Sundials!

Above is an analemmatic sundial. The idea is to orient the sundial facing south, and then place a vertical pointer on the central figure-8 track, in a position corresponding to the date. The sundial above shows a simulated shadow for 2:15 PM yesterday. It can be seen that the sundial tells the time reasonably well, thanks to the inbuilt adjustment for variation in solar position.

For large-scale analemmatic sundials, like the one below, people can stand on the central figure-8 track and act as a human pointer. A sundial like this is fun to have in the garden.

Here are blank sundials for some Southern Hemisphere cities:

The ShadowsPro software will also generate sundials like these, if anyone is particularly enthusiastic.


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Nature journals

Having said something about phenology wheels, I thought that I should mention nature journals too. Some years ago, I blogged about the professional aspects of this, but nature journals are a powerful educational tool, because of the way that they focus observational attention. John Muir Laws has good advice on getting started, including “Do not focus on trying to make pretty pictures. That just leads to journal block. Open your journal with the intention of discovering something new. Use the process to help you slow down and look more carefully.



Mother and child nature journaling examples from Nature Study Australia Instagram and website

The very useful Nature Study Australia website also has good advice and several examples, as well as other nature study resources for Australians. Artist Paula Peeters, aiming more at adults, runs nature journaling workshops around Australia, and offers an introductory book for sale or free download.


Nature journaling example from Paula Peeters, who runs workshops around Australia

Nature journals need not only contain pictures and text: a spiral-bound sketchbook will easily accomodate flat objects such as leaves, pressed flowers, feathers, and sun prints. Drawings are an essential aspect, however.


The CNPS curriculum

The California Native Plant Society offers a superb nature journaling curriculum for free download. It includes the observational prompts “I notice… I wonder… It reminds me of…” It advises parents and teachers not to say things like “that is really pretty” or “what a good drawing,” but instead to say things like “Oh, you found a spider on top of the flower! Great observation.” It also provides excellent practical advice on drawing, poetry, and other activities.

With so many excellent guides to nature journalling, why not get started on your own?


A drawing of mine (from quite some time ago)


Phenology wheels

Recently, somebody pointed me at phenology wheels, which are a popular tool for nature study among teachers and homeschoolers. Nature study is all about careful observation and finding patterns, and phenology wheels help with both. Every month, students draw a picture of what they see in the garden or on a nature walk, and the completed phenology wheel then shows an annual pattern. Other activities are possible – see this University of Wisconsin-Madison Arboretum document.

The picture below shows a pair of partially complete mother/daughter phenology wheels from the very useful Nature Study Australia website (they are using the central circle to show indigenous seasons). It is helpful to outline each month’s section in felt-tip pen:


Mother and daughter phenology wheels from naturestudyaustralia.com.au

I’ve generated blank wheels for the Northern Hemisphere and for the Southern Hemisphere, and produced a partially complete wheel of my own (from a European perspective):

Like nature journals, this is an activity both fun and educational!

Credits: lavender watercolour painting by Karen Arnold, sunflowers by Vincent van Gogh, butterfly from here, font is Jenna Sue, wheel constructed using R (with DescTools::DrawCircle, rasterImage, and the showtext package).


Mathematics in Action: Vehicle Identification Numbers

Motor vehicles have a 17-character Vehicle Identification Number or VIN on a metal plate like the one below, usually on the driver’s side dashboard, or on the driver’s side door jamb, or in front of the engine block:


A Vehicle Identification Number (VIN) plate (Photo: Michiel1972)

VINs offer an interesting example of check digit calculation. The central digit (or an X representing 10) is a check digit (calculated modulo 11) used to detect errors. Any letters in the rest of the VIN are decoded like this:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
1 2 3 4 5 6 7 8 1 2 3 4 5 7 9 2 3 4 5 6 7 8 9

The check digit calculation involves decoding the VIN, and multiplying the resulting numbers by the weights shown in blue, giving the products in purple:

VIN L J C P C B L C X 1 1 0 0 0 2 3 7
Decoded 3 1 3 7 3 2 3 3 10 1 1 0 0 0 2 3 7
Weights 8 7 6 5 4 3 2 10 0 9 8 7 6 5 4 3 2
Product 24 7 18 35 12 6 6 30 0 9 8 0 0 0 8 9 14

These products are added up modulo 11 (meaning the sum is divided by 11 and the remainder taken). In this case, the sum is 186 = 10 = X (mod 11), which makes the VIN valid, because it matches the original central X. What about the VIN on your vehicle?


