# Binomials Lesson 1 (Teachers)

### Understanding Area

Emily and Mauricio recall the formula to find the area of a rectangle: Area equals length times width (A = L • W). Then they go beyond the formula to understand the meaning of area measurement units (such as a square inch or a square centimeter), the relationship between linear and area measurement units, what the 24 in 24 in2 represents, and the meaning of multiplication in the area formula.

##### Episode 1: Making Sense

Mauricio and Emily distinguish between area and perimeter in the context of a rectangular pizza, as the amount of pizza one gets to eat versus the distance around the pizza.

##### Episode 2: Exploring

Emily and Mauricio find the area of two differently-sized rectangular pizza slices in order to determine whether each slice has the same amount of pizza.

##### Episode 3: Reflecting

The students reflect on the meaning of 24 in2 as the area of a rectangular piece of pizza, by drawing 24 of something inside the pizza and by considering the meaning of an inch squared.

##### Episode 4: Exploring

Emily and Mauricio revise their drawing from Episode 3, this time showing the meaning of 24 in2 as 24 units where each unit is a square with side length of 1 inch.

##### Episode 5: Reflecting

Mauricio and Emily reflect on the meaning of two other area measurement units—cm2 and ft2

##### Episode 6: Exploring

The students create and use a drawing, rather than a formula, to find the area of a rectangle that is 7 inches long and 4.5 inches wide.

##### Episode 7: Repeating Your Reasoning

Emily and Mauricio apply their understanding of area to create four different shapes, each with an area of 8 cm2

##### Episode 8: Reflecting

Mauricio and Emily reflect on the meaning of multiplication in the formula for the area of a rectangle.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Go beyond their procedural knowledge of the area formula for a rectangle (i.e., A = L × W).
• Understand that area is a measurement of coverage in 2-dimensional space. It is not the measure of the boundary of a figure.
• Recognize that it does not make sense to report an area with a linear unit, e.g., 8 feet cannot be an area. You need to measure area with some 2-dimensional unit, something that will cover space.
• Understand the meaning of area units. For example, a square inch (in2), also called an inch squared, is the space covered by a square with sides of 1 inch.
• Understand the meaning of area measurements. For example, to say that the area of a figure is 8 cm2, means that that figure covers the same amount of space as 8 squares with sides of 1 cm.

Common Core Math Standards

• CCSS.Math.Content.HSG.MG.A.3Apply geometric methods to solve design problems.

Textbook area problems often display figures with dimensions and ask students to compute the area. In contrast, in this lesson the students create drawings to find areas, which reinforces the meaning of both the numeric and unit components of area measurements. They also tackle problems in which constraints are given, such as the desired area of a figure, and they draw multiple figures that fit the constraint.
• CCSS.Math.Content.3.MD.C.5A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.

Like many secondary school students, Emily and Mauricio remember and can apply the area formula for a rectangle, A = L x W. However, after determining, for example, that a 6 inch by 4 inch rectangle has an area of 24 in2, they were unsure what an in2 was or how to draw 24 of something inside their rectangle. In this lesson, they resolve both of those challenges and deepen their understanding of the meaning of area.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP6Attend to precision.

According to the CCSSM, “Mathematically proficient students try to communicate precisely to others…They are careful about specifying units of measure.” This lesson provides an opportunity for students to think about the meaning of the units used to measure length versus area and communicate about them precisely. In Episode 3, Mauricio and Emily work on understanding the difference between 24 inches and 24 square inches. While both use the unit “inches,” 24 inches measures length, and 24 square inches measures area [Episode 3, 7:44]. They also reconcile the difference between measuring with standard units (e.g., squares with side length of 1 inch versus rectangles with dimensions 2/3 inch by 1.5 inches; Episode 3, 9:30). In Episode 5, the students draw accurate representations of different area units—square inches, square centimeters, and square feet—using rulers. This hands-on activity helps them build greater understanding of units of measure for two-dimensional quantities.

In Episode 6, Emily and Mauricio discover that the area of a rectangle with dimensions 7 inches by 4.5 inches can be calculated as either 63 units or 31.5 units, depending on whether they use a measurement unit of 1 inch by 0.5 inch or an inch squared. This helps them better understand the necessity for precise and standard units of measure.