Evaluating expressions (Why technology can help)

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Probability

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I agree that understanding comes from practice, practice and more practice, but sometimes its easy to be lead down the garden path and if there is no method of checking a result, then a person could believe the result even though that result is incorrect. Look at this example;

[MATH]\frac{5}{2}+a=[/MATH]
Where a = [MATH]-{2}[/MATH]
I understand from arithmetic that a [MATH]{+}{-}[/MATH] becomes a [MATH]{-}[/MATH]
So I rewrite the expression;

[MATH]\frac{5}{2}-{2}=[/MATH]
I now look at the expression and think [MATH]{5}-{2}=\frac{3}{2}[/MATH]
Then I ask myself, how do I know that the answer is correct?

Do I wait for next weeks lesson if I were part of a class?

Using the calculator here can give guidance.

[MATH]\frac{5}{2}-{2}=\frac{5}{2}-\frac{4}{2}= \frac{5-4}{2}=\frac{1}{2}[/MATH]........................................corrected

Now I can start to think about why my answer was wrong, and when I understand that I must cross multiply the fraction I see that I end up with a minus 4 in the expression rather than a minus 2.
 
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[Sorry, @lev888....stepped on your toes.]
I now look at the expression and think [MATH]{5}-{2}=\frac{3}{2}[/MATH]
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Think is where you're wrong. You need to move the -4 to the numerator in order to have cross multiplied correctly.

But don't get them started on cross-multiplication. :LOL:
 
No its not where I was going wrong, on paper it is the numerator but in latex it looks like its in the wrong place. Maybe there is a different command for that position?
 
Probability said:
No its not where I was going wrong, on paper it is the numerator but in latex it looks like its in the wrong place. Maybe there is a different command for that position?
Click to expand...
-4 should be together with 5 wherever in latex code it is.
 
Probability said:
I agree that understanding comes from practice, practice and more practice, but sometimes its easy to be lead down the garden path and if there is no method of checking a result, then a person could believe the result even though that result is incorrect. Look at this example;
...
Now I can start to think about why my answer was wrong, and when I understand that I must cross multiply the fraction I see that I end up with a minus 4 in the expression rather than a minus 2.
Click to expand...

It looks like a Latex error, and/or a non-ideal example, has derailed the conversation you hoped to start, which was about the role of a calculator in learning. Let's have that discussion!

I absolutely agree that a calculator can be a valuable tool to check one's work. And finding that the check fails may get you started in finding the error in your work. In particular, if you were to practice over and over without realizing that your method is wrong, you have just dug a deeper rut. As has been said, "Practice doesn't make perfect -- it just makes permanent."

Another benefit of having a calculator is that you can experiment with math and have some fun trying out new ideas.

There are, of course, several dangers. The best-known is that students get lazy and just use the calculator for everything, never learning to think for themselves. But you're diligent enough that you won't do that.

Another danger is that students use the calculator wrong; that is very common, for example, to enter something like [MATH]\frac{2 + 3}{5}[/MATH] as [MATH]2 + 3\div 5 = 2.6[/MATH]. They might think they were wrong when they were right, and get discouraged.

Or, as I see too often when students check the back of their book (which is good -- too many don't know to do that in the first place), rather than just go back and look at their work, they try to reverse-engineer the answer, looking for any way to get that answer from the numbers they were given. They are then using the minds the wrong way, and are likely to discover wrong methods.

We could probably make a list of pros and cons of calculators; but the outcome should be, not a blanket call to use them or not, but insight into how to use them, and how to help students use them effectively.

But what you suggest is (part of) the right way to go about it.
 
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@Probablity: My motto: Whenever possible learn how to do the problem and gain some skill before you even think about the calculator. (Yes, feel free to check your work but don't learn it by using the calculator.) I know this viewpoint is not currently in vogue but look at what happened here: the method you are using to subtract [math]\frac{5}{2} - 2[/math] you have called "cross-multiplying." This is not cross-multiplication which is done when you have [math]\frac{a}{c} = \frac{b}{d} \implies ad = bc[/math]. What you have is a subtraction you need to find the common denominator. This mistake tells me that you need more grounding in this. I'm not trying to insult you, you are still learning this, just making a point.

-Dan
 
topsquark said:
@Probablity: My motto: Whenever possible learn how to do the problem and gain some skill before you even think about the calculator. (Yes, feel free to check your work but don't learn it by using the calculator.) I know this viewpoint is not currently in vogue but look at what happened here: the method you are using to subtract [math]\frac{5}{2} - 2[/math] you have called "cross-multiplying." This is not cross-multiplication which is done when you have [math]\frac{a}{c} = \frac{b}{d} \implies ad = bc[/math]. What you have is a subtraction you need to find the common denominator. This mistake tells me that you need more grounding in this. I'm not trying to insult you, you are still learning this, just making a point.

