math

My #MCTM Sub Stuff

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.

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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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It’s probability time in my 9th grade prob and stats class.  Call me crazy for giving 9th graders dice and pennies with a month left of school, but it’s how I roll.  (Ha! I’m cracking up over here!)

I like to start with the Game of Pig, similar to the game used in the IMP curriculum.  I adapted it a little to have kids compare strategies for when playing with their own dice (or separate from their partner) to playing with the same dice as their partner.

It’s interesting to see their strategies develop here.  Some use very solid ideas like “I stopped when my round score reached 20.”  But I also get to see misconceptions like believing that a “one” will be rolled relatively soon after a “two” is rolled.  Having them share their strategies helps me to see where these misconceptions lie and deal with them before we start calculating any concrete probability.

Tomorrow, we’ll start by discussing which of these are legitimate strategies and which of them are not.

photo 1 (3) photo 1photophoto 4 (1)photo 2

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Chipotle for Everyone

I’m hard pressed anymore to find a classroom of high school kids who don’t absolutely adore Chipotle’s menu options.  They all have a favorite, and they own it as THEIR burrito.  (I like Chipotle in particular because as a vegan, I can get a delicious meal, as can any non-vegan meal companion.)

I came across this article from Vox claiming Chipotle’s menu calorie disclosures were inaccurate.  I’m going to give Chipotle the benefit of the doubt here because their website contains a very detailed nutrition calculator which allows you to determine the number of calories for your  customized burrito.

The article references a study from the Journal of Public Health Nutrition which reviews a study in which customers are asked to estimate the calorie content of their meal. Some groups were given no information at all.  Some groups were given a range of calories in which burritos in general fell.  Last, additional groups were given example burritos containing the low and high values in the calorie spread.

I had a randomly selected student create a burrito.  Each class was obviously something different which made it kind of fun.

First, I had them estimate the number of calories in the chosen student’s burrito.

Second, I gave them the calorie range of 410-1185 claimed in which Chipotle’s burritos are claimed to land.  I had them adjust their estimate and give reasoning for their adjustment based on the additional information.

I then showed them the calorie range with an example from the Journal article’s study:

IMG_5054

Third, I wanted them to use the examples above to adjust their estimate once more.

We then talked about how the range of our estimates changed and why.  We also had a discussion about ‘averaging bias’ and how healthy ingredients make us assume that certain food are lower in calories than they actually are.

We were able to discuss the surveying methods done for the study and the demographics of participants, which led to a nice discussion about sampling.  (Evidently high school 9th graders find it odd and quite a bit creepy that participants in the survey were given a “flavored ice pop” in exchange for 5 minutes of their time.)

As long as I had their attention with food, I asked them to estimate whether the student’s burrito had more or less calories than my vegan burrito.  I’ll let you decide:

Student’s Burrito:  chicken, white rice, pinto beans, tomato salsa, cheese, and lettuce

My Burrito:  brown rice, fajita vegetables, black beans, tomato salsa, corn salsa, guacamole, and lettuce.

 

 

 

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A Desmosian Gem

I finally had a chance to do the Function Carnival with my classes.  Thank you to Desmos, Christopher Danielson, and Dan Meyer for their work on this project.

As David Cox captured in his blog previously, the real power of this activity is the immediate feedback.

 

When the graph looks like the one below and 8+ rocket men burst out of the cannon, the students see that right away and adjust for it.

Rocketman

 

Dan had mentioned in a blog post a while back that “this stuff is really difficult to do well.”  After seeing students work through this activity today, I can appreciate the difficulty in creating an online math activity that gives both students and teachers detailed feedback in real time.

Some observations:

  • Students don’t realize at first that you can see their work live.  I allowed them to “play” for a minute, but some may need more encouragement.
  • A tool to allow you to communicate digitally with the class would be nice.  Google chat, for example?
  • Some students don’t realize that the bumper car SHOULD crash and make their graph to avoid it.
  • A student or two misunderstood the graph misconception questions and went back and changed their graphs to look like the misconception graphs.
  • It was interesting to see which students wanted their graphs to be perfect versus which ones said there’s was “good enough.”  It would be interesting to have a discussion about which is appropriate in the particular situation.

Bravo, Dan, Christopher and the Desmosians.  Thank you for creating an online math activity that gives me some faith in online math activities for the future.

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Class: 9th grade prob and stats. Topic:  Linear regression.  Enter: the Laundry Data.

The data sheet seemed to spark a LOT of curiosity.  In retrospect, I wish I would have given them some time to Notice and Wonder about the detergents.  Probably I’d also add some estimation first about these bottles of detergent rather than just handing them the data.  I should have known better.

Still, an interesting discussion ensued about ounces of detergent and loads of laundry.  We plotted the points on Desmos and wanted to choose two of them to create our linear model.  I teach three sections of this class and all three classes picked different points to make their equation.

One class picked (50, 33) and (200, 140), and after determining that they needed to find the slope in order to write the equation of the line, I posed that question to them.  How would we find the slope between these two points.  Crickets.

