## Two-step Linear Equations with Rational Numbers

Materials: Large grid paper or transparent grid for overhead projector (this grid should go at least from 10 to 10 on both axes), grease pencil or marker, ruler or yardstick

Preparation: Since students will be reading points from graphs and graphing lines from lists of points, they (and you) will need to be prepared to use a straightedge to generate accurate straight lines.

Prerequisite Skills and Concepts: Students need to be able to plot points on a coordinate plane and should be familiar with the various ways to indicate multiplication and division in an equation. They should also be familiar with the order of operations and the equality properties.

• Carefully draw a line through (0, 0) and (2, 2) on the grid. Be sure to extend it in both directions so there are plenty of easy-to-name points on it.

• Say: Name some points on this line.
Students should easily come up with a list of the points named by integer coordinates. If not, spend some time naming points on a grid before continuing with this lesson.

• Ask: Can you give me a rule for how to find y when we know x on this line?
Discuss how the coordinates are related, then ask students to write the rule in equation form. The equation for this line is y = x.

• Say: I'm thinking of a line. It has an equation of y = x + 3. How can you draw that line for me?
Most students will be relatively unsophisticated in terms of the use of the slope-intercept form of the equation to define the line. An easy way to draw the graph of a line is to substitute some values for x into the equation, find the corresponding values for y, and then plot those coordinate pairs. Two points give you enough information to draw the line, but it is probably safer to generate at least three points. This will help students spot computational errors. Work with students to make a T-table, assign some values to x, compute the related y values, and draw the graph of this line.

• Say: My new line has an equation of y = 2x + 3. Find some points and draw the line for me.
This equation requires students to compute twice. Remind them about the order of operations (multiply or divide left to right, then add or subtract left to right) if they forget to multiply their x-values by 2 before adding 3. It would be a good idea to ask different students to generate different points, talking through their reasoning as they go.

• Ask: Can someone give me a number between 5 and 5? Now, how about a number between 10 and 10?
Use these numbers to generate linear equations. The first number will be the coefficient of x, and the second will be added to the x term. Spend a good deal of time generating equations, finding points, and drawing the lines. If you've been working with slope, these problems will give you a chance to reinforce that concept as well (Do you think the slope of this line will be positive or negative? Do you think it will be very steep or not so steep? Will this line go through the origin?)
Slope        Two-step Linear Equations
with Rational Numbers.