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jamonwindeyer

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12 Avoidable Mistakes Students Make in Math Exams
« on: March 02, 2016, 02:35:38 pm »
+11
How many marks could this have lost you in the HSC? Let's go through common things to look out for, especially in pressure exam situations!  :o

1. Sign Errors

We’ll begin with the most common one that everyone will know about. Sign errors are the bane of a Math student’s existence. You could very easily lose 5+ marks to sign errors in your HSC Exam. Most commonly, this pops up when you are subtracting a multi-term expressions:



To minimise the chance of sign errors, you need to write all your lines of working. Trying to multitask increases the chance of making a silly error. Be especially careful when expanding brackets.

2. Incorrect Form/Units

Not putting your answers in the correct form affects numerous areas of the Mathematics course. Using radians instead of degrees, giving an equation in gradient-intercept form when general form is specifically requested, incorrect number of decimal points, etc. Make sure to address the specific question and put your answer in the correct form. There is nothing more frustrating than losing a mark because you didn’t convert your answer from meters to kilometers (and yes, you would lose a mark for this).

3. Integration Constant and Notation

We’ve all been guilty of this at some point (or maybe I was just especially bad at it).Two things to remember:

a) Remember the ‘dx’ or ‘dy’ at the end of your integral. This may not cost you a mark, but it very well could, so don’t risk it.
b) Make sure you remember to put the integration constant for indefinite integrals. This will definitely cost you a mark, and every year the marking center warns us about it. Don’t forget it!

4.Not Testing Nature of Points

This is a common error, especially under pressure. Remember that you MUST test the nature of stationary points, using either the ‘point either side’ method, or more commonly, the ‘second derivative’ test. Further, you also MUST test your points of inflexion using the ‘point either side’ method. Not doing so will cost you a mark. Consider the las question from my HSC (the scenario itself isn’t important):

Find the values of x and y that maximize the amount of light coming through the window under test conditions.

Many students found a value of x and y which gave a stationary point, but did not prove it was a maximum! Just because the question says you are looking for a maximum, you cannot overlook these texts.

5. Eliminating Solutions by Dividing by Zero

See if you can catch what is wrong with my solution to this trigonometric equation.



Looks okay, but look at the first step. We divide by sinx. What if sinx had been equal to zero, we have removed some solutions! The correct method is instead to factorise the sinx out of the expression, which will give additional solutions.

Make sure that you only divide by valid constant terms, and factorize pro-numerals/unknown expressions. This will ensure you never lose any solutions.

6. Not Fully Simplifying/Factorizing

An extremely common mistake in exams is not expressing the solution in the simplest possible form. This may involve simplifying a fraction, grouping like terms, or rationalizing a denominator, among others. This also includes not fully factorizing a solution, when required.

Make sure you take 5 seconds to check for these sorts of mistakes. For factorizing questions, check how many marks it is worth. If you have done 1 line of working for 3 marks, chances are there is more to be done.

7. Rounding Too Early

This one is simple; never round any values until the final answer. Or, if you do, make sure you take enough decimal places (I used to take four by default). I could go into the proper theory behind how many points to take, based on the error in the values and the question (first year university students who take Physics will know all about this), but it’s much easier to just give a blanket rule. Round only at the end, and use your calculator to store preliminary values.

8. Taking a Question ‘Too Far’

Many students will look at a question and just start working based on what they assume the question to be. Then, they read it back and go: “Oh, I only had to find the first maximum value, not all three.” Or, “Hmm, I didn’t even have to find inflexion points here…”

Make sure you only do the work required for a question. Part of this comes from experience, knowing exactly the data you need for an answer. For more difficult questions, some aimless working may be inevitable. However, minimize wasted time by reading a question fully before commencing work. It seems like a silly reminder, but exams are pressure scenarios, and common sense can sometimes fall out the window.

9. Incorrect Assumptions

This primarily concerns geometry questions, though it can apply for calculus questions, probability questions, and a few other areas: Never make any assumptions based on what the diagram looks like! Unless it specifically says that angle ABC is a right angle, then it isn’t a right angle until you prove it. Unless it says that three points are collinear, then they aren’t collinear until you prove it. Use only the information given to you, never create your own.

10. Square Rooting Sign Errors

Another common mistake. Make sure that when you square root (or any even power) a value, you acknowledge that it can be either a positive or negative result. Even if the answer could only be positive (EG – it is a distance value from the Pythagorean formula), you absolutely must acknowledge the negative answer, then disregard it based on the situation at hand. Also remember that you can’t square root a negative number (unless you are a 4 Unit Student, in which case you can, but 2U students need not worry about such complex things).
 
