Active Learning Notebooks for College Mathematics

Jordan Hasler
Wolfram Technology Conference 2022

Abstract:
Modern learning spaces transcend mechanical pencil-and-paper calculations by combining computation, visualization and concept exploration. Wolfram|Alpha Notebook Edition facilitates all three of these while quietly teaching programming in the Wolfram Language along the way. The breadth and depth of pedagogy tools for college mathematics are demonstrated through standalone student activities. Topics range from calculus, linear algebra and differential equations to probability and statistical and data analysis.

Summary:
Wolfram|Alpha Notebook Edition may be used to create an active learning classroom environment that may be used to promote active learning and inquiry. The ability to interactively create and work with various plots and diagrams allows the student to see the connection between the algebra and the visual. One benefit of this approach is to able to follow the concepts while the computational processing does the algebra. Today, Calculus, Linear Algebra, Differential Equations, Probability, Statistics, and Data Science will be discussed.

Guiding Questions

1.) What is the goal of the lesson? Does this goal vary in a traditional classroom setting compared to a interactive, technology driven classroom?

2.) In designing an interactive Wolfram|Alpha Notebook Edition lesson, one can format the course similar to a standard course. What are the benefits and drawbacks of developing an interactive course in the same way as a traditional course?

3.) What should the balance between conceptual, mechanical, data, and visual be in a course?

4.) Does the student still need to learn the standard algorithms to individually do the mechanical calculations? Why or why not? If the student should learn the standard algorithm, in what ways does technology help in this process?

5.) Spend 5 minutes discussing with the person next to you a course that you would like to develop as an active learning, technology driven classroom. Describe what challenges you think you will face. What material that goes beyond the traditional classroom do you plan to include?

6.) How will you use technology driven active learning in the classroom? Will you use it for adding to your existing structure? Will you use active learning to create inquiry driven lessons or to reinforce concepts taught in a traditional lecture?

Calculus

A variety of activities are introduced for the Calculus classroom. The activities can be used to develop courses that use technology at various entry points. The activities could be used to prepare for final exams and placement exams. The activities could be completed as standalone, review, or as entry points that are further developed to create an entire course using technology. Note the Calculus result definitions and theorems that are under development will be available soon.

Review

The student will review and discover interactive ways to think about linear, quadratic, and trigonometric functions.

Position / Velocity (Function / Derivative)

The student will discover the connection between functions and derivatives and position and velocity. Interactive plotting and student driven visualization will reinforce the concepts encountered.

One of the topics that is encountered in introductory Calculus is the concept of a derivative. Visualizing the connection between a function and its derivative may be thought of as position and velocity.

The slope of the position graph at a point is the velocity at that point.

Determine the “position” and determine the “velocity” in the graphs below:

Use a Canvas to sketch the velocity of the function above:

Canvas[Plot[(1/(x^2)),{x,-5,5},PlotRange->{-10,10},Axes->False]]

The plot may then be saved and emailed to an instructor.

Velocity and Position (Function and Antiderivative)

Students will discover how to plot the “antiderivative” of a function by considering velocity and position.

Reversing the process above is referred to as integration. In this case, one starts with the derivative and wants to find the original function. For instance, given an initial velocity, one wants to find a corresponding position.

Tangent line

The tangent line is reviewed. Students will see how to compute the tangent line using the limit definition and derivative rules in the next sections. The student will make a connection between the graph of the original function and tangent line. Next, the student will see various applications of tangent lines.

