Algebraic Expressions Lesson 1 (Teachers)

Using Algebra to Express a Generalization (Method 1)

Haleemah and ET create a 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. 

Episode 1: Making Sense

ET and Haleemah make sense of a swimming pool context and create a method for finding the number of tiles in the border of the pool.  

Episode 2: Repeating Your Reasoning

The students apply their method for finding the number of tiles in the border to swimming pools of different sizes.  

Episode 3: Exploring

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. 

Episode 4: Reflecting

The students work to explain what each symbol in their equation means in the pool context and encounter a challenge. 

Episode 5: Reflecting

ET and Haleemah resolve a struggle from the previous episode and connect each part of their algebraic equation to its meaning in the swimming pool context. 

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.BF.A.1. Write a function that describes a relationship between two quantities. 

    In this lesson, Haleemah and ET analyze square pools, each with a border made of tiles. They initially investigate pools with known side lengths before generalizing their method. They relate the total number of tiles in the border to the side length of the pool.
  • CCSS.M.HSA.SSE.A.1.B. Interpret complicated expressions by viewing one or more of their parts as a single entity.

    ET and Haleemah are asked to link expressions from their equation (x • 2) + (x – 2) • 2 = B (where x is the number of tiles on one side of the pool border and B is the total number of tiles in the border) to the diagram of the pool. Initially they only see x – 2 as the process of finding x (the number of tiles in one side of the border) and then removing each corner from that side to account for the “minus two.” Eventually, they are able to quantify the expression x – 2 as a single entity, namely the number of tiles along the part of the border that is between two adjacent corners.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP2Reason abstractly and quantitatively.

According to the CCSSM, “Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.” In this lesson, Haleemah and ET identify a visual pattern in the Pool Task, which allows them to determine the number of tiles in the border without counting each tile [Episode 1, 3:48]. As they apply their initial pattern to borders of different sizes, they begin to notice what varies and what stays the same in each case [Episode 2, 6:42]. From this generalization, ET and Haleemah create an equation with variables, which represents their reasoning abstractly. They are asked to link expressions from their equation back to their diagram [Episode 4, 1:40], which is an example of reasoning quantitatively. When this proves to be problematic, they re-contextualize the equation, using known side lengths to help them make sense of the expression x – 2 [Episode 5, 0:52].