11. Diagrams!

This is a big area that is identified by the Marking Centre every year. Make sure your diagrams are clear, labelled, and take up at least a third of the page. No more to be said here, just give the diagrams time and space and they are (normally) easy marks to be had.

12. Forgetting to Swap the Inequality Sign

Finally, don’t forget to swap the sign of an inequality when you multiply or divide by a negative. This should be second nature, but there is always somebody who forgets. Don’t be that person.

On inspection of the HSC Papers for 2014 and 2015, these errors had the potential to pop up at least once, and usually, multiple times. This could cost you upwards of a dozen marks in your HSC.

My biggest tip to avoid these unnecessary mark losses in any Mathematics exam is to take your time. Move carefully through the questions, don’t rush, it will cause silly mistakes.

If you know of any other traps, feel free to share them in the comments below! ;D

Ali_Abbas

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #1 on: August 21, 2016, 10:14:38 pm »
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Hi Jamon,

What you have done is excellent but I just thought I'd suggest an additional method for testing for the nature of a stationary point, which you outlined in point number 4.

There are actually three methods to determine the specific type of a particular stationary point, two of which you already mentioned, and the third being a first derivative test. This is done as follows:

Suppose we have a point which maps the first derivative to zero and that this occurs at the point x = c. We select a small interval about the point c of radius h, h > 0, such that no other neighbouring stationary points lie within its set. Then for all x not equal to c, the first derivative exhibits uniform monotonicity on either side of c. Given this property, it suffices to select just one point on both sides of c and pass each one through the first derivative to evaluate the exact sign it yields. Since there are only two possible signs it can take, there are a total of 2×2 = 4 possibilities. We outline each as follows:

f'(x) < 0 to the left of c, f'(x) > 0 to the right of c. This indicates a minimum stationary point.

f'(x) > 0 to the left of c, f'(x) < 0 to the right of c. This indicates a maximum stationary point.

f'(x) < 0 to the left and right of c. This indicates a horizontal point of inflection.

f'(x) > 0 to the left and right of c. This also indicates a horizontal point of inflection.

Remark: The test presented above is exclusive of ordinary inflection points and should only be used under the hypothesis that the point in question maps the first derivative to zero, as stated at the beginning.

Remark 2: The interval I chose was symmetric about c, however, in practice there is no real requirement to make it so. Also, a symmetric interval like the one demonstrated is known as an 'open ball' or 'open sphere'. There are multiple ways to denote an open ball, one such notation is:
B(x,e) which means the open ball centred at x, of radius e, and is the set: {y in R^n : || x - y || < e }. Thus, open balls can be defined on any Euclidean space.
« Last Edit: August 21, 2016, 10:23:16 pm by Ali_Abbas »

RuiAce

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #2 on: August 21, 2016, 10:20:36 pm »
+1
Hi Jamon,

What you have done is excellent but I just thought I'd suggest an additional method for testing for the nature of a stationary point, which you outlined in point number 4.

There are actually three methods to determine the specific type of a particular stationary point, two of which you already mentioned, and the third being a first derivative test. This is done as follows:

Suppose we have a point which maps the first derivative to zero and that this occurs at the point x = c. We select a small interval about the point c of radius h, h > 0, such that no other neighbouring stationary points lie within its set. Then for all x not equal to c, the first derivative exhibits uniform monotonicity on either side of c. Given this property, it suffices to select just one point on both sides of c and pass each one through the first derivative to evaluate the exact sign it yields. Since there are only two possible signs it can take, there are a total of 2×2 = 4 possibilities. We outline each as follows:

f'(x) < 0 to the left of c, f'(x) > 0 to the right of c. This indicates a minimum stationary point.

f'(x) > 0 to the left of c, f'(x) < 0 to the right of c. This indicates a maximum stationary point.

f'(x) < 0 to the left and right of c. This indicates a horizontal point of inflection.

f'(x) > 0 to the left and right of c. This also indicates a horizontal point of inflection.

Remark: The test presented above is exclusive of ordinary inflection points and should only be used under the hypothesis that the point in question maps the first derivative to zero, as stated at the beginning.
I don't know about you, but I interpret "point on each side" as exactly that. Testing left and right of f'(x) at x=c.

Using the original function f(x) is a bad method and should be avoided. I can't argue it's wrong but it shows a limited understanding.
« Last Edit: August 21, 2016, 10:26:50 pm by RuiAce »

Ali_Abbas

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #3 on: August 21, 2016, 10:26:06 pm »
0
I don't know about you, but I interpret "point on each side" as exactly that. Testing left and right of f'(x) at x=c.