I want to note that a good minute of silence and eye-contact avoidance went by before one brave student spoke up.

S:  You FOIL them.

Me:  Can you explain what you mean by that?

S:  (coming to the board) You multiply them like this.

IMG_5025

Me:  What do we think of what S just wrote up here?  (at least 8 hands shot up in the air)

Me:  Please put your hands down and let’s discuss this.  What I like about what S just did here is he got us started somewhere.  He was willing to take a guess and risk being wrong.  Before S showed us his idea, no one was willing to volunteer their method.  Now that S has broken the silence, lots of you seem ready to discuss.  Thank you S for starting us somewhere.

After this student broke the ice, we came up with about 4 ways to determine the slope of this line and about 8 ways overall to figure out the equation of the line between these two points.  In the past, I would have said to this student, “No, we don’t FOIL, who has another idea?”  Now I know that allowing this student to explain his method does multiple things.  First, it helps the other students practice patience and courtesy when listening and responding to this student whose solution they know is incorrect.  Second, it is a great opportunity for students to engage in SMP #3: Construct viable arguments and critique the reasoning of others.  Third, it provides an opportunity to praise the value in providing the wrong answer.  So much of math class for these students has been about getting the right answers.  I’m glad this teachable moment came about for students to learn from the wrong one.

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Alright, Mr. Stadel. We’ve Got Some Bacon Questions

Greetings, Mr. Stadel.  We know that you are very busy.  We appreciate your brief attention.  Rather than bombard you with tweets, we decided to bloggly address our questions and comments about your Bacon Estimates.

First of all, bravo.  You dedicated an entire section of your estimation180 blog to a culinary wonder some refer to as “meat candy.”  Even our vegan teacher felt compelled to engage us with these estimates.  (She says it is for the sake of the learning.)

Second, the time lapse videos of the cooking are pretty sweet.  Too bad the school internet wouldn’t stop buffering.  But nice touch, Mr. Stadel.  Nice touch.

A question:  Did you know that the percent decrease in length of bacon is 38% after cooking, but the percent decrease in width is only 23%?  We figured that out adapting your “percent error” formula to the uncooked/cooked bacon.  Do you have any initial thoughts about that discrepancy?  Is it bacon’s “fibrous” fat/meat striped makeup that allows it to shrink more in length than width, inch for inch?

Also, did you know that the percent decrease in time from the cold skillet to the pre-heated skillet is 29%?  That one was a little harder for us to calculate, because we figured out that we needed to convert the cooking times to seconds rather than minutes and seconds.

To summarize, we wanted to thank you, Mr. Stadel.  Our teacher tells us that you dedicate your time and energy to the estimation180 site so that WE don’t have to learn math out of a textbook.  We wanted to tell you that we appreciate it.  And the bacon.  We appreciate the homage paid to bacon.

Sincerely,

Mrs. Schmidt’s Math Class

St. Francis, MN

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When the Answer is E: He Falls Off the Roof and Breaks His Neck

Our annual state testing season is almost here. The juniors will partake in the Minnesota Comprehensive Assessments in Mathematics a week from Tuesday. Our department decided issuing a practice test to all of our juniors would help re-familiarize them with long lost skills. After distributing copies during our monthly staff meeting, I’m always curious if any teachers in other disciplines look at the practice materials. Much to my delight, the choir director approached me at lunch on Friday, test in hand.

Mr. Warren: Is this test just like the MCAs?
Me: Most likely similar. Why?
Mr. Warren: Ok, well look at this one.

 

Mr. Warren: I think the answer is E, Xai s going to fall and break his neck.

The conversation went on for another few minutes, with me agreeing  that what’s been called “math education” includes ignoring the context of situations and focusing on a procedure.  In fact, I was curious how many juniors who completed this practice test even noticed that the situation was outrageous.

Since we were running on a 2-hr delay schedule Friday, I thought it would be the perfect opportunity to present the problem to my algebra class. They are mostly juniors who have been continually frustrated with a mathematics curriculum that doesn’t make any sense in the real world.

Me: Read through this problem. Does it make sense?

Student: ok, it looks like 32.

I didn’t expect any of them to apply any trigonometry, so I thought we needed to approach the problem differently.  In fact, I wasn’t even concerned about the angle measure.  I wanted them to look at the scenario itself.

Me: Imagine this scenario. We’ve done a lot of estimating in here. We need to envision a 20-foot ladder, three feet away from a house. Does this seem reasonable?

Unfortunately, it did seem reasonable to most of them. I needed another approach.

Me: ok, how could we simulate this in classroom-scaled size?

Student: Get a ruler.

Me: Perfect. How close does it need to be to the wall?

Students: (a chorus of answers)

After exploring multiple methods of calculating exactly how far, we arrived at 1.8 inches.  With as much drama as possible, I set the ruler against the wall, exactly 1.8 inches away.

Me:  Does this look like a ladder that any of you would want to stand on? (of course, a few did).  Keep in mind, this is a TWENTY foot ladder, not a 12 inch ruler.

