This activity prepares students to write their own linear equations by providing a pattern to examine in the same style they will use in upcoming activities. They are given an image of lines on a coordinate plane and equations for two of the lines. They are then asked to write equations for the other lines. Although the given equations and solutions are in slope-intercept form, it is not important that students use that form. Any equivalent expressions should be accepted.

The purpose of the discussion is to help students write equations from lines. Consider asking students:

• “What methods did you use to write additional equations?” (Because the lines are parallel, I knew they would have the same slopes and only needed to use the vertical intercept to write the equation in slope-intercept form.)
• “Could you write the equations of the lines if the problem did not specify that they are parallel?” (Yes. Each line goes through at least two points on the coordinate grid, so I could find the slope and intercept from that.)

This activity gives students an opportunity to apply what they have learned from the concepts in this unit. In particular, students use linear equations to create a pattern, then use those equations to find intersection points using a method of their choosing.

The purpose of this discussion is to explore methods for finding equations to fit the vision that students have for a pattern.

Invite students to share their patterns and the methods that they used to find particular lines. Consider asking these questions:

• “Did you prefer to write your equations in standard or slope-intercept form? Explain your reasoning.” (It was easier for me to write equations in slope-intercept form because I could find one point and then guess an appropriate slope to match my pattern.)
• “Can you think of a situation in which you would prefer the other form?” (If I knew both the - and -intercepts, it might be easier to figure out the constants for standard form than to figure out the slope for slope-intercept form.)
• “How did you find a second line that did not intersect the first?” (I knew that non-intersecting lines are parallel and have the same slope but different intercepts, so I used the slope-intercept form to keep the slope, but I changed the intercept.)

This activity uses the patterns that students created in an earlier activity to create systems of equations to solve by using the algebraic methods that they learned in this unit. Students also write a system of equations with infinitely many solutions, using an equation given to them by their partner. Then students use the graph of their partner’s equations to check their solutions to the systems they solved.

The purpose of this discussion is to review solving systems of equations algebraically and examining systems with 0, 1, or infinitely many solutions.

Invite students to share any interesting systems they solved and the methods they used to solve it. Consider asking:

• “Why did you select the 2 equations to create your first system?” (They were both in slope-intercept form, so I knew it would be easy to use substitution on them.)
• “Do you prefer to use elimination or substitution to solve systems of equations?” (I prefer to use elimination in general, but substitution can be easier if a variable is already isolated.)
• “How did you find another equation to create a system with infinitely many solutions?” (I knew it had to be an equivalent equation, so I multiplied each side of the equation by 2.)
• “If you added your new equation from the third question to your partner’s pattern, how would it change?” (It would not change. It would graph on top of one of the other lines already in the pattern.)