Compare equivalent expressions for quadratic eunctions
• Understand that different ways of seeing a pattern give rise to different but equivalent expressions for the function arising from the pattern (A-SSE.B.3$^\star$).
• Reason through the equivalence of expressions (A-SSE.A.2).
This section explores equivalence of quadratic expressions. Visual patterns that are more complicated than in the previous section lead to different ways of expressing a quadratic function and prompt a discussion about the meaning of equivalence (equivalent expressions define the same function) and what purpose each of the equivalent forms might be useful for. Various equivalences can be explored in this section. Students draw on their previous knowledge of the distributive property to multiply binomials (the principle of multiplying “each by each”), including the special case of expanding the square in an expression given in vertex form. They also begin to think about how these processes might be reversed (factoring and completing the square).
WHAT: Students show that two expressions are equivalent in two ways, first by relating them to an arrangement of dots and a second by means of algebraic manipulation.
WHY: The purpose of this task is to draw attention explicitly to the reasoning about equivalent expressions that arose out of the previous activity (MP2). It can serve as a capstone to the sequence of visual pattern activities in this and the previous section, or it can serve as a formative assessment at this point in the unit.
WHAT: Students are asked for the value of the coefficients in a quadratic expression in standard form that will make it equivalent to a given expression in factored form.
WHY: This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression. This activity provides an opportunity to reinforce student knowledge of the distributive property. Using the distributive law twice to expand a product of binomials can be connected to multiplication of two digit numbers and mixed numbers in previous grades, emphasizing that this is not a new procedure but rather an extension of something students have already mastered (MP7). Working backwards through this procedure by factoring may be brought up here, but is not necessitated until later in this section.
WHAT: Students continue their work with visual patterns using Visual Patterns 101–106, 117, and the handout in the link. (The activity may be extended by also considering the number of edges.) The expressions for the $n$th pattern can be written in various forms depending on how the pattern is viewed. By the end of this activity, students should have created expressions for each pattern and worked through the equivalence of the expressions using the distributive property and other reasoning strategies (MP3).
WHY: The visual representation reinforces the fundamental principle that equivalent expressions have the same value for all values of the variables in them, because it is visibly evident that no matter how you express the number of squares that it is the same number. Students are motivated to convert between equivalent forms because they will come up with different forms as they see the visual pattern in different ways (MP7, MP8). The conversation leads naturally to the question of whether the various representations are the same and students will have to reason with their expressions and visual patterns to show that they are (MP2). It is important for students to reason through the equivalence step by step. The steps may include techniques like distributing products of binomials and factoring trinomials. Students do not learn a systematic approach to factoring until the next unit (A-SSE.A.2, A-SSE.B.3$^\star$).