Today my students will have a sub since I am attending our state’s math teacher conference (#mctm). Given the overall success of our Desmos Carnival activity from Monday, I decided that a computer lab activity might be fitting. Since we are starting a unit on probability, I took the opportunity to use some Nrich probability simulations.

I’m also attempting something new with Google Forms. I’ve observed my colleague, Dianna Hazelton, incorporate Google Forms, Sheets, and Docs quite seamlessly into her trigonometry and prob/stat classes. Her success with these apps made me eager to try them out as well. I like that I’m able to “see” what they did via the google form responses right away rather than have a pile of papers waiting for me on Monday.

# Nrich

# 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:

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.

# Pattern Power

If you have little kids and you’ve been privy to an episode of Team Umizoomi, then perhaps the title of this post evoked a little jingle in your head. You’re welcome; I’m here all day.

My daughter, although she doesn’t choose Umizoomi over Mickey Mouse as often as I’d like, picked up on patterns relatively quickly after watching this show a couple of times. She’s 3 years old, and she finds patterns all over the place. Mostly color and shape patterns, but a string of alternating letters can usually get her attention as well. These observations of hers made me realize that pattern seeking is something that is innate and our built-in desire for order seeks it out.

High school students search patterns out as well. For example, I put the numbers 4, 4, 5, 5, 5, 6, 4 so that the custodian knew how many desks should be in each row after it was swept. It drove students absolutely CRAZY trying to figure out what these numbers meant. I almost didn’t want to tell them what it really was as I knew they’d be disappointed that it lacked any real mathematical structure.

I’m not as familiar with the elementary and middle school math standards as perhaps I should be, but I’m confident that patterns are almost completely absent from most high school curriculum. Why are most high school math classes completely devoid of something that is so natural for us?

Dan Meyer tossed out some quotes from David Pimm’s Speaking Mathematically for us to ponder. This one in particular sheds light on this absence of pattern working in high school mathematics:

*Premature symbolization is a common feature of mathematics in schools, and has as much to do with questions of status as with those of need or advantage. (pg. 128)*

In other words, we jump to an abstract version of mathematical ideas and see patterns as lacking the “sophistication” that higher-level math is known for. To be completely honest, this mathematical snobbery is one of the reasons I discounted Visual Patterns at first. Maybe it was Fawn Nguyen’s charisma that drew me back there, but those patterns have allowed for some pretty powerful interactions in my classroom. I’ve used them in every class I teach, from remedial mathematics up to college algebra because they are so easy to differentiate.

I think high school kids can gain a more conceptual understanding of algebraic functions with the use of patterns. For example, this Nrich task asks students to maximize the area of a pen with a given perimeter. The students were able to use their pattern-seeking skills to generalize the area of the pen much more easily than if they had jumped right from the problem context to the abstract formula.

I also notice that the great high school math textbooks include patterns as a foundation for their algebra curriculum. For example, Discovering Advanced Algebra begins with recursively defined sequences. IMP also starts with a unit titled Patterns. I think these programs highlight what a lot of traditional math curriculums too quickly dismiss: patterns need to be not only elementary noticings of young math learners but also valued as an integral part of a rich high school classroom.

# Puzzling Perseverance

School mathematics has a bad reputation for being intellectually unattainable and mind-numbingly boring for many students. Proclaiming the falsity of these beliefs is usually not enough to convince kids (or people in general) of their untruth. Students need to experience their own success in mathematics and be given the opportunity to engage in curiosity-sparking mathematics. For me, one of the very best moments in a classroom is when a self-proclaimed math hater fully engages in a challenge and is motivated to work hard to arrive at a solution.

Enter January 2nd and 3rd. Students are back for a two-day week which they view as punishment and a rude-awakening from a restful winter break. To boot, the Governor Dayton announced today at about 11 am that all Minnesota schools will close Monday, January 6th due to impending dangerously cold weather. You can imagine where the motivation level was in school today.

As the CEO of room 114, I decided to make an executive decision and do a puzzle from Nrich (shocking, I know) in my probability and statistics class. Technically, the students could use the mean or median to help solve the problem, so I wasn’t veering too far off of what I had previously planned.

The Consecutive Seven puzzle starts like this:

Initially, one student began by explaining to me that she took one number from the beginning of the set, one from the middle and one from the end. Then she figured the other consecutive sums needed to be above and below that number. (*Spoiler alert*: These numbers actually end up being the seven consecutive sums, so I was very interested in her explanation of how she arrived at those particular answers. )

It’s worth noting that this student’s first words to me at the beginning of the trimester term were, “I hate math and I hate sitting in the front.” So you can imagine my excitement when she dove in head first into this particular task, happily and correctly.

