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.

Postscript: Since writing this, I have noticed that Duran has only averaged 2.97 plate appearances in the Red Sox’s last 40 games. According to the hyperbolic relationship shown above, 2.97 puts him in that too-many-games-are-needed-zone. However, he just had a great series against Toronto, going 5 for 5 yesterday with 4 doubles. Meanwhile, since returning from injury, his competitor for the center field position, Duvall, has struggled at the plate. If Duran plays almost every day for the next 33 games and averages 4 plate appearances (PA) per game, he would then qualify for the batting title. If he currently had enough PAs he would be tied with his teammate Yoshida for 5th place in the American League. If you make a variable out of the number of PA required by Duran from now to the end of the season, then he would need to average 3.50 PA for the Sox’s remaining 77 games.

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.