Somewhere between Concrete Sequential and Abstract Random

It occurred to be relatively early in during the trimester this past fall that my college algebra students (generally) have no idea what I mean when I say “quadratic function.” This isn’t because they have never heard it, learned it, or used it. But that technical of a term simply has not stuck around in their long term memory.

So, similar to linear functions, we start with a pattern:

from youcubed.org and visualpatterns.org

I then had them make posters including how it is growing, what the 10th, 100th, 0th, and -1st cases would look like, table, graph, expression, and relationship to the pattern. Instead of one large poster, they use 4 smaller pieces of paper and tape them together. That way each group member can contribute simultaneously.

I noticed:

It is difficult for students to describe how an irregular shape is growing.
It is even more difficult for them to describe something abstract like the -1 case.
Many of them expressed the overall growth as “exponential.”
Most could easily see the two rectangles formed and determine the dimensions with respect to “n.”

I wondered:

If they could connect the work with patterns to other quadratics.
How to have a meaningful discussion around the “exponential growth” issue.

Their homework was to answer similar questions for this pattern from You Cubed’s Week of Inspirational Math:

growing

Spoiler alert: the rule for the pattern is f(n) = (x + 1)^2 or f(n) = x^2 + 2x + 1

So where do we go from here, two days before Winter Break? My goals are to review some specifics on quadratic functions and simultaneously help the students make connections between different representations. I know what I must do. I must channel my inner Triangleman.

[Backstory: Christopher Danielson and I go way back. At least to 2014. Maybe even 2013. Seriously though, I strive to organize my college algebra class the way Professor Danielson describes in his blog. I have picked his brain on more than a few occasions and he is gracious enough to give me advice in certain curricular areas. In short, his philosophy titled “They’ll Need it for Calculus” is the foundation of my College Algebra course. ]

Ok, back to room C118.

Me: Write down everything you know about the function y = (x+1)^2

(Most write down the expanded form, some start to graph, but not many)

Me: What other ways can we represent this function?

Students: Tables! Graphs! Pictures! Words! Patterns! Licorice!

Me: Sweet! Let’s do all of that, minus the Licorice.

(I give them a few minutes to create a table and a graph.)

Me: NOW, write down everything you know about this function.

I circulate and hand each group a half sheet of paper.

Me: Write down the most important thing on your groups list.

At first I wasn’t really concerned what exactly they wrote down, but how they defended their choice. Then I came across this in all of my classes:

We came up with a pleasing list of attributes of a positive parabola that included vertex placement, end behavior, leading coefficients, and rate of change.

Next up for discussion: Parabolas grow exponentially.

Me: Turn to your partner and tell them whether you agree or disagree with this statement and defend your choice.

Students: Yes, words, words, words.

Me: Ok, now the other partner, say how you know something grows exponentially.

Students: Multiplied every time, more words, blah blah blah.

So we agreed that 2, 4, 8, 16, 32, 64… is an example of something that grows exponentially.

Me: Numerically, how can we tell how something is growing?

Students: (eventually) Rate of Change!

We came up with this table and agreed that these two functions were definitely NOT growing in a similar way.

Now on to helping them understand what it means for something to grow quadratically…

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Stringing Students Along

If I’ve done one thing consistently this year, it has been Number Talks in my Probability and Statistics classes. I have seen students who, at the beginning of the trimester, told me flat out, “I can’t do math in my head.” Now that Trimester 1 is coming to an end, those same kids are volunteering multiple strategies in these mental math challenges.

During the trimester, we started with the dot image below and have moved through the four operations, onto decimals, and even dabbled in fractions and percents.

How many dots are there? How did you count them?

What’s important to me with these number talks is the visible improvement I saw in my students’ confidence and flexibility with numbers.

I’ve shared before about my experience with number talks and I plan to continue these throughout the rest of the school year. But at the NCTM Regional conference in Minneapolis a couple of weeks ago, I had the pleasure of attending Pam Harris’s session on Problem Strings. I found that problem strings are very useful when wanting to elicit certain strategies or move toward generalization of a strategy.

Here are my notes from a problem string I did recently with the same group of students I have been doing number talks with.

I noticed:

Many students did not use “17 sticks in a pack” to figure out sticks in 10 packs
Many more strategies than expected were shared to find the number of sticks in 6 packs of gum.
Most students were able to generalize about number of sticks in n packs.
Participation increased with the multiple opportunities to volunteer their strategies.
Students could see relationships between the numbers and find the solution in multiple ways because of that relationship.
There are many implications of these problem strings in secondary mathematics. In this example, the slope formula can be easily elicited through further exploration of the table we made.

I’ve read all of Pam’s books, but getting to see her present problem strings in person really illuminated how these can be useful in my classroom. Thanks, Pam, for opening my mind to this and letting me fangirl you. I’m looking forward to doing more of these, including recording them. Stay tuned.

