Title: Example_questions
Course: Demonstrating STACK (v4.3.7)

Write the following in partial fractions. \[[value]\]

Integrate your expression with respect to \(x\).

Find

\[ \int_{0}^{1} [value] \mathrm{d}x.\]
Feedback: Writing [value] in partial fractions we have \[ [value]\] Integrating this with respect to \(x\) gives \[ [value]+c\] so that \[ \int_{0}^{1} [value], \mathrm{d}x = [value].\]

Multiple properties

In this question STACK establishes multiple separate properties.  Here they are

1. Continuous at x=0
2. Differentiable at x=0
3. Gradient is zero at x=2.

Typically, the answer to a mathematical question will need a number of separate things, e.g. equivalent to the correct answer, appropriate written form.  The teacher has to decide what these properties are, and design feedback if some, or all, are not satisfied.

The teacher can also choose to test for properties which arise as a result of common misconceptions.

Try (x-2)^2+1

Specific feedback

In this, more advanced, question the student is required to find a cubic polynomial which matches up with the given function f(x).  In particular it should connect smoothly with both given ends.

There are are five separate properties that characterize a correct answer

1. p(x) is a cubic (the highest power of x is 3).
2. p(-2)=1 (connects at the left)
3. p(0)=0 (connects at the right)
4. p'(-2)=0 (slope is correct at the left)
5. p'(0)=1 (slope is correct at the right)

In this case there is, up to algebraic equivalence, only one correct answer.  In fact, it is more helpful to students to check each property separately and give more specific feedback.

Try x^2*(x+3)/4 which is only partially correct.

(The correct answer is x^3/2+7*x^2/4+x)

Entering mathematical expressions into STACK

When you are answering the following questions please enter mathematical expressions in the form that you would enter them into a spreadsheet.

For example what you might write as 5x^2 you should type as 5*x^2. You will find that the computer will interpret what you have typed and reflect it back to you in mathematical format.

In practice, almost all students will need to learn how to type in mathematical expressions into some kind of technology, and after a relatively short period the majority of students are able to their answers without difficulty.

Please try the "syntax quiz" before attempting at this.  The purpose of the syntax quiz is to teach the syntax!

Establishing mathematical properties

STACK uses a computer algebra system to establish the mathematical properties of student's answers.  In some cases this is equivalence (as was the case with the railways question previously).  A much wider range of properties can be established, and in many cases there will be a whole range of acceptable answers which satisfy them.

A function f(x) is even if f(x)=f(-x) for all values of x.  For example, x^2, 3cos(x) and x sin(x) are all even.  Try getting these questions wrong!

The teacher always has to specify what properties are relevant to the question.

Lastly, try polynomial examples for the odd function and even function. The "formative feedback" does not contribute to the student's score.

Modelling

In this question the student is asked to write down a system of equations which represent the situation described.  There are many equivalent ways to do this.  STACK uses computer algebra to check if the student's answer is equivalent to the correct system.

E.g. try one of the following

v*t=90
(v+5)*(t-1/4)=90

d=90
v*t=d
(v+5)*(t-1/4)=v*t

The system can also check common mistakes (where the teacher knows about them in advance, which is commonly the case).  E.g. forgetting about units of time is relatively common.  Try using 15 minuites instead of 1/4 hour.

v*t=90
(v+5)*(t-15)=v*t

Randomly generated versions

STACK generates random question versions for each attempt at the quiz.  This is done in a sophisticated mathematical way, not by just putting random numbers into boxes which might lead to impossible questions.

Practice of methods and feedback

This symbolic integration question is typical of many mathematical methods.

In this case get the question wrong.  E.g. by typing in something like (3*x^2+1)^3+c as an answer.

Feedback is given to students which includes a calculation of the student's answer.  This is designed to help students check their own work, and hence improve on this task.

Multipart questions

Sometimes, particularly when first learning a topic, it is helpful to students to break a complex question down into parts.  STACK can implement follow-through marking, taking an incorrect answer from the student in the first part and using it in the second.

For example, try typing in the following answers to the first two parts

1. x+10=56
2. {46}

(The answer in part (2) needs to be a set which is entered using curly brackets and commas as { ?, ?, ?, ...}.  Here there is one element in the set.).  The system is able to establish that the second answer is correct, given the first one, even though the first answer is incorrect.

