Another old problem solved with a different method

[Steven G]

I believe that a student should know what is being asked!
Consider this problem->x(x+5)=24. Most people would say that (1) x^2 + 5x -24 =0. Then (2)they would try to find two numbers that multiply out to -24 and add up to 5. ...

How about this: x is some number and x+5 is 5 more. So I want a number (ok, numbers) that when I multiply it by 5 more gives me 24. This is very similar to step 2 above (same rate of difficulty) and you do not even have to do step 1!! One pair of numbers are 3 and 8 and the other pair (we can get a quadratic eq afterall) is -8 and -3. So x=3 and x= -8.
Students can learn to do this!

 

[lookagain]

So x=3 and x= -8.

So x = 3 or x = -8.

 

[stapel]

I believe that a student should know what is being asked!
Consider this problem->x(x+5)=24. Most people would say that (1) x^2 + 5x -24 =0.

How does what "most people would" do somehow mean that they don't "know what is being asked"? They were asked to solve an equation, and they are, using appropriate algebraic tools. How is this wrong?

How about this: x is some number and x+5 is 5 more.

Are you proposing that students should be taught (by implication) that all solutions are whole-number values? How then will you justify the Quadratic Formula and its frequent generation of irrational solutions? How is your method anything but misleading? :shock:

 

[Steven G]

How does what "most people would" do somehow mean that they don't "know what is being asked"? They were asked to solve an equation, and they are, using appropriate algebraic tools. How is this wrong?


Are you proposing that students should be taught (by implication) that all solutions are whole-number values? How then will you justify the Quadratic Formula and its frequent generation of irrational solutions? How is your method anything but misleading? :shock:

When students see a quadratic equation, fair enough they know it is a quadratic equation and they know that they have to solve it. Sometimes the answer is obvious but students don't see the answer. Like in x(x+1)=6, a student should at least think 2*3=6 and so x=2 is one solution. I feel this is very important for students to learn and for the most part it is not taught in high school. As far as my implying that all solutions to quadratic eqs are integers I did not really want that the reader to feel that way. The steps to solve a quadratic eq, especially when a=1, is to first try to factor. My technique above just just removes a step in the process whether there are 2 integer solutions or not. But in the end they know exactly what they are looking--something like some number times 3 more than that number equals 40.
Another thing is that when the student understands what the equation is asking it become fun for them to figure out the puzzle.
I remember when my algebra teacher told the class that to solve x+1=2 you subtract 1 from both sides. Fine it works but is there any understanding what the problem is asking? I feel that a student should ask themselves what plus 1 is 2? Isn't that better for the students instead of learning that you should subtract 1 from both sides.
Another problem. (x+1)^2 =(x+1). Fine, you can square the left side and ultimately use the quadratic formula. But it is better to see that they are asking for a number that when you square it gives you back that same number and they are calling the number x+1. Well 0^2=0 and 1^2=1. So x+1 =0 or x+1=1. Isn't that better? Not because it is quicker but because it shows understanding of the problem.
Last question. x + 1/x = 5/2. You once again can get a quadratic equation but then you have no idea what the problem is asking at all. This problem is asking what number plus its reciprocal give you 2 1/2 (5/2). Well x can be 2 or 1/2. I see no reason why this problem should take less then 4 seconds to solve--the time it takes a student to figure out that 5/2= 2 1/2. Just my thoughts.

 

[lookagain]

The steps to solve a quadratic eq, especially when a=1, is to first try to factor.

No, not necessarily. If the coefficient of the linear term is zero, then the root method may be preferred.

Last question. x + 1/x = 5/2. You once again can get a quadratic equation but then you have
no idea what the problem is asking at all. This problem is asking what number plus its reciprocal give you 2 1/2 (5/2).
Well x can be 2 or 1/2. I see no reason why this problem should take less then 4 seconds to solve--
the time it takes a student to figure out that 5/2 = 2 1/2. Just my thoughts.

You're demonstrating an opposite point. Happening to see a solution by inspection doesn't count.
The student has to take the time, effort, etc. of learning to solve that type of problem and then actually write
down proper algebraic steps that support each other.

 

[Steven G]

No, not necessarily. If the coefficient of the linear term is zero, then the root method may be preferred.



You're demonstrating an opposite point. Happening to see a solution by inspection doesn't count.
The student has to take the time, effort, etc. of learning to solve that type of problem and then actually write
down proper algebraic steps that support each other.

I politely disagree with you. Sure the student should know how to do the problem if they do not see the answer immediately. I agree with that 100%. But the student should know what is being asked. We ask student to translate words to math. Well I feel that students should be able to go from math to words. I have seen student have no idea what to do when they needed to know what times 7 is 9 even though they can solve any linear equation flawlessly. I knew this student who was able to solve 2*x=5. She quickly said x=5/2. Great, she knew the steps! I then asked this student 2 times what equals 5 and she said 2*nothing =5. She got the correct answer to 2x=5 but had no idea what the answer meant. In my opinion, regardless of how well this student could solve eqs of the form ax=b, she knew nothing about linear equations. I had a classmate years ago who was able to find the derivative of almost any function. I asked this student what a derivative was and she could not answer me. So in the real world, although she would know how to calculate the derivative of a function, she would never compute the derivative since she would not know that she needed to. This is not the way to learn math! Analyzing the problem, thinking about the problem in different ways, asking what if we changed this, what if we look at it in this way...this is the way to learn math.

