The PAT Curriculum:

These sample lessons illustrate some of the types of skills that we are currently teaching to students using intelligent tutors. These are sample lessons for the Algebra Word Problem Tutor demo (PAT 14.1) 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=mx, y=mx+b or y=mx-b. As students work through the sections in lesson one, 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 for lesson two 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 : The Pattern Finder

In lesson three, students will concentrate on finding the formula. They are REQUIRED to use the Pattern Finder Window. They should first label the columns and fill in the units row. They should then complete the Pattern Finder by typing in the expression for 2, 3, and 4 and then for any number of items. They should then insert the variable and the formula in the Formula row, and complete the worksheet by putting in the given quantities and computing where necessary the unknown quantities.

A major concern at this stage is ensuring that students can find the appropriate formulas.

Lesson Four : Equation Solver

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

Lesson Five : Solving Start-Unknowns

In lesson five, 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 is completely filled out.

Lesson Six : Four Quadrant Graphing

Problems in lesson six 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.

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 Seven : Equation Solver

In lesson seven, students will again use only the Equation Solver to formally solve more complex equations including ones with the distributive property.

Lesson Eight : Distributive Problems

In lesson eight, students are given problems where some manipulation of the slope or intercept information is needed to accurately write the formula. For example, "We work for a company that makes and sells hot air balloons. We currently have an inventory of balloons ready to sell, of 20 balloons. We are planning to sell all the hot air balloons for $3500 a piece." The student's goal in this case is to calculate the income from balloon sales. To do so, the sale price of the balloons must be applied to the initial inventory, to obtain the y-intercept. Students continue to use the worksheet and grapher tools to solve this type of problem.

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 Nine : Point-Intercept and Point-Point Problems

Problems in lesson nine present information which indirectly indicates the slope and intercept values for a problem. For example, "A tank is filling with water from 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. Students use the worksheet and the equation solver.

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

Lesson Ten : Slope Intercept Graphing

Problems in lesson ten involve a mixture of y=mx and y=mx+b equations with positive and negative m and b values. 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 Eleven : Equation Solver

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

Lesson Twelve : Systems of Equations

In lesson twelve, 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 formulae for the two equations, the intercept of the lines is labeled, and all points from the worksheet are displayed on the graph.

Lesson Thirteen : Percentage Problems

Lesson thirteen focuses on a specific subclass of y=mx and y=mx+b problems: percentage problems. Percentage problems involve calculations such as "An acceptable tip in a restaurant is 15% of the bill..."

Lesson Fourteen : Proportional Reasoning Problems

Lesson fourteen focuses on a specific subclass of y=mx and y=mx+b problems: proportional reasoning. Proportional reasoning problems involve calculations about situations such as batting averages: "Suppose a player gets two hits for every five times he bats..."

Lesson Fifteen : Equation Solver

In lesson Fifteen, students will again use only the Equation Solver to formally solve many different types of more complex equations.

Lesson Sixteen & Seventeen : Review Problems

Maintainer: tutor-help@andrew.cmu.edu
(last updated September 17, 1996)