This lab presents experience with angles. Begin with rays forming angles and how to compare them. These rays are then anchored at the origin on the Cartesian plane where they intersect the unit circle. With this setup measurement is established via the circumference of the unit circle.

After investigating this correspondence students hypothesize on the measurement system and identify markers for the quadrants. With this measurement system angle addition and subtraction is explored and hypothesized by the student.

Finally, the labs move around the unit circle exploring the various aspects of this correspondence: angles, arcs, coordinates, and right triangles. This offers a preview of trigonometric functions available in other labs.

This lab presents experience with mathematical modeling. Here the most basic model is investigated...fitting a line (linear model) to data (points). At the crux of this modeling is the idea of error measurement. The emphasis here is on the least squares error measurement. Students will create a least squares line fit to three points.

The data is only three points, which allows students to measure error, graphically represent it, and total it over and over and over to experience what the error is and how to reduce it.

This lab presents experience with discontinuities (breaks). The three most common types of discontinuities encountered in calculus are removable discontinuities (holes), jump discontinuities, and asymptotes. These are the subjects of these labs. We call them breaks.

Asymptotes and holes are somewhat natural for students as they occur in rational functions. However, jump discontinuities must be manufactured and the most straightforward method is via piecewise defined functions. This is where the first lab begins.

Piecewise functions are explored with emphasis on what is happening near and at the defining interval endpoints. With this tool available the three breaks are introduced and explored for defining characteristics.

This lab presents experience with complex numbers. The labs begin with a deficiency in the real number. There is no real number 'x' such that x² = -1. This number is thrown into the mix, represented as 'i', and its arithmetic properties are investigated.

With 'i' thrown into the real numbers all of the arithmetic operations are re-examined: addition, subtraction, multiplication, division, and fractions. Each time the student is presented an interactive situation where they can perform these operations quickly in order to observe patterns and rules. The rules of arithmetic are then hypothesized by the student.

Algebraic manipulation is further explored with the complex conjugate and its multiplicative effects. The complex conjugate also acts as a bridge to an extended absolute value called the modulus. The labs constantly tie ideas, rules, and skills back to the real numbers to demonstrate that the complex numbers act the same with simply another number available.

With complex numbers now available for arithmetic the labs tackle square roots, cube roots, and fourth roots, computationally and graphically. Although a polar representation seen in other lessons is better suited for this analysis the current form brings the analysis of roots closer to the real numbers. Students can verify in a familiar manner that the system is indeed working as advertised.

The same idea guides the investigation of complex logarithms. They are supposed to work the same for complex numbers as the do with real numbers except fill in holes that the real numbers were unable to handle. Again the algebraic and graphical connection is investigated and hypothesized by the student.

This lab presents experience with conic sections. Conic sections can be viewed in at least two seemingly unconnected ways. The first viewed is that is cross-sections of a double cone. The second view is that of point collections from the xy-plane. Both views are considered in these labs.

The first set of labs considers a double cone. One cone atop another with their peaks touching in the middle. Students can then pass a plane through this object and create cross-sections. Depending on the angle of the plane these conic sections form different shapes, which can then be categorized as circles, ellipses, parabolas, hyperbolas, including their extreme examples.

After experiencing conic sections and their shapes the labs move onto point sets from the xy-plane. Here collections of points are created according to rules about distances. These collections turn out to be the same shapes as the conic sections. The rules or definitions are not passive ideas in these labs, but fully interactive and controllable by the student. The labs allow the students to build the conics, animate the construction, and hypothesize on defining equations.

This lab presents experience with DeMoivre's Theorem. DeMoivre's Theorem provides an alternate form for complex numbers, which lends itself quite nicely to powers of complex numbers. This has its historical roots with Euler's Formula and that is where the first lab begins.

Euler's Formula expresses the relationship between complex exponentials and trigonometric functions. Student gather information on a very weird function and then hypothesize on this relationship. Students compare the work required in raising complex numbers to whole number powers using each part of the relationship and then rate which is quicker.

DeMoivre's Theorem takes the spotlight now. Students are guided to DeMoivre's formula from their previous work and hypothesize on how to operate the formula. The processes emphasizes the algebraic and geometric connections and uses both views to advance the other.

This lab presents experience with factoring polynomials. The labs begin by examining roots of polynomials and how they relate to linear factors. The polynomial's constant term is considered next by expanding factorizations and hypothesizing on the factorization of the constant term. Students test their theory so far and then move onto the leading coefficient.

Examining the leading coefficient brings in rational roots and students hypothesize on the factorization of the rational root's numerator and denominator.

