problem solving

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!

 

 

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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.

 

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Nrich’s Digit Doozy

If you are a math teacher who hasn’t taken some time to get lost in the problems on Nrich, stop reading this and go there  right now.  You’ll need to finish reading this post tomorrow because that’s how long you will be immersed in its seemingly endless array of engaging problems.

Today, my intention was to do a little starter activity with my 9th graders to help support their number sense.

Here’s the basis of the problem:

american billions C

For two out of three of my classes, it turned into a whole-class period problem-solving extravaganza.  Seriously.  30 minutes later, the brain sweat is still palpable in the room.  There were so many calculators in use, I think the smartphones were starting to get jelous.

Some chose to use whiteboards, some choose numbered cards 0 – 9 while some wanted to use paper.  It was so interesting to me to see them figure things out that must be true about the different number places.  A few remembered the divisibility rules for 3 and shared them.  Then they were able to put the divisibility rules for 2 and 3 together to get divisibility for 6.  I didn’t even know that there was a divisibility rule for 4 and 8!

Some student observations:

  • The 2nd, 4th, 6th, and 8th numbers need to be even.
  • The last number must be 0.
  • The 5th number must be five, since the last number must be 0.
  • The first three numbers have to add up to a multiple of 3.
  • The first 9 numbers need to add up to a multiple of 9.

I even had a student say, “How much longer do we get to play this game?”  Music to my ears.

It’s difficult to give students a task that you know most of them won’t solve which is why I’ve shied away from this one in the past.  I made sure to praise the efforts of those that were able to get their numbers to work for all except one of the digits.   (For example, their 2, 3, 4, 5, 6, 8, 9, and 10 digit numbers worked, but their 7 digit number didn’t).

Nrich gives another variation on this task by making it a game.  Basically, students take turns creating 1, 2, 3…digit numbers by choosing from the 0 – 9 digit cards until someone can’t use any more of the cards.  I think having them play this activity as a game would help alleviate some of the discontent of feeling like this problem was too difficult to solve.

 

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Vegan Teacher Crazy about Cheeseburgers

A year and a half ago, I made the best dietary decision of my life and decided to try a vegan diet for 30 days.  Fast forward to now, I love the vegan lifestyle and I’d never go back to a diet filled with animal products.  I know too much.  But that’s a story for another post.

A couple of weeks ago, I logged into Robert Kaplinsky’s presentation on Global Math Department.  He started off with a visual, which is usually good to draw listeners into the presentation.  However, this visual was a cheeseburger.  And he went through more and more visuals, and the cheeseburgers kept getting bigger and bigger until finally I’m face to screen with 100×100 cheeseburger from In N’ Out burger.  I try very hard not to be one of those ‘enlightened and superior’ vegans who constantly judge the dietary choices of others, but these burger pictures were not how I envisioned spending my Tuesday evening.  His methodology had my attention however.

After explaining his problem solving process and distributing his problem solving template, he threw this photo into the mix and asked,

“How much would that 100×100 cost?

Now I was hooked and needed to figure out how much that 100 x 100 cost.  I didn’t care if it was a cheeseburger or a truckload of kale.  The wizardry of Robert Kaplinsky drew this vegan teacher into the problem solving process and made me care how much this monstrosity of a cheeseburger cost.  Brilliant.

Then Robert Kaplinsky threw down the dynamite:

That’s right.  The actual receipt of this 100×100 cheeseburger.  A boatload of kudos to Mr. Kaplinsky for presenting something that was simple, with some great mathematics to go with it.

I’m glad this weeks ExploreMTBos mission was LISTEN and learn.  This was a great presentation, a great lesson, and a great resource.  I’m glad I took the time to listen to Robert Kaplinsky’s presentation, even if it wasn’t so appetizing on the outside.

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Visual patterns with a side of awesome sauce

Regular old Wednesday turned amazing today when I posed pattern #2 to my math recovery class, a remedial math class for kids to recover credit from a previously failed course. It may not need mentioning, but just to be clear, these kids hate math and think they’re no good at it. In pattern #2, the kids need to find how many cubes are in step 43 and the surface area of step 43. Side note:  My kids wondered, why 43, Mrs.Nguyen?


Anyway, finding the surface area was where the magic started to happen. I had 4 or 5 kids out of this class of about 15 get seriously invested in finding out the answer. They were drawing pictures, explaining their thinking to one another, figuring out different ways to think about the problem. It was inspiring and motivating for both them and me.

