Understand solving equations as a process of reasoning and explain the reasoning.
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Equation solving
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2 - Explain (steps for equation solving)
5 - Construct (viable argument to justify solutions)
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2 - I can explain steps for equation solving.
5 - I can construct viable argument to justify solutions.
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A. Steps for solving equations are based on the properties of real numbers.
B. Sometimes one must justify a solution to a problem, or justify the method of solving.
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A.1 Are there specific steps in problem solving?
A. 2 How are properties of equality of real numbers used in equation solving steps?
B.1 How do I justify my answer?
B.2 Why do I have to justify my answer?
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Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Equation solving
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2 - Solve (simple rational equations)
2 - Solve (simple radical equations)
2 - Give examples (showing how extraneous solutions may occur)
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2 - I can solve simple rational equations.
2 - I can solve simple radical equations.
2 - I can give examples showing how extraneous solutions may occur in rational equations.
2 - I can give examples showing how extraneous solutions may occur in radical equations.
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A. One needs to check for the reasonableness of solutions to rational and radical equations, as some equations may have extraneous solutions.
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A.1 How can rational equations be solved?
A. 2 How can radical equations be solved?
A. 3 What are extraneous solutions?
A. 4 Why do extraneous solutions occur?
A. 5 How can I recognize when extraneous solutions may occur, and provide examples?
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Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Equations
Inequalities
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2 - Solve (linear equations in one variable)
2 - Solve (equations with coefficients represented by letters)
2 - Solve (inequalities in one variable)
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2 - I can solve linear equations in one variable.
2 - I can solve equations with coefficients represented by letters.
2 - I can solve inequalities in one variable.
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A. Solving equations and inequalities in one variable is a building block for solving equations and inequalities in two variables.
B. Equations with coefficients represented by letters are solved by a method similar to that of solving linear equations
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A.1 How can I solve linear equations in one variable?
A.2 How can I solve linear inequalities in one variable?
A.3 How can I solve equations with coefficients represented by letters?
A.4 Why do I need to be able to solve equations and inequalities in one variable?
B.1 How can equations with letter coefficients be solved?
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Solve quadratic equations in one variable.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Equations
Quadratic equations [a]
Quadratic equations [b]
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2 - Solve (quadratic equations in one variable)
3 - Use (completing the square [a])
3 - Transform (quadratic equation into [x-p]^2=q with same solutions [a])
6 - Derive (quadratic formula [a])
2 - Solve (quadratic equations by completing the square [b])
2 - Solve (quadratic equations by taking square roots [b])
2 - Solve (quadratic equations by quadratic formula [b])
2 - Solve (quadratic equations by factoring [b])
2 - Recognize (when quadratic formula gives complex solutions [b])
2 - Write (complex solutions in a+bi or a-bi form [b])
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2 - I can solve quadratic equations in one variable.
3 - I can use completing the square [a].
3 - I can transform a quadratic equation into [x-p]^2=q form with the same solutions [a].
6 - I can derive the quadratic formula [a].
2 - I can solve quadratic equations by completing the square [b].
2 - I can solve quadratic equations by taking square roots [b].
2 - I can solve quadratic equations by the quadratic formula [b].
2 - I can solve quadratic equations by factoring [b].
2 - I can recognize when the quadratic formula gives complex solutions [b].
2 - I can write complex solutions in a+bi or a-bi form [b].
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A. Quadratic equations are often found in real world problems.
B. Quadratic equations have several forms.
C. There are several methods by which a quadratic equation can be solved.
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A.1 What are the characteristics of a quadratic equation?
A.2 When will I see quadratic equations?
B.1 Does the form of the quadratic equation tell me which method to use for solving?
C.1 How do I solve quadratic equations?
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Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Equations
Inequalities
|
2 - Solve (linear equations in one variable)
2 - Solve (equations with coefficients represented by letters)
2 - Solve (inequalities in one variable)
|
2 - I can solve linear equations in one variable.
