A Realistic Baseball Algebra Problem

Too often you hear adults complaining that they were tortured with algebra for no reason. That’s an emotionally loaded statement that should never influence schools to teach algebra to less people. For starters, the brain grows at a very rapid pace until age 25, so no one should be deprived of the chance to become a scientist or engineer. Algebra is a backbone of other branches of mathematics like trigonometry, linear algebra, and the various branches of calculus that all depend heavily on algebra and which are useful tools in the occupations mentioned. And contrary to popular belief, algebra can come up in everyday situations, as it did today with this baseball fan.

(Photo by Frank Jansky/Icon Sportswire via Getty Images)

Prior to May 20th’s games, the Boston Red Sox’s much-improved Jarren Duran was hitting 0.356. Unfortunately, if you look him up among Major League Baseball’s leaders, he’s not 2nd, as the uninitiated might expect. That’s because to qualify for the batting title, a player must have 3.1 plate appearances per team’s scheduled games. So far, Duran only has 114 plate appearances in his team’s 45 games. If he averages between 4 and 5 plate appearances in the team’s upcoming games, in how many games will he qualify?

The minimum overall average of 3.1 is equal to the number of plate appearances divided by the total games played. If you let y equal to the number of games that must be played to qualify, then

If the Red Sox score more runs and Duran averages 5 plate appearances in future games, then he will only need to play.

Finally if we make the average plate appearances for upcoming games a variable, x, then we can plot y versus x to see how Duran’s future plate appearances will affect the additional number of games he has to play before we see his name among the leaders.

This is known as an inverse relationship. As one variable increases —in this case the number of plate appearances per game, the other variable decreases. (Less extra games have to be played to qualify.) A bicycle pump creates another inverse relationship. As you apply more force per area (more pressure), we squeeze the area into a smaller volume. Other inverse relationships include that between truth and the certainty in the tone of the voice of many politicians and public relations people. The more self-assured they sound about an issue, the less likely they are to be telling the truth.


Where Dandelions Bloom First

For several years, I’ve noticed that in many different locations, dandelions located on a southern slope are the first to bloom in the springtime. It happens because the slope receives more direct rays from the sun, causing the soil to heat up faster. This presumably signals the plants that it’s time to grow and reproduce. It’s related to the reason why our noses, if they are sloped, get sunburnt before the rest of our faces. For centuries, people have been aware of the southern slope advantage, given that it’s been used as an optimal location for vineyards.

The dandelions on the left are growing on a ~30 degree slope. They bloomed first and are now at the seed-producing stage. On the right hand side are nearby dandelions growing on a flatter area. They are still in bloom, having blossomed after their sloped counterparts.

It’s not the kind of thing I could have noticed in my youth, given that municipalities would apply weed-killer to get rid of dandelions. Nowadays, to the dismay of herbicide manufacturers, many cities have bylaws prohibiting the use of such compounds, so dandelions abound. There are also less people who are obsessive about upkeeping weedless lawns, so in May especially, dandelions abound, and many children’s favorite color is in full display for a few weeks before they all turn white.

The seeds that are attached to the white fluffy part that helps them spread like wildfire are actually the result of parthenogenesis or virgin birth. That implies that all the seeds that a plant produces are clones —genetic copies of one another. Just last year biologists isolated the parthenogenesis (PAR) gene from the asexual dandelion that triggers embryo development in unfertilized egg cells. When the gene was deleted from dandelions by the CRISPR technique, they still produced the white sphere but one without seeds, as shown below in the picture to the right:

Image from https://www.nature.com/articles/s41588-021-00984-y

In my front yard, I enjoy having a variety of plants among my Kentucky bluegrass, but the dandelion is not my favorite. It grows at the expense of other plants that I prefer such as white clover, black medic, yellow clover, poppies and oregano. Since dandelions won’t naturally delete the PAR gene that allows them to multiply so efficiently, the key is to remove the plants before they flower. Even that can be a challenge. If you are like me and are aversive to using herbicides, then you have to pull them , making sure to pull out the entire root. Just like the lawn mower has acted as a selective force, which has shifted the dandelion population towards blooms with shorter stems, pulling weeds by hand has also had an impact. In my yard, it’s gradually given a selective advantage to plants that produce very long and deep-reaching roots. Those are more likely to leave a surviving fragment allowing the more common form of vegetative reproduction to keep propagating the weed

If dandelions don’t reproduce sexually, how do they create the variety that natural selection can then act upon? When pulling out dandelions, one observes a spectrum of root thickness among plants that are side by side in identical soil, reducing the likelihood of an environmental cause. When botanists checked for mutations among dandelions, they found a rate that ranged from 7 to 15 per 1000 individuals. A mutation is a change in the DNA code caused by either nature or by man-made products. The study from the same link concludes that the mutation rate in dandelions is directly proportional to concentrations of chromium, iron, manganese, and nickel in leaf-tissue. Another study reveals that some metals themselves cause dandelions to evolve. They don’t exclude the pollutants, but sequester them and then tolerate their presence, again in direct proportion to the amount of pollution they are exposed to.

