Create your own Questions for Revision

[Saracen Rue]

This is an idea I've been thinking of for a while. While providing a question for an individual a question to complete works fine as revision, numerous studies have proven that creating a question is, in itself, a more beneficial way to revise. Instead of simply recalling a process or equation to solve the problem, you have to craft the problem itself - doing this requires a much more in-depth understanding of the overall concepts of the topic.

This is why I'm creating this thread; to prompt people out there to create their own questions for other users to answer. Not only will the person forming the question achieve a better understanding of the topic than simply revising, they will also be exposed to the thought processes of other people. There are countless ways to solve a problem; by putting a question out on the internet to be solved by others you will not only be aiding other people in revising certain areas, but you will also achieve a greater overall understanding of the topic and will be exposed to problem solving process you had never even thought of before.

I'll pose a mathematical related question which addresses multiple year 12 course areas as an example:

Question
A function, $f(x)=2ax^3-a^2x$ intersects its inverse at the origin, point $S(-b,f(-b))$ and point $T(b,f(b))$. A probability density function, $p(x)=f(x)-f^{-1}(x)$, can be formed over the domain $[0, b]$. Determine, correct to 4 decimal places:
a) The value of the constant, $a$, and the coordinates of points $S$ and $T$.
b) The mean, variance and standard deviation of $p(x)$
c) The probability that the contentious random variable $X$ lies within $|a|$ standard deviations either side of the mean (i.e. $Pr(μ-|a|σ≤X≤μ+|a|σ)$)

Answers

Spoiler: Click here to see answers

a) $a=-0.2253, S\left(-1.4515,\ 1.4515\right),\ T\left(1.4515,\ -1.4515\right)$
b) $μ=0.6692, Var(X)=0.5673,$ $σ=0.7532$
c) $Pr(μ-|a|σ≤X≤μ+|a|σ)=0.3147$

 

[Andy Resnick]

This is an idea I've been thinking of for a while. While providing a question for an individual a question to complete works fine as revision, numerous studies have proven that creating a question is, in itself, a more beneficial way to revise. Instead of simply recalling a process or equation to solve the problem, you have to craft the problem itself - doing this requires a much more in-depth understanding of the overall concepts of the topic.

This is an excellent idea- I sometimes suggest to my students that they study by designing test-like questions. They often don't realize how difficult that is, but they do see the value very quickly.

 

[martane]

Wow, this is a really interesting idea! I've never thought of creating my own questions for others to answer as a way to revise. I can definitely see how it would be more beneficial in terms of understanding the topic and exposing yourself to different problem-solving processes.

As for the question you posed, it's definitely a challenging one. I'm not sure if I have the skills to solve it, but I'll give it a try.

a) To find the value of $a$, we can set the function equal to its inverse and solve for $a$. This gives us the equation $2ax^3-a^2x=x^3-2ax$. Simplifying, we get $x^3-2ax=0$. This is true for all values of $x$, so we can choose any value to solve for $a$. Let's choose $x=1$. Substituting, we get $1-2a=0$, or $a=1/2$.

To find the coordinates of points $S$ and $T$, we can use the fact that the function and its inverse intersect at these points. Substituting $a=1/2$ into the function, we get $f(x)=x^3-x$. Setting this equal to $-b$ and solving for $x$, we get $x=-b$ and $x=b$. Therefore, the coordinates of points $S$ and $T$ are $(-b, f(-b))$ and $(b, f(b))$ respectively.

b) To find the mean of $p(x)$, we can use the formula $\mu=\int_{0}^{b}xp(x)dx$. Substituting in the formula for $p(x)$, we get $\mu=\int_{0}^{b}x(f(x)-f^{-1}(x))dx$. Using the fact that $f(x)=x^3-x$ and $f^{-1}(x)=x^3+x$, we can simplify this to $\mu=\int_{0}^{b}x(x^3-x^3-x)dx$. Solving this integral, we get $\mu=-b^4/4$.

To find the variance, we can use the formula ##\sigma^2=\int_{0}^{b}(