Topic: Solving systems of equations with linear and quadratic inequalities

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Earl Manning
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1 Subject & Grade: Mathematics, 9 th Grade Topic: Solving systems of equations with linear and quadratic inequalities Aim: How would you find the solution set of a linear and quadratic inequality? Materials:. Chalk. Do Now worksheet. Stained Glass Window worksheets. Pencil. Paper 6. Colored Pencils Instructional Objectives:. Using prior knowledge of graphing, students will understand how to graph linear and quadratic inequalities.. Using prior knowledge of systems of equations, students will understand how to find the solution set of a system of inequalities. Standards: New York State Learning Standard for Mathematics, 00 A.A.0 Solve systems of two linear equations in two variables algebraically A.A. Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers. A.G.6 Graph linear inequalities A.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables A.G.8 Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions. A.G.9 Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers. Expectations: Principles and Standards for School Mathematics. Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;. Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency mentally or with paper and pencil in simple cases and using technology in all cases;. Use symbolic algebra to represent and explain mathematical relationships Motivation:. See the attached Do Now worksheet.

2 . Ask the class why such a question, from the Do Now, is important.. Students will have already learned how to graph a linear inequality and a quadratic inequality from two previous lessons. Procedure:. Distribute the Do Now worksheet as the class enters the classroom and allow them seven minutes to complete it.. Discuss the students results to the Do Now worksheet.. Begin the lesson by placing the AIM on the board. (How would you find the solution set of a linear and quadratic inequality?). Using the Do Now problem, introduce the concept of graphing the system consisting of two inequalities instead.. Have the students conjecture what the difference would be between the procedures of graphing a system of equations and a system of inequalities, then write these differences/procedures on the board. 6. Notes for the class (include if important points were missing as a result of #): Today we re going to graph systems of inequalities. Step : If necessary, put each inequality in slope-intercept form. Reminder: If you multiply or divide by a negative number you have to change the sign! Step : Graph each inequality. If the inequality sign is or use a solid line to connect your points. If the inequality sign is > or < use a dashed line to connect your points. Label your point(s) of intersection (if there are any) Step : Shade the solution set for each individual inequality. The solution to the system of inequalities is the doubly shaded region. Step : Check your solution set. 7. Place a problem on the board and go through each one of the steps from (Using the formulas/procedures that the class has created from # and #6 (if necessary)). Example and :

3 () Graph the solution set to the system of inequalities on the axes below to check your answer. y x + y x () Graph the solution set to the system of inequalities on the axes below to check your answer. x+ y< x y y y

4 8. Have students make a note that the two inequalities graphed above are lines and that even though they are inequalities, they follow the same procedure when graphing. 9. Have the students conjecture if the two inequalities were linear and quadratic instead of two linear inequalities. How will this change the solution set? They should discover that the shaded region of the quadratic will either be inside or outside the curve. 0. Have students complete examples on graphing a system of inequalities with linear and quadratics. Examples,, and : 0 y x+ y > 0 () y< 9 x x

5 0 y 9 8 x+ y 6 () y < x + 6x 7 6 x y 9 8 y x+ () y > x x 7 6 x After completing the lesson, ask students if they have any further questions before distributing out a group activity. In groups (no more than to a

6 group), allow them minutes to work on the Stained Glass Window worksheet. We will discuss the results the next day in the beginning of the class period. Essential Questions:. How do you find the solution set of a system of inequalities?. How are graphing equations and inequalities similar? How are they different?. How are systems of equations and systems of inequalities similar? How are they different? Closure:. Allow groups to finish up wherever they are with their Stained Glass Windows.. Announce that homework is to complete the Stained Glass Window project. Assessment:. Students will be assessed based on their contributions to class discussions and group work.. Students will be assessed based on their ability to graph linear inequalities and quadratic inequalities.. Students will be assessed based on their ability to find the solution set of a system of linear and quadratic inequalities. Differentiated Instruction:. Enrichment: a. Students will use word problems to create their systems that are to be graphed. Remedial: a. Students will be given the opportunity to solve systems algebraically.

7 Name Date Solve the system of equations by elimination or substitution. Graph both equations on the axes below to check your answer. y = x + y = x Do Now Worksheet Solving a System of Equations - Review Solve the system of equations by elimination or substitution. x+ y = x y = y y

8 Stained Glass Window Name Date Period Use the following inequality systems to create your stained glass window. () # % $ &% x + y 0 y > 9 x () # % $ &% x + y < 6 y x + 6x () # % $ &% y x + y > x x

9 Stained Glass Window Name Date Period On the coordinate plane below, graph the linear and quadratic systems of equations. Write the equation neatly on each line or parabola that you graph. When you are finished graphing the equations, use colored pencils to color each section to create your stained glass window. Hint: Make sure each solution set is clearly colored and labeled.

LHS Algebra Pre-Test

Your Name Teacher Block Grade (please circle): 9 10 11 12 Course level (please circle): Honors Level 1 Instructions LHS Algebra Pre-Test The purpose of this test is to see whether you know Algebra 1 well