ALEX Resources

   View Standards     Standard(s): [MA2015] AL1 (9-12) 16 :

16 ) 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. [A-REI1]

[MA2015] AL1 (9-12) 17 :

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 21 :

21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

[MA2015] AL2 (9-12) 27 :

27 ) 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.* [A-REI11]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).

[MA2015] ALT (9-12) 21 :

21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

[MA2015] ALT (9-12) 27 :

27 ) 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.* [A-REI11]

[MA2019] ACC-7 (7) 24 :

24. 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). Limit to linear equations. [Algebra I with Probability, 19]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).

[MA2019] AL1-19 (9-12) 9 :

9. Select an appropriate method to solve a quadratic equation in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as x² = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

[MA2019] AL1-19 (9-12) 10 :

10. Select an appropriate method to solve a system of two linear equations in two variables.

a. Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.

b. Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.

[MA2019] AL1-19 (9-12) 12 :

12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.

[MA2019] AL1-19 (9-12) 14 :

14. Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane.
Note: The graph of a relation often forms a curve (which could be a line).

[MA2019] AL1-19 (9-12) 15 :

15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.

b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions.

[MA2019] AL1-19 (9-12) 19 :

19. 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).

a. Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate.
Note: Include cases where f(x) is a linear, quadratic, exponential, or absolute value function and g(x) is constant or linear.

[MA2019] AL1-19 (9-12) 20 :

20. 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, using technology where appropriate.

This video lesson builds on the idea that both graphing and rewriting quadratic equations in the form of expression = 0 are useful strategies for solving equations. It also reinforces the ties between the zeros of a function and the horizontal intercepts of its graph, which students began exploring in an earlier unit.

Here, students learn that they can solve equations by rearranging them into the form expression = 0, graphing the equation y = expression, and finding the horizontal intercepts. They also notice that dividing each side of a quadratic equation by a variable is not reliable because it eliminates one of the solutions. As students explain why certain maneuvers for solving quadratic equations are acceptable and others are not, students practice constructing logical arguments (MP3).

   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 12 :

12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.

[MA2019] AL1-19 (9-12) 23 :

23. Identify the effect on the graph of replacing f(x) by f(x)+k,k·f(x), f(k·x), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph, using technology as appropriate. Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.

[MA2019] AL1-19 (9-12) 29 :

29. 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. Limit to linear, quadratic, exponential, and absolute value functions.

[MA2019] AL1-19 (9-12) 30 :

30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.

   View Standards     Standard(s): [MA2015] AL1 (9-12) 5 :

5 ) Define appropriate quantities for the purpose of descriptive modeling. [N-Q2]

[MA2019] AL1-19 (9-12) 11 :

11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.

[MA2019] AL1-19 (9-12) 12 :

12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.

[MA2019] AL1-19 (9-12) 28 :

28. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and linear piecewise functions.

Module 5, Topic A focuses on the skills inherent in the modeling process: representing graphs, data sets, or verbal descriptions using explicit expressions (F-BF.A.1a) when presented in graphic form in Lesson 1, as data in Lesson 2, or as a verbal description of a contextual situation in Lesson 3. They recognize the function type associated with the problem (F-LE.A.1b, F-LE.A.1c) and match to or create 1- and 2-variable equations (A-CED.A.1, A-CED.2) to model a context presented graphically, as a data set, or as a description (F-LE.A.2). Function types include linear, quadratic, exponential, square root, cube root, absolute value, and other piecewise functions. Students interpret features of a graph in order to write an equation that can be used to model it and the function (F-IF.B.4, F-BF.A.1) and relate the domain to both representations (F-IF.B.5). This topic focuses on the skills needed to complete the modeling cycle and sometimes uses purely mathematical models, sometimes real-world contexts.