-Dan
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Thank you for your advice. When I introduced the idea of using a calculator I thought (could be wrong) that I'd explained that I wanted the use of the calculator to help me understand if I'd worked out the maths correctly, hence using the calculator to check my answers. To me the brain storming of maths is not the answer, but working my way through the example to gain an answer. All I need to understand then is two things, did I use the correct procedure to reach my answer, and is the answer correct!

Interesting that I said I cross multiplied to move forwards in the example you mention. I did cross multiply to get the correct numerator, however I can see now that the terminology I'm using is incorrect.

To save me painstakingly soul searching through maths books at this time to find the correct terminology and techniques to use for that type of math, would you be so kind please to show me by example. This I can then remember for the future. Thank you.
 
firemath said:
Let's ask Jomo about calculators......
Click to expand...
OK, lets ask Jomo.
Jomo thinks that if you know how to do the calculation but the numbers are not friendly, then use a calculator. Jomo thinks that if you use a calculator to compute 7*8 then you should not be allowed to use a calculator anymore. Jomo agrees with Dr Peterson that there are pros and cons for using a calculator.
 
The point about calculators is, in my opinion, just half right.

Calculators, if used properly, can confirm when a numerical answer is correct. Similarly, a graphing calculator, when used properly, can confirm (or obviate the need to find) the locations of a function's zeroes and local extrema. Please notice the qualification about used properly. Furthermore computer tools such as derivative calculators can be great in confirming results. Use them, but do not rely on them. (In one case, I used a statistical package and got a result that seemed very odd to me. Eventually, I sent an email to the creator and told him that I thought I had found an error. He was most thankful because I had indeed found an error that arose only in a special case and he was eager to eliminate it.)

A large part of mathematics is not about computing specific numeric results, but rather about determining general results. A calculator is of no use whatsoever in such cases because there is no numeric answer. If you get in the habit of using a calculator as a checking device, you will be lost the moment you step beyond basic arithmetic and the most routine uses of elementary algebra.

The most general method of checking results in arithmetic and elementary algebra is inserting your result back into the original problem. Does your answer agree with the problem as originally stated. If the two do not agree, you have made one or more errors and got the wrong answer. If they do agree, you have got A correct answer (there may be more than one).

Let's take your original problem.

[MATH]\dfrac{5}{2} + a = x \text { and } a = - 2.[/MATH]
You solve as follows

[MATH]\dfrac{5}{2} + a = x \text { and } a = - 2 \implies\\ 5 + 2a = 2x \implies\\ 5 + 2(-2) = 2x \implies \\ 5 - 4 = 2x \implies\\ 2x = 1 \implies\\ x = \dfrac{1}{2}.[/MATH]Now to check we go back to the original problem to check the answer.

[MATH]\dfrac{5}{2} + a = \dfrac{5}{2} - 2 = \dfrac{5}{2} - \dfrac{4}{2} = \dfrac{5 - 4}{2} = \dfrac{1}{2}. \checkmark[/MATH]
There is no need for a calculator whatsoever, and you have now seen a technique that will work for OTHER problems where a calculator is useless for checking results.

In short, calculators are FABULOUS for confirming arithmetic, but they completely distract from general methods of confirming mathematical results in any but the most basic mathematics. If you want to learn mathematics, use calculators to do arithmetic, but even then check your results some other ways because people make mistakes even when using calculators.
 
I use a calculator (or computer) to help me to find the line where I made a mistake in a question. For example, the question might be to simplify:-

[math] \frac{\sin\left(3a\right)-\sin\left(a\right)}{2\cos\left(2a\right)} [/math]
I do 5 lines of manipulation and I end up with something that doesn't agree with the answer at the back of the book, which is:-

[math] \sin\left(a\right) [/math]
But where did I go wrong? To find out I simply plug in an arbitrary "a" value like 0.12 into the original. (Choose an "a" value that is a bit random!) Plug this number into the original expression, and you get approx 0.1197122

Then plug this number into the text book's answer - it also gives 0.1197122. Therefore the text book is probably correct!

I plug a=0.12 into my 5th line and I get a number that is NOT 0.1197122. Therefore I have made a mistake :(

But where did I make a mistake? To find out I can use a "binary search" technique to find the line where I went wrong. Just check the line that is half way between the line that I know is correct and the line that I know is wrong. In this case I check my 3rd line (half way between the 1st and 5th lines). To do this I plug a=0.12 into my 3rd line on a calculator. I get 0.1197122. So my lines 1 to 3 are probably correct. My mistake is probably in lines 4 to 5. So now I can test line 4 on the calculator. Hopefully you get the idea!
 
The one place where I most strongly recommend using a calculator is in checking a numerical answer to an algebra problem. The reason I recommend it is so that students will actually do the check! If a calculation scares them off (say, putting a fractional answer into a big equation, that will take too long by hand, and will likely be wrong), using the calculator makes it less scary, and they are more likely to try.

But I also tell them that if they do that on a test, they need to have used the calculator enough to be sure they enter the calculation correctly. I don't want them thinking they were wrong when they were right, because they failed to use parentheses where they were needed, or something like that.

And, of course, it is far better to check each step you write to see that what changed makes sense. A calculator can't do that.
 
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