 

[stapel]

When students see a quadratic equation, fair enough they know it is a quadratic equation and they know that they have to solve it.

Not always. Which is why misrepresenting the situation to them is likely only to be harmful.

Another thing is that when the student understands what the equation is asking it become fun...

Not true.

I see no reason why this problem should take less then 4 seconds to solve...

So your methods are intended to make things take longer? :shock:

 

[lookagain]

Happening to see a solution by inspection doesn't count. The student has to take the time,
effort, etc. of learning to solve that type of problem and then actually write down proper algebraic steps that support each other.

I politely disagree with you. Sure the student should know how to do the problem if they do not see the answer immediately.

It's irrelevant whether the student ever sees the answer immediately. Regardless, the content in the last ten words
in the above quote have to be adhered to.

 

[Dale10101]

?

I believe that a student should know what is being asked!
Consider this problem->x(x+5)=24. Most people would say that (1) x^2 + 5x -24 =0. Then (2)they would try to find two numbers that multiply out to -24 and add up to 5. ...

How about this: x is some number and x+5 is 5 more. So I want a number (ok, numbers) that when I multiply it by 5 more gives me 24. This is very similar to step 2 above (same rate of difficulty) and you do not even have to do step 1!! One pair of numbers are 3 and 8 and the other pair (we can get a quadratic eq afterall) is -8 and -3. So x=3 and x= -8.
Students can learn to do this!

Yes, I do see you have introduced a new way to look at x(x+5)=24.

The first key is to note that you are looking for two numbers whose product is 24. This is especially useful if x is an integer, the solution set is easily listed. If then you are telling me that the two numbers differ by 5, then the list is rapidly reduced to usually one or two solutions.

But why bother, who cares, just apply the rules, solve the problem, learn to become Mathematica.

Hmmm, this reminds of the division of the problems at the end of the chapter, algebra drill problems vs word application problems. The word problems are often dreaded, they require a different kind of thinking.

It seems to me that what Jomo is suggesting is that before solving a drill problem one might learn to see a shorter or longer method based upon a sort of overview of what you basically attempting to do. Perhaps what he is suggesting is to try and introduce a third type of problem, problems using the same type of verbal reasoning and understanding of the whole of the problem required to solve a world problem, to solve algebra drill problems.

It seems to me that offering different perspectives about how to solve even algebraic manipulation problems, at the very least, invites closer inspection of a problem.

Finally, these insights, these shortcuts, isn't this part of algebraic manipulation as it stands. I mean, why teach how to factor a quadratic any other way then by using the quadratic formula, in essence aren't the other methods irrelevant to the case in hand? Of course I do not believe that but for the same reason that Jomo offers his insights, these additional methods are often shorter and provide a perspective that is useful in a larger context.

(I think I now see what is going on Jomo, sound pedagogy requires keeping things basic and simple, straight forward. This is essentially junior school plus, if you introduce skipping before walking then you might invite confusion. There is a point there. That said, it does seem to me that here in the odds and ends forum some might find your insights useful.)

 

[Steven G]

Yes, I do see you have introduced a new way to look at x(x+5)=24.

The first key is to note that you are looking for two numbers whose product is 24. This is especially useful if x is an integer, the solution set is easily listed. If then you are telling me that the two numbers differ by 5, then the list is rapidly reduced to usually one or two solutions.

But why bother, who cares, just apply the rules, solve the problem, learn to become Mathematica.

Hmmm, this reminds of the division of the problems at the end of the chapter, algebra drill problems vs word application problems. The word problems are often dreaded, they require a different kind of thinking.

It seems to me that what Jomo is suggesting is that before solving a drill problem one might learn to see a shorter or longer method based upon a sort of overview of what you basically attempting to do. Perhaps what he is suggesting is to try and introduce a third type of problem, problems using the same type of verbal reasoning and understanding of the whole of the problem required to solve a world problem, to solve algebra drill problems.

It seems to me that offering different perspectives about how to solve even algebraic manipulation problems, at the very least, invites closer inspection of a problem.

Finally, these insights, these shortcuts, isn't this part of algebraic manipulation as it stands. I mean, why teach how to factor a quadratic any other way then by using the quadratic formula, in essence aren't the other methods irrelevant to the case in hand? Of course I do not believe that but for the same reason that Jomo offers his insights, these additional methods are often shorter and provide a perspective that is useful in a larger context.

(I think I now see what is going on Jomo, sound pedagogy requires keeping things basic and simple, straight forward. This is essentially junior school plus, if you introduce skipping before walking then you might invite confusion. There is a point there. That said, it does seem to me that here in the odds and ends forum some might find your insights useful.)

Dale, thanks for your response. I agree with you up to a point. I have a problem with when you said if you introduce skipping before walking then you might invite confusion. I guess because of my poor k-12 education I learn towards the other extreme. I went to public school in the Brownsville and East New YorK section of Brooklyn. This was an area where some of the people in power and the some of the teachers did not care about the students at all (code for they looked different then the people in power). I heard that my assistant principal said that 'his students are not garbage, how dare someone call my students garbage. My students are recycled garbage'. I truly felt that my teachers went out of their way to not let their students think. I have the opposite feeling regarding this. I believe that students can understand math if it is explained to them very clearly and carefully. The way for the most part that education is taught is not the way to teach math. Please understand that the volunteers of this forum are not the average teachers. The educators here know and understand math as opposed to memorizing it. I absolutely respect the math ability of all the volunteers after seeing them in action after just a few weeks.