ALEX Classroom Resources

ALEX Classroom Resources  
   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 4 :
4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

Example: Interpret the accrued amount of investment P(1 + r)t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t.
[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) 6 :
6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
[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.
Subject: Mathematics (9 - 12)
Title: Completing the Square (Part 1): Algebra 1, Episode 20: Unit 7, Lesson 12 | Illustrative Math
URL: https://aptv.pbslearningmedia.org/resource/im20-math-ep20-712/completing-the-square-part-1/
Description:

Previously in this video series, students saw that a squared expression of the form (x + n)2 is equivalent to x2 + 2nx + n2. This means that, when written in standard form ax2 + bx + c (where a is 1), b is equal to 2n and c is equal to n2. Here, students begin to reason the other way around. They recognize that if ax2 + bx + c is a perfect square, then the value being squared to get c is half of b, or (b/2)2. Students use this insight to build perfect squares, which they then use to solve quadratic equations.

Students learn that if we rearrange and rewrite the expression on one side of a quadratic equation to be a perfect square, that is if we complete the square, we can find the solutions of the equation.



   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 4 :
4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

Example: Interpret the accrued amount of investment P(1 + r)t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t.
[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) 6 :
6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Subject: Mathematics (9 - 12)
Title: Algebra I Module 4, Topic A: Quadratic Expressions, Equations, Functions, and Their Connection to Rectangles
URL: https://www.engageny.org/resource/algebra-i-module-4-topic-overview
Description:

Module 4, Topic A introduces polynomial expressions. In Module 1, students learned the definition of a polynomial and how to add, subtract, and multiply polynomials. Here, their work with multiplication is extended and connected to factoring polynomial expressions and solving basic polynomial equations (A-APR.A.1, A-REI.D.11). They analyze, interpret, and use the structure of polynomial expressions to multiply and factor polynomial expressions (A-SSE.A.2). They understand factoring as the reverse process of multiplication. In this topic, students develop the factoring skills needed to solve quadratic equations and simple polynomial equations by using the zero-product property (A-SSE.B.3a). Students transform quadratic expressions from standard form, ax2 + bx + c, to factored form, f(x) = a(x - n)(x - m), and then solve equations involving those expressions. They identify the solutions of the equation as the zeros of the related function. Students apply symmetry to create and interpret graphs of quadratic functions (F-IF.B.4, F-IF.C.7a). They use the average rate of change on an interval to determine where the function is increasing or decreasing (F-IF.B.6). Using area models, students explore strategies for factoring more complicated quadratic expressions, including the product-sum method and rectangular arrays. They create one- and two-variable equations from tables, graphs, and contexts and use them to solve contextual problems represented by the quadratic function (A-CED.A.1, A-CED.A.2). Students then relate the domain and range for the function to its graph and the context (F-IF.B.5).



   View Standards     Standard(s): [MA2015] AL1 (9-12) 3 :
3 ) Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. [N-RN3]

[MA2019] AL1-19 (9-12) 4 :
4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

Example: Interpret the accrued amount of investment P(1 + r)t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t.
[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).
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Algebra I Module 4, Topic B: Using Different Forms for Quadratic Functions
URL: https://www.engageny.org/resource/algebra-i-module-4-topic-b-overview
Description:

Students apply their experiences from Module 4, Topic A as they transform quadratic functions from standard form to vertex form, (x) = a(x - h)2 + k in Topic B. The strategy known as completing the square is used to solve quadratic equations when the quadratic expression cannot be factored (A-SSE.B.3b). Students recognize that this form reveals specific features of quadratic functions and their graphs, namely the minimum or maximum of the function (i.e., the vertex of the graph) and the line of symmetry of the graph (A-APR.B.3, F-IF.B.4, F-IF.C.7a).  Students derive the quadratic formula by completing the square for a general quadratic equation in standard form, y = ax2 + bx + c, and use it to determine the nature and number of solutions for equations when y equals zero (A-SSE.A.2, A-REI.B.4). For quadratics with irrational roots, students use the quadratic formula and explore the properties of irrational numbers (N-RN.B.3). With the added technique of completing the square in their toolboxes, students come to see the structure of the equations in their various forms as useful for gaining insight into the features of the graphs of equations (A-SSE.B.3). Students study business applications of quadratic functions as they create quadratic equations and graphs from tables and contexts, and then use them to solve problems involving profit, loss, revenue, cost, etc. (A-CED.A.1, A-CED.A.2, F-IF.B.6, F-IF.C.8a). In addition to applications in business, students solve physics-based problems involving objects in motion.  In doing so, students also interpret expressions and parts of expressions in context and recognize when a single entity of an expression is dependent or independent of a given quantity (A-SSE.A.1).



   View Standards     Standard(s): [MA2019] AL1-19 (9-12) 4 :
4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

Example: Interpret the accrued amount of investment P(1 + r)t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t.
[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).
Subject: Mathematics (9 - 12)
Title: Algebra I Module 1, Topic B: The Structure of Expressions
URL: https://www.engageny.org/resource/algebra-i-module-1-topic-b-overview
Description:

In middle school, students applied the properties of operations to add, subtract, factor, and expand expressions (6.EE.3, 6.EE.4, 7.EE.1, 8.EE.1). Now, in Module 1, Topic B, students use the structure of expressions to define what it means for two algebraic expressions to be equivalent. In doing so, they discern that the commutative, associative, and distributive properties help link each of the expressions in the collection together, even if the expressions look very different themselves (A-SSE.2). They learn the definition of a polynomial expression and build fluency in identifying and generating polynomial expressions as well as adding, subtracting, and multiplying polynomial expressions (A-APR.1). The Mid-Module Assessment follows Topic B.



   View Standards     Standard(s): [MA2015] AL1 (9-12) 4 :
4 ) Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. [N-Q1]

[MA2019] AL1-19 (9-12) 4 :
4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

Example: Interpret the accrued amount of investment P(1 + r)t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t.
[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.
Subject: Mathematics (9 - 12), Mathematics (9 - 12)
Title: Algebra I Module 1, Topic D: Creating Equations to Solve Problems
URL: https://www.engageny.org/resource/algebra-i-module-1-topic-d-overview
Description:

In Topic D, students are formally introduced to the modeling cycle through problems that can be solved by creating equations and inequalities in one variable, systems of equations, and graphing (N-Q.1, A-SSE.1, A-CED.1, A-CED.2, A-REI.3). The End-of-Module Assessment follows Topic D.



ALEX Classroom Resources: 5

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