Using Algebra to Express a Generalization (Method 2)
Haleemah and ET create a new method for finding the number of tiles in the border of a swimming pool. Then they apply their method to pools of different sizes and generalize their method. Finally, they write an algebraic equation to express their generalization and explain what each part of the equation means in the pool context.
ET and Haleemah make sense of their previous solution and think of a way to correct their answer, creating a new method for finding the number of tiles in the border of the pool.
Haleemah and ET generalize their method and write an algebraic equation that shows the relationship between a border with any number of tiles on one side and the total number of tiles in the border.
The students work to explain what each symbol in their equation means in the pool context and encounter a challenge.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Generalize a relationship between an independent variable and a dependent variable, and then express that generalization using an algebraic equation.
Justify algebraic equations by appealing to the quantities in and structure of a visual pattern.
Understand algebraic expressions from both process and product perspectives. For example, 2x can mean 2 times x number of tiles (in the pool context) or the amount of tiles in the left and right sides of the border together.
Explain what each symbol and number in an algebraic equation means in terms of the quantities in the context.
Common Core Math Standards
CCSS.M.HSF.IF.C.8.Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
In this lesson, Haleemah and ET return to the problem of expressing the number of tiles in the border of a square pool. Like before, they analyze pools with known side lengths and then generalize their method to count the tiles in the border of any pool with side length that is unknown. Instead of using their previous equation (x • 2) + (x – 2) • 2 = B, they develop the new equation (x • 4) – 4 = B.
CCSS.M.HSA.SSE.A.1. Interpret expressions that represent a quantity in terms of its context.
ET and Haleemah make sense of their new equation (x • 4) – 4 = B by linking the arithmetic operations they used in their equation to the diagram of the pool. Specifically, they work to develop the quantitative meaning of multiplication as “groups of” and connect subtraction to the need to account for double counting the corners when they multiplied.
According to the CCSSM, “Mathematically proficient students …look both for general methods and for shortcuts….As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.” This lesson builds upon the efforts in Lesson 1 to identify different ways to see a relationship between an independent variable (here, the number of tiles in one side of the border of a square pool) and a dependent variable (the number of tiles in the border), generalize the relationship (across pools of different sizes) and express the relationship using an algebraic equation. Haleemah and ET generalize their new method, opting to use variables to represent the number of tiles along one side of the pool and the total number of tiles in the border of the pool [Episode 3, 1:20]. The students also evaluate the reasonableness of their method by making frequent connections between their symbolic representation (their equation) and their visual representations (the pool diagram) [Episode 4, 0:50 and 2:11].