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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:

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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Bingo Lingo

This time of year, the standards used to measure the success of a lesson may look different than they do at other times of the year. For example, some teachers might consider “Students not using worksheets to have paper airplane throwing contest” to consistute a lesson well executed. To a certain extent, I am joking, but there’s a thread of reality there. Think back to a time when your excitement for a future event prevented you from doing anything productive. Now imagine leading a room full of 32 people with that same excitement and handing them a manual for their new scanner/copier. You get the idea.

I can usually distinguish between my being pleased with a lesson based on lowered expectations and my being pleased with a lesson because of a high level of learning and collaboration. Today was the latter, with my 9th grade probability and statistics class again.

I found this on Don Steward’s website. If you have seen his blog and are not fascinated, or at least intrigued, we cannot be friends. He comes up with some amazingly simple, yet elegant classroom problems.

We started this yesterday. They are in groups of 4; the oldest student in the group got to choose first and so on. Then they played three “games” using a pair of dice and a whiteboard with their numbers on it. Today, they worked on figuring out why the “6,7” card was the best and determining how to rearrange the numbers on the cards to make them all equally likely to win.

I’ve had this glossy paper in my room forever, so I decided to have them make a mini-poster with their solution and some reasoning. Here are my two favorites:

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Talking Pizza and Pennies

Today was a banner day in my ninth grade probability and statistics class.

First, our number talk was a bite out of the real-world and not the “you and 5 friends share 8 pizzas” kind of real-world. When my daughter has a babysitter, as she did last night, I usually spring for pizza. (Yes, our vegan lifestyle maintains a real iron-grip on nutrition when mom and dad are gone.) I even splurge on the good stuff: $5 Pizza.

With tax, my vegan-less vice cost $5.36. I gave the cashier $20.11. How much change did I receive?

Lots of great strategies: counting up, counting down, counting to the middle even. It’s worth noting that the two students in each class that insisted on stacking the numbers and borrowing were not able to do so correctly. I say this not to discount the standard algorithm. Rather I wish to point out that in this case, when it’s necessary to borrow three times, the standard algorithm is blatantly inefficient.

The students had to know why on earth I would give the cashier $20.11 rather than just $20. The answer: Quarters. Because if you’re at the store with a 4 year old and you do not have a quarter for a gumball machine, god help you.

The main portion of the lesson was the real magic. This problem is from Strength in Numbers by Ilana Horn:

Imagine that you have two pockets and that each pocket contains a penny, a nickel and a dime. You reach in and remove one coin from each pocket. Assume that for each pocket, the penny, the nickel, and the dime are equally likely to be removed. What is the probability that your two coins will total exactly two cents?

They sit in groups of three or four. I gave each group a large piece of paper, had them put a circle in the middle for their final solution and then divide the paper into 4 sections for their individual work. When looking through my pictures of student work, I noticed that I have a tendency to capture correct work (but differing methods), but I do not take photos very often of incorrect work. Today, I changed that.

Here is a sample of their strategies for determining the number of outcomes:

The level of discussion was exquisite. But what’s more important was that they were able to work together to organize their thinking and to make sense of their solution. They built on what they knew an gained conceptual understanding as a result. In addition, they were able to focus on understanding their path to the solution rather than simply being satisfied with the solution itself. I’m very proud of them.

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Math is Messy. So Are Gender Roles.

I have been absolutely humbled by all of the positive feedback I have received from my previous post. Thank you to infinity for taking the time to read, write, and share. I believe that it is our common humanity that makes it possible for us to learn from one another, not necessarily our knowledge of content. There is so much of my sobriety that goes into my teaching. It is an incredibly freeing feeling to be able to be honest about that part of my life as I blog.

Rose Eveleth wrote a great piece about the roles that girls find themselves taking on in group work. In short, Eveleth focuses on acknowledging that girls often self-assign the “recording” role, absolving (and downright excluding) themselves from a problem solving opportunity. The end result, career-wise, may lead women away from high-profiled positions. As teachers, it’s easy for us to overlook this discrepancy because girls, generally speaking, are neater and more organized, and may seem like the best fit for the job. In a related article, Dale Baker does a great job of asking teachers to examine gender preferences that exist in our classrooms in order to help encourage all students to step into the “lime light.”

On Friday, I tried a simple version of this. First, students were presented this scenario (taken from the Math Forum POW section):

The Student Council at Rahkenrole High School is planning a concert. They’ve hired the Knox Mountain Boys, a popular local band, for $340. A poll among the students has shown that if tickets cost $5, 140 people will come to the concert. For every dollar the ticket price goes up, 10 fewer people will come, and for every dollar it goes down, 10 more people will come.

