Fractions in Old Algebra Book

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Thursday, November 23, 1911: Am working at my algebra in the evening so I can make a better mark than I did last month. If it isn’t any better I will be beyond all hope.

 Her middle-aged granddaughter’s comments 100 years later:

In October Grandma struggled mightily with algebra—the topic was least common multiples (L.C.M.) and highest common factors (H.C.F.)—and she ended up getting a 68% on the exam.

I’m not sure what Grandma was working on in November—but in one early 20th century algebra book—Durrell’s School Algebra, the chapter after L.C.M. and H.C.F. was Fractions.

The book says:

In algebra, a fraction is often useful in expressing a general formula

Here are a couple of exercises from the book:

1. If three boys weigh a, b, c pounds respectively, what is their average weight?

2. If sugar is worth a cents a pound, how many pounds can be obtained in exchange for b pounds of butter worth c cents a pound?

3. If coal is worth c dollars a ton, how many tons can be obtained in exchange for f bushels of wheat worth h cents a bushel and for w bushels of corn worth y cents a bushel?

Colleges and Public Service a Hundred Years Ago

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Friday, November 3, 1911: Nothing very much doing today. Didn’t get any of my lessons out this evening. I wasn’t in a very studious mood.

Her middle-aged granddaughter’s comments 100 years later:

Grandma  so often worried about school—though she often seemed to not quite get around to studying. I wonder if Grandma ever considered going to college after she graduated from high school.

I suppose college seemed beyond the realm of possibilities to a farm girl in rural central Pennsylvania a hundred years ago. Less than 3% of the people were college graduates back then—and the rate would have been much lower than that for women.

There was an article in the November 6, 1911 issue of Youth’s Companion about why men—the article didn’t mention women—attended college.

Excerpts from

The College in the Service of the Nation

by

Arthur Twining Hadley (President of Yale University)

The American college serves the nation in three conspicuous ways: first, by training men for public office; second by establishing standards of professional success in private business which lead men to do what the public needs, instead of trying merely to make money for themselves; third, by promoting the search for the truth and the spirit of discovery and invention that are necessary for national progress. . .

When we think of public service, we naturally think of these meanings. So did the founders who established the earliest colleges. The founders of the collegiate school at New Haven [Yale] stated in the charter of 1701, that it was the purpose of their institution to fit youth for employment in church and state. . .

Every man, whatever his business can conduct it in such a way as to serve the public. The lawyer who pleads in the courts ought to be doing the same sort of service to the public as the judge who decides the cases. The physician can render and ought to render the same service in providing for public health that the watchman or the signalman provides for public security against accidents.

Any business however simple in its character, where a man thinks first of the work that he is doing and only secondarily of the pay that he is going to get deserves the name of profession.

One of the most valuable things that our colleges can do is to emphasize this ideal of public service, so that the professional element will count for more in men’s lives and the trade element will county for less.

A third way in which our colleges can render public service is by keeping alive the spirit of exploration and discovery-the spirit which leads men to test new methods of action and to pursue new lines of truth. I believe that this is the most important and necessary service of all.

So far as our colleges teach their students the love of pursuing truth for truth’s sake, without regard to the material reward, they fulfill their highest and most necessary duty in the service of the nation.

What Is Rhetoric?

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Thursday, November 2, 1911: Am now taking up the study of Rhetoric, so if my English is not all together proper now is the time to expect a change for the better.

Recent photo of building that once housed McEwensville High School.

Her middle-aged granddaughter’s comments 100 years later:

Rhetoric is the art of writing and speaking. It includes the study of writing rules. I found a 1911 Rhetoric textbook and it includes sections on proper sentence structure (no dangling participles!) and punctuation (sentence structure must be understood to punctuate correctly!).

Oh dear—my English probably is not all together proper . . .

Lowest Common Multiples and Highest Common Factors

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Thursday, October 26, 1911: Have such difficult algebra problems. So much work it is to find the H.C. F. and L.C.M. Good bye for me if we happen to get one of these in examination.

Her middle-aged granddaughter’s comments 100 years later:

Important: If you aren’t into math—skip my comments today and come back tomorrow.  Suffice it to say that Grandma was doing some fairly difficult algebra.

But, if you enjoy math here’s my take on what this diary entry is talking about–

First I’ll give an example of the L.C.M. (lowest common multiple) and H.C.F. (highest common factor) of two whole numbers (integers);  then I’ll explain how it’s done for algebraic expressions.

Integers

The L.C.M. is the smallest integer that two whole numbers can be divided by.  1 would always be the L.C.M.

For example, for 8 and 12  the L.C.M. would be 1.

The H.C.F.(highest common factor) is the largest integer that two whole numbers can be divided by.

For the same two numbers (8 and 12), the H.C.F. would be 4.

If the H.C. F. is 1, it is a prime number.

Algebraic Expressions

The basic idea is the same as for algebraic expressions. For example, for H.C.F. of 2ab and 4a2b is 2a.

But it quickly gets complicated. I’m going to give you directions and examples from a 1911 algebra textbook below for H.C.F. [An aside:  If you really want to understand this concept you might find the information on the algebrahelp.com website helpful.]

Now, here are the directions for finding the H.C.F. in  Durrell’s School Algebra (1912):

The method of finding the H.C.F. is to:

Factor the given expressions, if necessary:

Take the H.C.F. of the numerical coefficients:

Annex the literal factors common to all of the expressions, giving to each factor the lowest exponent which it has in any expression.

