“If a boy can pick one basket of strawberries in an hour and a girl can pick two, any mathematician will tell you that the two of them can pick three baskets in an hour.
Any farmer can tell you they won’t pick any.”
I always liked the geometry problem that begins, “Two planes intersect in a fiery crash…”
If that is a college textbook, then I don’t think it supports the observation that math books get dumbed down every year. This problem is fairly simple and should not be a major problem for high school students. I would think/hope that at today’s college level, mathematics gets more complicated than that (and I’m not saying that to sound condescending - I’m certainly not a maths whiz).
If you can reach the point of writing 65(x+2)=75x then the rest is trivial algebra that any student should be able to do by that point in the course. But how did you come up with that equation?
See, that’s was the perennial problem with “word problems” that most students had conniptions with. How do you develop an equation from the written conditions given in the probem? Where did the (x+2) come from? And you are writing an equation saying that something is equal to something. What are you saying is equal to what, and how did you know to write that?
It’s the thinking that leads from the statement of the word problem to writing that equation that perennially befuddles beginning algebra students. You just started your solution with that equation, but you didn’t “show your work” in showing how you got to that point.
The student who looks at your solution will either get it or he won’t. For those who don’t, the work you showed doesn’t help. And most teachers I’ve heard tend to gloss over that.
In your equation 65(x+2)=75x you are expressing that the two trains’ R[sub]1[/sub]T[sub]1[/sub]=R[sub]2[/sub]T[sub]2[/sub] (that is, their rate X time is equal). I don’t think that’s “obvious” to a lot of beginning students. Why are they equal, and why does the idea even occur to you to think that? Well, because they are the two trains’ D[sub]1[/sub] and D[sub]2[/sub] respectively, and those two D’s are equal. The beginning student, first learning how to approach these problems, needs this sequence of thinking explained. You and Dsorgnzd skipped that, and jump right into the equation and the algebra. You skip over demonstrating the very part of the problem that a great many students have trouble with.
There is a further problem with the solutions suggested by Dsorgnzd and GoodOmens. You somehow see the solution at a glance. You didn’t explain how you arrived at the logic that you did. (It took me a while to see your logic.)
That kind of intuitive thinking might work for the simplest of the problems, which that problem was. What happens why you get a more complicated version of such a problem?
The student needs to start with the simplest problems, but learn an organized and methodical way to work them. And this includes especially developing a skill at developing that initial equation from the stated conditions of the problem. If the student is able to see the problem at a glance, that may not work when you get to the more complicated problems.
Here is another one is the same problem set:
As with all of these word problems, the real problem is in “setting up the problem”, that is, developing an equation(s) from the stated conditions. Once that is done, the algebra ought to be straightforward.
I agree. The entire challenge is understanding the problem well enough to come up with the equation. I never said it was obvious. In fact, as a kid, I dreaded these kind of problems. The only point I wanted to make is that once you grasp it, you only need a single equation, rather than multiple equations which I thought you were implying.
I don’t mind showing my work. I wrote down a small table like this to get an understanding of the problem.
Time Train1D Train2D
2 Hrs 130 0
3 Hrs 195 75
4 Hrs 260 150
Which made me realize, I was looking for X where the two Ds are equal:
Time Train1D Train2D
X D D
I then wrote the formula as: 65x = 75(x-2) which give me X = 15
Time Train1D Train2D
15 Hrs 975 975
but I realized I needed to subtract 2 hours from 15 to get the correct answer which is 13.
So I decided to rewrite the equation, so the 2 hour adjustment was built in: 65(x+2)=75x
and this is really all the work I did.
I disagree with that, because I’ve seen the entire book and you haven’t. This book is a college algebra book, but seems to be written for the student who has absolutely zero background in algebra and apparently barely any background in basic arithmetic.
It begins with a review that looks more like third grade arithmetic, then gets into algebra with the bare-bones commutative, associative, and distributive laws (I think we got that in 3rd or 4th grade), but ultimately progresses to advanced topics like ratio, proportion, and variation; the binomial theorem; progressions; complex numbers; a whole chapter on inequalities; a chapter titled “Advanced Topics in Quadratic Equations”; Theory of Equations; systems involving quadratics; a whole chapter on determinants; and permutations, combinations, and probability. Every topic that was covered in this book and is also covered in a typical modern book, is covered in much more depth and detail in this book.
Being written in 1948, before everyone had calculators, it also has extensive practice in numerical calculations: A whole chapter on working with approximate numbers, and an entire chapter on computations with logarithms.
The whole book is apparently intended to take one school year. I think that’s probably an intense course to cover in one year.