I’ve been a huge fan of the Math Forum, long before I joined Twitter (and got to fangirl Max Ray at TMC14). The reasons might not seem obvious from this scenario, but kids noticed right away that there was no question asked at the end. What’s brilliant here is that there is literally an infinite number of questions that we could ask here. Granted, some questions are more important than others, but I framed the task in a way that elicited what I needed.

I handed out a big white piece of paper to each group of 4 and had them divide the paper up into sections. This way each person in the group was both the recorder and the problem solver. I asked them to write down 2 questions they think that I would ask about the scenario and one question (anything) that they would ask. They identified their group’s most important question and put it up on the whiteboards on the wall.

Low and Behold! They READ MY MIND! They asked about maximizing profit, income, and people, and also requested modeling equations for each. The excellence in this scenario (and the Math Forum in general) is that it can be applied to so many levels of math for so many reasons. For example, most high school kids can make a table and figure out a reasonable answer for the maximization questions, and kids with more know-how can develop mathematical models.

Some great things happened:

They knew they needed ONE set of answers in the center of their paper. This meant they had to communicate the work in their section. The traditional group roles dissipated, and they all had equal stake in solving the problem.
They solved the problem in so many different ways. (Do you remember these types of questions from Algebra 1/2? I’m sure they have a trendy textbook label that alludes me at the moment. But they are solved by making the variable “number of price increases. Interestingly, very few students solved it that way successfully.)
They were messy. And I loved it. In fact, I made the second class use markers exclusively so that they could not erase.
They were uncomfortable leaving some of the questions unanswered. When I didn’t label certain questions as “bonus” or “extension” they felt that all were necessary to be successful. My goal was for them to collaborate with ownership in their individual contribution. I may have gotten more joy out of this part than I should have 🙂

Here are some fun photos of their work:

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Matchmaker, Matchmaker: The Algebra Way

I’m trying to make my blog less OMG-you’ve-got-to-try-this-amazing-activity-that-I-found-cuz-it’s-awesome and more analysis of my teaching and an examination of where improvements can be made. That being said, this post is going to be a little of both.

Today, my Algebra students did another Talking Points activity on linear functions. I used the same format I did with Number and Operations where I gave them 5 minutes to look over the TP’s and make any notes they needed for the group activity. Here’s the link if you are interested in seeing what they chatted about. As I honed in on their exploratory talk, I noticed that many more of them were convinced by the reasoning of their tablemates and changed their “unsure” to “agree/disagree.” I’m not sure if that was because the topic was a tad more difficult, or if they are getting better at listening to one another. Of course, I am hoping it’s the latter.

At the end, after they wrote their self-assessments, we talked as a group about some of the points that they were still not convinced of agreement or disagreement (which included The opposite reciprocal of zero is zero). I tried to do this using the Talking Points rules, even though the whole class was able to participate in the discussion. I feel that this was a positive addition to the process.

Okay, onto the real star of today: An Nrich Task.

Each group of 4 receives a pack of 16 cards with algebraic expressions on them.

They cannot take cards from other group members; they may only give cards to others. Each person must have a minimum of two cards in front of them at all times, to alleviate the temptation of having one person sort the cards. To complete the task, each group member must have four cards in front of them that have the same simplified expression. Caveats: no talking, no non-verbal communication, no writing, no taking others cards.

I used the Glenn Waddell #TMC14 Smartphone Camera Hack to position my camera in the back of the room and I took a time-lapse video with my old iPhone. Although I can’t post that video online because it contains images of students, I did make a screenshot with blurred faces:

I mean, you can practically SEE the brain sweat pouring from their ears!

In our debriefing, we discussed what was hard, what was easy, and what strategies they developed. Here are some highlights:

It was nice to be dealt a card that was already simplified.
Besides not being able to talk or non-verbally communicate, it was difficult to simplify some of the expressions mentally.
It was difficult to not take cards from others, knowing where they belonged.
A good strategy when starting was to see if there were any matches to begin with.
Another good strategy: give unsorted cards to players who have completed sets so that they can help divvy those out.

I was so impressed with these kids today. I know that they will learn more together by working with one another. I’m so thrilled that THEY are beginning to see the truth to that.

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The Corn Sandbox

For an entire year, I’ve been anticipating our family’s return to Stade’s Shades of Autumn Festival. My excitement has been building for one reason: To estimate the amount of corn contained in their corn play area (or the Corn Sandbox as my 4 year-old has named it).

