Essential Questions and Standards
1.How can you determine sequences are functions? How can you write formulas using sequences? MAFS.912.F-IF.1.3
2. How do you determine slope as a rate of change? MAFS.912.S-ID.3.7
3. How do you determine the slope and y intercept of a linear line? MAFS.912.S-ID.3.7
4. How do you find a solution to a system of equations when graphing? MAFS.912.A-REI.4.10
5. How do you determine a solution to a system of equations using the substitution method? MAFS.912.A-REI.3.5
6. How do you determine the solution to a system of equations using equivalent equations? MAFS.912.A-REI.3.6
7. How do you solve system of equations using the elimination method? MAFS.912.A-REI.3.5, MAFS.912.A-REI.3.6
8. How do you create solution sets when solving inequalities with two variables? MAFS.912.A-REI.4.12, MAFS.912.A-CED.1.3
9. How do you solve systems of linear inequalities? MAFS.912.A-CED. 1.3, MAFS.912.A-REI.4.12
MAFS.912.F-IF.1.3- Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
MAFS.912.F-IF.2.6- Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. MAFS.912.F-LE.1.2- Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input/output pairs (including reading these from a table). This section focuses on linear functions and arithmetic sequences.
MAFS.912.F-BF.1.1- Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context.
MAFS.912.A-CED.1.2- Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MAFS.912.A-CED.1.3- Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of
different foods.
MAFS.912.A-REI.3.5- 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.
MAFS.912.A-REI.3.6- Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
MAFS.912.A-REI.4.10- 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).
MAFS.912.A-REI.4.12- Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes.
MAFS.912.S-ID.3.7- Interpret the slope (rate of change) and the intercept (constant term) of a
linear model in the context of the data.
1.How can you determine sequences are functions? How can you write formulas using sequences? MAFS.912.F-IF.1.3
2. How do you determine slope as a rate of change? MAFS.912.S-ID.3.7
3. How do you determine the slope and y intercept of a linear line? MAFS.912.S-ID.3.7
4. How do you find a solution to a system of equations when graphing? MAFS.912.A-REI.4.10
5. How do you determine a solution to a system of equations using the substitution method? MAFS.912.A-REI.3.5
6. How do you determine the solution to a system of equations using equivalent equations? MAFS.912.A-REI.3.6
7. How do you solve system of equations using the elimination method? MAFS.912.A-REI.3.5, MAFS.912.A-REI.3.6
8. How do you create solution sets when solving inequalities with two variables? MAFS.912.A-REI.4.12, MAFS.912.A-CED.1.3
9. How do you solve systems of linear inequalities? MAFS.912.A-CED. 1.3, MAFS.912.A-REI.4.12
MAFS.912.F-IF.1.3- Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
MAFS.912.F-IF.2.6- Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. MAFS.912.F-LE.1.2- Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input/output pairs (including reading these from a table). This section focuses on linear functions and arithmetic sequences.
MAFS.912.F-BF.1.1- Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context.
MAFS.912.A-CED.1.2- Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MAFS.912.A-CED.1.3- Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of
different foods.
MAFS.912.A-REI.3.5- 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.
MAFS.912.A-REI.3.6- Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
MAFS.912.A-REI.4.10- 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).
MAFS.912.A-REI.4.12- Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes.
MAFS.912.S-ID.3.7- Interpret the slope (rate of change) and the intercept (constant term) of a
linear model in the context of the data.
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Lessons
Arithmetic SequencesArithmetic Sequences represent a linear pattern. A linear pattern shows a constant change between each term in the sequence. The common difference is the constant change between each term. In order to find the next term in the sequence, you can use the recursive formula. A(n-1) represents the previous term followed by the common difference. In order to find a future term in the sequence, you need to use the explicit formula. The explicit formula includes the first term in the sequence, the common difference, and the previous term number of the term we're looking for.
www.virtualnerd.com/algebra-1/relations-functions/arithmetic-sequence-definition.php |
Rate of ChangeRate of change represents a linear relationship of how something is changing over a period of time. In order to find the rate of change, you will need to find the change in y and divide it with the change in x (y - y)
(x - x). It does not matter which two points are chosen from a linear line, as long as you subtract the points in the same order. Rate of change is also known as slope. The slope of a line is measured by how steep the line is in both the positive direction and negative direction. www.virtualnerd.com/pre-algebra/linear-functions-graphing/rate-of-change-definition.php |
Slope and Y-InterceptIn order to find the slope of a line, you can find the change in y and divide it with the change in x. Or you can use rise
run. The rise is the change is y values between two points and the run is the change in x values between the same two points. The y-intercept is a constant value, or where the line crosses the y-axis. In order to graph the linear line from an equation, the equation must be written in slope intercept form (y = mx + b). Linear equations written in standard form (Ax + By = C) does not accurately show the slope and y-intercept. www.virtualnerd.com/pre-algebra/linear-functions-graphing/graph-line-given-slope-intercept.php |
Graphing Systems of Equations |
Equivalent Equations |
Finding a Solution to a System of Equations Using Elimination |
The solution to a system of equations is the point of intersection of two or more linear lines. When finding the solution to a system, you will either have one solution, no solution, or infinitely many solutions. In order to have no solution, that mean two or more linear lines will not cross at a point. This means the lines are parallel to each other. In order to have infinitely many solutions, the lines will intersect at all points, which means they are the same line. A linear line cannot bend to cross any other way, therefore the lines will overlap each other.
www.virtualnerd.com/algebra-2/linear-systems/equations-solution-by-graphing.php Finding a Solution to the System of Equations Using SubstitutionThe substitution method can be used when one equation is set equal to a variable. The expression that is equal to a variable is substituted back into the other equation in order to solve for another variable.
www.virtualnerd.com/algebra-2/linear-systems/equations-solution-by-substitution.php |
In order to create an equivalent equation, a linear line needs to be multiplied by another number. Each term in the equation, needs to be multiplied by that number for it to be equivalent. The solution will still make the equation true.
Solving Linear InequalitiesSolving a linear inequality is very similar to graphing a linear equation. However, a solution to the linear inequality has multiple solutions that are in the shaded region. Ordered pairs that are not in the shaded region are not solutions. Also, whenever the inequality symbol is > or < that means the solutions can be included on the solid line. When the symbol is > or < that means the solutions cannot be included on the dotted line of the graph. If the solutions are greater than y, the solutions lie above the line. If the solutions are less than y, the solutions lie below the line.
www.virtualnerd.com/pre-algebra/linear-functions-graphing/linear-inequality-definition.php |
The elimination method is another way you can find the solution to a system. The elimination method is used to eliminate a variable when either added or subtracted by another equation. Once a variable is eliminated, the other variable is used to solve the equation. Once you find what one variable is equal to, its substituted back into an equation to solve for the missing variable. Sometimes, you will need to create an equivalent equation in order to eliminate a variable.
www.virtualnerd.com/algebra-2/linear-systems/equations-solution-by-elimination-multiplication.php Solving a System of Linear InequalitiesSolving for the system of linear inequalities is much like graphing a single linear inequality. However, the solutions lie where the shaded regions overlap and will prove true for both inequalities. Above is an example of where the inequalities have an overlapping shaded region.
www.virtualnerd.com/algebra-2/linear-systems/inequality-solve-by-graphing.php |