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:

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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.

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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.

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:

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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:

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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.

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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.

 

 

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:

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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:

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

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

 

 

The Un-Puzzle

I’ve heard this said a thousand different ways:  a task does not need to apply to the real world in order to be engaging.  Dan Meyer’s version seems to be thrown around most often:  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.

Today is Homecoming Friday.  It’s tough to get students engaged today, as their minds are on the game and the glitter (oh, the glitter).

Here’s a very short video clip of the noise level in my Algebra class.  Crickets.

No, I’m not giving a test.  I gave them a puzzle called Quadruple Sudoku:

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In short, besides regular Sudoku rules applying, the four small numbers are clues as to what goes into the boxes touching them.

And both classes, all period, the brain sweat was palpable.  Why, on such a wild, exciting school day would these kids be so focused and so engaged?  The answer I come up with every time is puzzling and unpuzzling.  unnamed (7) unnamed (6) - Copy unnamed (1) - Copy unnamed (2) - Copy unnamed (3) - Copy unnamed (4) - Copy unnamed (5) - Copy

By the way, Nrich has tons of these fun, puzzling, engaging variations on Sudoku.  Check them out.

A Twist on Old Venn

How many of you went nuts over the Google Doodle for John Venn’s 180th Birthday?  I have no shame in admitting I spent more than a few minutes messing around with it.

These not-so-modern overlapping circles of wonder have fascinated mathematicians, scientists, and even linguists alike.  When searching for rich tasks for my college algebra classes, I came across this new twist on the traditional Venn diagram:

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This activity can be applied to all kinds of topics with the main task being to find an equation to fit into all eight of the Venn diagram regions.  Since we are working with systems of equations, I offered this challenge to my classes:

Can you find three graphs that all intersect and also each intersect one another at unique points?  Also, is there a 4th graph that does not intersect the first three?  

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Out came the iPads and Desmos.  Here are a few highlights:

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Some of my observations during their work time:

  • A few of them assumed we were creating an actual Venn Diagram with Desmos. I made sure the expectation was more clear the next period.
  • Attention to precision was important.  Some students assumed that if the three graphs appeared to cross one another, their task was complete.  They were mistaken when I zoomed in to examine the intersection points.
  • Students assumed that if a graph did not intersect another in their viewing window, it didn’t intersect at all.  We had some good conversation about where graphs might cross as the x and y approached infinity.
  • Using sliders in Desmos makes this task more doable in one class period.
  • I wonder if they would be able to solve for their intersection points algebraically.

Side note:  these RISPs (Rich Starting Points created by Jonny Griffiths) are all available on this website, and are excellent starters for college level mathematics.  

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:

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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

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:

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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.