If students are not familiar with or have not recently practiced plotting points on the first quadrant using ordered pairs, review that concept before continuing.
Distribute the Coordinate Geometry sheet (M-5-3-3_Coordinate Geometry and KEY.docx) to each student. On a copy of the First Quadrant worksheet from Lesson 2 (M-5-3-2_First Quadrant and KEY.docx), have students plot the points in Figure 1 in order, connecting each point to the previous point and connecting the last point to the first to make a complete, closed shape. Ask students to come up with any words they can think of to describe Figure 1. They may only come up with “triangle.”
“There are a lot of names we can call this figure. Shapes have many descriptions, just as you might. You are a person, a boy or girl, maybe a brother or sister, a student, maybe a baseball player, right- or left-handed, and so on. Just as there are many ways to describe you, there are many ways to describe shapes. So, this shape is a triangle since it has three sides, but we can also call it a polygon.”
Depending on the class, you can break the word polygon down into two parts: poly- and -gon, and examine each part of the word, explaining that poly- means “many” and -gon means “angles,” and so the word polygon literally means “many angles.” This approach is useful when dealing with other terms like hexagon or octagon (and continues to be useful in higher mathematics when dealing with terms like polynomial).
After explaining that a polygon is a figure that has many sides, tell students the sides must be straight lines and the figure must be closed. In other words, they have to connect the last point they plotted back to the first point with 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, ask them to describe it. Students may respond with rectangle and 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?” Here, students should focus on the four right angles in the figure.
“How many sides does our rectangle have?” (Four) “Just like we have a general name for shapes with three sides—triangle—we also have a general name for shapes with four sides. We call them quadrilaterals.” Possibly write “quadrilaterals” on the board so students can see the term.
Again, depending on the class, breaking down the word quadrilateral into parts might be helpful: quad- means “four” and -lateral means “sides.” If students are familiar with football, they may have heard of a lateral pass, which is a pass that goes sideways (as opposed to backward or forward).
“So far, then, our shape has a few names. It is a polygon, it is a quadrilateral, and it is a rectangle. It actually has at least one more name. Look at the two long sides that go straight up and down. What word do we have for line segments that will never cross one another no matter how long they are?” (Parallel)
“And what about the two short sides on 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, write this word on the board so students can see it written out, pointing out the word parallel inside the word parallelogram. Have students write all the terms associated with a parallelogram next to the rectangle.
Have students graph Figure 3 on the same coordinate plane as Figures 1 and 2. Ask them to describe it. They should note that it’s a square, a polygon, a quadrilateral, and a parallelogram. If not, ask them if any of the previous terms that applied to rectangle also apply to it. Ask students to explain why the figure is a polygon, quadrilateral, and parallelogram. Lastly, ask them to explain how they know it’s a square. Here, students should focus on both the four right angles and the four sides of equal length.
“Now, you said it’s a square because it has four sides of equal length and four right angles. Since it has four right angles, can we also call it a rectangle?” (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 the terms that apply to the square.
Give each student a copy of Quadrilateral Venn diagram sheet (M-5-3-3_Quadrilateral Venn Diagram.docx). Describe how to interpret the diagram (i.e., all squares are rectangles, all rectangles are parallelograms, and all parallelograms are quadrilaterals). Make sure to emphasize that even though all squares are rectangles, for example, there are definitely rectangles (like the one they plotted) that are not squares. On the diagram, illustrate this by identifying the region that is inside the rectangle part of the figure but is outside the square part of the figure.
“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 portion but not the square portion, while 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 if we want to be able to organize all our polygons. This diagram just organizes quadrilaterals, which have how many sides?” (Four)
Before plotting Figure 4, ask students what shape they think it’s going to be. If they aren’t sure (they may be trying to visualize it in their heads), ask them how many points they have to plot. They may at least guess it will have five sides even if they aren’t sure what the figure is called. After discussion, have students 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, explaining the meaning of the prefix penta- (five) may be helpful to students. They may also be familiar with the Pentagon in Washington, D.C. (This image: http://www.sciencephoto.com/image/357691/350wm/T8350265-Pentagon_building-SPL.jpg shows the Pentagon from overhead so students can clearly see that it has five sides.) Have students label their pentagon appropriately. Also, explain that whether a polygon is a pentagon is determined only by the number of sides it has. Even though the pentagon in Figure 5 isn’t exactly the same as the Pentagon, they both have five sides and so are both classified 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, so the word literally means “six angles.” Have students label Figure 5 appropriately.
Finally, have students plot Figure 6. “How many sides does it have?” (Eight) “What do we call an eight-sided polygon?” If students don’t know, guide them toward the realization that an octopus has eight arms and the prefix oct- means eight, and have them guess what we might call a polygon with eight angles.
Have students label the octagon on their coordinate plane appropriately.
The Coordinate Plane worksheet can be collected at the end of class and checked against the key to ensure understanding. (Students may use it for reference in Activity 3.)
Have students work in pairs for Activity 3.
Each student should draw a pattern or design on a coordinate plane that incorporates at least two different polygons, at least one of which should be a quadrilateral.
Students should plot each part of their design and include coordinate instructions to provide to their partner. They should label each “set” of coordinates with the name or names of the appropriate polygon. (If they are drawing, for example, a square, they should label the set of coordinates describing the square with the terms square, rectangle, parallelogram, quadrilateral, and polygon.)
Once students have listed the coordinates and double-checked their work, they should give their instructions to someone else.
“Now, you have the instructions to make someone else’s design. Go ahead and start with the first point on the list and graph each set of points in order. Make sure that the shape you graph matches the name or names the instructions have listed. If you graph something and it’s not a square but the instructions say it is, for example, work with your partner to figure out if you made a mistake in graphing it, your partner made a mistake in writing down the coordinates, or if you both graphed it correctly and it just has the wrong description.”
Once students are finished, they should compare their drawings and identify the source of any errors and correct them.
This Activity can be repeated if students struggle with writing accurate instructions.
Depending on time, to engage students further, they can color and decorate their designs.
Use the following strategies to tailor the lesson to meet the needs of your students throughout the year.
- Routine: As students explore other geometry topics throughout the year, they can graph the shapes on the coordinate plane, including regular polygons, rhombuses, and even circles (with a designated point as the center and a given radius). They can also explore polygons with more than eight sides, describing them through the use of coordinates.
- Small Group: Using larger coordinate planes, students can work in groups to create elaborate designs, with each student responsible for creating the instructions (i.e., listing the coordinates) for part of the design. This activity can be done on large rolls of butcher paper (the coordinate plane can be drawn with a meterstick or yardstick) to create large murals.
- Expansion: When working with parallelograms, students can be encouraged to make shapes with parallel sides that are not horizontal or vertical lines. They can explore the idea of slope in the context 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,” etc. Students can also be introduced to the distance formula and/or Pythagorean theorem when working on the coordinate plane.
Students can also explore the idea of convex and concave polygons through graphing.