# Discovering the Area Formula for Parallelograms

Lesson 2 of 4

6th grade

45–60 minutes

**Description**

Students use prior knowledge of the area formula for rectangles and triangles to discover the formula for the area of parallelograms.

**Materials**

### Introduce

In partners or small groups, provide students with Pattern Blocks Activity Sheet (download from Materials section above) or pattern blocks of these shapes: square, **rhombus**, **parallelogram**, and **rectangle**. Teachers could also use Shape Tool for students to create the shapes on their own devices.

Have students compare the attributes of these shapes.

### Explore

To begin the lesson, have students look at a U.S. map. Ask students, "What state is in the approximate shape of a **parallelogram**?" Although not exactly, Tennessee is roughly a **parallelogram**. Review the **area** formula for** rectangles**, A = bh. Working in groups of three, students should discuss strategies that could be used to determine the **area **of Tennessee.

Distribute the Rectangles and Parallelograms Activity Sheets. Give students time to determine the area of shapes A-E.

For each **rectangle,** students can use multiple methods for finding **area**. Some methods can include counting the number of squares, multiplying the length by the width, or cutting shapes into multiple pieces and rearranging the pieces. For each **parallelogram**, students may need to count the squares to determine the **area; **they will need to combine partial squares to form full squares when making their estimates. Students should share their strategies with each other or as a whole class.

### Synthesize

In their partners or small groups, have students create a formula for determining the** area** of a **parallelogram**. Have them explain their reasoning and justify that their formula works. To promote discussion, you may need to ask questions about the relationship between the **area **of a **parallelogram** and the **area **of **rectangle**. Be sure to include a discussion about the **base** and **height** of the **parallelogram** and **rectangle**. These ideas will help students discover the formula for **area** of **parallelogram** on their own.

### Assessment (optional)

At the end of the lesson, return to the motivating problem: Students should now use measurements from the map to determine the **height** and** base**, and then they should use the formula they've discovered to find the **area**.

### Extension

**Activity 1 (Technology Option)**

Open the
Shape Tool using a tablet or computer. Students can create either a **rectangle** or **parallelogram,** make an appropriate cut, and then rearrange the pieces. Further, students could even make other cuts to show that two non-rectangular **parallelograms** with the same **base** and **height** have the same **area**, as shown below. (If the red piece were moved to the other side, notice that a different **parallelogram** would be formed.)

**Activity 2**

Students can create **parallelograms** by giving coordinates on the coordinate plane; another student can draw the **parallelogram** and determine its **area**. Students can use the distance formula to calculate the **base** and **height** if the **parallelogram **is not in a typical orientation.

**Activity 3**

Students can create **parallelograms** of varying sizes using geoboards. In pairs, one student can create a **parallelogram**, and the other student can determine its **area** using the **area** formula. To verify the result, students can use Pick's theorem (requires use of a geoboard or grid paper): I + (½P) - 1, where I is the number of points in the interior of the **polygon** and P is the number of points on the perimeter of the **polygon**.

Note that an online geoboard is also available to make this technology-based.

### Teacher Reflection

- What concepts did your students struggle with the most? How did they persevere?
- How did using technology change students’ level of engagement?
- What alternative methods did students use to calculate the
**area** of **parallelograms**? Will they always work? Did students clearly explain these methods?
- How did the questions you asked help lead students to making connections between finding
**areas** of **rectangles** and **areas **of **parallelograms**?

Leave your thoughts in the comments below.