Give an example of a function \(f(x)\) with a stationary point at \(x=[value]\) and which is continuous but not differentiable at \(x=0\).

\(f(x)=\)
Feedback: One way to construct such a function is to start with a continuous curve (the first property) with a stationary point at \(x=[value]\) and then to "bend" the curve at the point at which we need differentiability to fail. This can be done using the absolute value function \(|x|\). We start with a quadratic with roots at \(x=0\) and \(x=2\times [value]\) which has a stationary point at the required \(x=[value]\). For example \(x(x-[value])\). Now we can "bend" it with the absolute value function as follows: \[ [value]\] This is plotted below. [value]

Consider the real function

\[ f(x) :=\left\{ \begin{array}{ll} [value] & \mbox{for } x\leq [value], \\ p(x) & \mbox{for } [value] < x < [value], \\ [value] & \mbox{for } [value]\leq x.\end{array} \right. \]

[value]

Find the cubic polynomial \(p(x)\) which makes \(f(x)\) continuously differentiable.

\(p(x)=\)
Feedback: First we need to find the information which \(p(x)\) needs to satisfy. This is \[ p([value])=[value].\] \[ p([value])=[value].\] \[ p'([value])=[value].\] \[ p'([value])=[value].\] If we define \(p(x)\) to be the cubic \[ p(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0,\] We then set up the matrix equation \[ [value] [value] = [value] \] Solving this gives the coefficients, from which we get the polynomial. [value]. We can view this as [value]

In a railway journey of \(90km\) an increase of \(5\) kilometers per hour in the speed decreases the time taken by \(15\) minutes.

Write a system of equations (one equation per line) to represent this situation using \(v\) as the speed of the train and \(t\) as the time.

Eliminate \(t\) from your equations, writing a single equation only in \(v\) which represents this situation.

What is the velocity?
Feedback: Let \(v\) be thespeed of the train, \(d\) the distance travelled and \(t\) time. If \(t\) is the time (in hours) then \(15\) minutes is \(1/4\) hour. We need to use the equation \(d=v\times t\) twice. For the original journey we have \[ 90 = v\times t.\] For the faster journey we have \[ 90 = (v+5)(t-1/4).\]

Find the equation of the line tangent to \(p(x):=\)[value] at the point \(x=[value]\).

1. Differentiate [value] with respect to \(x\).

2. Evaluate your derivative at \(x=[value]\).

3. Hence, find the equation of the tangent line to \(p\) at \(x=[value]\). \(y=\)
Feedback: 1. The derivative of [value] with respect to \(x\) is \[ [value].\] 2. If we subsitiute \(x=[value]\) into \(p'(x)\) we have \(p'([value])=[value]\). 3. The tangent line is given by the equation \[ y = m(x-a)+c\] where \(m\) is the gradient of the line, \(a\) is the point at which the tangent line is being contructed and \(c:=p(a)\). In our case we have \(a=[value]\), \(m=[value]\) and \(c=p([value])=[value]\). Combining these we have \[ y=[value] = [value].\]

Find \[ \int [value] d[value]\]
Feedback: We can either do this question by inspection (i.e. spot the answer) or in a more formal manner by using the substitution \[ u = ([value]-[value]).\] Then, since \(\frac{d}{d[value]}u=1\) we have \[ \int [value] d[value] = \int u^[value] du = \frac{u^[value]}[value]+c = [value]+c.\]

A rectangle has length [value]cm greater than its width. If it has an area of \([value]cm^2\), find the dimensions of the rectangle.

1. Write down an equation which relates the side lengths to the area of the rectangle.

2. Solve your equation for the dimensions. Enter your answer as a set of numbers.

3. Hence, find the length of the shorter side.
cm
Feedback: If \(x\)cm is the width then \((x+[value])\) is the length. Now the area is \([value]cm^2\) and so \[ [value].\] \[ [value]=0\] \[ [value] \] So that \(x=[value]\) or \(x=[value]\). Since lengths are positive quantities \(x>0\) and we discard the negative root. Hence the length of the shorter side is \(x=[value]\)cm.

In this question use the data captured from your moving cart in the speed lab.

How far did the cart go?

How long did it take?