2 - I can solve equations with coefficients represented by letters.
2 - I can solve inequalities in one variable.
|
A. Solving equations and inequalities in one variable is a building block for solving equations and inequalities in two variables.
B. Equations with coefficients represented by letters are solved by a method similar to that of solving linear equations
|
A.1 How can I solve linear equations in one variable?
A. 2 How can I solve linear inequalities in one variable?
A. 3 How can I solve equations with coefficients represented by letters?
A. 4 Why do I need to be able to solve equations and inequalities in one variable?
B. 1 How can equations with letter coefficients be solved?
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Solve quadratic equations in one variable.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Equations
Quadratic equations [a]
Quadratic equations [b]
|
2 - Solve (quadratic equations in one variable)
3 - Use (completing the square [a])
3 - Transform (quadratic equation into [x-p]^2=q with same solutions [a])
6 - Derive (quadratic formula [a])
2 - Solve (quadratic equations by completing the square [b])
2 - Solve (quadratic equations by taking square roots [b])
2 - Solve (quadratic equations by quadratic formula [b])
2 - Solve (quadratic equations by factoring [b])
2 - Recognize (when quadratic formula gives complex solutions [b])
2 - Write (complex solutions in a+bi or a-bi form [b])
|
2 - I can solve quadratic equations in one variable.
3 - I can use completing the square [a].
3 - I can transform a quadratic equation into [x-p]^2=q form with the same solutions [a].
6 - I can derive the quadratic formula [a].
2 - I can solve quadratic equations by completing the square [b].
2 - I can solve quadratic equations by taking square roots [b].
2 - I can solve quadratic equations by the quadratic formula [b].
2 - I can solve quadratic equations by factoring [b].
2 - I can recognize when the quadratic formula gives complex solutions [b].
2 - I can write complex solutions in a+bi or a-bi form [b].
|
A. Quadratic equations are often found in real world problems.
B. Quadratic equations have several forms.
C. There are several methods by which a quadratic equation can be solved.
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A.1 What are the characteristics of a quadratic equation?
A.2 When will I see quadratic equations?
B.1 Does the form of the quadratic equation tell me which method to use for solving?
C.1 How do I solve quadratic equations?
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Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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3 - Prove (in a system of 2 equations in 2 variables, replacing one equation with sum of that equation and multiple of the other gives system with same solution)
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3 - I can prove in a system of 2 equations in 2 variables, that replacing one equation with sum of that equation and a multiple of the other produces a system with same solution.
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A. A system of equations can be replaced by the sum of one of the original equations and a multiple of the other equation, without changing the solution of the system.
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A. 1 Why can a system of equations be replaced by the sum of one of the original equations and a multiple of the other equation without changing the solution to the system?
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Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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2 - Solve (systems of linear equations in 2 variables exactly)
2 - Solve (systems of linear equations in 2 variables approximately)
2 - Solve (systems of linear equations in 2 variables by graphing)
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2 - I can solve systems of linear equations in 2 variables exactly.
2 - I can solve systems of linear equations in 2 variables approximately.
2 - I can solve systems of linear equations in 2 variables by graphing.
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A. Systems of equations can be solved graphically.
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A.1 How can I solve a system of equations?
A. 2 How do the graphs of systems with exact solutions differ from the graphs of systems with approximate solutions?
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Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
2 - Solve (simple system consisting of 1 linear and 1 quadratic equation in 2 variables algebraically)
2 - Solve (simple system consisting of 1 linear and 1 quadratic equation in 2 variables graphically)
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2 - I can solve simple system consisting of 1 linear and 1 quadratic equation in 2 variables algebraically.
2 - I can solve simple system consisting of 1 linear and 1 quadratic equation in 2 variables graphically.
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A. Systems of equations can be solved algebraically and graphically.
B. The solution to a system of equations is a set of ordered pairs.
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A.1 How do I solve a system of equations graphically?