Student:  Yea, I don’t think anyone is climbing up that ladder and coming down in one piece.

Another Student:  What if they had a spotter?

A spotter!  Now we’re talking.  To be honest, I have no idea if a spotter could hold a 20-foot ladder so that it could be placed three feet from the wall.  But now I’m interested to find out!

I know Mathalicious investigated a similar scenario using a claim from Governor Janet Napolitano.

In my mind, these are the questions that should be circulating Facebook and aggravating parents.  This is the kind of math that should rile up Glenn Beck and company.  Our state of Minnesota opted not to adopt the Common Core State Standards in Mathematics, but requiring this kind of math instead is what is actually dumbing down the curriculum.  It assumes that the real world doesn’t apply, only rote procedure does.  “Just figure out the answer, don’t question the situation,” is what kids read and do over and over when problems like this are solved without real context.  A richer classroom experience for both teachers and students comes when we ask students to assess the reasonableness of situations, create new scenarios that are more appropriate, and solve the new problems they develop.  The CCSS Standards for Mathematical Practice tell students that it’s vital that they “construct viable arguments and critique the reasoning of others.”  I don’t think “critique the reasoning of others” should be reserved for only reasoning created in the classroom.  I’d like my students to critique the reasoning of the creator of these types of problems and others like it that have been deemed a necessary component of high school math success.

Thank you, Mr. Warren for igniting the exciting conversation in my classroom.

 

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Creative Craziness

I teach a lot of 9th graders this trimester. We offer a class called probability and statistics 9 and it is open to 9th grade students who also will have had the quadratic portion of algebra 1 this year. I really enjoy this class for multiple reasons. First, it lends itself very well to applying math to real-world scenarios.  Secondly, the hands-on opportunities are endless.
One of the issues I have been committed to improving with my own professional demeanor is the way I deal with 9th grade boys. Nothing brings out my sarcastic, short-tempered, disagreeable side like the antics of freshman boys. There’s something about the decision to play soccer with a recycling bin that just invokes the my inpatient side. Regardless, I need to develop more patience with this demographic. Boys are unique, both in the way that they act and the way that they perceive acceptable behavior. I’m not talking about “I’m bored” acting out. I’m talking about the “I really need to see if this eraser will fit in this kids ear” kind of acting out. I think that my short fuse has more to do with my failure on my part to  fully understand them rather than gross misbehavior on their part. What I’m really trying to grasp here is not “why can’t these kids sit still?” But more “when they can’t sit still, what makes them want to kick a recycle bin around the room or toss magnets at the learning target?” I think if I had a better understanding of what drives those behaviors, I could deal with them more productively. Suggestions?

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Class Commences – an hour I won’t soon forget

Recently, Michael Pershan unearthed a Shell Centre gem straight from the 80’s (literally).  This collection of materials is fantastic, and hopefully demonstrates to both students and teachers that engaging in rich tasks and high-level thinking is timeless.

I decided to give the function unit a shot in my Algebra 2 class today.  Some background on this group of students:  there are 38 juniors and seniors, last hour of the day, in a class geared toward lower-level students.   So far though, the only thing that’s been “lower” in this class is the number of empty desks I have.   I handed out this task, gave minimal directions and let them go for a few minutes on their own:

 

from: Shell Centre for Mathematical Education, University of Nottingham, 1985

from: Shell Centre for Mathematical Education, University of Nottingham, 1985

It was so interesting to watch the different ways each of them started.  Some began with 7, since that was the first you saw when reading the graph from left to right.  Others insisted to work from 1 to 7, identifying the corresponding people along the way.  A few worked the other way around, from the people to the graph.

I walked around to make sure each student was able to get started and that those who thought they had determined a solution also supported their claims.  Then, I wrote the numbers 1 – 7 on the dry-erase board, stepped back, and let these kids amaze me.
One student volunteered an answer, and then handed the marker off to another.  I intervened only briefly to make sure that every student had an opportunity to contribute if he or she wanted.  Once 7 names were completed, I knew a couple of them were out of place.  I sat and said nothing, and this entire class showed me what they are capable of.  Here was a class full of students labeled mathematical underachievers completely nailing SMP #3.  Their arguments were viable, their critiques constructive, their discussion productive.  It bothered a few of them that I wouldn’t let them know if/when they were correct.   But most of them are starting to understand that my main focus here is not the correct answer, but the incredibly rich and interesting process they used on their journey to finding it.  They came up with multiple ways to support their answers and noticed tiny details about the people that supported their findings.  For example, did you notice that Alice is wearing heels? According to my students, that is perhaps why she appears slightly taller than Errol.

I had a heart-to-heart with this group when we were done about how proud I was at how they conducted themselves throughout this task.  I’m really thoroughly looking forward to a fantastic trimester with this special group of kids.  Their work on this task gives both of us the confidence that they can tackle something more difficult next time, and they are capable of mastering high-level mathematics this trimester.

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