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# Algebra 1 Unit Descriptions & Learning Targets

Unit 3: Systems of Linear Equations

In the previous unit, it was determined that the graph of a linear equation is a line. Parallel and perpendicular lines will be revisited to demonstrate systems
of equations. Next, it is a natural to then investigate where two lines might intersect in the coordinate plane. A simultaneous system of linear equations is
then defined in order to answer this question. The solution set of a system of linear equations is defined as the set of ordered pairs that satisfy both
equations. This concept is connected to the intersection of two lines in the coordinate plane. Examples of the intersection of two linear functions are
discussed, including cases where the graphs of the linear functions are parallel lines (no solution), or the same line (infinitely-many solutions).

The equations used in a linear system often are written in standard form, which is defined as Ax + By = C. The substitution and linear combinations
methods are then explored as a way of solving systems of linear equations. Empty and infinite solution sets are discussed in the context of substitution and
linear combinations. These methods are also used to solve systems of linear equations in a real world context, and the solutions of these systems are
interpreted in that context.

As stated in the previous unit, a root of a function is a value of x that makes the function have a value of 0, or in other words, the value that makes y = 0.
A more geometric interpretation is that a root is the x-value of the intersection of the given function and the horizontal line y = 0. This concept is an example
of the solving part of system of linear equations. In other words, the solution to any equation in one variable can be viewed as the x-value of the intersection
of two functions (e.g. the solution to 2x -1= 5 - x is the x-value of the intersection of y = 2x -1 and y = 5 - x). A wide variety of examples of this
concept are explored.

 # Learning Targets Stand ard Textbook Active Practice 3A Explain the meaning of a system of linear equations and a solution to a system of linear equations 9.0 8.1 pg 358 3B Use knowledge of slopes to identify systems of parallel and perpendicular lines as well as write equations for lines that are parallel or perpendicular and explain 9.0 7.8 pg 338-341 # 1- 41 3C Explain the reason why not all systems of equations have the same number of solutions. 9.0 8.1 pg 359 # 21 - 23 3D Graph a system of linear equations and estimate the solution based on the graph. 9.0 8.1 pg 358 #15 – 20 3E Verify that an ordered pair is a solution to a system by substituting the x and y coordinate into each equation of the system. 9.0 8.1 pg 358 #1 - 8 3F Solve a system of equations by substituting one equation into the other and solving for x and y coordinate of the solution and explain the process 9.0 8.2 pg 365 #1 – 21 3G Solve a system of linear equations using linear combinations and explain the process of doing so. 9.0 8.3 pg 371 #1 - 30 3H Solve real world problems by setting up algebraic systems of equations and solving for the two unknowns. 15.0 8.2 pg 365 22 – 27 8.3 pg 371 31 – 36 8.4 – 8.6 all 3I I can justify my reasoning for choosing one method of solving a system of equations over another method, while still explaining how all methods could lead to the same solution. 9.0, 15.0 n/a Mini White Boards

Essential Standards:
9.0: Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve
a system of two linear inequalities in two variables and to sketch the solution sets.
15.0: Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

Supporting Standards:
6.0: Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear
inequality (e.g., they sketch the region defined by 2x + 6y < 4).
7.0: Students verify that a point lies on a line, given an equation on the line. Students are able to derive linear equations by using the point-slope formula.
8.0: Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a
line perpendicular to a given line that passes through a given point.
24.0: Students use and know simple aspects of a logical argument.
25.0: Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements.

Unit 4: Introduction to Quadratic Equations
Another important type of function studied in this course is a quadratic function. A quadratic function is a function defined by an equation of the form
y = ax2 + bx + c. This is called the standard form of a quadratic function. Function tables are used to build the basic shape of the graph of quadratic
functions and demonstrate that not all graphs are linear. Basic components of the graph are identified, including the vertex, axis of symmetry, and roots.
The difference between the number of roots of a linear function and a quadratic function is identified and the idea of “double root” and “no real roots” are
introduced. Simple transformations of the parent graph are shown to illustrate the relationship of the equation and the graph. The factored form of a
quadratic function is then introduced as y = a(x - p)(x - q) . It is observed that p and q are the roots of the function, and that the x-coordinate of the vertex
of a quadratic function lies halfway between the roots of the function. Given a quadratic function in factored form, the roots and vertex are found, and the
graph of the function is drawn. Once factored form of the quadratic is introduced, polynomial multiplication is demonstrated as a means to move from
factored form to standard form of the quadratic.

Essential Standards (CA):
21.0: Students graph quadratic functions and know that their roots are the x-intercepts.
10.0: Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these
techniques.

Supporting Standards:
16.0: Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information
23.0: Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
24.0: Students use and know simple aspects of a logical argument
25.0: Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements.

The concept of a solution to an equation in one variable is once again linked with the intersections of the graphs (e.g. the intersection of and is
the solution to the equation ). In particular, it is discussed that the solutions to represent the roots of the function . The quadratic formula is then introduced as a way of determining the solutions to a quadratic equation, or the roots of a quadratic
function. It is emphasized that the equation must be written in standard form to use the quadratic formula. The quadratic formula is used to solve a wide
variety of quadratic equations, including problems modeling the motion of a launched or falling object. The zero product property is introduced and used as a
means to find the solutions to quadratic equations that are in factored form. Using the found roots of a quadratic function, sketches of the graph of the
function are created, and the relationship among solutions, roots, and x-intercepts is communicated.

Essential Standards (CA):
14.0: Students solve a quadratic equation by factoring or completing the square.
19.0: Students know the quadratic formula and are familiar with its proof by completing the square
20.0: Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
23.0: Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

Supporting Standards:
21.0: Students graph quadratic functions and know that their roots are the x-intercepts.
22.0: Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in
zero, one, or two points.
24.0: Students use and know simple aspects of a logical argument.
25.0: Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements.