2015 – Page 2 – Montessori Muddle

Chickens 911

Initial reports indicated a chicken with a broken leg; some rumors suggested the chicken had gone missing as well. These reports instigated an investigation by the Chicken Committee. They determined that the chicken was there, but something was wrong. They sent out a call for medical assistance.

Help came in the form of Dr. Emily Leonard from the Cherry Hills Veterinary Hospital (who happens to be a mom at our school). She took the chicken in for examination.

Based on the X-ray, there were no bones broken, so the issue must have been something else. The large egg that showed up on the radiograph suggested that the chicken could have been egg-bound, however, 20 minutes later, the chicken laid the egg.

So, the chicken is still under observation.

After the initial examination, Dr. Leonard brought the chicken back to school. It needed to be isolated and observed–which is something we now know to do in the future in any other case of injury–and the head of the Chicken Committee (the Chicken Head) made the call that the animal should go back to the hospital for the weekend.

Dr. Leonard deals mostly with pets, so she had to do quite a bit of research. “I learned a lot about chickens today,” she told me afterwards. This is a message I hope the students internalize. With the ready access to information we have today, it’s not so much about the facts you have memorized, but more about having the flexibility and ability to deal with new challenges by doing research and then applying what you learn are essential skills.

Dr. Leonard takes the chicken in for observation.

Transformations and 3D Printing

My pre-algebra class is working on transformations, so it seemed a good opportunity to try some 3d printing. I showed them how to create basic shapes (sphere, triangle, and box) and how to move them (transform) in a three-dimensional coordinate system using OpenScad.

Students create their car models using OpenScad.

Then we started printing them out on our 3d printer. Since it takes about an hour to print each model, we’re printing one or two per day.

The students were quite enthused. The initial OpenScad lesson took about 15 minutes, and I gave them another 55 minutes of class-time to work on them with my help. Now they’re on their own, more or less: they’ll be able to make time if they get their other math work done. A couple of them also came back into the classroom at the end of school today to work on their models.

And I think they learned a bit about transformations in space.

Toothpick Shapes’ Sequences

Using a sequence of connected shapes to introduce algebra and graphing to pre-Algebra students.

Make a geometric shape–a square perhaps–out of toothpicks. Count the sides–4 for a square. Now add another square, attached to the first. You should now have 7 toothpicks. Keep adding shapes in a line and counting toothpicks. Now you can:

make a table of shapes versus toothpicks,
write the sequence as an algebraic expression
graph the number of shapes versus the number of toothpicks (it should be a straight line),
figure out that the increment of the sequence–3 for a square–is the slope of the line.
show that the intercept of the line is when there are zero shapes.

Then I had my students set up a spreadsheet where they could enter the number of shapes and it would give the number of toothpicks needed. Writing a small program to do the same works is the next step.

Coding Online on the Coding Ground

For some of my students with devices like Chromebooks, it has been a little challenging finding ways for them to do coding without a simple, built-in interpreter app. One interim option that I’ve found, and like quite a bit is the TutorialsPoint Coding Ground, which has online interfaces for quite a number of languages that are great for testing small programs, including Python.

Screen capture of Python coding at Tutorial Point's Coding Ground. — Screen capture of Python coding at Tutorial Point’s Coding Ground.

Synthetic Muscle Fibers from Fishing Line

I need some students to try this at school. Muscle fibers that contract on heating sounds like a great way to open and close vents for air circulation (in the chicken coop to start with).

ᔥ Popular Mechanics

Teaching Creativity

Tina Seelig teaches creativity and innovation at Stanford. She has a new book out on the topic called Insight Out. She answered questions about the book recently and gave this answer (I’ve reformatted it into bullet points for clarity):

The model I present in Insight Out describes a series of steps from ideas to actions. Each step requires more effort than the one before.

Imagination requires engaging and envisioning what might be different.

Creativity is applying your imagination to solve a problem. This requires motivation and experimentation.

Innovation is applying creativity to come up with unique solutions. This requires focus and reframing. And,

entrepreneurship is applying innovations to bring the to the world. This requires persistence and inspiring others.

–Tina Seelig on Reddit.com.

In response to a question about testing students:

I don’t test my students. In fact, I tell them to never ask about their course grade. 🙂
I want the students to be internally motivated. I tell them that I expect them to put as much work into the course as I do, and that they should “never miss an opportunity to be fabulous.” Guess what? It works! They are waiting for someone to give them this freedom to tap into their own motivation, not respond to an external motivation.

Our Microbial Symbionts

Sometimes I ask my students if we’re not just giant mechs for our microbial symbionts. After all, they outnumber us by about 10 to 1–in our own bodies. Rob Knight’s TED talk stokes my curiosity.

