Nrich-ing New School Year

I have made no secret of my unwavering devotion to Nrich.  This University of Cambridge-based website is a treasure trove of rich tasks.  The best part about this website is the ability to engage students with abstract algebra concepts.  Very infrequently does Nrich apply their problems to any real-world concept.  They let the arithmetic itself set the hook.  And the algebra solidifies the concept.

Example:

Special

As a bonus, now that school is underway, they have many “live” problems which are open for class submission.  I’m hoping that my class might want to submit solutions for the Puzzling Place Value problem.

There are a lot of great problems here that I’ve used in my classes.  The only problem with available new problems is now I want to devote all my time to solving and implementing them!

 

 

Nrich – For What It’s Worth

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:

board3

 

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.

Untitled

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

Sitting in a Circle, Talking about Numbers

“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:

Multiplication Square C thumb (1) thumb

 

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.

 

The Anti-Answer-Getter

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.

Pair Products – An Nrich Favorite

In a few short weeks, I will be making a presentation at Twitter Math Camp on my favorite Nrich Tasks.  I know a lot of teachers have reservations about integrating rich mathematical tasks into their regular routines so I want to focus on problems that have that “traditional” feel while still allowing students to explore mathematical relationships more deeply.

Pair Products is an amazing offering by Nrich and its low barrier to entry makes it accessible for all students.  After working through the problem myself, Nrich offers additional questions to raise the ceiling.

Pair Products C

Additional Questions to Ask:

  1. What happens when you use 4 consecutive even or odd numbers? 5? 6? n?
  2. What happens when you use 4, 5, 6, n consecutive multiples of 3? Multiples of 4? 5? 6?
  3.  (My Favorite) What happens when you use n consecutive multiples of w?
  4. Does your generalization from #4 hold for numbers that increase by .5?  (For example: 3, 3.5, 4, 4.5)

My favorite Nrich pair, Charlie and Alison, offer two different approaches.  Charlie explains a clear algebraic manipulation to arrive at two expressions with a numerical difference.  Alison, on the other hand, represents the product of numbers with an area model.

Alison

An interesting challenge might be to ask students to show the area model that Alison employs for some of the additional questions.

 

#TBT Math Style – SMILE Cards

While perusing UK’s National Stem Centre website recently, I came across something called SMILE.  Here’s what the website has to say about them:

SMILE (Secondary Mathematics Individualised Learning Experiment) was initially developed as a series of practical activities for secondary school students by practising teachers in the 1970’s. These mathematics books are intended to be not only a source of ideas but to be a flexible resource that can be adapted to different circumstances and ability groups. 

Not that it takes much to ignite my mathematical excitement, but the 1970’s got my blood moving.  I was sold.

Here’s a sample:

photo

 

It sort of shocks me when I use these kinds of resources and kids ask, “why is color spelled wrong?” I wonder what they’d say if they knew the rest of the world says “maths” instead of “math.”

Anyway, I could spend about a day looking through the National Stem Centre.  If you’re going to check it out, make sure you have Evernote ready!

 

If you’re decently competent in the area of probability, you might know that your chances of winning fall below things like “death from a vending machine” and “having identical quadruplets.”  This doesn’t stop many people from playing.  I think playing the lottery is more about the chance to dream of what our lives would be like with that much money rather than actually believing we could win.

In the UK, the lottery consists of picking 6 numbers between 1 and 49.  Any player to match all 6 numbers is the grand prize winner.  The chances of this are certainly astronomically low.  A fun question to ask a class of students:  If we bought a lottery ticket for every different combination of 6 numbers to ensure we’d win, how high would that stack of tickets reach?

In the task Do You Feel Lucky, Nrich tackles the idea of evaluating advice given on raising your chances of winning this seemingly impossible lottery. Students are asked to comment on the validity of the advice given and one in particular caught my eye:

When picking lottery numbers, choose numbers that sum between 100 and 200 because the total is rarely outside this range.  

Whoa.  There are so many ways we could evaluate the validity of that claim.  So I sent my students off to the races. Most of them wanted to use a random integer selector and then gather the data from the class’s trials.

IMG_5135

GeoGebra Results:

Lottery

Lots for them to talk about here.  Lots of questions for them to ask as well.  Does the range seem too wide?  Do we have enough trials?  What do we make of the dip in the middle?  Should we change the bar graph to have different class sizes?  Would a box plot have been more appropriate?  What about the descriptive statistics?  Would those help us out?

I’m hoping next year to extend this into more of a class activity rather than an impromptu discussion.