# Binomials Lesson 2 (Teachers)

### Effect of Increasing Length and Width on Area

Emily and Mauricio compare the areas of two home offices. One office was formed by increasing both the length and width of the other office. The students create a visual representation of the area of the new office that breaks the room into four parts. They write arithmetic equations that express different ways to find the area of the new office.

##### Episode 1: Making Sense

Mauricio and Emily make sense of floor plans that architects use to represent rooms in a house.

##### Episode 2: Exploring

The students make a drawing to find the area of a home office and explain its meaning.

##### Episode 3: Reflecting

Emily and Mauricio reflect on the meaning of multiplication when finding the area of a rectangular room. They show multiplication as “groups of” in their drawing.

##### Episode 4: Making Sense

Mauricio and Emily make sense of a situation in which both the length and width of a rectangular office are increased by 3 feet. They make a prediction about how these increases will affect the area of the new room.

##### Episode 5: Exploring

The students create a drawing to explain why increasing the length and width of a room increases the area so much more than they predicted in Episode 4.

##### Episode 6: Exploring

Emily and Mauricio create a drawing that splits the area of the new room (from Episodes 4 and 5) into four parts. They determine the area of each part.

##### Episode 7: Reflecting

Mauricio and Emily reflect on their drawing from Episode 6 and write equations to express different ways to find the area of the new room.

##### Episode 8: Repeating Your Reasoning

The students apply what they learned in the previous episodes to explore a new situation in which the length of a media room is increased by 3 meters and the width is increased by 2 meters. They make a drawing and write equations to find the area of the new media room in different ways.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Understand that when you increase the length and width of a rectangle (e.g., when you increase both the length and width of a 10 ft by 8 ft rectangular office by 3 ft to create a new office), then the area is increased by the sum of the following areas: 10 ft × 3 ft = 30 ft2; 8 ft × 3 ft = 24 ft2; and 3 ft × 3 ft = 9 ft2.
• Create a drawing that shows what happens to the area of rectangle when both its length and width are increased.
• Represent the area of the new rectangle through a 2-dimensional drawing that decomposes the new area into four parts, one of which is the area of the original rectangle.

Common Core Math Standards

• CCSS.MATH.CONTENT.HSN.Q.A.2Define appropriate quantities for the purpose of descriptive modeling.

Emily and Mauricio define many quantities in the modeling tasks in this lesson, which are set in the context of determining the area of rooms in a home using an architectural blueprint. These quantities include: the length and width of a given home office, the amount of increase in the length, the amount of increase in the width, the area of the original office, the area of the new office, and the length, width and area of three rectangular spaces that are created when the length and width of the original office are increased.
• CCSS.MATH.CONTENT.3.MD.C.7.DRecognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

In this lesson, Mauricio and Emily consider the effect of increasing both the length and width of a 10 ft by 8 ft rectangular office by 3 ft to create a new office. They predict that the area of the original office will be increase by 9 ft2 (3 ft × 3ft), when, in fact, that increase should be 63 ft2. The students create a drawing to explain why increasing the length and width of a room increases the area so much more than they predicted. Their drawing decomposes the area of the new office into four parts: the original office with an area of 80 ft2, the increase of 9ft2 that they predicted, and two other rectangles of area 30 ft2 and 24 ft2

Common Core Math Practices

CCSS.MATH.PRACTICE.MP4Model with mathematics.

According to the CCSSM, mathematically proficient students “can apply the mathematics they know to solve problems arising in everyday life…. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams…They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.” Throughout this lesson, Emily and Mauricio model with mathematics in the practical situation of finding the area of rooms in a house from architectural blueprints, and of determining the effect of changing the length and width on the area of the rooms. In Episode 1, they start by making sense of the floor plans that architects use to represent rooms in a house. Later, they tackle the challenge of creating a single drawing that shows both an original home office and a new office, which is formed by increasing the length and width of the original office [e.g., Episode 5, 3:11]. In Episode 6, they refine their drawing from Episode 5 to better analyze how the changes in length and width affect the area [1:30]. Finally, in Episode 7, Emily and Mauricio write equations to express different ways to find the area of the new office and carefully interpret each number and term in their equations in the context of architectural floor plans.