Definition of a tangent line:

The slope of the tangent line to (x^2)+3 may be found by computing the aforementioned limit:

The tangent line to a function:

The plot of the above tangent line:

%156["Plot"]

Verify the slope of the above tangent line:

The connection between the slope of a tangent line and the derivative to a function may be visualized:

Plot two parabolas:

Find the derivative of the two parabolas:

Find conditions to determine equivalent slopes at two points:

Solve for one of the values:

With (1, -7) and (5, 1), find the secant line that is tangent to both parabolas:

Find the intersection of the two parabolas:

Find the area between (x^2)-8 and the tangent line:

By symmetry, the second integral will be the same as the first integral:

Note that Wolfram|Alpha Notebook Edition can also compute the integral separately as:

Plot a function:

Determine the tangent line at -1/2:

Determine the tangent line at 1/2:

Solve for the intersection of the two tangent lines:

Plot the two tangents and the original function:

Find area between tangents and the function:

N[-(3/8)+Log[(3/2)]]

N[-(5/8)+Log[2]]

Derivative at a point

Students will discover what the derivative is. The connection to the limit is made. A discussion of differentiable functions then follows.

Definition of a derivative at a point:

Derivatives

Technology will be used to discover a property for derivatives of monomials. Next, various special derivatives are discussed. The chain rule is approached through observing the rule being applied in action. Students may then formulate a conjecture about the derivative of a composition of functions will be. Then an example of a nondifferentiable function is presented and visualized.

Notice the pattern with the following derivatives:

Is a pattern visible in the derivatives of the above monomials?

Important derivatives:

What is the derivative of a polynomial?

Compute the derivative for a composition:

Another example of a composition of two functions:

A derivative of composition with logarithms:

Another example:

Do you notice any patterns? Suppose that f(g(x))is a composition of two functions. What do you predict the derivative of the composition will be?

Is the exponential function differentiable?

Is the sum of two differentiable functions differentiable?

A function which has a point that is non-differentiable?

Is there a function that is nowhere differentiable?

Optimization Example

Advanced activities can be processed and visualized using technology. The student will work through an advanced application of an example of optimization using technology to help in the visualization and computational parts of the process.

Reference: California Mathematics Framework Second Field Review (Chapter 11) See: http://cde.ca.gov/ci/ma/cf

The following calculus problem is adapted based on a similar algebra problem from the California Mathematics Framework

Suppose that the Euclidean plane represent two separate types of terrain which a robot is searching on Mars. The control tower is located at point (10, 10). The robot starts at the point (0,0). If the robot can travel at 10 km/hr for points y < 5 and at 15 km/hr at points y > 5, determine the optimal path and time.

N[%102]

Integral Calculus

The student will discover what an integral is. The connection with limits and derivatives is presented. Next, the student discovers the fundamental theorem of calculus and how the fundamental theorem simplifies the calculation of integrals if we have information about the derivative.

What is an integral?

Set up an limit that represents the value of the integral between 0 and 1 for the quadratic:

Verify the integral:

What is the fundamental theorem of Calculus?

Thus, knowing about derivatives reveals information about antiderivatives:

Instead of computing the complicated limit of a summation above, one may use the antiderivative and the fundamental theorem of calculus:

Area under a curve (AP BC Fall 2021 Free Response)https://apcentral.collegeboard.org/media/pdf/ap21-frq-calculus-bc.pdf

The student will work through and visualize finding the area under a curve.

Visualize the function:

Find an integral which represents the area under the curve

The area under a curve is the area between the graph and the x-axis can be explicitly calculated:

Show the steps for this computation:

Generalize to arbitrary coefficient

Find where the maximal radius of the curve when revolved about x - axis exist if y = 1.2 inches:

Simplify[-((c (x^2))/(Sqrt[4-(x^2)]))+c (Sqrt[4-(x^2)])]

When the volume when revolved about the x-axis is 2 π (in^3), find the value of the constant c:

Solve[2 π==((64 (c^2) π)/15),{c},(Reals)]

Last[{{c->-((Sqrt[(15/2)])/4)},{c->((Sqrt[(15/2)])/4)}}]

Series

The student will learn about convergence and divergence of series. The connection with the integral is made. Various series tests are discussed and examples are worked through.

Is (1/(n!))<(1/(2^n))?