As if that wasn’t enough to make it a great day, I decided to pose the problem to my College Algebra class as a starter and try my hand at the 5 Practices for Orchestrating Productive Mathematics Discussions. My expectation was that they found the number of cubes and surface area of step ‘n.’ What was gorgeous about this problem was not necessarily the answer, but the numerous ways they came up with to arrive at the nth step. Here are a few:

n + n + (n-1) + (n-1) + n + (n-1) + n + (n-1) + 2

4(n-1) + 4n +2

4(2n – 1) + 2

6 + 8(n-1)

4[n+(n-3)] +10

6(n-1) + 2n + 4

8n – 2

What was even more powerful was, as Ben Blum-Smith calls, an effing game changer.  He’s right, and this was beautiful.  I used the tactic he lays out in his blogpost where students are asked to summarize the ideas of someone else.  I had a few try to slyly summarize their own ideas, but alas, I would have none of it.  As a result, I had more engagement, more involvement, and more buy-in that this problem solving process is helping them to understand the mathematics more deeply.

Here is an exchange between two students (T and C) that is worth highlighting.  T is the student who came up with 6 + 8(n-1) as the surface area for step n:

C: Oh, I see.  T just used the arithmetic sequence formula.  The first term is 6 and it goes up by 8.

T:  Actually, that’s not what I was thinking.  I thought that there were 8 sides of the figure that had ‘n-1’ squares and then 6 squares left over, two on the caps and 4 in the corner.  OH, you’re right, it is the formula.

Then the lights came on.  This girl who had probably only known mathematics and algebra to be a long list of rules, procedures, formulas, and practice was able to experience that developing a conceptual understanding of this pattern help her to create the arithmetic sequence formula.  It was the bottom-up approach that I’d been talking about all trimester where developing conceptual foundations are where real math learning happens.

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Nrich Love Affair: MTBoS challenge #1

I told my husband that if we weren’t already married, I’d run away with nrich in a heartbeat.
That being said, it’s probably no surprise that my favorite problem comes from nrich: consecutive sums and is my response to the explore the mtbos mission 1.
I’ve used this problem a few times, with high level students, low level, and in between as well.

Here’s a poster with a general overview of the problem.  The link from nrich provides starting help as well as teacher notes and a solution.

Some things I love about it:
1. So many points of entry and a low barrier.
2. So many paths. I’ve had so many different conjectures arise from this problem because of the open-ended nature of it and its ease of exploration. The numbers are not intimidating so students are unafraid to explore some of their findings.
3. Multiple extensions. For example, do consecutive differences work similarly? What about consecutive products? Or better yet, the difference of consecutive products!
4. Students organize their work in so many different ways. It’s completely fascinating to see it happen.

When doing this with a lower level class, I usually have them make a list of noticings and/or wonderings. This way, the patterns they learn to communicate what they believe to be true in their head. I may challenge them to generalize a little, especially with odd numbers always being a consecutive sum.

The most exciting thing that happens when I do this problem is the “what ifs” that students can’t help but think up themselves.  For example, What if we took the difference of consecutive numbers?  What if we took the sum of consecutive odd or even numbers?  Consecutive square numbers?  Triangle numbers?  Negative numbers?  It’s pretty amazing to be part of.

If you have tried or try this problem in the future, I’d love to collaborate on it.

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Correlation Investigation

A benefit of having advanced students for probability and statistics is that there is often a level of curiosity in the room that can drive an entire class period of discussion. Sometimes.
We were about to start a unit on linear modeling and correlation, and I decided to start with giving them the data and the best-fit line and then having them determine the correlation coefficient based on some examples. We used a data table of 21 common cereals, all with calories, sugar, fat, and carbs per 1 cup portion.
The students first were given time to notice and wonder and I had them determine what two columns of data were most closely correlated and least related. We came up with a list for both and then we started number crunching.
The NCTM website has a nice set of core math tools . The Java-based app lets you put in data and then will have students “guess” the correlation coefficient from a list of 4 choices. Some kids made conjectures about how correlation coefficients were calculated. Others refuted these with counter examples. It was a beautiful discussion. In the process, we compiled the correlation coefficient for all of the data sets and this answering our other question about which sets were most and least related.
Finally, I set them off on the map shell lesson about devising a method for measuring correlation. This was just the intro to this lesson. The real fun started the next day…

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