Using the original function f(x) is a bad method and should be avoided. I can't argue it's wrong but it shows a limited understanding.

Oh okay, yeah I originally interpreted the 'point on either side' method as using the orginal function, f, and not its derivative.

Mei2016

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #4 on: August 28, 2016, 12:09:31 am »
+1
To add to this nice list of common mistakes that are easily avoidable, I'd say:

1) read the question properly. :) For multiple choice questions, you might want to read it twice to make sure you that actually know what they're asking for, instead of reading half of the questions and assuming half of it, which will lead to mistakes.
2) I like to 'rush' through my exam so I know that time wouldn't be an issue for me, but in doing so, you really need to look carefully at the paper, and don't skip any questions (which can be extremely easy marks to get)

jamonwindeyer

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #5 on: August 28, 2016, 07:22:54 am »
0
To add to this nice list of common mistakes that are easily avoidable, I'd say:

1) read the question properly. :) For multiple choice questions, you might want to read it twice to make sure you that actually know what they're asking for, instead of reading half of the questions and assuming half of it, which will lead to mistakes.
2) I like to 'rush' through my exam so I know that time wouldn't be an issue for me, but in doing so, you really need to look carefully at the paper, and don't skip any questions (which can be extremely easy marks to get)

Great additions! Thanks Mei2016 ;D

Mei2016

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #6 on: August 28, 2016, 10:53:48 pm »
+2
3) Also, make sure your calculator is in the correct mode of radians or degrees. :)  If not, this could lead up to some errors in calculations.

studybuddy7777

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #7 on: September 01, 2016, 06:55:29 am »
+1
3) Also, make sure your calculator is in the correct mode of radians or degrees. :)  If not, this could lead up to some errors in calculations.

If you dont have your calculator in degrees, then thats rad!!
Spoiler
(Lol soz I had to go there)

RuiAce

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12 Avoidable Mistakes Students Make in Math Exams
« Reply #8 on: September 01, 2016, 07:55:04 am »
0
If you dont have your calculator in degrees, then thats rad!!
Spoiler
(Lol soz I had to go there)
Stealing the meme much :P

AngelicOnyx

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #9 on: October 10, 2016, 10:16:21 am »
0
I can relate so much to 7. and 8. :P In a way that's kinda sad.

RuiAce

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12 Avoidable Mistakes Students Make in Math Exams
« Reply #10 on: October 10, 2016, 12:35:46 pm »
0
I can relate so much to 7. and 8. :P In a way that's kinda sad.
Well seeing as though you can relate to point 7, you should now know that it's something to not walk into in the exam. If you feel like you rounded too early, stop. And also make sure you leave a mark as to where you rounded and where you did not.

Point 8 is interesting. It's different for every person - how do you over complicate?

BPunjabi

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #11 on: October 10, 2016, 12:41:57 pm »
0
How many marks could this have lost you in the HSC? Let's go through common things to look out for, especially in pressure exam situations!  :o

1. Sign Errors

We’ll begin with the most common one that everyone will know about. Sign errors are the bane of a Math student’s existence. You could very easily lose 5+ marks to sign errors in your HSC Exam. Most commonly, this pops up when you are subtracting a multi-term expressions:



To minimise the chance of sign errors, you need to write all your lines of working. Trying to multitask increases the chance of making a silly error. Be especially careful when expanding brackets.

2. Incorrect Form/Units

Not putting your answers in the correct form affects numerous areas of the Mathematics course. Using radians instead of degrees, giving an equation in gradient-intercept form when general form is specifically requested, incorrect number of decimal points, etc. Make sure to address the specific question and put your answer in the correct form. There is nothing more frustrating than losing a mark because you didn’t convert your answer from meters to kilometers (and yes, you would lose a mark for this).

3. Integration Constant and Notation

We’ve all been guilty of this at some point (or maybe I was just especially bad at it).Two things to remember:

a) Remember the ‘dx’ or ‘dy’ at the end of your integral. This may not cost you a mark, but it very well could, so don’t risk it.
b) Make sure you remember to put the integration constant for indefinite integrals. This will definitely cost you a mark, and every year the marking center warns us about it. Don’t forget it!

4.Not Testing Nature of Points

This is a common error, especially under pressure. Remember that you MUST test the nature of stationary points, using either the ‘point either side’ method, or more commonly, the ‘second derivative’ test. Further, you also MUST test your points of inflexion using the ‘point either side’ method. Not doing so will cost you a mark. Consider the las question from my HSC (the scenario itself isn’t important):

Find the values of x and y that maximize the amount of light coming through the window under test conditions.