Surfaces and graphing

In Vector Calculus, the student will learn about various surfaces. The quadric surfaces are introduced. Both 3D plots and contour plots are discussed. Interactive parameter variation will provide visualization as the parameter changes.

Determine a value of the constant c such that there is a local maximum at the origin:

Does c = -5 have a local extrema at (0, 0):

Div/Grad/Curl

The student will encounter gradient, divergence, and curl. Plotting of vectors in a vector field will help to visualize each operation. Various connections with fluid mechanics are discussed.

Plot the vector field (y, -x):

Plot the contours:

Show that (-y, x)has circular flow lines:

Find all possible stream lines for the vector field (y, x):

Determine whether the velocity vector field is incompressible:

Find the curl of a vector field:

Determine the equation of motion of an incompressible fluid flow:

Partial derivatives

The student will see examples of computing partial derivatives.

Multiple Integrals

The student will see examples of computing multiple integrals to verify solution techniques discussed during class:

Practice Problems

Find the derivative to the following: sin(x), cos(x), tan(x), ((tan(x))^10),exp(tan(sin(x))):

Find the integral to cos(x), x log(x), 1/(1+(x^2)):

Use a Canvas[] to plot the velocity of the function below if the graph represents the position:

Linear Algebra

One can build an active learning environment by exploration of concepts. For instance, using the natural language input, a student can compute the calculations to notice patterns and build intuition. For instance, in the activities below, one can start with numeric cases and explore a variety of inputs. Note, in the activities below, only one or two inputs have been used, but in building up such an active learning environment, a complete lesson could involve 5 or 6 examples to notice the pattern. Then once the pattern is noticed, the formal definitions and properties may then be stated in their full generality.

Solving systems of linear equations

In this lesson, students will recognize solutions to linear systems of equations through visualization. The set of equations will be represented as an array of values called a matrix. Technology will be used to verify the solution to the system of equations. Interactive variation of parameters will be used to describe how the solution to a system of equations varies as the parameter is varied. The student will discover the 3 types of solution sets that are possible: no solution, one solution, and infinitely many solutions. The student will discover in 3 dimensions both graphically and algebraically that infinitely many solutions can be formed by intersection of planes to a line.

1.) Use the plot to determine the solution set of the following equations:

Verify the solution:

By focusing only on the coefficients, rewrite the system as an augmented matrix:

By varying the parameter, does the following set of equations always have a solution? Describe your reasoning.

Varying c, does the following set of equations always have a unique solution:

Does the following always have a unique solution?

Determine an equation without a solution:

Why did the above not have a solution? Verify the above has no solution:

Write an augmented matrix for the system:

Notice the last row after applying various operations to reduce the matrix to the simplest form has 0 x + 0 y = 1:

In three dimensions, what are the possibilities when two planes intersect. Consider the following example:

When 3 planes intersect, how many solutions are there? How many different ways can the planes intersect?

Varying the parameter k, describe what happens to the planes. Note whether or not the number of solutions changes:

Describe the line of solutions to the above equation by noting x = y = t and solve for z:

Collecting this information into the parameterized form:

Describe the intersection of the following planes. Is there ever a value of k where there is a solution? Is the solution unique?

Notice the last row here has 0 x + 0y + 0z = 1:

Addition of vectors and introduction to matrices

Students will discover how to add vectors using inquiry based methods applied to both numerical and symbolical points. This will be reinforced using the parallelogram method of addition. Next students will see how to multiply a vector by a constant. This will be presented graphically and numerically. Next a discussion of similar ideas of multiplication by a constant and addition will be applied to matrices. Technology will be used to determine the dimension of a matrix and verify whether a matrix is a square matrix. Students will discover the parts of a matrix. Students will have an introduction to what a row reduced matrix looks like which will allow the student to see how this compares to the solution of a system of equations.