So what is the speed?
Feedback: Remember that \[ d=st.\] So if you have a distance of \(d\) meters, in a time of \(t\) seconds, then the speed \(s\) will be \(d/t\).

Live data capture and scientific units

STACK can be used in conjunction with live data capture in science labs.  For example, here is a sample question which confirms if students have calculated a speed correctly from their own data.

STACK also has full support for scientific units and numerical precision.

Write the following expression in an equivalent form in which all common factors have been cancelled.

\[ [value]\cdot [value]\]
Feedback: The first step is to write all expressions in factored form \[ \frac[value][value]\frac[value][value].\] Next we look for common factors in the numerator and denominator and cancel these to give \[ [value].\] (Note, this question is very famous, and has appeared in the majority of algebra texts since Van Schooten introduced it in his textbook in Principia Matheseos Universalis (1651). See Heller, H.F. (1940) Concerning the Evolution of the Topic of Factoring in Textbooks of Elementary Algebra Publish in England and the United States from 1631 to 1890. PhD, Columbia University.)

1. Give an example of an odd function by typing an expression which represents it. \(f_1(x)=\) .

2. Give an example of an even function. \(f_2(x)=\) .

3. Give an example of a function which is odd and even. \(f_3(x)=\) .

4. Is the answer to 3. unique? (Or are there many different possibilities.)
Feedback: A function \(f\) is odd if \[ f(x)=-f(-x) \mbox \forall x.\] An example is \(f(x)=4x^3\). Indeed, polynomials with only odd powers are fine. A function \(f\) is even if \[ f(x)=f(-x) \mbox \forall x.\] An example is \(f(x)=5x^4\). Indeed, polynomials with only even powers are fine. It is possible to have both \[ f(x)=f(-x)=-f(-x) \] in which case \(f(x)=0\) for all \(x\). This example is unique.

If you choose an answer to this question at random what is the probability you get it correct?

Justify your answer to the previous question.

(You may use \(\LaTeX\) mathematics in the above question, e.g. \( \pi r^2 \), but don't use $...$ signs for delimiters.)

If you choose an answer to this question at random what is the probability you get it correct?
Feedback: Clearly the multiple choice question is self-referential nonsense! If you have a good answer to this riddle, please email the developers and we will add it here!

MCQ and human marked questions

STACK does support multiple choice questions, and random generation of mathematical MCQ or MCQ as part of a multi-part question, certainly has its place.

Note that we can also use STACK (and other question types) to gather students' answers and, where appropriate, to mark their answers by hand. Of course, the automatic marking of online assessment is the typical use, but we can also just use the quiz to gather data. The Moodle essay question can be used to gather longer free-form answers. The STACK "notes" input can give instant validation of students' LaTeX within their answer (if they choose to use LaTeX).

If a single part of the question uses the "manual grading" option, then the whole question becomes manual grading. You need to separate out questions if you want part to be automatically marked. Hence, the MCQ here will not be automatically assessed!
Feedback: The complete proof is below Let \(P(n)\) be the statement {@p0@} Since \( {@ev(ak,k=1)@} = {@ta1@}\) and \( {@ev(rh0,n=1)@} = {@ta2@}\) it follows that \(P(1)\) is true. Assume that \(P(n)\) is true. \({@lh1@} = {@ta3@}\) \( = {@rh0@} + {@ev(ak,k=n+1)@}\) (using the induction hypothesis) \( = {@factor(rh0+ev(ak,k=n+1))@}\) \( = \) {@ta5@} Since \(P(1)\) and \(P(n)\Rightarrow P(n+1)\) it follows that \(P(n)\) is true for all \(n\in\mathbb{N}\) by the principal of mathematical induction. More generally, consider the following when writing an induction proof to establish the finite sum of a series of the form \[ {@sum(p_k,k,1,n)=P(n)@} \] - The base case is often very easy to prove, so easy in fact that it looks pointless or trivial. The base case is just as important as the induction step! - At the start of the induction step, separate the \(n+1\)th term from the sum, so that you have \[ {@sum(p_k,k,1,n+1)=sum(p_k,k,1,n)+p_(n+1)@} \]

Type in an expression with variables \(A\) and \(B\) using only the operations and, or and nor which has the following truth table.

[value]
Feedback: The true table needed is [value]