This lab presents experience with fractions. Fractions are a system of representing numbers and this system has its own rules for algebraically manipulating the representations around. Each lab here investigates an algebraic operation on fractions. Students will examine many examples to hypothesize on the algebraic rules and then test the system they have described.

This lab presents experience with the concept of function. A function is a partnership between three objects.

• a domain
  For us this will consist of real numbers, but the elements of a domain could be anything.
• a range
  For us this will consist of real numbers, but the elements of a range could be anything.
• a mapping
  a mapping is a set of instructions that tell you how to associate each member of the domain with exactly one member of the range.

It doesn't take a very large domain or range for things to get out of control. Therefore, it is of utmost urgency to create a representation that can handle domains and ranges of large size. And, represent the mapping in such a way that it is easy to encode and decode the mapping in the representation. This is called graphing and it is the focus of all these labs.

The representations begin with familiar number lines and students are guided through a series of improvements that culminate with the Cartesian plane. Along the way the representation will use many different tools and symbols and students are given a feel for what works and where the problems occur.

Finally, the representation is extended a dimension to represent multivariable functions.

Representations of functions are graphical and these labs are graphics heavy with plenty of student interaction directly aimed at designing, building, and exploring the graphical structures.

This lab presents experience with graphing. Graphing is a representation of information and there can be many methods of graphing. Each would have its own methods of encoding and decoding information. These labs explore graphing on the Cartesian plane.

Graphing on the Cartesian plane is a process of collecting points out of the plane to designate as 'on' or 'included in' the graph. The actual picture that is displayed is an exaggeration of this collection because points are really small and our pencils are not. The points that are included in the graph include coordinates that satisfy a particular relationship. And, it is with coordinates that these labs begin.

After investigating 2D and 3D coordinate systems the labs present graphs, their main features, and different points of view.

This lab presents experience with the greatest integer function. The greatest integer function strips off the fractional part of a real number leaving only the integer part. The labs investigate the greatest integer function computational as well as graphically. They then move on to composing with other functions and investigate the resulting graphs.

After exploring the effect on numbers and functions of one variable the composition increases in dimension and the labs explore functions defined on the Cartesian plane as well as paths through the plane.

This lab presents experience with indeterminate forms. There are several well known algebraic rules. One to any power is one. A fraction with a numerator of zero is itself zero. A fraction with a denominator of zero is not a fraction at all. Indeterminate forms here are fractions and exponential expressions that are close to following these rule. So, close that they defy the rules all together. This is an unexpected result until you think about it for a while and then it unravels and is quite natural.

The labs here investigate each of the situations described above. Students are guided through the experimentation and asked to create prescribed fractions and exponential expressions, change components, and then compensate with the other components of the expression. In this way the student controls the very nature of the indeterminacy.

This lab presents experience with inverse functions. The first lab established a difference between the reciprocal and the inverse although notation may be confusing. The lab continues with input and output reversal and computational experimentation. This leads directly to an investigation of coordinates on the graph of a function and an inverse.

This lab presents experience with lines. The labs begin with an examination of the Cartesian, points, and coordinates. Using this structure the lab prompt a student's natural idea of linear patterns and from there establishes guides students to the defining characteristic of lines: constant growth rate or slope.

Slope between points is the test for membership in a particular line and students are put in charge of this test in many situations. From here all points satisfying the test are gathered to form a line and the main features of lines are identified.

After approaching lines graphically, algebraic representation is added via equations. Y-intercept and standard forms of linear equations are presented with plenty of investigation and hypothesizing. This leads to comparison of lines including perpendicularity.

This lab presents experience with logarithms. Logarithms can be viewed in three different ways. First they are exponents. They are numbers that are used as powers to create other numbers. The notation for logarithms is actually the only important part when viewed this way. Secondly, logarithms follow the arithmetic of exponents. Here the rules for algebraically manipulating logarithms needs to fit into the notation. Finally, logarithms can be viewed as functions. Now domain, range, and graphs are added to complete the story.

This lab presents experience with matrices and their arithmetic. The labs begin with the structure of matrices: rows columns, and entries and how to interact with them in LiveMath Maker. Transposition and dimension are also introduced. Once the matrix object has been explored the labs guide students to hypothesize on an arithmetic system for matrices.

Students are able to investigate as many instances as they need to arrive at their predictions. Predictions on what works and what does not work. In particular multiplication is several possible definitions and the labs work through some of these to lay the groundwork for the final operation. During this process students will visit linear systems of equations and complex number representation. These are some of the motivating desires for matrix multiplication.

This lab presents experience with the equation of the parabola. The definition of a parabola dictates which points are on the parabola by their distances from a focus point and a directrix line. This is where the labs begin. After getting familiar with the definition, students are lead through the steps from the geometry to the algebraic equation via the distance formula.

