These lessons at-a-glance illustrate some of the types of skills that we are currently teaching to students using intelligent tutors. These are lessons for the Word Problem Tutor (PAT 15.0) and for the Equation Solver as a standalone tutor.

Lesson One : The Worksheet

In Lesson One, students are introduced to word problems. Each word problem involves a single linear equation of the form y=ħmx, y=ħmxħb. As students work through the sections in this lesson, they progress from equations in which numbers are simple (integers or halves) and small (between -100 and 100) to equations involving numbers which are difficult (decimals and fractions) and large. (This type of progression also occurs in later lessons.)

Students begin their work with word problems by reading a problem statement and filling out a tabular worksheet. Steps in filling out the table include:

1. identifying important quantities or concepts in the problem and using them to label the column headings
2. indicating the units of measurement for each quantity
3. solving for values of y (result-unknowns), given values of x in the questions for the problem
4. writing a formula for the relationship in the problem, by entering a variable for x, and an arithmetic expression for y expressed in terms of x.

The student has solved the word problem once the table is correctly filled out.

Lesson Two : Graphing Worksheet Points

In Lesson Two, students continue to use the worksheet, as in Lesson One. They also learn to use a new representation, graphs. Steps in completing a graph include:

1. identifying the quantities and/or units of measurement to be used to label the axes of the graph
2. identifying lower and upper bounds for the graph, based on the smallest and largest numbers in the columns of the worksheet
3. identifying a suitable scale for the graph, given the chosen bounds
4. creating points and placing them at the coordinates of the table entries
5. drawing a line to connect the points

A major concern at this stage is ensuring that students can map from numbers in the table to locations on the graph. Students tend to find problems especially difficult when they involve large numbers, decimals (as opposed to whole numbers), and negative numbers (as opposed to positive numbers).

The student has solved the word problem when the table has been filled out, all points from the worksheet are displayed on the graph, and a line is drawn to represent the formula for the equation.

Lesson Three : Equation Solver

In Lesson Three, students will be introduced to the Equation Solver and learn how to formally solve simple one and two step equations.

Lesson Four : Four Quadrant Graphing

Problems in Lesson Four include a variety of y=mx and y=mx+b problems, with slopes and intercepts which are both positive and negative. Students are required to set bounds and graph points for both positive and negative values. Solution values are found in all four quadrants of the graph. Students use the worksheet, grapher, and equation solver tools.

In Lesson Four, in addition to being asked to solve for y, given x, students are asked to solve for x, given y. Students are given a new tool for finding values of x when given values of y This tool is the Equation Solver, which they can use to write and solve equations such as 230x+4550=0. Students continue to use the worksheet.

The student has solved the word problem when the table has been filled out, all points from the worksheet are displayed on the graph, and a line is drawn to represent the formula for the equation.

Lesson Five : Equation Solver

In Lesson Five, students will use the Equation Solver and learn how to formally solve equations by combinlng like terms with both integer and decimal coefficients.

Lesson Six : Slope Intercept Graphing 1

Problems in Lesson Six involve a mixture of y=ħmx and y=ħmxħb. Students use the worksheet to write an equation describing the problem, and are introduced to a new graphing tool, which they use to graph points based on the y-intercept and slope of the line. Steps in completing a graph for lesson ten include:

1. identifying the quantities and/or units of measurement to be used to label the axes of the graph.
2. identifying lower and upper bounds for the graph, based on the smallest and largest numbers in the columns of the worksheet
3. identifying a suitable scale for the graph, given the chosen bounds
4. correctly placing the y-intercept on the graph
5. correctly placing at least two more points on the graph, using the slope to calculate their desired location. Students must calculate desired target locations by moving over from the previous point (adding the scale size to the previous x-value), and then up (adding the slope times the increment to the previous y-value.)
6. identifying the value of the y-intercept
7. identifying the value of the slope
8. drawing a line to connect the points

The purpose of the lesson is to get students to identify the values of the slope and intercept, and to reason about rates of change in the graph using the concepts of slope and intercept. Students must explicitly place points on the graph using the slope and intercept. Students tend to have difficulty with the concepts of x- and y-intercepts. They may not understand what they mean in relation to an equation or what they mean in relation to a graph. The purpose of the lesson is to focus on these areas.

The student has solved the word problem when the table has been filled out, the slope and intercept are represented on the graph both as points and in numbers, and a line is drawn to represent the formula for the equation.

Lesson Seven : Equation Solver

In Lesson Seven, students will again use only the Equation Solver to formally solve more complex equations, specifically ones with variables on both sides of the equality which will be used in Lesson Eight with the systems problems.

Lesson Eight : Systems of Equations 1

In Lesson Eight, students begin to work with systems of linear equations. They solve word problems which involve reasoning about 2 equations of the form y=mx+b.

Students continue to use the worksheet and equation solver. The worksheet now involves three columns, rather than two. The equation solver allows students to solve questions about values of x, given y; and about values of x, given that two equations y and z are equal to each another. Students use a new form of the grapher to graph their equations, focusing on the idea of intersection of equations. Steps in completing a graph for lesson twelve include:

1. identifying the quantities and/or units of measurement to be used to label the axes of the graph.
2. identifying lower and upper bounds for the graph, based on the smallest and largest numbers in the columns of the worksheet.
3. identifying a suitable scale for the graph, given the chosen bounds
4. identifying the two lines to be drawn, by labeling the line markers
5. displaying the two lines
6. entering the x- and y-z values for the intersection of the two lines
7. displaying the rest of the points in the table (Students do not have to explicitly create and place the points. They just tell the computer to do so, after setting the bounds.)

The student has solved the word problem when the table has been filled out, lines are drawn to represent the two equations, the intercept of the lines is labeled, and all points from the worksheet are displayed on the graph.

Lesson Nine : Equation Solver

In Lesson Nine, students will again use only the Equation Solver to formally solve more complex equations, specifically ones that require distribution to solve.

Lesson Ten : Distribution

In Lesson Ten, students will write expressions and equations that contain distribution. They will have the opportunity to utilize the complex equations solving techniques in context that they practiced during Lesson Nine.

Lesson Eleven : Proportions

Students will solve proportion problems in Lesson Eleven by expressing a total sample as a variable. Then they will express another quantity, a dependent portion of the total sample, as a function of the total sample.

Lesson Twelve : Point-Intercept and Point-Point Problems 1

Problems in Lesson Twelve present information which indirectly indicates the slope and intercept values for a problem. For example, "A tank is filling with water form a natural spring Two days ago the water was 10 feet deep, and yesterday the water was 12 feet deep. Assume that the water continues to rise at this rate." The student must determine the slope and intercept from the description and write a formula describing the equation.

The student has solved the word problem when the table has been completely filled out.

Lesson Thirteen : Systems of Equations 2

Students will solve proportion problems in Lesson Thirteen as they did in Lesson Eight, but the difficulty level will have increased.

Lesson Fourteen : Quadratics

Students will write second degree expressions with the option of using the quadratic formula to solve for the independent variable when they are set equal to a given y-value in context.

Lesson Fifteen : Point-Intercept and Point-Point Problems 2

Students will revisit problems like Lesson Twelve's with increased difficulty.

Lesson Sixteen : Slope-Interecept Graphing 2

Problems in Lesson Sixteen will involve various linear equations with both positive and negative values for the slopes and y-intercepts that are of greater difficulty than those in Lesson Six. They will use the same graphing tool that they used in Lesson Six as well.

Lesson Seventeen : Systems of Equation 3

Here students will be challenged with the most difficult systems problems in the curriculum. The features will be the same as they were in Lessons Eight and Thirteen.

Lesson Eighteen : Equation Solver

In Lesson Eighteen students will again use only the Equation Sovler to formally solve many different types of more complex equations.

Lesson Nineteen : Review Problems

Students can get any equation, but need complete no graphing.

Lesson Twenty : Review Problems

Students can get any equation, and they will plot points to graph the functions.

Lesson Twenty-One : Review Problems

Students can get any equation, and they will graph using the slopes and y-intercepts.

Lesson Twenty-Two : Review Problems

Students can get any equation, but they will not need to graph.