loved the attention to precision. I also loved that they were so savvy with Google Earth.

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The Welding of Metal Tech and Math

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An early morning text from my brother prompted this tweet:

We estimated a little, but the Slinky didn’t yield much discussion until someone shared: I wonder how much wire is in one of those Mig welder spools!

If you are like me and have no idea what a mig welder is, here’s a photo:mig_welder

These things hold massive spools of 1 mm thick wire.

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Luckily, the welding instructor was willing to part with one of these for the hour.  Unluckily, the only information we had was the  1 mm thickness of the wire and the spool’s weight of 44 pounds.

Our initial thought was to weigh a snippet of wire and then scale it to the entire spool, but I was sure that the welding teacher wouldn’t have appreciated the rogue math teacher messing with his supplies.   I praised them for the interesting method anyway.  They then began measuring:  height, diameter of the spool, diameter of the inner circle, diameter of the wired portion, number of wires going up the spool.  It was math-magical.

Here are their calculations:

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Now, I realize that these are a little off because we needed to take into account that the circumference of the spool is getting smaller as we move inward.  But I was very pleased with their work thus far and their vigor in posing this problem and then working to solve it.  I’m excited about where we can go with this type of problem posing.  There are only eight students in this class, all boys that have metal in their bloodstream.  I am hoping that with these sorts of ideas, I can engage them in math that excites them more often.  Maybe I can even get them excited about my Slinky question.

Sitting in a Circle, Talking about Numbers

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“I feel like all we do is sit in a circle and talk about numbers.   It doesn’t even feel like work.”

“This class is more exhausting than my PE class!”

“It’s nice to be confused and then un-confuse ourselves.”

These are words I’ve overheard from my college algebra students this year.  I couldn’t be more pleased with the strides they are making with my problem-solving framework.  I learned the hard way last year that you cannot just throw a problem solving scenario at a student and expect them to immediately persevere, even if they understand the underlying mathematics involved.  Having learned from my mistake, I sequenced the problems this year in a way that has worked to build on their Algebra problem-solving skills.  Furthermore, I’ve put them in groups of 3-4, which has helped tremendously in getting them to talk about their approaches.  Last year, while in pairs, the conversations didn’t occur as naturally as I had hoped.    Here are a few of the problems we’ve tried:

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Additionally, we’ve used other Nrich problems such as Odds, Evens, and More Evens.

And to add some non-dairy whipped topping to this algebra awesomeness, my students are breezing through visual patterns and having some great conversations about them.  Credit here is due to their fabulous algebra 2 teachers who began visual patterns with them last year and let them struggle with them.  The result has been deeper connections and a more thorough understanding.

 

Do You Let Yourself Fail?

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I sat down this weekend to do some recreational mathematics with a friend.  Maybe you know him; his name is Justin Aion.  He writes a pretty cool blog over at Re-Learning to Teach.

I made it a goal of mine this year to work on some geometry for a few reasons.  First, I’m not that great at it.  Second, the students at our school historically struggle with it as well.  Two of the problems we chose were from the Five Triangles blog.  And to be completely honest, I sucked.  I sucked a lot.  I sat there for much of the Google Hangout drawing and drawing the figures and then writing down what Justin had eloquently discovered.  And then nodding in agreement. Here are pictures of Justin’s and my respective work:

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Then we decided to work on something I thought was more my cup-o-mathematics tea.  Turning to the Math Forum, we tried this weeks scenario for Trig/Calculus.  How silly of me to assume that since this is just the beginning of the school year, perhaps the task could be solved using Algebra.  Of course Justin busted out the calculus seamlessly and like a pig in numerical-feces, excitedly worked his way to viable solution. (It turns out that applying algebra to this problem was not as straight forward as it might have seemed.) Again, I felt defeated by the mathematics.

The point here is that doing math that’s unfamiliar is hard.   Thinking deeply about problems is hard work.  Applying previous knowledge to a new situation is also taxing.  What I really took away from hours of difficult mathematics was an empathy for the anxiety of many of my students when I ask them to do the same.  It is disingenuous of me to expect my students to persevere through problems if I’m not willing to do the same.  So, I’m committing to being uncomfortable, mathematically, and I will get better.  My geometry skills will improve, and perhaps I’ll be able to revisit my long lost calculus pals, derivative and integral.  The important thing is that I’m willing to try and willing to fail.   In the long run, I think my students will benefit, and I know that I will as a teacher.

The Anti-Answer-Getter

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I must start off today saying that I have never experienced such a fantastic start to the school year than I have this year.  The energy within our department is almost palpable, and I know that the students are catching on as well.  Here’s an email I got from one of my co-workers this morning:Untitled

I want to give credit to Teresa and Dianna because they were more of the driving force behind encouraging the use of Plickers.  I’m thrilled with the result nonetheless.

The group that impressed me the most today was my first hour, math recovery.  These are kids who have previously failed a math class and are recovering credit.  You can imagine the lack of math love in the room.  Here was their prompt:

Make 37 1885 C

 

SPOILER ALERT:  I’m going to reveal the answer so if you’d like to try it for yourself, stop reading.

I had them come up with ways they could make 37 using different amounts of numbers.  It seemed that we could get 36 using 10 numbers or 38 using 10 numbers but couldn’t quite get 37.  Then we tried getting 37 using 9 numbers or 7 numbers.  We had some good discussion about which strategy seemed the most useful.

One student in particular mentioned that he wanted to add some and subtract some but he felt he would always be short without a 2.  I had them share their results on the board and I was very satisfied with the effort I’d seen.

I was nervous about the answer reveal because as it turns out, it’s impossible to make 37 with 10 numbers.  What we were able to do is focus our attention on what we DID discover, rather than the fact that there was no answer.  We discovered that Odd + Odd = Even, Even + Even = Even, and Even + Odd = Odd.  Because there is an even number of odd numbers, an odd sum is not possible.  I was more pleased with this result than any single answer they could have given me.  I expected a backlash from a group of students used to answer-getting but found that they were able to embrace a learning activity that didn’t one final answer.  I’ll mark that class period in the win category.

Talky, Talky, Talky. No More Talky.

Because I’m hyper-interested in helping to create a space where kids feel comfortable sharing ideas and making mistakes, I began my classes today with the Talking Points activity that Elizabeth Statmore (@cheesemonkeysf) shared at Twitter Math Camp this past summer.  Learning that a tight rule of No Comment was a cornerstone of the activity intrigued me to try it in my classroom.  Productive conversations in math class don’t happen automatically very often.  I’m hoping that using this process helps students to use exploratory talk around mathematics.

The No Comment was difficult for students, but I realized quickly, it was difficult for me as well.  For example, when debriefing with the whole class, I was tempted to comment…after each group presented.  I had to tell myself each time a group gave a summary that there wasn’t a need for my comment.  I was tempted to clarify thinking or give a follow up explanation.   I needed to let the groups own their experience.

This realization made me cognizant of the other times a comment by me is unnecessary following a student response.   How many times have I insisted on having the last word in the class?  How many times have I summarized a student’s thinking for him or her?  Hopefully, as students move toward being more exploratory with their discussions, I can move toward being less dominant in the conversation.