Starting an element collection

In the spirit of the wonderful photobook The Elements by Theodore Gray (which I have previously blogged about), starting a collection of elements is a great way of introducing yourself (or your children) to basic chemistry. Here are some suggestions, and a list of 24 elements to start with….

2: Helium (He)

Helium is lighter than air, so balloons are often filled with helium.

6: Carbon (C)

Carbon is most easily added to your collection in the form of charcoal. Zinc–carbon batteries have a carbon rod at the centre.

7: Nitrogen (N)

Air is about 78% nitrogen. To add nitrogen to your collection, just fill a small bottle with air.

8: Oxygen (O)

Air is about 21% oxygen. To add oxygen to your collection, just fill a small bottle with air.

9: Fluorine (F)

Fluorine is a toxic gas. But octahedral fluorite crystals (calcium fluoride, CaF2) make a great addition to a collection.

11: Sodium (Na)

Sodium is a reactive metal which will spontaneously catch fire when in contact with water. But sodium chloride (ordinary table salt, NaCl) is perfectly safe.

12: Magnesium (Mg)

Magnesium is a flammable metal, but you can substitute crystals of Epsom salts (magnesium sulfate, MgSO4), which can be obtained from a pharmacy.

13: Aluminium (Al)

Aluminium (aluminum in the USA) is most easily available as aluminium foil.

14: Silicon (Si)

Silicon is widely used in transistors and integrated circuits (chips).

15: Phosphorus (P)

The side of a box of matches is largely composed of phosphorus.

16: Sulfur (S)

Sulfur powder, also called “flowers of sulfur,” is available from pharmacies.

17: Chlorine (Cl)

Chlorine is a toxic yellowish-green gas. But sodium chloride (ordinary table salt, NaCl) is perfectly safe.

20: Calcium (Ca)

Calcium is a reactive metal, but you can substitute crystals of calcite (calcium carbonate, CaCO3) or gypsum (calcium sulfate, CaSO4).

24: Chromium (Cr)

Chromium is used for plating (“chrome plating”) to prevent rusting. Also, “stainless steel” is between about 16% and 25% chromium.

26: Iron (Fe)

Iron is one of the most widely used metals. Iron nails are easy to add to your collection. Like nickel and cobalt, iron is attracted by a magnet.

28: Nickel (Ni)

The United States “nickel” coin is actually only 25% nickel (and 75% copper), but objects made of pure nickel can be found. Indeed, Canadian “nickel” coins from 1955–1981 are almost pure nickel.

29: Copper (Cu)

Copper pipes are widely used in plumbing. You can buy copper plumbing fittings, or get offcuts of pipe from a plumber. Copper electrical wire is also easy to find.

30: Zinc (Zn)

Galvanised iron is coated with zinc to prevent rusting. Also, filing off the copper coating on a US penny reveals a coin made mostly of zinc.

47: Silver (Ag)

A silver coin, or a piece of silver jewellery, would make a fine addition to your collection.

53: Iodine (I)

Iodine is a dark solid, but is sold in pharmacies as a brown solution in alcohol, called “tincture of iodine.”

60: Neodymium (Nd)

Neodymium is one of the “rare earth” elements. Neodymium magnets are the most common form of strong magnet. They are made of an alloy of neodymium, iron and boron (Nd2Fe14B).

74: Tungsten (W )

The filament in an incandescent light bulb is made from tungsten (but because of the danger of broken glass, only an adult should attempt to remove the filament, and then only with very great care).

79: Gold (Au)

A gold coin, or a piece of gold jewellery, would make a truly wonderful addition to your collection. Alternatively, for under $10, science museums will sell impressive-looking bottles of gold leaf floating in liquid.

82: Lead (Pb)

A fishing sinker is probably the easiest lead object to find.

So there you are. Those could be the first 24 elements in your collection!


Fibonacci and his birds (solution)

In the previous post, we described Fibonacci’s “problem of the birds” (“the problem of the man who buys thirty birds of three kinds for 30 denari”). In English:

“A man buys 30 birds of three kinds (partridges, doves, and sparrows) for 30 denari. He buys a partridge for 3 denari, a dove for 2 denari, and 2 sparrows for 1 denaro, that is, 1 sparrow for ½ denaro. How many birds of each kind does he buy?”

