Piet Mondrian is an artist famous for creating his masterpieces out of line art that utilized clean lines through rectangles. This activity will help us to create our own “Mondrian” by using our knowledge of factoring Quadratic trinomials through the use of Algebra tiles and area models.
This activity was created as a result of the ALEX Resource Development Summit.
In this video lesson, students revisit some situations that can be modeled with quadratic functions. They analyze and interpret given equations, write equations to represent relationships and constraints (MP4), and work to solve these equations. In doing so, students see that sometimes solutions to quadratic equations cannot be easily or precisely found by graphing or reasoning.
In this video lesson, students apply what they learned about transforming expressions into factored form to make sense of quadratic equations and persevere in solving them (MP1). They see that rearranging equations so that one side of the equal sign is 0, rewriting the expression in factored form, and then using the zero product property make it possible to solve equations that they previously could only solve by graphing. These steps also allow them to easily see—without graphing and without necessarily completing the solving process—the number of solutions that the equations have.
In this video lesson, students contrast visual patterns that show quadratic relationships with those that show linear and exponential relationships. To analyze the patterns, students generate tables of values, write expressions, and create graphs. They also encounter the term quadratic expression and learn that a quadratic relationship can be written using an expression with a squared term.
In this video lesson, students learn about the zero product property. They use it to reason about the solutions to quadratic equations that each have a quadratic expression in a factored form on one side and 0 on the other side. They see that when an expression is a product of two or more factors and that product is 0, one of the factors must be 0. Students make use of the structure of a quadratic expression in factored form and the zero product property to understand the connections between the numbers in the form and the x-intercepts of its graph (MP7).
In this video lesson, students begin to rewrite quadratic expressions from standard to factored form.
Students relate the numbers in the factored form to the coefficients of the terms in standard form, looking for structure that can be used to go in reverse—from standard form to factored form (MP7).
(This lesson only looks at expressions of the form (x + m)(x + n) and (x – m)(x – n) where m and n are positive.)
Earlier in this video series, students transformed quadratic expressions from standard form into factored form. There, the factored expressions are products of two sums, (x + m)(x + n), or two differences, (x – m)(x – n). Students continue that work in this video lesson, extending it to include expressions that can be rewritten as products of a sum and a difference, (x + m)(x – n).
Through repeated reasoning, students notice that when we apply the distributive property to multiply out a sum and a difference, the product has a negative constant term, but the linear term can be negative or positive (MP8). Students make use of the structure as they take this insight to transform quadratic expressions into factored form (MP7).
In this video lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form. Through repeated reasoning, students are able to generalize the equivalence of these two forms: (x + m)(x – m) and x2 – m2 (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other. Students also consider why a difference of two squares (such as x2 – 25) can be written in factored form, but a sum of two squares (such as x2 + 25) cannot be, even though both are quadratic expressions with no linear term.
This video lesson serves two main purposes: to reiterate that some solutions to quadratic equations are irrational, and to give students the tools to express those solutions exactly and succinctly. Students recall that the radical symbol (√) can be used to denote the positive square root of a number. Many quadratic equations have a positive and a negative solution, and up until this point, students have been writing them separately. Here, students are introduced to the plus-minus symbol (±) as a way to express both solutions. Students also briefly recall the meanings of rational and irrational numbers. They see that sometimes the solutions are expressions that involve a rational number and an irrational number—for example, x = ±√8 + 3. Students make sense of these solutions by finding their decimal approximations and by solving the equations by graphing. The work here gives students opportunities to reason quantitatively and abstractly (MP2).
Students relate the numbers in the factored form to the coefficients of the terms in standard form, looking for a structure that can be used to go in reverse—from standard form to factored form (MP7).
In this video lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form.
Through repeated reasoning, students are able to generalize the equivalence of these two forms: (x + m)(x – m) and x2 – m2 (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other.
Students also consider why a difference of two squares (such as x2 – 25) can be written in factored form, but a sum of two squares (such as x2 + 25) cannot be, even though both are quadratic expressions with no linear term.
This video lesson has two key aims. The first aim is to familiarize students with the structure of perfect-square expressions. Students analyze various examples of perfect squares. They apply the distributive property repeatedly to expand perfect-square expressions given in the factored form (MP8). The repeated reasoning allows them to generalize expressions of the form (x + n)2 as equivalent to x2 + 2nx + n2.
The second aim is to help students see that perfect squares can be handy for solving equations because we can find their square roots. Recognizing the structure of a perfect square equips students to look for features that are necessary to complete a square (MP7), which they will do in a future video lesson.
Previously in this video series, students used area diagrams to expand expressions of the form (x + p)(x + q) and generalized that the expanded expressions take the form of x2 + (p + q)x + pq. In this video lesson, they see that the same generalization can be applied when the factored expression contains a sum and a difference (when p or q is negative) or two differences (when both p and q are negative).
Students transition from thinking about rectangular diagrams concretely, in terms of area, to thinking about them more abstractly, as a way to organize the terms in each factor. They also learn to use the terms standard form and factored form. When classifying quadratic expressions by their form, students refine their language and thinking about quadratic expressions (MP6).
In this video lesson, students return to some quadratic functions they have seen. They write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward (MP5).
The work in this lesson—writing equations, solving them, and interpreting the solutions in context—encourages students to reason quantitatively and abstractly (MP2).
In this video lesson, students learn about the zero product property. They use it to reason about the solutions to quadratic equations that each have a quadratic expression in the factored form on one side and 0 on the other side. They see that when an expression is a product of two or more factors and that product is 0, one of the factors must be 0. Students make use of the structure of a quadratic expression in factored form and the zero product property to understand the connections between the numbers in the form and the x-intercepts of its graph (MP7).
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.
In Module 3, Topic D, students apply and reinforce the concepts of the module as they examine and compare exponential, piecewise, and step functions in a real-world context (F-IF.C.9). They create equations and functions to model situations (A-CED.A.1, F-BF.A.1, F-LE.A.2), rewrite exponential expressions to reveal and relate elements of an expression to the context of the problem (A-SSE.B.3c, F-LE.B.5), and examine the key features of graphs of functions, relating those features to the context of the problem (F-IF.B.4, F-IF.B.6).
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).
