Comparing Quadratic Relationships Using Visual Patterns
Haleemah and ET explore the context of cobblestone patterns from the Czech Republic. They make sense of another student, Amir’s, way of seeing the relationship between a Size Number and the Haleemah and ET are given another way of seeing a pattern in a cobblestone figure, this time starting with an algebraic equation. Given Nicole’s algebraic equation, the students will work to make sense of how she may have been seeing the relationship between the size number and the number of gray stones. Then, they will compare and explore the equality between Amir and Nicole’s methods.
ET and Haleemah make sense of the cobblestone context and think ET and Haleemah make sense of Nicole’s algebraic and arithmetic equations and create a drawing that they believe describes how she was seeing the pattern.
ET and Haleemah reflect on their generalized algebraic expression and make sense of each portion of their expression in the cobblestone context in two ways.
Haleemah and ET explore the equality between Amir and Nicole’s methods by discussing the similarities and resolving the differences between the two methods.
The students look at someone else’s reasoning about a key difference between Amir’s and Nicole’s methods. They use this to explain what parts of the algebra equations mean in the cobblestone pattern.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Make connections between drawings, verbal generalizations, and algebraic expressions and be able to reverse these connections (e.g., interpret an algebraic equation in terms of a visual pattern in a drawing).
Develop quantitative meaning for the distributive property of multiplication over addition as used in algebraic expressions; for example, understand why x(x + 1) = x2 + x and why x(x + 1) ≠ x2 + 1.
Common Core Math Standards
CCSS.M.HSA.SSE.B.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
In the previous lesson, ET and Haleemah examined, utilized, and explained features of an equation for the total number of gray stones in a growing cobblestone pattern. In this lesson, they work with a different equation, one created by a new fictional student named Nicole. First, they use specific examples of the cobblestone pattern (Size 3 and Size 9) to make sense of Nicole’s equation, connecting expressions in the equation to features of the visual pattern [Episode 1, 1:52; Episode 3]. They move beyond this empirical approach when comparing Nicole’s equation to Amir’s equation. In addition to noting that the two equations produce the same total given the same input [Episode 4, 1:30], they also analyze algebraic expressions across the two equations [Episode 4, 4:39] and reason about how those expressions are equivalent by linking them to the same quantities in the cobblestone context [Episode 4, 7:10].
CCSS.M.HSA.SSE.A.2. Use the structure of an expression to identify ways to rewrite it.
As part of this lesson, Haleemah and ET must make sense of a new equation for the cobblestone pattern context. Haleemah and ET claim that one of the expressions, x • (x + 1), is equivalent to the expression x • x + x. They initially justify this by examining a specific instance for x = 5 [Episode 4, 7:10]. Later, they reason more abstractly by examining the structure of the cobblestone pattern and linking it to the expressions. They also consider the thinking of another student, who claims (incorrectly) that x • (x + 1) = x2 + 1 [Episode 5]. Haleemah and ET note that, in the cobblestone context, x2 + 1 does not represent the number of stones in the middle section because it omits some of the stones in the extra column [Episode 5, 5:49]. In doing so, they provide another justification for why x • (x + 1) can be rewritten as x • x + x or x2 + x.
Common Core Math Practices
CCSS.MATH PRACTICE.MP3. Construct viable arguments and critique the reasoning of others.
According to the CCSSM, “Mathematically proficient students…can recognize and use counterexamples… justify their conclusions, communicate them to others, and respond to the arguments of others. They…distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.” In this lesson, Haleemah and ET construct many viable arguments. In the first part of this lesson, they support their claim that Nicole’s equation is valid for the cobblestone pattern, first by examining the equation for specific cases [e.g., Episode 1:44; Episode 2, 2:43], and later by explaining how each expression in the equation matches features of the cobblestone pattern [e.g., Episode 3, 0:52]. They also critique the reasoning of others. Their arguments about Nicole’s equation serve as analysis of Nicole’s assumed reasoning, with Haleemah and ET evaluating that reasoning as valid and accurate. Moreover, they critique the reasoning of another friend who demonstrates incorrect reasoning. For this friend, ET and Haleemah first offer a counterexample for why x • (x + 1) ≠ x2 + 1, showing that the two expressions are not equal for when x = 5 [Episode 5, 1:52]. They then reason more abstractly by linking the algebraic expressions directly to the visual features of the cobblestone pattern to show why the two expressions are unequal.