Add two vectors:

Add the following vectors and describe in terms of the vector components how vector addition works:

Plot a set of vectors:

Use the parallelogram method to add two vectors:

Describe what constant multiple multiplication does to a vector:

Describe geometrically how the two vectors compare after constant multiplication:

Describe the multiplication both algebraically and geometrically for the vector:

Generalize addition and constant multiplication of vectors:

Plot the addition of two vectors and form the parallelogram:

Describe the two following two vectors geometrically:

Describe the following vectors geometrically:

The same ideas can be applied to matrices:

Generalize to arbitrary constants:

The dimensions are the number of rows and the number of columns of a matrix. For example,

Find an example of a 2 by 3 matrix:

A square matrix is one which has the same number of rows and same number of columns:

Find parts of a matrix:

Put a matrix in reduced row echelon form:

Matrix multiplication

In this lesson, the student will use technology to explore how matrices multiply together. First an example of matrix multiplication is applied. Next, this is generalized to the symbolic case. A definition of matrix multiplication is then produced. Finally, the student will see how an identity matrix can be multiplied with another matrix.

Multiply two matrices of the same dimensions:

Can we multiply a 2 by 2 matrix with a 2 by 3 matrix?

Can we multiply a 2 by 3 matrix with a 2 by 2 matrix?

What do we know about the number of columns of the first and the number of rows of the second matrix for matrix multiplication to be defined?

Multiply arbitrary 2 by 2 matrices:

The identity matrix for a 3 by 3:

What does multiply by an identity matrix do to a matrix?

What are the dimensions of the following matrix:

Notice that the matrix multiplied to form a 3 by 4 system. If one multiples a m by n matrix with a nby  p matrix what will be the dimensions of the resulting matrix?

Matrix arithmetic

In this lesson, the student will work with various matrices and vectors with different dimensions. The student will discover that some calculations are computable and others are not computable due to incompatibility of dimensions. A student will encounter the concept of a symmetric matrix which will be formally defined later. Students will see that matrices and vectors can sometimes be multiplied together and sometimes not.

Is 2a.b computable?

Is a.b + c computable?

Is b.a + a.b computable?

Symmetric matrices will be discussed later. For now, let’s determine whether or not this matrix is symmetric:

Find a.d - 3d:

Find a.c + b.c:

Are a.c + b.c and (a=b).c equivalent?

Transpose

Students will learn about the transpose of a matrix. The

Inverse

The student will learn about the inverse of a matrix. Students will then see that a matrix multiplied by its inverse returns the identity matrix. Next, the student will see the connection to the determinant and the adjoint matrix for 2 by 2 matrices.

Inverse of a matrix:

Verify that the matrix is actually invertible:

Symbolically find an inverse:

Verify the connection between inverse and the adjoint and determinant:

Symmetric matrix

Students will see the definition of a symmetric matrix and two examples.

What is a symmetric matrix?

Normal matrix

Students will see the definition of a normal matrix. Next, examples and how to verify whether a matrix is normal will be discussed.

What is a normal matrix?

Inverse properties

Students will discover properties of the inverse. Students will discover matrices that are not invertible.

Is the following invertible? If not invertible, what went wrong here?

Is the following matrix invertible? If not, what went wrong?

Is the following invertible? If not, what went wrong?

Is the following invertible? If not, what went wrong?

Is a matrix invertible if it has a.) a row of 0s? b.) a column of 0s? c.) two rows are equal? d.) two columns are equal?

Am example of an invertible matrix:

What happens if one takes the inverse of the inverse when applicable?

Define a matrix:

What happens if a matrix is multiplied by a constant and the inverse is applied?

Conjecture a property for the inverse of the matrix c A:

Conjecture the inverse of a matrix (A B):

Antisymmetric matrix

Students will see an example of an antisymmetric matrix. The student will encounter the definition of an antisymmetric matrix.

Trace

Students will discover the trace operator. Next, the formal definition is presented after many examples are presented.

Find the trace of the following:

The formal definition:

Idempotent Matrix

Students will learn about an idempotent matrix. Students will see a standard example and then an example defined by using algebraic conditions.

