Monday, November 13, 2023

First Algebra Lesson (Cumberland Saga)

“Good afternoon Ladies and Gentlemen. Welcome to Algebra One” Blain said to his assembled class of six students. Two girls sat along one side of Sarah’s parlor and four boys sat on the other side. Blain was working with a small blackboard, about two feet on a side. Each student brought their own slate to perform calculations.

At Sarah’s insistence, he asked each student their name and then he repeated it back to the student. Sarah “suggested” that he do it three times during the first class for each student.

Then he had them warm up by reciting the times-tables to 12-times-12. Sarah said that is how they started out every math class.

Blain started the class off with a story problem. “I will be surprised if anybody gets the answer but this is the best way to make algebra interesting. That is, to show you it can be useful.”

“Suppose land sells for \$1000 an acre.” Blain said.

One of the boys, Gregory, objected “Land costs a lot more than that.”

Blain replied, “I am not going to quibble. I am using round numbers to make the math simple so you focus on the ideas.”

“So where was I...Oh, land sells for \$1000 an acre and a neighbor comes up to you and offers to sell you some” Blain continued.

“He says, “I don’t know exactly how much it is but its gotta be at least forty-acres so I am askin’ \$40,000.”

“So you ask, “What are the dimensions?”

“Your neighbor says, “Well, its a triangle and one side measures 440 yards east and 440 yards north and the other side measure 880 yards south and another 880 yards west.”

"You know that 40 acres is a square that is 440 yards on a side, so a triangle that stretches as far as this one does has to be at least 40 acres, right?” Blain said.

“So take a few minutes and see if you can figure it out exactly HOW big it is and if it is worth \$40,000” Blain said...he looked over at Sarah’s wind-up clock on the mantle as he sat down. “I will give you five minutes to figure it out.”

Gregory, one of the boys said “I got 180 acres.”

Blain invited him to explain how he got that number.

“The plot measures 1320 yards east-west and the same north-south. If it were square, that would be 9, 40-acre-parcels or 360 acres. Since it is a triangle, it will only be half that or 180 acres” Gregory said.

“Impressive logic, but that is not the right answer and I am going to show you why but first I am going to teach you how to measure the area of a triangle using something called “Cross Products”.

He showed them the method using X-and-Y notation and he explained that North is usually positive Y, East is positive X, South is negative Y and West is negative X .
Then he invited them to try the method but to not share their answers for a minute.

Lliam finished in thirty seconds and was shaking his head. The others were slower than Lliam and one student didn’t finish.

Lliam said, “It don’t work. I got zero using Cross-Products.”

Three of the other students nodded their heads in agreement.

Two seemed unsure and the one who didn’t finish just watched.

“Well, let’s draw a picture and see” Blain suggested.

Then, using a ruler, he sketched out the “triangle” as close to scale as possible and the students could see that even thought the sides were very long, the points that defined the corners fell in a straight line that ran southwest-to-northeast.

“The method worked fine” Blain said. “Your neighbor was going to cheat you but the math didn’t lie.”

One of the girls said “I don’t see how I will ever use that.”

Blain asked “Do you sew? Do you buy fabric or ever have to scale a dress pattern?”

Enlightenment blossomed across her face.

“Let’s do a few more examples” Blain suggested.

They talked about how much fertilizer Sarah would have to put on one of her garden plots if Sally put 1500 pounds of fertilizer on his one-acre of field.

The talked about how many bags of dog-food Sally would have to buy after his bitch had puppies if he kept six of them.

Blain told them about his studies in Pharmacy Science and explained how medicine doses are based on so-many micrograms of active ingredient per kilogram of the patients weight. Then he showed them some examples of how an injected rate could be converted to pills.

At the end of the class, Gregory said “I still don’t see what this does that I can't do by messing around with numbers a bit at a time.”

Blain thought for a second before responding. “It does a bunch of things for you.”

“For one, because you are carrying around the units and USING them, you will quickly see if you are multiplying when you should be dividing or vice versa” he said.

“Another thing is that by using symbols instead of numbers, you can check your work. The example with the triangle that was a straight line...easy to get confused when every number was 440 or 880. It is hard to keep them straight” he added.

“Finally, Lliam was able to get his answer WAY quicker by using an algebraic equation than it took you to push the numbers around” Blain concluded, tactfully omitting the fact that Lliam got the RIGHT answer while Gregory got the wrong one.

The students left the lesson chattering like a flock of starlings. Algebra class was nothing like they had expected it to be. Gregory was DEFINITELY going see if his dad would get the right answer with the Triangle of Land problem.

1. Good application of math, ERJ.

One of the biggest losses of my high school years was the fact that knowledge was never connected to utility. Likely I could have had more interest in science or math but it was always in the context of "just the facts", not "this is how the facts can used". When I got into industry I found all kinds of pretty cool things science will do for you.

2. Maybe I would have stayed in school if I had had a math teacher like you. Could not see any utility in what they were teaching.

3. It being more than a half century since I’ve done any algebra my mathematical ability has atrophied. You lost me by defining the location of two points of the triangle but not the third. So could you please ‘show me your work’ with a diagram. Thanks.

1. My graphics program is not cooperating at the moment so you will have to suffer with a word-picture.

Imagine you are standing at Point A. That is the first point of the triangle. Then the person selling the "property" points to the northeast (45 degrees) and says, the second corner is 622 yards in that direction (440 east and then 440 north). Next, the seller points to the southwest (225 degrees) and says, the third corner is 1244 yards in that direction.

The three points, the one where you are standing, the one to the northeast and the one to the southwest are in a straight line. The "triangle" has an area of zero.

2. Thanks. I can see now that I missed the point, ‘A’, that is.

4. I truly wish that was the method I learned algebra- I struggled with it, but breezed thru geometry and trig. I'm older than dirt, but younger than sand ; )
irontomflint

5. And of course, seeing how Blain connected with the children had a profound effect on Miz Sarah...

Readers who are willing to comment make this a better blog. Civil dialog is a valuable thing.