Engaging Algebra: Exploring Linear Equations

1. Discovering Algebra Graphing Linear Equations by David A. Thomas and Rex A. Thomas

2. Brief Overview • Use technology (a computer graphing program) to provide for student directed exploratory learning and the development of mathematical thinking and reasoning • Students experimented with various linear equations and their graphs and the effect of changing variables on the graphs • Attempted to help students learn how to learn math

3. NCTM Standards Addressed • Build new mathematical knowledge through problem solving (problem solving) • Make and investigate mathematical conjectures (reasoning and proof) • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others (communication) • Recognize and use connections among mathematical ideas (connections)

4. NCTM Standards Addressed(Algebra) • Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior • Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations

5. NCTM Standards Addressed(Algebra) • Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency - using technology in all cases

6. The Lesson • A three day lesson split between a computer lab and classroom each day • Students worked in pairs to determine where a line would appear on a graph depending on the coefficients of AX + BY = C • Students were encouraged to experiment and make conjectures with few instructions or directions given • Students then shared their findings

7. The Lesson

8. The Lesson • Day 2 required critical thinking about a few given conjectures and a need to communicate their thoughts to the class • Conjecture – Changing only the coefficient of x causes all lines to cross the y-axis in the same place • The final part of the lesson was making predictions based on what they had learned

9. Strengths • Addresses higher order thinking and learning • Allows for an emphasis on strategies of learning and thinking and not just on the correctness of an answer • Encourages participation by all students as there is no “right” answer only the students’ thoughts and observations

10. Strengths • Provides for an in depth learning of a particular topic which if used at the beginning of a unit can give a good foundation for topics to come • A great opportunity to introduce students to student directed learning

11. Issues or problems • If this is the first time students have done a student directed learning experience there will be confusion and resistance • The time it takes to fully implement the lesson is prohibitive to material coverage • Lesson didn’t provide any real life connections

12. Implementation in the Classroom • Could be done with students in the regular classroom with graphing calculators • New TI technology allows students to share information with the teacher electronically in the classroom from graphing calculators • Student ideas could be shared by showing graphs on overheads, smartboard, etc. • Could be scaled back in scope or expanded depending on time to be devoted to the lesson

13. Discussion Questions • In what ways could you build on this lesson to make connections for the students to real life applications? • How could this lesson be shortened or simplified and still accomplish some of the same objectives and outcomes?

14. Field experience – A simpler lesson • Students worked in pairs on a calculator activity • Students used graphing calculators to determine the effect of changing m and b in an equation in slope-intercept form • Students required to answer higher level questions including making predictions and explaining their thinking to the class

15. Citations Thomas, David A. and Thomas, Rex A. Discovering Algebra – Graphing Linear Equations. The Mathematics Teacher, Vol. 92, No. 7, October 1999