What is an idempotent matrix?

Is {{a, b}, {c, 1-a}} with (a^2) + b c = a idempotent?

Determinant

Students will see many examples of a determinant. The student will see how this is connected to cofactor expansion. Next, students will learn about singular matrices and how determinants are connected to a singular matrix. A method for computing the determinant of a matrix inverse given the original matrix is presented.

What is the determinant of the following:

Dot product, Norms, and Distance

Dot product examples are presented. The formal dot product for a 2 by 2 is presented. Students will learn through inquiry and discovery how to think about dot products. Next, the student will see the distributive property for a dot product. The connection with the Cauchy-Schwarz inequality will be discussed. Then connections with distance are formulated. The angle between two vectors will be discussed.

Find the dot product of the following:

Find a vector orthogonal to (3,-1):

5 vectors orthogonal to (3,-1):

Angle between vectors

In this lesson, students will see how to compute the angle between vectors. This will be connected with the law of cosines from precalculus. Technology will be used to give a formal derivation to analytically solve the algebra involved in finding the angle.

Find the angle between two vectors:

Solve for the angle in the law of cosines definition:

Suppose two sides of the triangle to solve for are represented by vectors u and v:

Thus note that the numerator of the RHS above:

Thus, simplifying to 2u.v:

Vector Space

Students will read about vector spaces and related operations including the span. Next a visualization of the span of vectors will be presented. A connection between span of column vectors and solutions to systems of equations will be discussed.

What is a vector space?

Plot the unit vectors:

Plot the span of the two vectors restricted to [-1, 1] by [-1, 1]:

Compute properties of the span:

\[LeftAssociation]"Region"->ParametricRegion[{(Subscript[\[FormalX], 1]),0},{(Subscript[\[FormalX], 1])}],"Basis"->{{1,0}},"OrthonormalBasis"->{{1,0}},"Dimension"->1\[RightAssociation]["Region"]

Visualize the span of the two vectors:

Recall discussing systems of equations:

Solve the system above:

Represent the original system as a matrix:

Find the span of the column vectors v1, v2, v3:

A consistent system should have the target vector be part of the column vectors span. 
Test whether the region specified above contains {1, 0, 0}:

Can you find an inconsistent system (one that does not have a solution with the column vectors above)?

Linear Transformation

The student will discover linear transformations. The student will work through examples that demonstrate how to prove whether a transformation is linear. Properties used to prove or disprove whether a transformation is linear are presented. Connections with matrices and matrix operations are presented. Visualizations are used to derive and analyze further properties of linear transformations. Rank-Nullity theorem is discussed. Review and further exploration of the various spaces of linear algebra are discussed.

Is S(u, v) = (u, 2) a linear transformation?

Does this transformation send lines to lines?

Determine whether the matrix A is invertible. In other words, the linear transformation associated to the matrix is one-to-one or not:

Write (4, 5) in terms of the standard basis and the basis specified by {{(3, 4}, {-4, 5}}

Suppose that (2, 1) is given, write the vector u with respect to basis by Q:

Let T(x, y) = (-x, y), find the type of transformation this represents:

Normal[CoefficientArrays[{-x,y},{x,y}][[2]]]

What type of transformation is the following?

Visualize the graphics:

A one to one linear transformation is one which has at most one solution for every target vector. Equivalently, the matrix representation has a pivot in every column:

Is T(x, y) = (x+y, x-y) one-to-one?

Normal[CoefficientArrays[{x+y,x-y},{x,y}][[2]]]

Equivalently the columns of A are linearly independent:

Equivalently the m by n matrix A is such that the range of T is of dimension n:

Is the following a one-to-one linear transformation where T(x) = A.x?

