This lesson allows students to investigate different two-dimensional figures by plotting their vertices on the coordinate plane in the first quadrant. Students will: 
- graph polygons and discover their names. 
- investigate the relationship between different polygon types (quadrilaterals, parallelograms, rectangles, and squares). 

- How do spatial relationships, such as shape and dimension, help to create, construct, model, and portray real-world situations or solve problems? 
- How may geometric properties and theorems be utilized to describe, model, and analyze problems? 
- How may using geometric shape features help with mathematical reasoning and problem solving? 

- Coordinate Plane: Formed by the intersection of two number lines (called axes) that meet at right angles at their zero points. Used to locate points in the plane or in space by means of two numbers that represent the distance the point is from the horizontal axis and the vertical axis.
- Origin: The point at which the number lines of a coordinate plane intersect. As an ordered pair, the point (0,0).
- x-Axis: The horizontal number line of a coordinate plane. Used to show horizontal distance.
- x-Coordinate: The first number in an ordered pair, it designates the distance a point is along the horizontal axis.
- y-Axis: The vertical number line of a coordinate plane. Used to show vertical distance.
- y-Coordinate: The second number in an ordered pair, it designates the distance a point is along the vertical axis.

- at least two copies of the First Quadrant worksheet (M-5-3-2_First Quadrant and KEY) per student
- one copy of the Coordinate Geometry sheet (M-5-3-3_Coordinate Geometry and KEY) per student
- one copy of the Quadrilateral Venn Diagram (M-5-3-3_Quadrilateral Venn Diagram) per student

- In Activity 2, students check each other's work, which is also collected for assessment. 
- You can collect the designs in Activity 3 to validate the polygon coordinates and descriptions, as well as walk around during the activity to ensure that everything is working properly. 

Scaffolding, Active Engagement, and Modeling
W: The student will learn about the categories of polygons, including quadrilaterals, while also practicing using and becoming comfortable with the first quadrant of the coordinate plane. 
H: Students will be engaged by the concept that a quadrilateral might have multiple names or descriptors. The lesson begins by addressing the concept that each student can have a variety of descriptors and connecting it to mathematics. 
E: Students will plot points on the coordinate plane and connect them to create polygons. They are given blank coordinate planes and instructions to work on. (Most students will need to be encouraged to take their time and not plot all of the figures right away after receiving the instructions.) Students will learn about quadrilateral classifications using language, graphing, and a graphical organizer. 
R: The lesson assumes students have prior knowledge of triangles, squares, and rectangles. They will refine their classification of polygons by breaking down larger categories into smaller, more detailed classifications. They will be able to build and explain their own patterns, practicing their categorization and recognition skills (as well as point-plotting skills). 
E: Students will evaluate their work by collaborating with a partner to graph and describe the same figures. When their work does not match, the two students collaborate to identify and correct the problem. 
T: Use the Extension section to customize the lesson to match the needs of students. The Routine section includes suggestions for periodically reviewing lesson concepts with students throughout the school year. The Small Group section is designed to give additional learning opportunities and practice scenarios for students who may benefit from them. In addition, the Expansion section contains suggestions for challenging students who are willing to go above and beyond the standard standards. 
O: The class begins with a teacher-led discussion on terminology, followed by students charting and studying figures independently (with your supervision). The student is immediately engaged when plotting simple figures that they are likely familiar with. As students gain confidence in plotting polygons and identifying them by name, they are encouraged to do more on their own (or with a partner), but in an organized manner. Finally, students are free to build and explain their own designs that incorporate geometric figures. 

Activity 1

If students are unfamiliar with or have not recently practiced plotting points in the first quadrant using ordered pairs, they should review the subject before proceeding. 

Hand out the Coordinate Geometry sheet (M-5-3-3_Coordinate Geometry with KEY) to each student. On a copy of the First Quadrant worksheet from Lesson 2 (M-5-3-2_First Quadrant and KEY), students should layout the points in Figure 1 in order, linking each point to the previous point and the last point to the first to form a complete, closed shape. Ask students to come up with any words they can think of to describe Figure 1. They might simply come up with "triangle."

"There are numerous names for this figure. Shapes, like you, can be described in many different ways. You are a person, whether a boy or a girl, a brother or sister, a student, a baseball player, right- or left-handed, etc. Shapes, like people, can be described in a variety of ways. So, this shape is a triangle since it has three sides, but we may also refer to it as a polygon."

Depending on the class, you can separate the term polygon into two parts: poly- and -gon, and explain that poly- means "many" and -gon means "angles," therefore the word polygon literally means "many angles." This method is useful when dealing with other concepts such as hexagon or octagon (and remains useful in higher mathematics when dealing with terms such as polynomial). 

After explaining that a polygon is a form with several sides, instruct students that the sides must be straight lines and the figure must be closed. In other words, they must connect the last point they plotted to the starting point using a straight line.

"Next to the triangle you graphed, write the words polygon and triangle, and then graph Figure 2 on the same coordinate plane on which you graphed Figure 1." 

After students have plotted Figure 2, have them describe it. Students may respond using rectangle or polygon. (or incorrect answers). 

"What makes this shape a polygon?" (It has many sides, the sides are straight, and the figure is closed.) 

"What makes this shape a rectangle?" Students should focus on the figure's four right angles.

"How many sides does our rectangle have?" (Four) "Just as we have a general term for shapes with three sides (triangle), we also have a general name for shapes with four sides. We call them quadrilaterals." Consider writing "quadrilaterals" on the board so that students can see the phrase.

Again, depending on the lesson, dividing the term quadrilateral into sections may be useful: quad- means "four" and -lateral means "sides." Students who are familiar with football may have heard of a lateral pass, which is a pass that goes sideways (as opposed to backward or forward). 