Adding to my excitement about the class’s progress, another girl (who was equally enthusiastic about math at the beginning of the term) was the first one to arrive at a correct solution. And although she probably wouldn’t admit it, she was thrilled when I took a picture of her work. And I am more than thrilled to display it here:

If you were wondering about how math-love girl #1 fared in completing the task, she persevered and impressed her skeptical cohorts:

This phenomenon fascinates and excites me that students, when confronted with a puzzle, highly engaged and motivated throughout the lesson. Dan Meyer summarized this idea nicely on his blog recently:

*“The “real world” isn’t a guarantee of student engagement. Place your bet, instead, on cultivating a student’s capacity to puzzle and unpuzzle herself. Whether she ends up a poet or a software engineer (and who knows, really) she’ll be well-served by that capacity as an adult and engaged in its pursuit as a child.”*

And who knows. Maybe one of the girls featured above will become a puzzling poet.

# Curiosity Driven Mathematics

In my very first years of teaching, I used to have students ask me, in that age-old, cliche teenage fashion, “When are we ever going to use this?” I vividly remember my response being, “Maybe never. But there are plenty of other things we do in life, like play video games, that have no real-world application. That doesn’t seem to bother us too much.”

In fact, if every moment of our lives needed to apply to the bigger picture, the REAL-world, when would we do anything for pure enjoyment? or challenge? or even spite? I know kids are capable of this because some of them spend hours upon hours a day engaging not only with a video game but also collaborating with other people through their game system.

And furthermore, where do we think this resentment for learning math really comes from? I have a guess…probably adults who have realized that through the course of their lives, being able to solve a polynomial equation algebraically is not all that useful! News flash, math teachers: Our secret is out!

There are many kids across all levels of achievement that will not engage in the learning process simply because the state mandates it or the teacher swears by its real-world relevance. Students (and arguably people in general) are motivated by immediate consequences and results and cannot easily connect that the algebra they are learning today will be the key to success in the future. They do not care that if they don’t nail down lines, they’ll never have a prayer understanding quadratics. If they are bored to death by linear functions, I can’t imagine that they have even an inkling of desire to comprehend the inner workings of a parabola.

What does resonate with learners is the satisfaction of completing a difficult task, puzzling through a complicated scenario, or engaging in something for pure enjoyment. Kids are naturally problem-solving balls of curiosity. There are ways to provoke curiosity and interest while simultaneously engaging in rich mathematics. I think many teachers assume that in mathematics, especially Algebra, curiosity and deep understanding need to be mutually exclusive, and I’m positive that mindset is dead wrong. For example, show this card trick to any group of kids, and you’d be hard-pressed to find a group who isn’t trying to figure out how it works. I also think you’d be hard-pressed to find the real-world relevance to a card trick. It’s still no less amazing, as well as algebraic.

# Nrich – Factors and Multiples Puzzle

Nothing gets me more excited about teaching mathematics than a task that can engage my lower level students while simultaneously challenge my high achieving students. The Factors and Multiples Puzzle from Nrich did just that. (Thanks to @drrajshah for posting this on twitter.)

I’m glad I used this in multiple classes because if nothing else, it gave students the opportunity to learn about triangular numbers! What a testament to the fact that we don’t allow students to explore with numbers nearly enough: I’ll bet only one student out of 60 had any idea what triangular numbers were. A fantastic, interesting set of numbers, arithmetically and visually, was unbeknownst to 99% of my students.

My math recovery students were intrigued by the puzzle portion of it. In fact, I have one student in particular who is not particularly motivated by much . He’s a ‘too cool for school’ kind of kid, and he’ll tell you as much. When I bust out a puzzle, he’s all in. And when I say ‘all in,’ I mean 100%, until he solves it. It’s pretty awesome stuff to have been able to catch his attention and see how cleverly he thinks through things. Amazing.

I also gave this task to a group of advanced students. An interesting strategy these students developed was to grab a whiteboard to work out some patterns in groups of numbers. I loved walking around and hearing their strategies. As some groups finished, they started walking around and giving tips (not answers) to other groups. It was wonderful.

One of my particularly eager students taped his together uniquely. I appreciated his humor. 🙂