Normal[CoefficientArrays[{x+2 y+3 z,4 x+5 y+6 z},{x,y,z}][[2]]]

Note there is not a unique solution:

A linear transformation is onto if there is at least one solution for a specified target vector:

Normal[CoefficientArrays[{x+4 y,2 x+5 y,3 x+6 y},{x,y}][[2]]]

Equivalently a linear transformation is onto if the matrix representation has a pivot in every row. Is T a onto mapping?

Note we see that there is no solution:

The matrix is of size m by n. Note that the columns do not span since the dimension is 1 fewer than would be required:

Let T be the matrix that represents rotation by Pi/4. Is this a one to one and onto mapping?

Normal[CoefficientArrays[{(x/(Sqrt[2]))-(y/(Sqrt[2])),(x/(Sqrt[2]))+(y/(Sqrt[2]))},{x,y}][[2]]]

Is this matrix and the corresponding linear transformation represent an invertible mapping?

There should be 4 pivots if the matrix is invertible:

The determinant is going to be 0 since there are less than 4 pivots:

The null space will contain more than the 0 vector:

Exercise: Determine if T is one-to-one or onto using the ideas discussed above.

Invertible matrix equivalences:
1.) A is invertible. 
2.) T is invertible
3.) T is one-to-one
4.) T is onto
5.) A has n pivots
6.) Nullspace(A) = {0}
7.) The columns span the domain of T.
8.) Ax = b has a unique solution for any target vector b.
9.) The columns of A are linearly independent.

For transformations where m = n:
1.) Row reduces to identity matrix.
2.) Ax = 0 has only the trivial solution. 
3.) The dimension of the column space is n. 
4.) The columns for A form an n-dimensional basis.
5.) The system is consistent for any target vector.
6.) The rank of the matrix A is n.

Exercise: Determine if whether 0 being an eigenvalue can be added to the list above.

Differential Equations

Using technology helps in the visualization of system of equations. This allows the student to see the connection between the system of equations, the solution of the system of equations, and the qualitative theory behind the system of equations.

Solve and plot an ODE

Students will learn about the exponential function and the differential equation for exponential growth and decay. The equation will then be visualized. Next the student will use technology to look at the slope field of a given equation.

Solve a differential equation:

Solutions to ODEs

Technology is used to discover the solution to system of differential equations. Then visualization and limiting operations are applied.

Solve an linear, constant coefficient equation with forcing term Exp(t):

Use the predictive interface to get the solution and then plot:

%23[[1,1,2]]

Plot[%24,{t,0,40}]

Find the limiting behavior:

Linear homogenous constant coefficient ODEs

Students conjecture and inquire into systems of equations and make connections with the exponential terms and solving roots of equations.

Solve a differential equation:

Think about the corresponding linear equation:

Think about the corresponding quadratic equation:

Do you notice any patterns? What do you think is a way to determine the appropriate exponential that appears in the solution?

What ansatz should be made to determine the form of solution for the homogenous part of the solution? What ansatz should be made for the inhomogeneous part of the solution?

Does this method work if the coefficients are not constant:

Linear systems of ODEs

The student will work through and visualize a system of linear ordinary differential equations. Connections are made with the equilibrium solution and discussion of stability of such a fixed point.

Solve a system of equations:

Consider the linear system of differential equations:

Find when x'(t)=0:

Find when y'(t)=0:

The equilibrium point is at the intersection of these points:

The system of differential equations correspond to a matrix system of the form x'(t) = A x

Find the eigenvalues:

Solve the system analytically:

Since the eigenvalues are real, distinct, and of opposite sign, the system will usually not converge to the equilibrium:

ParametricPlot[%165,{t,-2,2}]

ParametricPlot[{-(1/4) (E^(-(Sqrt[2]) t)) (-2-(Sqrt[2])-2 (E^(2 (Sqrt[2]) t))+(Sqrt[2]) (E^(2 (Sqrt[2]) t))),(1/4) (E^(-(Sqrt[2]) t)) (-4-3 (Sqrt[2])-4 (E^(2 (Sqrt[2]) t))+3 (Sqrt[2]) (E^(2 (Sqrt[2]) t)))},{t,-2,2}]

As can be seen above, when the initial conditions are such that y (0) is not 0, then the region is "saddle" shaped and is thus an unstable equilibrium point . What happens when y (0) is 0?