Many students found a value of x and y which gave a stationary point, but did not prove it was a maximum! Just because the question says you are looking for a maximum, you cannot overlook these texts.

5. Eliminating Solutions by Dividing by Zero

See if you can catch what is wrong with my solution to this trigonometric equation.



Looks okay, but look at the first step. We divide by sinx. What if sinx had been equal to zero, we have removed some solutions! The correct method is instead to factorise the sinx out of the expression, which will give additional solutions.

Make sure that you only divide by valid constant terms, and factorize pro-numerals/unknown expressions. This will ensure you never lose any solutions.

6. Not Fully Simplifying/Factorizing

An extremely common mistake in exams is not expressing the solution in the simplest possible form. This may involve simplifying a fraction, grouping like terms, or rationalizing a denominator, among others. This also includes not fully factorizing a solution, when required.

Make sure you take 5 seconds to check for these sorts of mistakes. For factorizing questions, check how many marks it is worth. If you have done 1 line of working for 3 marks, chances are there is more to be done.

7. Rounding Too Early

This one is simple; never round any values until the final answer. Or, if you do, make sure you take enough decimal places (I used to take four by default). I could go into the proper theory behind how many points to take, based on the error in the values and the question (first year university students who take Physics will know all about this), but it’s much easier to just give a blanket rule. Round only at the end, and use your calculator to store preliminary values.

8. Taking a Question ‘Too Far’

Many students will look at a question and just start working based on what they assume the question to be. Then, they read it back and go: “Oh, I only had to find the first maximum value, not all three.” Or, “Hmm, I didn’t even have to find inflexion points here…”

Make sure you only do the work required for a question. Part of this comes from experience, knowing exactly the data you need for an answer. For more difficult questions, some aimless working may be inevitable. However, minimize wasted time by reading a question fully before commencing work. It seems like a silly reminder, but exams are pressure scenarios, and common sense can sometimes fall out the window.

9. Incorrect Assumptions

This primarily concerns geometry questions, though it can apply for calculus questions, probability questions, and a few other areas: Never make any assumptions based on what the diagram looks like! Unless it specifically says that angle ABC is a right angle, then it isn’t a right angle until you prove it. Unless it says that three points are collinear, then they aren’t collinear until you prove it. Use only the information given to you, never create your own.

10. Square Rooting Sign Errors

Another common mistake. Make sure that when you square root (or any even power) a value, you acknowledge that it can be either a positive or negative result. Even if the answer could only be positive (EG – it is a distance value from the Pythagorean formula), you absolutely must acknowledge the negative answer, then disregard it based on the situation at hand. Also remember that you can’t square root a negative number (unless you are a 4 Unit Student, in which case you can, but 2U students need not worry about such complex things).
 
11. Diagrams!

This is a big area that is identified by the Marking Centre every year. Make sure your diagrams are clear, labelled, and take up at least a third of the page. No more to be said here, just give the diagrams time and space and they are (normally) easy marks to be had.

12. Forgetting to Swap the Inequality Sign

Finally, don’t forget to swap the sign of an inequality when you multiply or divide by a negative. This should be second nature, but there is always somebody who forgets. Don’t be that person.

On inspection of the HSC Papers for 2014 and 2015, these errors had the potential to pop up at least once, and usually, multiple times. This could cost you upwards of a dozen marks in your HSC.

My biggest tip to avoid these unnecessary mark losses in any Mathematics exam is to take your time. Move carefully through the questions, don’t rush, it will cause silly mistakes.

If you know of any other traps, feel free to share them in the comments below! ;D

- The first one is right
- Jamon I am really bad with geometrical proving (got no idea how to find and attempt it) and also integration with fractions, once I intergrate it I dont know how to put it back to its original form
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RuiAce

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #12 on: October 10, 2016, 12:50:49 pm »
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- The first one is right
- Jamon I am really bad with geometrical proving (got no idea how to find and attempt it) and also integration with fractions, once I intergrate it I dont know how to put it back to its original form
- Yeah

What do you mean by integrate and put back to its original form?

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #13 on: October 10, 2016, 12:52:44 pm »
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- Yeah

What do you mean by integrate and put back to its original form?
i dont know how to explain it, basically you integrate a fraction, make it negative and everything, then make it back into a fraction
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RuiAce

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Re: 12 Avoidable Mistakes Students Make in Math Exams
« Reply #14 on: October 10, 2016, 12:53:45 pm »
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i dont know how to explain it, basically you integrate a fraction, make it negative and everything, then make it back into a fraction
So can you please provide a hand written photo example?