This lab presents experience with curves described parametrically. The normal notion of graphing is to move 'forward' along the x-axis and plot points vertically. This method works well for functions, but there are many curves which are not associated with functions. To graph these curves the idea of 'forward' must be disengage from the x-axis and move to an auxiliary axis.

With a third axis keeping track of 'forward' movement both the 'x' and 'y' coordinates can move 'forward' while still moving up and down and left and right. This is the idea behind parametric descriptions.

The labs explore this idea right from the beginning with the difficulties upfront to motivate the extra parameter. In particular students explore the movement of a point on the unit circle and are guided to hypothesize on a suitable parameterization.

This lab presents experience with polar coordinates. Here students play a game of entering in numbers to move a point plotted on the plane. The numbers are to represent angle and distance. After gathering some information students can then hypothesize on what is happening and compare the process to rectangular coordinates.

This lab presents experience with square roots. Square roots are often represented with a radical sign and this sign as its own algebra, which is explored in these labs. The goal of these labs is very straightforward. Students can see which algebraic rules work and which don't by observing as many examples as they can think up.

This lab presents experience with the step function. The step function is a very useful function in advanced mathematics. It is the mathematical model of a switch and thus has many applications in science and engineering. As a switch it is zero for negative inputs and then its output changes to one for zero and positive inputs.

By shifting the step function the switching can happen for any set input value. Together with the multiplicative properties of zero and one the step function becomes the foundation for piecewise defined functions. In the labs here this characteristic is presented as a cookie cutter. Students begin with a function defined on the real line and then cut-out pieces of it.

This lab presents experience with linear systems. The solution to a linear system is basically the same as the solution to a single linear equations. The goal is to find numbers that can be substituted into the equations and satisfy all of the equations simultaneously.

Of course there are many algebraic methods for discovering solutions, but they are all after the same numbers. This lab is concerned not with the steps to locating solutions, but with the solutions themselves? How do you know if a number is a solution or not? How is the system represented graphically and where are solutions positioned when represented as points?

This lab presents experience with the Taxicab Metric. Students understandably come away from precalculus with the idea that 'distance' is an absolute idea. There is THE distance and there are mathematical situations where you would like to know THE distance. But, there are many distances. In fact the study of distance is a wide topic in advanced mathematics. But, you don't need high level math to experience other distances.

If you have heard the phrase 'as the crow flies' then you know there is more than one distance commonly in use. Since taxicabs are restricted to driving on the streets they more than likely will not travel the shortest distance possible from one location to another. To do so they would need to drive through buildings, parks, and lakes. Their distance is measured along horizontal and vertical lines rather than 'as the crow flies'.

The Taxicab distance (metric) follows all of the rules that the 'normal' distance follows and this can lead to some unexpected situations. In these labs students will explore conic sections using the taxicab distance.

Conic sections can all be describe via distance. Points are included or excluded from the collection according to a distance rule. Here the distance rule is the taxicab distance and the conic sections can be very surprising.

This lab presents experience with the founding functions of trigonometry: sine and cosine. One approach to trigonometry is the analysis of right triangles. Trigonometric functions are built from ratios of lengths of sides of the triangles. This approach has immediate applications, but not many. In fact, another viewpoint is much more effective and useful for mathematics beyond precalculus.

Instead of beginning with right triangles and then specializing these to coincide with the unit circle for later development, the unit circle can pose the motivation for trigonometric functions and lay a stronger foundation for advanced mathematics. This is the approach pursued in these labs.

The labs here begin with a survey of the unit circle: points, angles, and distances. Following the connection between angles and points on the unit circle information about the coordinates is collected and examined. Students build the sine and cosine function as data from their travels around the circumference.

After revealing sine and cosine the labs investigate several relationships (identities) between the functions. Students now have a view of sine and cosine as functions instead of ratios and are in a much better position to use them post-precalculus.

This lab presents experience with the basics of vectors. Vectors are the directions of math. You might tell a friend to drive 3 miles north, then drive 2 miles west, then drive 3 miles north. This is information about movement. They do not contain any information about position. Depending on your starting position you could end up in quite different places. The directions only tell you a direction to move and how far. Same with vectors.

Vectors encode information about movement: what direction and how far. It turns out that you can do a lot with directions and these labs explore the algebra of vectors as well as graphical representations and geometric interpretations of the algebraic operations.

Vectors can also be used as tools to investigate other geometric objects. For this type of analysis there are further algebraic calculations and formulas that encode information. The dot product is one such calculation and is introduced here.