Lorenz equation

Students will learn about a system of equations that represent chaotic dynamics. The non linear system of equations is then visualized. The student will then discover the stability a fixed point of this system.

Is the origin a stable or unstable fixed point?

Exponential Growth:  ODEs & Regression

Exponential growth is formally discussed and a connection with regression techniques is made.

Solve the following differential equation:

Imagine you have data from the lab testing the relative surface area of bacteria cultures on an agar plate:

Notice that the prefix was near unity despite the noise in the data. This indicates surface area growth relative to the amount of bacterial surface area at a time t = 0.

Probability, Statistics, and Data Science

Applying active learning methods to statistics helps the student to visualize the data, perform experiments, summarize information, and discover related properties. Interactive plot quizzes help the student to think about how data and regression parameters are connected.

Random numbers and Histograms (with contributions from Nick Brunk)

The student will generate random sets of data. Then connections on measuring whether the data is normally distributed or not are made. Observations are then made to formulate a conjecture that the data is log normally distributed.

Let’s look at some randomly generated data for the normal distribution and look at the convergent behavior of repeated measurements.

Starting with ten points generated from a normal distribution with (m = 10, s = 2.0):

We see the suggestion of symmetry, but it also looks fairly skewed given the finite number of times we sampled the population distribution.
Let’s repeat our exercise, measuring the same underlying population distribution 100 times:

This looks reasonably symmetric, though let’s drive the point home with 1000 points:

With 1000 sample points, this is beginning to look like the standard normal distribution from which we know it was sampling.

Let’s look at the box-and-whisker chart comparing the variably-sparse datasets:

Box-and-whisker charts enable visualization of the five-number summary of a dataset:  the inner three quartiles and the extrema (min/max).

Indeed, we can tell the skew in the sparser two samples (left, center) by virtue of the true centrality of the central tendency metric, the median.  However, when you sample the same distribution 1000 times, the 25/75% quartiles appear symmetric about the median, as they should be for a normal distribution.

It’s not uncommon, even in research, for researchers to assume a sample dataset to be sampling a population dataset that is inherently normally distributed when, in fact, it is not.  Let’s take the same 1000 samples shown to be sufficient to reveal a distributions “true character” above, and sample the same number of times from a log-normal distribution

Let’s first compare the histograms, since they both have the same number of data points:

Clearly, these differ.  One is bounded at zero, a characteristic of the log-normal distribution, whereas the other is not.

Let’s look at the box and whisker chart:

The skew can be visualized in another way using distribution charts, wherein the symmetry of only one distribution is apparent:

This clearly shows the underlying distributions are not the same.  Upon mouse-over, both the BoxWhiskerChart and the DistributionChart provide the five-number summary of the data.  It can be seen from this 5-number summary, and additions thereto, that on distribution is reasonably symmetric whereas the other is skewed.

Assume we do not already know that the latter was sampled from a log-normal distribution.  

What can we add to this 5-number summary to deduce the log-normality?

Let’s look at alternative indicators of central tendency and compare them to the five-number summary’s median.

In a normal distribution, the mean and median are approximately equal measures of central tendency:

In a log-normal distribution, the mean and median are not approximately equal (differing proportional the standard deviation):

However, the geometric mean is approximately equal to the median (here, by an even better margin than the median and mean agreed for the normal case):

The agreement of the geometric mean and median as approximately equal measures of central tendency suggest this to be log-normal.

If this is indeed log-normally distributed, we should be able to take the logarithm of each individual number, look at the transformed distribution, and see the same symmetry in the transformed data was was observed for the untransformed data sampled from the normal distribution.