We went last Friday, armed with a measuring tape and a measuring cup. The sign seemed to give away the answer of 800 bushels, but I wasn’t satisfied given that their was no mathematics to back up their claim. We needed to attend to precision. Here are some of the photos I took:

My favorite is the bottom photo, my parents counting kernels of corn in 1 cup. (There are 579, by the way)

I love when I present a problem to my class, and it takes longer than I anticipate for them to solve. There was supposed to be time for solving inequalities for the group that worked with this, but that will just have to wait until tomorrow. I’m sure they were crushed.

The essential question we wanted to answer was: How many kernels of corn are there in this corn sandbox?

Initial estimates were very low. I let them revise after I revealed that 1 cup contained 579 kernels.

One approach: Use the number of bushels to calculate kernels.

After looking up on Wolfram Alpha that 1 bushel = 9.31 gal, we determined that a reasonable calculation of the number of kernels, based on our 1 cup count, was 69,000,000.

Second Approach: Use the volume of the enclosure to calculate kernels.

This was a little trickier given the irregular shape of the sandbox. Numerous calculations and conversions later, we arrived at 81,667,931 kernels of corn.

We were uncomfortable with the over 12 million kernel discrepancy between our two methods. It remains unclear which is more accurate given the fact that one includes actual measurements and assumptions and the other is provided by the farm. Perhaps Stade’s Farm should expect a call from Mrs. Schmidt’s 2nd hour Algebra 2 class in the near future to clear this up.

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Spiders Everywhere!

Steven Leinwand has a huge influence on how I approach a math lesson. In my experience, one of the easiest ways math can be extrapolated from almost any task is by asking the questions: How big? How far? and How Much?

This weekend, I came across this picture on social media, posted by David Roberts:

Of course the question I asked first was “How big is that tarp!?”

Luckily, David was willing to make an estimate and allowed me to share his reasoning:

I thought it might be interesting to present the photo to my first hour and see what questions they would ask about the photo. Of course, the surface area of the tarp was on their list.

The original article including the photo added more depth to their questions. As it turns out, the house is being fumigated after a spider infestation. It seems as though their curiosity surrounded more the spiders than the tarp and legitimately so. (The article estimates the house was infested with approximately 5,000 spiders)

I was pleased that my students used visual cues in the photo to make their estimates including the average height of a story of a house, the approximate height of the man in the photo, and the size of the window. Luckily, we found another photo that gave us a better understanding of how much tarp was needed for the other side of the house.

I’m glad this man’s golf game was not disturbed by a spider problem in the distance. (sarcasm)

Anyway, after making some calculations, my industrial-minded first hour realized that this type of tarp must have a somewhat standard size. After doing some Google searching and actually calling one of the companies that manufactures these behemoth plastic coverings (authentic! Yes!), they decided that there were four 84′ x 25′ tarps covering the house and the space surrounding it. (We were able to have a nice conversation about how multiplying each dimension by 4 was different that multiplying the area of the tarp by 4.)

In the end, their estimate was 8,500 square feet, approximately double of what David had estimated. We then critiqued David’s argument and decided that based on the picture only, his calculations were reasonable. Because we were able to dig for more information, my class believes that their estimate may be a tad more accurate. Thanks, David for sparking our curiosity this morning.

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What Questions Do They Have?

I’m always delighted by the extra wave of energy students put forth when they are asked to develop their own question to a scenario. I love my job, and this year has started amazingly. But today was probably my favorite day thus far.

College Algebra:

Since we are working on quadratics, we did the Many or Money scenario from the Math Forum Problems of the Week. It’s interesting (and almost entertaining) to watch them discover that there is no question. This is the first time we’ve done an activity where they developed the question so they came up with the questions I would have expected:

What price will maximize profit?
How many students would go if the price were $8?
How many students will attend at the maximum profit?
(My favorite) Can you write an equation that models Ticket price and Profit?

They were able to get started on answering some of these questions. I had them work on one large sheet of paper in order to share their work. The period ended before they could wrap up their work. Here is what one group has so far:

When talking with teachers about using the Notice and Wonder strategy is usually surrounding the unexpected “wonderings” that students will have. I think it’s important to allow them to have that creativity of asking outlandish questions like, what is the band’s favorite pre-concert meal? But to make sure that the math goals are met, shifting their focus on what we can mathematically deduce from the scenario. I usually ask what would I most likely ask about this scenario and what questions do you have about this scenario?

Algebra 2:

Last year, with this same class, we examined Val’s Values. The authentic, real-world awesomeness of that particular lesson was going to be impossible to re-create, but the scenario was still applicable and intriguing to this new group of students.

Last year, my students insisted that the ages of both Val and Amir were vital to answering the question Who spends more on jackets over their lifetime? Most fascinating to me was their estimations of Val and Amir’s ages:

Desmos made up a nice scatter plot for us that we could also Notice and Wonder about:

And Val, my students were slightly disappointed that they didn’t get to examine the entire $300 jacket. They are VERY curious about it. 😉

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