In each chapter, the text and problems lead into techniques beyond anything seen in any modern book. The chapter on Quadratic Equations, for examples, leads to problems like:
(x - 2)/(x[sup]2[/sup] - x - 6) - x/(x[sup]2[/sup] - 4) = 3/[2(x + 2)]
(Hint: One integer solution and one rational non-integer solution.)
And: (x[sup]2[/sup] - 1)[sup]2[/sup] - 5(x[sup]2[/sup] - 1) + 6 = 0
You don’t find problems like those in modern textbooks, even at college level.
When I was taking more advanced (lower division) math classes in college, I referred back to this book often. The chapter on working with inequalities was later vastly helpful when, in Calculus, I had to deal with epsilon-delta proofs. The techniques had a lot of overlap, and a solid understanding of inequalities is essential for understanding limits and epsilon-delta definitions and proofs.
This is also pretty straightforward. I doodled a little picture to visualize the problem.
I then wrote this equation: 3x+3(x+65) = 1755
Then plugged into Wolfram Alpha because I’m lazy :): 3x+3(x+65)=1755 - Wolfram|Alpha
x = 260 = speed of slow plane
260+65 = 325 = speed of fast plane
That is the only work I did for this one.
A fairly current citation from How to Solve Word Problems in Algebra https://www.amazon.com/gp/aw/d/0071343075?psc=1&ref=ppx_pop_mob_b_asin_image
Chapter 2, page 19 has a one train leaves Chicago for Boston and at the same time another train leaves Boston for Chicago problem as the first example of a distance problem.
In my first Algebra class, doodling the little picture was emphatically taught as the starting point in these problems. Everybody in the class, myself included, had a hard time understanding what to draw.
There are TWO trains (planes, boats, whatever). They aren’t standing still; they are moving. There is a starting point, or each train has a separate starting point or a separate starting time. The problem may end at one moment (when the trains meet), or at different times (When is the first train 450 miles away from the second train)? What are you supposed to put in the drawing? How do you draw moving trains? Do a simple animation 
?
In every case, the teacher showed the right drawing to start the problem, but always left the students wondering: How did you know to draw that ?
I start these problems with two equations, but I didn’t mean to imply that the beginning student, at this stage, should. Maybe I ought to imply that. At the stage where these problems are introduced, the student typically has learned to deal with linear equations in one variable, but has NOT been introduced to systems of two (or more) equations in two (or more) variables.
So a LOT of time is spent dealing with the idea of finding one variable that is related to another and thus writing two unknowns in terms of one variable. For example, in the first problem, you used x for one time and (x+2) for the other time, and you implicitly noted that both trains had the same D in order to make the equation. In the second problem, you used x for one speed and (x+65) for the other speed, and noted that they both have the same time. (And furthermore, you realized that you had to work in the fact that D[sub]1[/sub]+D[sub]2[/sub]=1755 somehow.)
In starting with two equations, and writing out several steps before reaching your starting equation, I was essentially just showing more of my work than you did – that is, more of my thought-process steps that you did but didn’t write down. And those were, probably, the very steps that the beginning student needs to be shown.
Spotting those kinds of relations, that allow you to write one unknown in terms of another, was a major object at that stage of the class, and was always a big stumbling block for a lot of students for a long time (middling so, for myself).
Much later in the class, when we get into systems to two equations in two unknowns, it became MUCH easier to set those equations up. Then, when you use various techniques to eliminate one variable and end up with one equations, it turns out you’re back exactly where you were six weeks earlier in the class, but it sure was easier to set up problems that way. In fact, I noticed that a whole lot of the word-problems for systems of equations were exactly verbatim the same problems that were given earlier in the book!
So my knee-jerk starting point for these problems is to begin with two separate D=RT equations, then fill in the known pieces, then see what it takes to merge them into one equation. For me, this seems to take the place of doodling the little picture, which I always had trouble with.
I’ll make another plug for that old Britton Snively 1948 book: It has whole section devoted simply to the process of writing an algebraic expression from a verbal description. It’s given at a level that modern books (and teachers) seem to expect students to just intuit.
After some explanatory text with some examples, here’s a sampling of they types of problems here. If I were teaching a beginning algebra class, I would definitely photocopy this section and hand it to everyone.
The later problems in the problem set get more complicated:
This is an excellent exercise for beginning algebra students, preliminary to working with actual word problems.
BTW: Avoid using the [noparse]
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[/noparse] tags in your posts. Depending on which theme the reader is using, these are either displayed perfectly as you intended, or are totally illegible.
