Lesson+Plan+Model


 * **Academic Standards**

**A-CED 2 **: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

**A-CED 6**: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

**A-CED 7**: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

**A-CED 10**: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). ||
 * **Key ideas for this lesson are:**

The key idea is for students to be introduced to systems of linear equations. Also, for students to recognize what the graphs of coinciding, independent, dependent, etc. look like and how their slopes and y-intercepts will either be similar or different with each type. By having students do the demonstration they should be able to make the connection so when the vocabulary is reviewed, they will already have a mental image of what each type look like. ||
 * **Essential Question:**

What do you notice about the equations as your adjust the lines in the graph? What similarities/differences do you recognize between the equations (slope and y-intercept) and the graphs (lines and intersections)? ||
 * **Solve the problems that students will be doing in the space below.**


 * x - 3y + 3z = 8 **
 * 2x +3y - 9z = 12 **
 * 5x - 6y + 7z = 5 ** ||
 * **What other ways might students solve these problems?**

**Students might attempt to use the method of substitution when solving three variable equations, when elimination should be the first method used in solving these systems. ** ||
 * **What misconceptions and challenges might students have?**

How will you incorporate academic language? Which key vocabulary, algorithms, terms, or concepts might be unfamiliar? How will you support students in developing their understanding of these?
 * Students often confuse dependent and inconsistent systems once they begin solving equations algebraically.
 * In order to address this misconception, we will reinforce the visual explanation as to why a dependent system of equations has an infinite number of solutions and why an inconsistent system of equations has no solutions through the use of the Mathematica demonstration.
 * <span style="font-family: Arial,sans-serif;">We will also emphasize that students can determine how a system will look by analyzing slopes and y-intercepts of equations through the use of the Mathematica demonstration.
 * <span style="font-family: Arial,sans-serif; font-size: 12px;">Students might not recognize that when both equations have one side equal to y, they can use a combination of symmetric and transitive properties to set the other sides of the equations equal to one another. ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Academic Language**

We will incorporate academic language by utilizing it frequently in our lectures and when explaining examples. In addition, the Mathematica demonstration will allow the students to explore various components of the academic language that we will aim for the students to know. Finally, we will emphasize the use of academic language in the students’ individual and group presentations in order to promote their understanding of a wide range of mathematical terms that will be essential to their success in this unit.

The following mathematical terms will be stressed throughout the unit, all of which may or may not be familiar with the students, depending upon their prior knowledge of algebra and systems of equations:


 * Slope
 * Y-intercept
 * Slope-intercept form
 * X- and y- axes
 * Variables
 * Equations
 * Inequalities
 * Systems
 * Substitution
 * Elimination
 * Graphing
 * Intersecting
 * Parallel
 * Coinciding
 * Infinite solutions
 * No solution
 * Algebraically
 * Constant
 * Multiply
 * Divide
 * Negative ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Key Academic Genres**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Systems of Linear Equations falls under the category of functions in the Algebra 2 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Which of the following goal(s) will receive attention during your lesson? <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">___Students will justify their work both in terms of why they selected a particular approach and reasons their answers are reasonable.__ _Students will ask each other questions that press for reasoning, justification, and understanding. <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> _x_ Students will compare strategies and results in order to learn from each other’s work. <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> ___Students will use mistakes as a site for new learning.
 * **Sociomathematical Goals:**


 * Explanation for your choice(s):**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Considering that our lesson is more of an investigation to introduce the different vocabulary that is associated with linear systems of equations, students will be able to compare what they noticed about the different graphs and equations with a partner to see what similarities and differences they recognize. They could even work with each other to have a dialogue will completing the demonstration. || =<span style="background-color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 80%;">In this learning segment, both the oral and written assignments that involve vast knowledge of mathematical vocabulary may cause difficulties for learners who have not been previously exposed to this language. These students will need repetition of mathematical vocabulary terms and further explanations of many concepts in order to gain a better understanding of how the topics build towards the unit as a whole. Even more, while working with the interactive demonstrations, the students may find difficulties, especially since we will expect them to utilize complex academic language to describe the systems of equations and different solutions to the systems after working with these demonstrations. Therefore, in order to account for diverse learners, we will need to provide a variety of scaffolds and differentiate certain aspects of instruction to account for the high demands of the academic language. Even more, by emphasizing the use of mathematical vocabulary and incorporating multiple representations of the material (through Mathematica demonstrations, graphs, charts, etc.), we will aim to account for the diversity of the learners in our classroom. = ||
 * **How will you account for diverse learners?**