Entries Categorized as 'Mathematics'

Some Geometry in Archery

August 29, 2016

Our PE teacher just started up an archery program at school (first classes today actually), and she shared this video with me that seems like it might be useful in geometry class. Specifically, it uses circle packing to estimate the relative difficulty for an archer to hit different sized circular targets.

Citing this post: Urbano, L., 2016. Some Geometry in Archery, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Volumes of Revolution

April 13, 2016

3D printed volumes.

3D printed volumes.

It can be tricky explaining what you mean when you say to take a function and rotate it about the x-axis to create a volume. So, I made an OpenScad program to make 3d prints of functions, including having it subtract one function from another. I also 3d printed a set of axes to mount the volumes on (and a set of cross-sections of the volumes being rotated.

The picture above are the functions Mrs. C. gave her calculus class on a recent worksheet. Specifically:

y = e^{-x}+2

from which is subtracted:

y = 0.5 x

Citing this post: Urbano, L., 2016. Volumes of Revolution, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

The Joy of Math

February 2, 2016

Regardless, if learning is to be as efficient and deep as possible, it’s essential that it be done freely. That means giving children a voice in which activities to participate, for how long, and also the level of mastery they want to achieve. (“This is the biggest clash with traditional curriculum development,” Droujkova notes.)
— Vangelova (2014): 5-Year-Olds Can Learn Calculus

This article provides a lot of evidence to support the notion that the conceptual aspects of calculus and other “higher level” forms of math should be taught at all age levels, not just at the end of high school or in college.

The presentation below elaborates:

Citing this post: Urbano, L., 2016. The Joy of Math, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Toothpick Shapes’ Sequences

August 27, 2015

Toothpick shape sequences.

Toothpick shape 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.

Citing this post: Urbano, L., 2015. Toothpick Shapes' Sequences, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Planting Probabilities

March 16, 2015

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.

Planting seeds in coconut pellets.

Planting seeds in coconut pellets.

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.

The Head of Gardening leads the planting of seedlings.

Citing this post: Urbano, L., 2015. Planting Probabilities, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

The Importance of Collecting (and Reporting) Good Data

March 9, 2015

Image capture from Ben Wellington's TED Talk on what can be done with data from New York City.

Image capture from Ben Wellington’s TED Talk on what can be done with New York City’s data.

I’m having my students collect all sorts of data for Chicken Middle, their student-run-business. Things like the number of eggs collected per day and the actual items purchased at the concession stand (so we don’t have to wait until we run out of snacks). It takes a little explanation to convince them that it’s important and worth doing (although I suspect they usually just give in so that I stop harassing them about). So this talk by Ben Wellington is well timed. It not goes into what can be done with data analysis, but also how hard it is to get the data in a format that can be analyzed.

Doubly fortunately, Ms. Furhman just approached me about using the Chicken Middle data in her pre-Algebra class’ chapter on statistics.

We’re also starting to do quarterly reports, so during this next quarter we’ll begin to see a lot of the fruits of our data-collecting labors.

Citing this post: Urbano, L., 2015. The Importance of Collecting (and Reporting) Good Data, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Math Gifs

September 2, 2014

Mike Schmidt passed along this link to Lisa Winter’s post collecting 21 GIFs That Explain Mathematical Concepts.

For example:

Converting a trigonometric function (sin curve) from Cartesian to polar coordinates.

Converting a trigonometric function (sin curve) from Cartesian to polar coordinates. Source: “Cartesian to polar” by KieffOwn work. Licensed under Public domain via Wikimedia Commons.

Citing this post: Urbano, L., 2014. Math Gifs, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Finding the Limit (Following up the Guitar Project)

August 31, 2014

Following up on the project to find the volume (and surface area) of a guitar, and the slope at a point along the outline of the guitar, I asked students to use the same techniques to estimate the area under a curve (y = x2) and find the slope at a point along the curve. Specifically:

y = x2

y = x2

  1. Draw the function y = x2
  2. Find the area bounded by the function, the lines x = 1 and x = 4, and the x-axis
  3. Find the slope of a tangent to the y = x2 function at the point where x = 3.

The point of the second question is to test if students have internalized the idea that they can approximate curved shapes with trapezoids, but they have to weigh the time it will take to do a lot of trapezoids, versus the reduction in error that will result from more trapezoids. It’s interesting to see students’ character come through in this assignment: some choose to make one big trapezoid and are done, while other will go so many trapezoids that they run out of time to get them done.

It just occurs to me, however, that an interesting way to assess this assignment would be to give them a fixed time, and tell them that their score will be the 100 minus the percent error in their calculations.

Limits

The point on the function where x = 3.

The point on the function where x = 3.

The third question–about finding the slope of a tangent line at x = 3–is our jumping off point into the mathematics of limits and calculus.

Some students do a single approximation–either forward or backward–, while others do both and take the average.

Finding the approximate slope using a forward difference.

Finding the approximate slope using a forward difference.

The forward approximation involves finding the values for the function y = x2 at x = 3 and x = 4 and finding the slope between the two points:

  1. when x = 3, y = 9, so we have the point (x1,y1) = (3, 9)
  2. when x = 4, y = 16, so we have the point (x2,y2) = (3, 16)

The slope (m) between two points is found with the equation they learned back in algebra:

\text{slope} = m = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1} = \frac{y_2-y_1}{\Delta x}

Where Δx = x2-x1 and Δy = y2-y1.

Using the two points above gives:

\text{slope} = m = \frac{\Delta y}{\Delta x} = \frac{16-9}{4-3} = \frac{7}{1} = 7

Those who use the backward approximation simply use the point when x = 2 instead of x = 4, and they end up with a value for the slope of 5.

Averaging the forward and backward approximations give a slope of 6.

Now, since they know that the closer you make the points the better the approximation, I ask them to make a table to see what happens as they do so. This means reducing the value of Δx. In both the forward and backward approximation shown above, Δx = 1.

This can be done very quickly in Excel (or any other spreadsheet program), however, this time at least, most students chose to do it by hand. They end up with a table that looks like this:

y=x2-dx-to-zero

As dx gets smaller the calculated slope approaches 6.

As dx gets smaller the calculated slope approaches 6.

As the difference in x gets smaller and approaches zero, the slope approaches 6.

As the difference in x gets smaller and approaches zero, the slope approaches 6.

As you plot slope versus the change in x (Δx), you can see that as Δx gets smaller and smaller and approaches zero, the slope gets closer and closer to 6. So we could say that:

the limit of the slope as Δx approaches zero is 6.

Mathematically this can be written as:

limit-slope

or using the equation for slope:

limit-dx

Now, we can work on taking the limit in a more general way to do differentiation.

Citing this post: Urbano, L., 2014. Finding the Limit (Following up the Guitar Project), Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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