Math Mondays

Math Mondays – A Farm Problem

So I have a real-life problem from our farm and I am wondering how to make it a three act task.  (If you are into organic farming you might want to look away…as I just said this is a real-life problem 😉 )

Dan Meyers describes these type of problems as tasks happening in three acts.

Act 1:  You introduce the central conflict of your story.   (Make it visual…use as few words as possible.)

IMG_0989
Acres and acres of weeds

Act 2:  The student looks for resources, asks questions and develops new tools…to overcome the problem.

Notice I said students, but for now I am just going to tell you the tools we have and the question we asked.

TOOLS:  200 gallon tank on a sprayer with a 30 foot span, 19 sprayer heads, each sprayer head disperses treatment at 0.36 gallons per minute, 3 pints of weed killer per acre, and the intention to drive 3.5 miles per hour

Our question:  How many pints of weed killer do we need to add to the 200 gallon tank?

Act 3: Resolve the conflict…and set up for a sequel.

                My work and Greg’s work

We both got the same answers.  YAY!!!!  And both used dimensional analysis…I love using dimensional analysis.

So what would be appropriate photos for acts 2 and 3?   What about photo 1?

And what would be a sequel?

And actually now that I think about it…it would be interesting to see the questions, resources and tools the more organic minded folks would ask/use.

Math Mondays

Math Mondays – Fractals

One of my favorite undergrad classes at North Central College was Chaos Theory.   I loved (still do) the idea that we can describe chaos.   One of the assignments of the class was to design our own fractal and then determine its fractional dimension.

Inspired by the Koch Snowflake, I set out to create something similar with a hexagon.   This was in the early 90’s so I had a computer, but none of the apps we have today.   So I st out to construct my “snowflake” by hand.  I used my protractor, a ruler and a very large sheet of paper and set out to create the next big (i.e., most important) fractal in the world!  FAIL.  Turns out it had a dimension of one.   Now if you are wondering why I didn’t know this before I spent a weekend drawing and redrawing the dang then…well…you just don’t know me!   I take my time and do things the hard way or the long way or the illogical way!   But I eventually get there!

Not sure why I shared this story other than this problem from my license review test had me reliving my glory days.  (Hey there are athletic glory days…why not math geek glory days?)

Here’s the problem:Koch

Here’s my work:

So many things have changed since my college days.   However, I am still fascinated by fractals and I still prefer to do my work in pen!   What can I say?  I live on the edge…of chaos!

Sigh Yup.   I just wrote that.

Oh and in case you are wondering…I selected D.

Math Mondays

Math Mondays – Back to Calculus

I am back to a calculus review problem today.  I am just not sure where I want Math Mondays to finally settle.  So for now I will just grab and solve whichever problem is at hand.

Here it is:

A merchants daily inventory over a 30 day period is given by

Capture 1, where x is the number of days from 0 to 30, inclusive.

What is the merchant’s average daily inventory over the 30-day period?

 

So I am thinking I want to find the integral to get the total….but I have some questions about that.   If I want daily, do I really want the area under the curve for the total inventory?   To me it makes more sense to use Sigma….for the summation from 0 to 30.   Regardless of how I find the sum, I will then divide by 30 to get the average.

So lazy student moment here.   I really just want to find the inventory on day 0…(1200) and on day 30 (487.88) and then find the average.  This is approximately $844.00.   Which is a choice.

My not at lazy approach was to find the integral of the function and divide by 30.

Capture 4

To get approximately 791. Which is also a choice.  And the choice I am making.

Long story short, I think I conceptually understand why I want the area.  I am picturing 30 rectangles with a width of 1 and an average height of the function value f(0) and f(1), f(1) and f(2), etc.  Finding the sum of these rectangles (or the total area under the curve to be more accurate) and dividing by 30 gives the average daily inventory.

Am I right?

 

 

Math Mondays

Math Mondays – Getting Dirty

Follow-up:  I found my error.   Apparently 6-2 = 4 not 2.   So yes.  Distance is the area under the curve.  So in this case, one can use the formula for the area of a trapezoid (just use the correct height) or one can calculate the integral.

IMG_0703

Do you see my error ?

So some questions for teachers:  How do you grade a student who makes this type of error?  (Submits just the top portion?  Submits both portions?) What messages do our grading practices send to student? How do we encourage students to move past the answer?   To realize the importance of understanding the concepts and to make sense of the process?   And to not give up when something goes wrong?   (Please share your thoughts in the comments.)