Without seeing that actual book, I can’t guess how good it is at what it tries to do.
But I certainly agree that an entire book on how to solve word problems would be a vastly valuable thing for the beginning algebra student, if it’s really done well.
I saw one section in a relatively modern college-level beginning algebra book that advised the students to draw some kind of a chart (like AnalogSignal did in Post #26 above), which was a new idea to me at the time. I certainly had not seen that technique when I was learning.
And yet, the book gave very little advice as to what to put into the cells of that chart! Oh, sure, it gave several fully worked-out examples. But it never really explained the thought process behind it. So, it pretty much omitted the very part that most students have a hard time with!
Anyway, I guess we have a consensus as to the answer to OP’s question by now, yes?
The answer is: Yes.
You may think, as many people do, that such problems are unnatural contrived problems that never occur in real life, generated only by sadistic math teachers solely to torment students.
In fact, for those not old enough to remember, this particular problem really did happen, in March 1977, with 583 fatalities and 61 non-fatal injuries. Two planes intersected on a runway in a fog, shedding light and heat on the situation.
This reminds me of sophomore year in college when my two roommates and I were taking the same class in formal logic. I could breeze through the homework doing all the proofs in a few minutes and they worked for hours as a team and would get stuck hitting brick walls on the harder problems. They would ask me for hints, help, suggestions. They didn’t want the answer, they wanted to understand how to get a little closer to the solution. And I tried to help but couldn’t because I was making leaps automatically that weren’t really teachable. I tried to explain but couldn’t because I was doing a lot of the heavy lifting automatically or subconsciously.
So I think what you are doing is great. You start from first principles and build the solutions from the ground up. Maybe you are not jumping immediately to the answer but you are not running into brick walls either. Slow and steady wins the race as they say.
This is a near-constant challenge! Not just to solve any one particular problem, but to understand, and be able to explain, the underlying thought process for approaching problems in general.
My idea for word problems is generally along the lines:
[ul][li] Identify every number, known and unknown, mentioned in the problem. Sometimes this could be subtle or hidden. For example, if the problem mentions nickels and dimes and quarters, those mean the numbers 5, 10, and 25.[/li][li] Find the phrases that describe (perhaps only partially) some of those numbers. For example, “John is as old now as Mary was three years ago”.[/li][li] Be aware that the phrases “more than” and “less than” do not refer to inequalities, but simply mean “add” or “subtract”. (For example, the potato weighs 8 oz more than the cucumber.)[/li][li] Look for a statement in the problem to the effect that one number (that is partially described) is the same number as another partially described number. This may be explicitly stated, or it may be somewhat hidden (as in those train problems, for example – it doesn’t outrightly say that train #1’s distance is the same as train #2’s distance, but you have to see that).[/li][li] Then, you should be able to write an equation, in the general form:[/li](Some partial description of a number) = (Some other partial description of the same number)[/ul]
It’s still going to take quite a few specific examples of this for a student to begin to get it, if ever.
Oh yeah. These were tricky when I was a kid. Let me show the work.
A’s rate of work = 1/12 of the job per day
B’s rate of work = 1/8 of the job per day
f= 1/123 + 1/83 = 5/8
This is just an abstraction of 14(a)
f= 1/xt + 1/yt = (1/x + 1/y) * t
That’s exactly the point of that whole exercise set – Do a simple(?) calculation with some given numbers, and then abstract the same calculation to produce an algebraic expression for the same. (Any student will agree that algebra = abstract!)
Interesting that you first thought of 1/xt + 1/yt with the t “factored in” and then factored it out. My thought went directly to having the t factored out:
– A does 1/x of the job in one day.
– B does 1/y of the job in one day.
– Therefore A+B do 1/x + 1/y of the job in one day.
– Therefore A+B do t(1/x + 1/y) of the job in t days.
Then I’d ike to contibute my thinking for arriving at the conclusion step-by-step.
I set up an equation D[sub]2[/sub], meant to calculate the distance (in miles) the second train has travelled as a function of the time t after its departure (in hours):
D[sub]2[/sub] = 75t
The first train goes at 65 mph rather than 75. In addition, it had a head start of two hours, during which it travelled 130 miles, before the clock for t even started ticking. So the equation for the distance travelled by the first train, as a function of the same t as for the second train, is:
D[sub]1[/sub] = 130 + 65t
Now since we want the point where the two trains meet, the distances at that point mus be equal, so we can equate D[sub]1[/sub] and D[sub]2[/sub]:
130 + 65t = 75t
It’s easy to build in a trap here, considering that there is a one-hour time difference between those cities.