Ex. 1:  Find the H.C. F. of 6x2y – 12xy2 + 6y3 and 3x2y2 + 9xy3 – 12y4

6x2y – 12xy2 + 6y3 = 6y(x – y)2

3x2y2 + 9xy3 – 12y4 = 3y2(x2 + 3xy – 4y2) = 3y2(x + 4y)(x – y)

H.C.F. = 3y(x – y)

Whew, I’m getting a headache just typing these expressions. But if you’re still with me, here’s a couple problems you could try from the 1911 textbook:

Find the H.C.F.

1. 4a2b , 6ab2

2. x2 – 3x , x2 – 9

3. x2 + x , x2 – 1 , x2 – x – 2

4. 4a3x – 4ax3 , 8a2x3 – 8ax4 , 4a2x2(a – x)2

5. 3a2 – 10a + 3 , 9a – a3 , (3 – a)3

More Odd, Unusual, and Strange Math Problems

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Monday, October 16, 1911: Nothing new at school or at home. Read several stories after I had worked some problems. Still have some for tomorrow though.

Her middle-aged granddaughter’s comments 100 years later:

I wonder what type of problems Grandma was working on.  I enjoy looking at hundred-year-old math books. The problems are so different from the ones in today’s books.

I’ve previously shared some problems with you.  Here are some more odd, unusual, and strange problems from 1911:

1. If 44 cannons, firing 30 rounds an hour for 3 hours a day, consume 300 barrels of powder in 5 days, how long will 400 barrels last 66 cannons, firing 40 rounds an hour for 5 hours a day?

2. A ditch 80 yards long, 10 ft. deep, and 9 ft. wide was dug by 20 men in 12 1/2 days of 10 hours each; and a ditch 76 yards long and 12 ft. wide was dug by 30 men in 7 1/2 days of 9 1/2 hours each. How deep was the latter ditch?

3. A speculator bought 10 village lots, and gave a 4-months’ note in payment. This note was immediately discounted in the bank at 8%, and the bank discount was $192. What was the average price of the lots?

4. A druggist bought 6 pounds of quinine at $11 per pound, avoirdupois weight, and sold it in 2-grain capsules at 10 cents per dozen. What was his profit?

Kimball’s Commercial Arithmetic: Prepared for Use in Normal, Commercial and High Schools and the Higher Grades of the Common School (1911)

A hundred years ago prescriptions weren’t required and druggists made their own medicines, men actually dug ditches by hand, and labor laws about how many hours a day a person could work had not yet been enacted.

If you want to do the quinine problem–and, for some reason never had a math class that taught you the conversion factors for apothecaries and avoirdupois weights 🙂 – here is the information you need:

Apothecaries Weight

20 grains = 1 scruple

3 scruples = 1 dram

8 drams = 1 ounce

12 ounces = 1 pound

Avoirdupois Weight

16 ounces = 1 pound

In case you missed the previous posts that contained math problems, here are the links:

Odd, Unusual, and Strange Math Problems

1911 Algebra Problems: The Lusitania and Molasses

Old Math Problems

Beliefs a Hundred Years Ago About High Quality Education for Young Children

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Wednesday, October 11, 1911: Don’t know what to write. Got my report today. Was better than what I expected.

Her middle-aged granddaughter’s comments 100 years later:

Yeah! Even though it’s silly to be happy about an event that occurred a hundred years ago, I’m glad that Grandma did well on her exams.

Two days ago I quoted from a 1911 book about the purpose of education.  Since Grandma didn’t write much today I’ll tell you about an interesting section in the book about the role of education for younger children (pre-school and primary grades).

In 1911 there was a lot of interest in kindergartens. Many believed that young children needed an enriching environment and that children should develop at a pace they set for themselves. There was a huge amount of interest in the ideas of Madame Maria Montessori. She believed that children spontaneously educated themselves based upon their experiences and environment.

Madame Maria Montessori (Source: Wikipedia)

Today much of the policy discussion for young children revolves around whether there should be universal pre-schooling and how to standardize educational experiences for children in grades k-2. The focus is on teaching children reading skills.  This is very different from what people believed about early education a hundred years ago:

There is good reason, however for believing that early childhood freedom is more important to good mental development than to good physical development. The mind of the child may be more injured by “thorough” mental training of any particular kind, than the body by any special form of physical training. . .

Children not only develop the power to perceive remember, imagine, reason, etc. without any special assistance, but they acquire knowledge without special teaching.

  The Making of the Individual (1911) by E.A. Kirkpatrick

The Purpose of Schooling a Hundred Years Ago

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Monday, October 9, 1911: Had examinations today. Weren’t as hard as I expected they would be.

Building that once housed McEwensville High School

Her middle-aged granddaughter’s comments 100 years later:

It’s always a good sign when you feel good after an exam. It sounds like Grandma probably learned what her teacher wanted her to know.  Today we worry so much about student performance—and whether they’ve learned what they were supposed to learn.

A hundred years ago people believed that schools had several purposes. According to a 1911 book:

The public school performs one of its greatest functions in developing common knowledge, habits, and ideals in its future citizens, a function that it could not perform if all school teaching and regulation of conduct were individual. A common standard of knowledge, power, and achievement, to which everyone is expected to conform, helps to mould the life of an individual in a normal way and to fix in his mind and character standards by means of which his achievements and ideals may be guided.

The Making of the Individual (1911) by E.A. Kirkpatrick