A. 2 How do I solve a system of equations algebraically?
B.1 What does the solution to a system of equations look like?
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(+) Represent a system of linear equations as a single matrix equation in a vector variable.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
System of Equations
|
5 - Represent (system of linear equations as single matrix equation in a vector variable)
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5 - I can represent a system of linear equations as a single matrix equation in a vector variable.
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A. Systems of equations can be written as a matrix equation.
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A.1 What is a matrix equation?
A. 2 What is a vector variable?
A.3 How can a matrix equation be solved?
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(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
2 - Find (inverse of matrix)
3 - Use (inverse of matrix)
2 - Solve (systems of linear equations)
3 - Use (technology if matrix is 3x3 or larger)
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2 - I can find the inverse of matrix.
3 - I can use the inverse of matrix.
2 - I can solve systems of linear equations.
3 - I can use technology to find the inverse of a matrix if it is 3 X 3 or larger.
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A. Matrix equations are solved by finding and using the inverse of the matrix.
B. Some matrix equations do not have inverses.
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A.1 Why do I need to be able to find the inverse of the matrix?
A.2 How do I find the inverse of a matrix?
A. 3 How is the inverse used to find the solution to a system of equations?
B. 1 If a matrix equation does not have an inverse, how is the problem solving process affected?
|
Represent and solve equations and inequalities graphically.
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).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Equation
|
2 - Understand (graph of an equation in 2 variables is the set of all solutions plotted in the coordinate plane)
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2 - I can understand that the graph of an equation in 2 variables is the set of all of its solutions plotted in the coordinate plane, often forming a curve/line.
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A. The solutions to an equation in two variables, when plotted in the coordinate plane, form the graphical representation of the equation.
B. The graphical representation of an equation in two variables often forms a curve, including lines.
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A.1 How are the solutions of a two-variable equation found?
A. 2. What is the connection between the graph of an equation and its solutions?
B. 1 How do the solutions of an equation in two variables form a curve(or line) when plotted?
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Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
2 - Explain (why x-coordinates where two graphs intersect are solutions of the equations)
2 - Find (solutions approximately)
3 - Use (technology)
3 - Graph (functions)
3 - Make (tables of values)
2 - Find (successive approximations)
3 - Use (linear, polynomial, rational, absolute value, exponential, and logarithmic functions)
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2 - I can explain why x-coordinates where two graphs intersect are solutions of the equation.
2 - I can find solutions approximately.
3 - I can use technology.
3 - I can graph functions.
3 - I can make a tables of values.
2 - I can find successive approximations.
3 - I can use linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
|
A. For two graphs y = f(x) and y = g(x), the solutions of f(x) = g(x) are the x-coordinates of the points of intersection of the two graphs.
B. Solutions of y = f(x) = g(x) can be found using technology, a table of values, and successive approximations for a combination of linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
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A.1 Why is the point of intersection of two graphs important?
B. 2 How can I use technology to find approximate solutions of a system of two equations?
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Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Linear inequality
|
3 - Graph (solutions to a linear inequality in two variables as a half-plane)
3 - Graph (solution set to a system of linear inequalities in two variables as intersection of half-planes, including strict linear inequalities)
|
3 - I can graph solutions to a linear inequality in two variables as a half-plane.
3 - I can graph the solution set to a system of linear inequalities in two variables as the intersection of half-planes.
3 - I can graph the solution set of a strict system of linear inequalities, with an excluded boundary, as the intersection of half-planes.
|
A. The solution to a linear inequality in two variables is the set of points found in a half plane.
B. The solution to a system of linear inequalities is the set of points determined by the intersection of two half-planes.
C. The solution of a strict inequality excludes the boundary of the half-plane.
|
A.1 What does the solution to a linear inequality look like?
A. 2 What is a half-plane?
B.1 How would I graph the solution to a system of two inequalities?
C. 1 How is the graph of a strict inequality the same and/or different from other inequalities?
|
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