Lesson objective: Determine the number of solutions a system of equations has.

Students bring prior knowledge of number of solutions of a system on a graph from 8.EE.C.8a. This prior knowledge is extended to equations as students use the slope and y-intercept to determine the number of solutions by inspection. A conceptual challenge students may encounter is not being able to recognize the slope when the equation is not in slope-intercept form.

The concept is developed through work with equations, which allow students to identify different cases and sort them into three categories: lines that cross, lines that will never cross, and lines that are the same. They are then challenged to form conclusions about how you can tell by the equations alone which category a system fits into. Finally they have to prove that understanding by writing a new system that fits into each category.

This work helps students deepen their understanding of equivalence because systems with equivalent slopes will either have no solution or infinitely many solutions.

Students engage in Mathematical Practice 2 (Reasoning abstractly and quantitatively) as they use the relationships between two equations to determine the number of solutions for the system.

**Key vocabulary:**

- infinitely many solutions
- no solution
- one solution
- rate of change
- system of equations
- slope
- Y-intercept

**Special materials needed:**

- sorting cards, found in the "Additional Materials" section of this lesson