"So far, our shape has several names. It's a polygon, a quadrilateral, and a rectangle. It actually has at least one other name. Look at the two long sides that run straight up and down. What do we call line segments that will never cross one another, no matter how long they are?" (Parallel)

"How about the two short sides at the top and bottom of our rectangle?" (They are also parallel.)

"Because our quadrilateral has two pairs of parallel sides, we call it a parallelogram." Again, put this word on the board so that students may see it, highlighting the word parallel within the word parallelogram. Ask students to write all the terms connected with a parallelogram next to the rectangle.

Have students graph Figure 3 using the same coordinate plane as Figures 1 and 2. Ask them to describe it. They should recognize that it is a square, a polygon, a quadrilateral, and a parallelogram. If not, ask if any of the terms used to describe the rectangle also apply to it. Ask students to describe why the figure is a polygon, quadrilateral, or parallelogram. Finally, have them explain how they know it's a square. Here, students should concentrate on both the four right angles and the four sides of equal length.

"Now, you stated that it is a square because it has four sides of equal length and four right angles. Can we call it a rectangle since it has four right angles?" (Yes) "If I ask you to draw a square, can you ever draw one that doesn't have four right angles?" (No) "So we know that every square is a rectangle." 

Have students write down all of the terms that relate to the square.

Give each student a copy of the Quadrilateral Venn Diagram sheet (M-5-3-3_Quadrilateral Venn Diagram). Describe how to interpret the diagram. Make it clear that, while all squares are rectangles, there are certain rectangles (such as the one they plotted) that are not squares. On the diagram, illustrate this by identifying the region that is inside the rectangle but not inside the square. 

"Write the words “Figure 2” and “Figure 3” on your diagram to show in which part of the diagram they belong." (Figure 2 belongs in the rectangle but not the square area, but Figure 3 belongs in the square portion.)

"Where does Figure 1 go on the diagram?" (Students may respond with Outside the quadrilaterals or Not on the diagram.) "We might need another diagram to manage all of our polygons. This graphic simply organizes quadrilaterals, which have how many sides?" (Four)

Activity 2

Before plotting Figure 4, ask students what shape they think it to be. If they are unsure (they may be visualizing it in their heads), ask how many points they need to plot. Even if they don't know what the figure is called, they can probably infer that it has five sides. After the discussion, ask students to plot Figure 4 on the second coordinate plane.

"Do any of the words we talked about with the figures on the first coordinate plane apply to this figure?" (Polygon) 

"We call a five-sided polygon a pentagon." Again, discussing the meaning of the prefix penta- (five) may be useful for students. They may also be familiar with the Pentagon in Washington, DC. (This image: http://www.sciencephoto.com/image/357691/350wm/T8350265-Pentagon_building-SPL.jpg shows the Pentagon from above, allowing students to clearly distinguish its five sides.) Have students label their pentagons correctly. Also, explain that the number of sides of a polygon determines whether it is a pentagon. Even though the pentagon in Figure 5 is not identical to the Pentagon, they both have five sides and are so categorized as pentagons.

"How many sides will Figure 5 have, based on the number of points that need to be graphed?" (Six) 

Have students plot Figure 5. 

"What is our six-sided figure called?" Write the word hexagon on the board and explain that hex- means six, thus the word literally means "six angles." Ask students to label Figure 5 appropriately. 

Finally, ask students to plot Figure 6. "How many sides does it have?" (Eight) "What do we call an eight-sided polygon?" Help students understand that an octopus has eight arms and the prefix oct- means eight. Then, ask them to identify a polygon with eight angles.

Ask students to identify the octagon on their coordinate plane correctly.

The Coordinate Plane worksheet can be collected at the end of class and compared to the key to ensure comprehension. (Students may use it as a reference in Activity 3.)

Activity 3

Have students work in pairs for Activity 3. 

Each student should create a pattern or design on a coordinate plane using at least two different polygons, one of which must be a quadrilateral. 

Students should plot each part of their design and provide coordinate instructions to their partner. They should label each "set" of coordinates with the name(s) of the corresponding polygon. (For example, if they are drawing a square, they should name the set of coordinates that describe it with the terms square, rectangle, parallelogram, quadrilateral, and polygon.)

After listing the coordinates and double-checking their work, students should give their instructions to someone else. 

"You now have the instructions to create someone else's design. Begin with the first point in the list and graph each set of points sequentially. Make sure the shape you graph matches the name(s) specified in the instructions. If you graph something and it is not a square but the instructions say it is, for example, work with your partner to determine whether you made a mistake in graphing it, your partner made a mistake in writing down the coordinates, or you both graphed it correctly but the description is incorrect."

This Activity can be repeated if students have difficulty writing accurate instructions. 

Students can color and paint their designs to better interest them depending on time.

Extension:

Use the strategies listed below to adjust the lesson to your students' needs throughout the year.

Routine: As students go through other geometry lessons during the year, they can graph shapes on the coordinate plane, such as regular polygons, rhombuses, and even circles. They can also examine polygons with more than eight sides and describe them using coordinates.

Small Group: Using bigger coordinate planes, students can work in groups to create intricate designs, with each student in charge of writing the instructions (i.e., listing the coordinates) for part of the design. To create big murals, conduct this project on large rolls of butcher paper (draw the coordinate plane with a meterstick or yardstick).

Expansion: When working with parallelograms, students may be encouraged to create shapes with parallel sides that are not horizontal or vertical lines. They can investigate the concept of slope in terms of "from this point I went to the right 5 units and up 2 units, so from this other point I have to do the same steps," and so on. When working on the coordinate plane, students can learn about the distance formula and/or the Pythagorean theorem. 
Graphing allows students to study the concept of convex and concave polygons.