Experiments for Demonstrating Different Types of Mathematical Functions

December 11, 2013

This is my quick, and expanding, reference for easy-to-do experiments for students studying different types of functions.

Linear equations: y = mx + b

  • Bringing water to a boil (e.g. Melting snow)
  • Straight line, motorized, motion. (e.g. Movement of a robot/Predicting where robots will meet in the middle)
  • Current versus Voltage across a resistor as resistance changes.

Quadratic equations: y = ax2 + bx + c

Exponential functions: y = aekx

  • Cooling water (ref.)

Square Root Functions: y = ax1/2

Trigonometric Functions: y = asin(bx)+c

Citing this post: Urbano, L., 2013. Experiments for Demonstrating Different Types of Mathematical Functions, Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

The Math of Planting Garlic

October 10, 2013

Planting a bed of garlic at the Heifer Ranch CSA.

Planting a bed of garlic at the Heifer Ranch CSA.

One of the jobs my class helped with at the Heifer Ranch was planting garlic in the Heifer CSA garden. The gardeners had laid rows and rows of this black plastic mulch to keep down the weeds, protect the soil, and help keep the ground warm over the winter.

Laying down the plastic using a tractor. The mechanism simultaneously lays down a drip line beneath the plastic for watering.

Laying down the plastic using a tractor. The mechanism simultaneously lays down a drip line beneath the plastic for watering.

We then used an improvised puncher to put holes in the plastic through which we could plant cloves of garlic pointy side up. The puncher was a simple flat piece of plywood, about one foot by three feet in dimensions, with a set of bolts drilled through. The bolts extended a few inches below the board and would be pressed through the black plastic. Two handles on each side of the board made it easier for two people to maneuver and punch row after row of holes.

Punching holes in the plastic.

Punching holes in the plastic.

As I took my turn punching holes, we did the math to figure out just how much garlic we were planting. A quick count of the last imprint of the puncher showed about 15 holes per punch. Each row was about 200 feet long, which made for approximately 3,000 heads of garlic per row.

We managed to plant one and a half rows. That meant about 4,500 garlic cloves. With ten people planting, that meant each person planted about 450 cloves. Not bad for an afternoon’s work.

Citing this post: Urbano, L., 2013. The Math of Planting Garlic, Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Embedding more Graphs (using Flot)

September 8, 2013

Here’s another attempt to create embeddable graphs of mathematical functions. This one allows users to enter the equation in text form, has the option to enter the domain of the function, and expects there to be multiple functions plotted at the same time. Instead of writing the plotting functions myself I used the FLOT plotting library.

Citing this post: Urbano, L., 2013. Embedding more Graphs (using Flot), Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

A Concept Map for Mathematical Functions

August 22, 2013

Analyzing functions.

Analyzing functions.

This year I’m trying teaching pre-Calculus (and it should work for some parts of algebra as well) based on this concept map to use as a general way of looking at functions. Each different type of function can by analyzed by adapting the map. So linear functions should look like this:

Adapting the general concept map for linear functions.

Adapting the general concept map for linear functions.

You’ll note the bringing water to a boil lab at the bottom left. It’s an adaptation of the melting snow lab my middle schoolers did. For the study of linear equations we’ll define the function using piecewise defined functions.

The rising temperatures in the middle of the graph can be modeled with a straight line. Graph by A.F.

The relationship between temperature and time on a hotplate. The different parts of the graph can be defined by a piecewise function. Graph by A.F.

Citing this post: Urbano, L., 2013. A Concept Map for Mathematical Functions, Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

What are the odds?

May 15, 2013

Three successful oregano plants out of four cuttings.

My middle school class stumbled upon a nice probability problem that I might just use in my pre-calculus exam.

Oregano

You see, four of my students took cuttings of oregano plants from the pre-school herb gardens when we were studying asexual reproduction, but only three of them grew roots.

That gives a 75% success rate.

P[success] = 0.75

The next time I do it, however, I want each student to have some success. If I have them take two cuttings that should increase the chances that at least plant will grow. And we can quantify this.

If the probability of success for each cutting is 75% then the probability of failure is 25%.