So I already have a change of plans.   I think math is all about getting dirty.   Struggling with the tough problems.   Discovering the boundaries and/or working within them.

Me sharing my mostly correct work on calculus review questions does not fit this description and really is not that interesting. (However, I am interested in exploring questions like those listed above.)  So I am going explore math tasks from all grade levels.

To begin with I want to share my son’s work on an Inside Mathematics 2nd Grade Performance Assessment Task

Sheep and Ducks.

Here are the questions:

  1. How many legs on 1 duck?
  2. How many legs on 4 ducks?
  3. How many legs on 5 sheep?
  4. Next to the barn is a pen with 2 sheep and 3 ducks?  How many legs altogether? Show how you know your answer is correct.
  5. One of the farmer’s pens has a high fence around it.  He can see 32 legs under the fence.  How many sheep and ducks are in this pen? Show one way to have sheep and ducks with 32 legs in all.  Show another way to have sheep and ducks with 32 legs in all.

Try the question yourself.

Where do you think your students or children might struggle?

According to Inside Mathematics the following are areas of difficulty:

  • Organizing their work.

  • Showing their work.

  • Working backwards from a given number of legs to create correct combinations of sheep and duck legs together

Here is my son’s work.  You will see most of these difficulties in his work. IMG_0770

I was surprised that I had to push him to explain/show his thinking.  He was quick to write the an answer, but did not want to record his thinking as he solved the problems.  To get me started, I recorded what he told me about the process.

I was also surprised how quickly he wanted to quit when he first encountered a problem.  In problem number four he very quickly concluded that the answer was 10.  When I  had him record how he solved the problem he was very disappointed when the answer was 14 not 10.  We had a conversation about how the process was helpful, but he still felt defeated by the fact that he was initially incorrect.  He carried this with him into the next question and did not want to try to solve it.   When I suggested he draw the legs he regained his confidence.  But once he had one answer, he did not want to consider (draw) other possibilities.  I asked him if he could have all ducks and he used the picture to explain why this was a possibility.  He was also able to explain why there could be all sheep.

So this little conversation takes me back to the questions above.  I think it really is about building those mathematical practices as habits.  But I want to hear from elementary teachers and parents.  How do you help students develop these habits?  How do you know if you are successful?  How do you help them embrace getting dirty?

fffhhh
Math Mondays

Math Mondays

Well I have been toying with the idea of a blog for some time now.   And despite my attempts to tell myself “now is not the time,”  the idea of writing a blog keeps coming back to me.  And yes in the form of cheesy hashtags…but, it is what it is.  And then today, on a podcast I heard several reasons why every teacher should keep a blog…and so it begins!

Math Mondays – I have been thinking about something my principal said in our last late start meeting.  She suggested that we teachers engage in reading to appreciate the many nuances of reading comprehension.  I believe the same applies to any other subject…writing, science, music, sports, etc.  And it certainly applies to mathematics. So for now, Math Mondays will feature me doing mathematics so that I can better appreciate the many nuances of mathematical practice.

I am in the process of preparing for the Oregon Educator Licensure Assessment in Mathematics.  So I am brushing up on my Calculus.   I remember most of it, but I have relied on Sal Khan to refresh my memory from time to time.  I am certain there is a conversation to had around the fact that I needed to review, but for now I just want to dive into the math.

This is the current question I am working on:

The velocity of a particle moving along a line is given by  v(t)=4.9t, where t is in seconds.  What is the total distance traveled by the particle in the interval from 2≤t≤6?

So I am certain…I think… this is an integral problem.   But integrals always amaze (i.e., scare) me. I know they give us the area under the curve…so I always hesitate when calculating a distance.  I mean don’t we always harp on the fact that area and distance are two very different things so we must always show our labels…blah.blah.blah.BLAH .blah?

But here we are using the area under the curve to find a total distance.

So I am not using an integral.

I will find the area using d=rt  where r = v(t).  Thus d=v(t)·t

IMG_0703

And so I have some questions.

Why twice as large?  Is the problem with my drawing (I think it is!!!)  or is it with my calculations?

And you can bet I will seek the answers tomorrow.

So tune in to Teaching Tuesdays.