Let’s check:

Indeed, these both now appear to have the same degree of symmetry, if not differing their underlying standard deviations!

The ability to log-transform to normality the dataset is a strong indication the untransformed data is log-normally distributed!

Normal distribution

The student will read about the normal distribution. Then various examples of normally distributed data are presented. Connections with expected value and standard deviation are recalled. Plotting of the probability density function is presented. Using technology, the area under the curve is then calculated. Finally, a connection is made to the central limit theorem.

N[(1/2) (1-Erf[(1/(Sqrt[2]))])]

N[Erf[(Sqrt[2])]]

Data analysis

Students will learn about skewed data and how to visualize skewed data using a histogram. Students are then going to create a data set that is skewed. Finally, students will learn about how to interactively produce data regressions with specified parameters using an interactive plotting quiz.

How is the data skewed?

Exercise: Create a set of data where the mean is larger than the median and a set of a data where the median is larger:

Coin tossing

Students will generate experiments to find the length of runs of coin tosses. The student will then plot the experimental data. Lastly, a histogram is made to help visualize the length of the longest runs when tossing coins.

Find the length of the runs:

Geometric distribution

Students will learn about geometric distributions. The definition will be presented and various calculations performed.

Note in general there is not one consistent meaning of geometric distribution. The definition that the WL uses is the one above. One may encounter other definitions depending upon the given text.

A coin is heads with probability pfind the probability that first heads is on the 4th toss:

Check that probability of first heads being for i coin tosses happening with probability (1-p)^(i-1) p is consistent:

Find information on a given distribution:

Find a property of a given distribution:

Common core

Final Exam (Problem Set) for Calculus

1.) Find the derivative of sin(cos(sin(log(exp(x)))))

2.) Compute the area under the curve y = x^3 from x = -1 to 1 by first setting up an integral for the area under the curve:

2.) b. Compute the area under the curve from x = -1 to 1:

3.) Find a 4 term Riemann sum for the (Sqrt[(x^2)+1])

4.) Evaluate the integral: ∫ x Cos(((x^2)))

5.) Find the maxima and minima of (5(x^3)-6(x^2))

6.) Find the intervals of concavity for (5(x^3)-6(x^2))

7.) Find the equation of the tangent line for y = 1 + (2^x) through x = 1

8.) Find the slope of the tangent line to x^2 + y^3 + x y = 1through x = 1

9.) Find the limit to ((x^2-4x+3)/x)as x →2

10.) Is y = (x^2)+ (2^(Cos(x))) continuous?

11.) Compute the integral

12.) Is y = (1/(1+(x^2)))convergent or divergent from 0 to infinity?

13.) Find the area of the surface generated by revolving y = Exp(x)about the y-axis:

14.) Find the following and determine whether it converges:

15.) What is the interval of convergence?

16.) Find the volume of y = 1 + (x-2)^2 and x = 2 when revolved about the y-axis:

17.) Determine convergence of

18.) Determine convergence of

19.) Determine the Taylor series of up to 5 terms for

20.) Determine the image of the parametric equation specified by

21.) Determine whether 3x + 2y + z = 0 is perpendicular to (3t, 2t, -t)

22.) Determine the cross product, dot product, and angle between vectors of (2, 1, 1) and (3, 4, 5):

23.) Use the contour plot of x^2+y^2 to determine the critical points:

24.) Does the limit of exist as (x,y) goes to the origin for

25.) Let u(x,y) = sin(x y)where x = 2+ t and y = s-t. Find the value of u(x,y)when (s,t) = (2, 3)

26.) Find the critical points of

27.) Classify the type of the critical points for the above:

28.) Compute the following:

29.) Compute the following:

30.) Compute the following:

31.) Find the maximum of (x^2)+(y^2)+2z on 2x+y-z = 1

32.) Find the tangent plane to z = Cos(x+(y^2)) at (2, 3)

Questions, Comments, and Resources

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