Generalizing and Representing a Quadratic Relationship
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 number of gray stones in a cobblestone pattern. They apply and then generalize Amir’s method using algebra.
Haleemah and ET explain what each symbol and operation of their general equation means in the cobblestone context.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Recognize a relationship between an independent variable and a dependent variable for a quadratic growth situation, generalize the relationship, and then express that generalization using an algebraic equation.
Explain what each symbol and number in a quadratic function means in terms of the quantities in the context.
Common Core Math Standards
CCSS.Math.Content.HSA.CED.A.1:Create equations and inequalities in one variable and use them to solve problems.Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
In this lesson, ET and Haleemah analyze a geometric context involving growing cobblestone patterns made from white and gray stones. First, they are asked to connect the Size Number of a growing pattern to the number of gray stones in the associated figure. Later they examine the work of a fictional student, Amir, who has created a method for determining how many gray stones are needed [Episode 2, 1:15]. Haleemah and ET connect expressions in Amir’s method to the diagram and use his method to find the total number of gray stones for different-sized figures [e.g., Episode 3]. This culminates in ET and Haleemah generalizing the method [Episode 4, 1:08], which results in the quadratic equation (x • x) + (x • x) + x + (x • x) = T, where x is the Size Number and T is the total number of gray stones.
CCSS.M.HSA.SSE.A.1.A.Interpret parts of an expression, such as terms, factors, and coefficients.
Having generalized the cobblestone pattern, ET and Haleemah work to interpret each part of expressions in their equation (x • x) + (x • x) + x + (x • x) = T. First, they reason quantitatively by replacing the variable with a specific value, corresponding to Size 11 [Episode 5, 2:53]. For example, they explain that 11 is both the number of gray stones in one column and the number of columns in two of the three sections of gray stones. They interpret 11 • 11 as the gray stones in each of two sections of the pattern, while simultaneously seeing the “+ 11” as the extra column of stones in the middle section. Later, in Episode 6, they reason more abstractly, by stating that x • x can be thought of as the number of gray stones in one column multiplied by the number of columns in a Size X figure. They also link the product itself to the total number of gray stones in one square section, and note that the “+ x” represents the extra column of gray stones in the middle section [2:57].
According to the CCSSM, “Mathematically proficient students look closely to discern a pattern or structure…They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.” This lesson features Haleemah and ET making sense of another student’s method for counting the number of gray stones in a cobblestone pattern. Rather than mimic Amir’s method, ET and Haleemah analyze the structure of the pattern to discern how Amir might have seen the figure to derive his method. For example, in Episode 1 they notice that in each of the figures, the columns contain a number of gray stones equal to the Size Number. Moreover, they notice that there are three sections of gray stones. For two of those sections, the number of columns is always equal to the Size Number. However, the middle column always had one additional column. ET and Haleemah connect that extra column to the addition of 3 and 9 in Amir’s method for finding the total number of gray stones for Size 3 and 9, respectively [Episode 2, 1:28 and 2:20]. By the end of the lesson, ET and Haleemah generalize this method, resulting in a quadratic equation that relates the Size Number to the total number of gray stones for that figure [Episode 4]. They then interpret each part of the equation in terms of the quantities in the context of the cobblestone pattern, simultaneously seeing the expression x • x as the process of multiplying the number of columns in a section of gray stones by the number of stones in one column and as the total number of gray stones in one section [Episode 6].
