Borland Educational News and Views

Friday, March 08, 2024

Hybrid Notes

Word problems make most math students a little nervous.

Generally, it’s the translation from English into algebra that poses the problem. Instead of getting stuck, consider taking “hybrid notes,” written partly in English and partly in math, at least initially. Once you gain more clarity, you can shift completely into algebraic sentences (i.e. equations).

Do translation in stages, in baby-steps, rather than a single leap. First write notes that mix English and math (e.g. Expense = Burgers * Price, or Total Time = Time Running + Time Walking), then translate fully into mathematics as you gain more understanding.

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Example

You own a boat rental business. Boats rent for $100 per hour plus a $150 security deposit. On average, your expenses amount to 30% of the hourly fee for each rental. Write an equation relating profit and rental hours for a typical boat trip.

It’s hard to translate that into algebra in a single step. Using hybrid notes will help.

Profit = Revenue – Expenses.

Revenue: 100*hours+150
Expenses: .3*(hourly fee)

Hours: h
Hourly fee: 100h

Profit = (100h+150)–.3(100h) = 100h+150–30h = 70h+150

Answer: P = 70h+150

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If you’re dealing with a particularly puzzling word problem, don’t let yourself get tripped up over language. Direct translation from English to math may be too much to ask.

Instead, use hybrid notes to get things going.

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Contact Chris

borlandeducational.com

Friday, March 01, 2024

U.S. News State Educational Rankings

U.S. News is famous for its annual college rankings, but it also ranks states on a number of educational benchmarks. Their recent College Readiness Rankings are a revelation.

Not surprisingly, Northeastern states occupy half of the first 10 spots.

California? Number 49. Ouch.

When I was schooled as a boomer kid, California public schools were the best in the country – and, therefore, the best in the world. Number 49 is very hard to take.

Annual U.S. News college rankings (and others) are notorious as less-than-stellar indicators of college caliber. But they do, at least, provide a sense of relative quality. Similarly, one should probably take these overall educational rankings of the 50 states with a large grain of salt.

Nevertheless, if you'd like to see them, click here for the complete state rankings.

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Contact Chris

borlandeducational.com

Thursday, February 08, 2024

GPS for Functions

Functions are of central importance in algebra, and problems based on functions form a major subset of questions appearing on the SAT/ACT.

Wouldn’t it be awesome if you had a easy, reliable way to navigate straight to the answer to any question involving functions?

"GPS" will do it for you.

The acronym outlines three options that will help solve any function-related question on the SAT/ACT.

“G” stands for “graph.” (Would a graph be helpful?)

“P” stands for “points.” (Could you use coordinates of specific points?)

“S” stands for “substitute.” (Does a simple substitution solve the problem?)

Looking at virtually any function question on the SAT/ACT through these lenses will quickly and easily reveal the path to the answer.

Many times solutions can be found through simple visualization or basic pencil and paper techniques.

Otherwise, Desmos can be used to carry out graphing, locating points, or doing substitution.

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For practice, search Google for worksheets covering “SAT ACT function problems,” pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered the GPS strategy.

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Contact Chris

borlandeducational.com

Thursday, February 01, 2024

Teachers are Quitting

It seems our schools are always in trouble for this reason or that one.

This was not at all the case when I was a boy in the 1960’s. California public schools, K-College, were unquestionably the best in the world, and all were levels were free or very low cost. So it’s especially hard for boomers like myself to see the kite hit the ground like this.

As a private academic coach for more that 45 years, I’ve tried to figure out what went wrong. I still don’t know, after all these years.

And now, a sobering new trend is emerging. Teachers are leaving K-12 schools in droves.

The reasons are many and various. But what I hear over and over again is that they boil down to several core issues: low salary, poor mental health, toxic work cultures, unsupportive administrations, lack of respect inside and outside school, lack of authority in the classroom, and out of control students and parents.

I have no answer, but I do have suggestions. At a minimum:

Pay teachers like doctors, require the same achievement, give them the same respect inside and outside schools, and weed out poor teachers (the Finnish approach); give teachers back-up they need and deserve within schools hierarchies; return classroom authority to teachers; allow sensible grading and disciplinary procedures, including compassionately but unapologetically holding back students who don’t meet grade level standards; a teacher-aid in every classroom; effective protection against parent-zillas think their little Johnny can do no wrong; make it unnecessary for teachers to buy learning supplies their students need; and provide enough public funding to pay for all this, realizing that doing so is less expensive than not.

For further information, I recommended the following:

I quit teaching for better mental health: former teachers share the jobs they got after teaching

Why Teachers Quit: Lack of respect, abominable working conditions, and more.

After Teaching For 11 Years, I Quit My Job. Here's Why Your Child's Teacher Might Be Next.

I quit my job as a teacher after 6 years to work in tech sales. I make $20,000 more ...and am so much happier now.

Teachers Who Quit Are Sharing The Moment They Realized It Wasn't For Them

Why Teachers Quit + Top Signs Quitting Teaching Is The Right Move

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Contact Chris

borlandeducational.com

Monday, January 08, 2024

Solving Literal Equations

To beginning algebra students, literal equations, those involving more than one letter, often seem inherently more difficult to solve than simpler, univariate ones. But this isn’t so.

The trick is to treat the extra letters like simple numerals using the same steps you would ordinarily.

Solving equations is a matter of “undoing” what’s been done to the variable, using inverse operations, starting as far from the variable as possible.

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Example 1

To solve 3(x+4)–8 = 19, we first undo subtracting 8 by adding 8, then divide by 3 to undo multiplying by 3, and finally subtract 4 to undo adding 4. The result is x = 5. [Note: we could first simplify by distributing 3 across (x+4) and adding like terms, but this would take four steps, not three.]

If the letters a, b, and c were to replace 3, 4, and 19 in the same equation, we’d carry out exactly the same series of steps. This time we’d start with a(x+b)–8 = c. We’d then add 8, divide by a, then subtract b to get x = (c+8)/a – b.

When solving univariate equations, it’s important to add like variable terms as soon as possible. Sometimes this isn’t possible when solving literal equations. In such a case, factoring out the variable leads to a simple solution.

Example 2

To solve (nx–mn)/q = x+p for x, we first multiply by q and then collect and isolate variable terms on the same side by subtracting qx and adding mn to yield nx–qx = qp+mn. Since nx and x are not like terms, we factor out the variable x to produce x(n–q) = qp+mn, and divide by n–q. The result is x = (qp+mn)/(n–q).

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For practice, search Google for “ solving literal equations worksheets,” pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this important topic.

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Contact Chris

borlandeducational.com