The man must buy at least one of each kind of bird, or he wouldn’t be buying “birds of three kinds.” Also, he must buy less than 10 partridges, because 10 partridges (at 3 denari each) would use up all his money. Similarly, he must buy less than 15 doves. We can thus make up a table of possible solutions:

Of those 126 possible solutions, only one works out correctly in terms of cost, and that’s the answer. But that’s an unbelievably tedious way of getting the answer, and you’d be rather foolish to try to solve the problem that way. The obvious approach is to use algebra. Write p for the number of partridges bought, d for the number of doves, and s for the number of sparrows. Because the man buys 30 birds, we have the equation:

p + d + s = 30

And because the costs add up to 30 denari, we have:

3 p + 2 d + ½ s = 30

Doubling that second equation gets rid of the annoying fraction:

6 p + 4 d + s = 60

If you’ve done any high school algebra, no doubt you want to subtract the first equation from this, which will eliminate the variable s:

5 p + 3 d = 30

But now what? That gives a relationship between the variables p and d, but there doesn’t seem to be enough information to get specific values for those variables.

Fibonacci solves the problem a different way. His solution is based on a key insight – the man buys 30 birds for 30 denari, so that the birds cost, on average, 1 denaro each. Fibonacci then makes up “packages” of birds averaging 1 denaro each. There are only two ways of doing this. Package A has 1 partridge and 4 sparrows (5 birds for 5 denari), and package B has 1 dove and 2 sparrows (3 birds for 3 denari). The solution will be a combination of those two packages.

Now the man can take 1, 2, 3, 4, or 5 copies of package A, leaving 25, 20, 15, 10, or 5 birds to be made up of package B. But the birds making up package B must be multiple of 3, so that the only possible answer is 3 copies of package A and 5 copies of package B. This means that the man buys 3 partridges, 5 doves, and 3×4 + 5×2 = 22 sparrows. That’s 30 birds and 3×3 + 2×5 + ½×22 = 30 denari.

Now it turns out that, had we kept on going with the algebraic approach, we would have gotten the same answer. We had:

5 p + 3 d = 30

Given that the numbers of partridges and doves (p and d) had to be positive whole numbers, that meant that p had to be a multiple of 3, and d a multiple of 5. That could only be achieved with p = 3 and d = 5.

We can also return to the diagrammatic approach. The equation:

5 p + 3 d = 30

describes the diagonal red line in the diagram below. That line only crosses one of the possible solutions, namely the dot corresponding to 3 partridges and 5 doves.

In mathematics, there’s more than one way to skin a cat. Or, in this case, a bird.


Fibonacci and his birds

The mathematician Leonardo of Pisa (better known as Fibonacci) is famous for his rabbits, but I was recently reminded of his “problem of the birds” or “the problem of the man who buys thirty birds of three kinds for 30 denari.” This problem appears in his influential book, the Liber Abaci.

The “problem of the birds” is expressed in terms of Italian currency of the time – 12 denari (singular: denaro) made up a soldo, and 20 soldi made up a lira. In the original Latin, the problem reads:

“Quidam emit aves 30 pro denariis 30. In quibus fuerunt perdices, columbe, et passeres: perdices vero emit denariis 3, columba denariis 2, et passeres 2 pro denario 1, scilicet passer 1 pro denariis ½. Queritur quot aves emit de unoquoque genere.”

In English, that translates to:

“A man buys 30 birds of three kinds (partridges, doves, and sparrows) for 30 denari. He buys a partridge for 3 denari, a dove for 2 denari, and 2 sparrows for 1 denaro, that is, 1 sparrow for ½ denaro. How many birds of each kind does he buy?”

How many birds of each kind does the man buy? It may help to cut out and play with the bird tokens below (click image to zoom). In a similar vein, what if the man buys birds as follows (still purchasing birds of all three kinds, and at the same price)?

  • 4 birds for 6 denari
  • between 6 and 10 birds for twice as many denari as birds
  • 8, 11, 13–14, 16–22, 24–25, or 27 birds for the same number of denari as birds
  • 8 birds for 12 denari
  • 12 birds for 18 denari
  • 16 birds for 12 denari
  • 28 birds for 21 denari
  • 6, 8–9, or 14 birds for 11 denari
  • 7–10, 12, 15, or 18 birds for 13 denari

Solution to the main problem here.