Planting Probabilities

The Gardening Department of our Student-Run-Business sowed seeds in little coconut husk pellets. The question was: how many seeds should we plant per pellet.

Since we’ll only let one seedling grow per pellet, and cull the rest, the more seeds we plant per pellet, the fewer plants we’ll end up with. On the other hand, the fewer seeds we plant (per pellet) the greater the chance that nothing will grow in a particular pellet, and we’ll be down a few plants as well. So we need to think about the probabilities.

Fortunately, I’d planted a some tomato seeds a couple weeks ago that we could use for a test case. Of the 30 seeds I planted, only 20 sprouted, giving a 2/3 probability that any given seed would grow:

$P[\text{grow}] = \frac{2}{3}$

So if we plant one seed per pellet in 10 pellets then in all probability, only two thirds will grow (that’s about 7 out of 10).

What if instead, we planted two seeds per pellet. What’s the probability that at least one will grow. This turns out to be a somewhat tricky problem–as we will see–so let’s set up a table of all the possible outcomes:

Seed 1	Seed 2
grow	grow
grow	not grow
not grow	grow
not grow	not grow

Now, if the probability of a seed growing is 2/3 then the probability of one not growing is 1/3:

$P[\text{not grow}] = 1 - P[\text{grow}] = 1 - \frac{2}{3} = \frac{1}{3}$

So let’s add this to the table:

Seed 1	Seed 2
grow (2/3)	grow (2/3)
grow (2/3)	not grow (1/3)
not grow (1/3)	grow (2/3)
not grow (1/3)	not grow (1/3)

Now let’s combine the probabilities. Consider the probability of both seeds growing, as in the first row in the table. To calculate the chances of that happening we multiply the probabilities:

$P[(\text{seed 1 grow}) \text{ and } (\text{seed 2 grow})] = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$

Indeed, we use the ∩ symbol to indicate “and”, so we can rewrite the previous statement as:

$P[(\text{seed 1 grows}) \cap (\text{seed 2 grows})] = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$

And we can add a new column to the table giving the probability that each row will occur by multiplying the individual probabilities:

Seed 1	Seed 2	And (∩)
grow (2/3)	grow (2/3)	4/9
grow (2/3)	not grow (1/3)	2/9
not grow (1/3)	grow (2/3)	2/9
not grow (1/3)	not grow (1/3)	1/9

Notice, however, that the two middle outcomes (that one seed grows and the other fails) are identical, so we can say that the probability that only one seed grows will be the probability that the second row happens or that the third row happens:

$P[\text{only one seed grows}] = P[(\text{Row 2}) \text{ or } (\text{Row 3})$

When we “or” probabilities we add them together (and we use the symbol ∪) so:

$P[\text{only one seed grows}] = P[(\text{Row 2}) \cup (\text{Row 3}) \\ = \frac{2}{9} + \frac{2}{9} = \frac{4}{9}$

You’ll also note that the probability that neither seed grows is equal to the probability that seed one does not grow and seed 2 does not grow:

$P[\text{neither seed grows}] = P[(\text{seed 1 does not grow}) \cap (\text{seed 2 does not grow}) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$

So we can summarize our possible outcomes a bit by saying:

Outcome	Probability
both seeds grow	4/9
only one seed grows	4/9
neither seed grows	1/9

What you can see here, is that the probability that at least one seed grows is the probability that both seeds grow plus the probability that only one seed grows, which is 8/9 (we’re using the “or” operation here again).

In fact, you can calculate this probability by simply taking the opposite probability that neither seeds grow:

$P[\text{neither seed grows}] = 1 - P[\text{neither seed grows}]$

Generalizing a bit, we see that for any number of seeds, the probability that none will grow is the multiplication of individual probability that one seed will not grow:

Probability that no seeds will grow

Number of seeds	Probability they wont grow
1	1/3	(1/3)¹
2	(1/3)×(1/3) = 1/9	(1/3)²
3	(1/3)×(1/3)×(1/3) = 1/27	(1/3)³
n	(1/3)×(1/3)×(1/3)×…	(1/3)ⁿ

So to summarize, the probability that at least one plant will grow, if we plant n seeds is:

$P[\text{at least one seed grows}] = 1 - P[\text{no seeds grow}]$

which is:

$P[\text{at least one of n seeds grows}] = 1 - P[\text{1 seed grows}]^n$

Which is something we may have seen before: What are the odds?

Finally to answer our question: how many seeds we should plant, we need to decide how high a probability we need of success:

Probability that at least one seed will grow

Number of seeds	Probability that at least one seed will grow	%
1	2/3	67%
2	8/9	89%
3	26/27	96%
4	80/81	99%
n	1-(1/3)ⁿ

The Head of Gardening leads the planting of seedlings.