P[failure] = 0.25

Given a probability of one plant failing of 25%, the probability of both plants failing is the probability of one plant failing and the other plant failing, which, mathematically, is:

P[2 failures] = P[failure] x P[failure]

P[2 failures] = 0.25 x 0.25

P[2 failures] = 0.252

P[2 failures] = 0.0625

So the probability of at least one plant growing is the opposite of the probability of two failures:

P[at least 1 success of two plants] = 1 – P[2 failures]

P[at least 1 success of two plants] = 1 – 0.0625

P[at least 1 success of two plants] = 0.9375

which is about 94%.

If I had the students plant three cuttings instead, then the probability of at least one success would be:

P[at least 1 success of 3 plants] = 1 – P[3 failures]

P[at least 1 success of 3 plants] = 1 – 0.253

P[at least 1 success of 3 plants] = 1 – 0.02

P[at least 1 success of 3 plants] = 0.98

So 98%.

This means that if I had a class of 100 students then I would expect only two students to not have any cuttings grow roots.

An Equation

In fact, from the work I’ve done here, I can write an equation linking the number of cuttings (let’s call this n) and the probability of success (P). I did this with my pre-calculus class today:

P = 1 - 0.25^n

so if each student tries 4 cuttings, the probability of success is:

P = 1 – 0.254

P = 1 – 0.004

P = 0.996

Which is 96.6%, which would make me pretty confident that at least one will grow roots.

However, what if I knew the probability of success I needed and then wanted to back calculate the number of plants I’d need to achieve that probability, I could rewrite the equation solving for n:

Start with:
P = 1 - 0.25^n

isolate the term with n by moving it to the other side of the equation, and switching P as well:
0.25^n = 1 - P

take the natural log of both sides (we could take the log to the base 10 if we wanted, or any other base log, but ln should work just as well).

\ln{(0.25^n)} = \ln{(1 - P)}

now use the rules of logarithms to bring the exponent down into a multiplication:

n \ln{(0.25)} = \ln{(1 - P)}

solving for n gives:

n  = \frac{\ln{(1 - P)}}{\ln{(0.25)}}

Note that the probability has to be given as a fraction (between 0 and 1), so 90% is P = 0.9. A few of my students made that mistake.

Lavender

Roots on two of three plants.

At the same time my students were making oregano cuttings, six of them were making cuttings of lavender. Only one of the cuttings grew.

So now I need to find out how many lavender cuttings each student will have to make for me to be 90% sure that at least one of their cuttings will grow roots.

Students can do the math (either by doing the calculations for each step, or by writing and solving the exponential equation you can deduce from the description above), but the graph below shows the results:

Probabilities of successfully growing a lavender cutting.

Citing this post: Urbano, L., 2013. What are the odds?, Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Graphing Polynomials

March 24, 2013

Try it. You can change the order and coefficients of the polynomial. The default is the second order polynomial: y = x2.

I originally started putting together this interactive polynomial app to use in demonstrating numerical integration, however it’s a quite useful thing on its own. In fact, I’ve finally figured out how to do iframes, which means that the app is embeddable, so you can use it directly off the Muddle (if you want to put it on your own website you can get the embed code).

This app is a rewritten version of the parabola code, but it uses kineticjs instead of just HTML5 canvases. As a result, it should be much easier to adapt to make it touch/mouse interactive.

Citing this post: Urbano, L., 2013. Graphing Polynomials, Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Working with Climate Data

March 14, 2013

Monthly climatic data from the Eads Bridge, from 1893 to the 1960’s. It’s a comma separated file (.csv) that can be imported into pretty much any spreadsheet program.

135045.csv

The last three columns are mean (MMNT), minimum (MNMT), and maximum (MXMT) monthly temperature data, which are good candidates for analysis by pre-calculus students who are studying sinusoidal functions. For an extra challenge, students can also try analyzing the total monthly precipitation patterns (TPCP). The precipitation pattern is not nearly as nice a sinusoidal function as the temperature.

Students should try to deconstruct the curve into component functions to see the annual cycles and any longer term patterns. This type of work would also be a precursor the the mathematics of Fourier analysis.

This data comes from the National Climatic Data Center (NCDC) website.

Citing this post: Urbano, L., 2013. Working with Climate Data, Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Flying Robotics

February 27, 2013

A fascinating demonstration of using modeling to maneuver flying robots (quadrocopters).

Citing this post: Urbano, L., 2013. Flying Robotics, Retrieved May 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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