An Easter Friday Reminder of the Shroud of Turin

In 1960 Willard Frank Libby was awarded the Nobel Prize for his method of using carbon-14 to find the age of objects ranging from ancient bows and arrows to trees buried in glacial ice. Since then, the technique of radiocarbon dating has been improved so that it can be used with much smaller samples—fractions of milligrams instead of the original 8 grams. This has made it less intrusive when dealing with precious art and in trying to figure out the authenticity of artefacts like the Shroud of Turin. Whereas, Libby modestly assumed that radiocarbon dating’s upper limit was 20 000 years old, it can currently estimate the age of objects as old as 55000 years old.

Carbon has 15 isotopes, of which only 3 are natural: 12C, 13C, 14C. The superscript refers to the sum of neutrons and protons in each atom. Any atom is characterized by its number of protons, so the trio of carbon’s isotopes 12C, 13C, 14C, have 6 protons each, but they contain 6, 7 and 8 neutrons, respectively. That last number seems to be too much for a nucleus to handle. Every 5730 ± 40 years, in half of 14C nuclei, a neutron transforms into a proton, an antielectron neutrino and a beta particle, becoming 14N in the process, the common isotope of nitrogen in the air. A beta particle is a highly energetic electron and is a form of radiation identical to what old “cathode ray” television sets and computer screens emitted from their long bulky bodies; it was blocked by lead (Pb) in the glass screen. The reason some 14C exists on earth is that neutrons released from violent cosmic ray-collisions in the Earth’s upper atmosphere cause a small portion of 14N in the air to become hydrogen and 14C.

This property of 14C has been very useful in dating old samples of bones, charcoal, seeds, wall paintings and in just about anything containing carbon.  While plants are still alive, through photosynthesis, they continuously absorb 14C from the small fraction of carbon dioxide which contains the isotope. This was first realized by Libby. Whether carbon dioxide contains 14C or the common 12C doesn’t affect the chemical properties of the vital gas. So plants can still use it to make glucose.

After organisms die, after bacterial decomposition, whatever 12C that remains behind does not undergo radioactive decay. Unlike 14C, 12C is a stable isotope. But 14C keeps transforming back into 14N. So the longer something has been dead, the less 14C it contains. The most efficient carbon-dating technique is accelerator mass spectrometry (AMS).

The 14C content is directly measured relative to the amount of 12C and to 13C present, which, like 12C, is also stable. From these ratios, the age of an object can be calculated. Unlike other methods, AMS does not measure the amount radiation emitted but determines the number of carbon atoms present in the sample and the age-dependent ratios. The 55000-year upper limit of radiocarbon-dating has only been reached in the past 15 years. As recently as 2008, the limit was a little less than half of that because sunspot activity affects the rate of cosmic rays reaching the earth, and hence the amount of 14C-formation in the atmosphere has not been constant. A calibration curve based on tree rings and their analyses takes the fluctuations into account. The improvement of the curve and the use of the AMS technique have been mainly responsible for giving the method more historical scope.

The Shroud of Turin is a  linen  cloth that Christian tradition associates with the crucifixion and burial of Jesus .   In 1988, scientists at three separate laboratories used carbon-14  to date samples from the Shroud to be from 1260 to1390 AD, coinciding with the first verified appearance of the shroud in the 1350s. Given that the burial of Jesus was in 30 or 33 AD, the evidence strongly suggests that the Shroud is not the real McCoy. 

35 years later, the 1988 tests continue to be debated. Is it because the conclusion clashed with longstanding belief about the cloth? About 10 years ago, Professor Christopher Ramsey of the Oxford Radiocarbon Accelerator Unit, one of three labs which carried out the research, said, “We’re pretty confident in the radiocarbon dates. There are various hypotheses as to why the dates might not be correct, but none of them stack up.” More recently, in 2019, Phillip Ball, former chief editor of Nature, wrote, “Nothing published so far on the shroud, including this paper, offers compelling reason to think that the 1989 study was substantially wrong – but apparently it was not definitive either.” Given ever-improving techniques and rigorous statistical analyses, the bar can always be raised.  But so far, more radiocarbon testing has not been given the OK by the Church. As Ball said, “As it stands, reticence looks more like fear of what further studies might reveal.”


https://www.nature.com/articles/s43586-021-00063-w.pdf   2021

Nobel Prize Winners in Chemistry. Eduard Farber. Abelard-Schuman. 1962