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 a 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 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).
Students will apply their critical thinking skills to learn about multiplication and division of exponents. This interactive exercise focuses on positive and negative exponents and combining exponents in an effort to help students recognize patterns and determine a rule.
This resource is part of the Math at the Core: Middle School collection.
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).
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.
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).
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).
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.