An expanded valence shell is where more than eight of an atom's electrons participate in covalent bonding. Eight of these electrons will always be valence shell electrons. The others will be electrons from another shell closer to the atom's nucleus. Generally not more than sixteen electrons per atom participate in covalent bonding. Finally, note that an expanded valence shell is also called an expanded octet

The most important note is to understand when an element can use an expanded octet based on its availability of electrons beyond the eight valence electrons. Memorizing the list of common molecules that use expanded octets is not important, and this is not the point that should be emphasized in teaching. 

 

Phosphorus

Phosphorus is one element that commonly participates in expanded octets. Here are the most common molecules that fit under this category:

Note the pattern here: in each case, phosphorus has a single bond to five of the same atom, and that other atom is always a halogen. The outside atoms all satisfy the Octet Rule, with three lone pairs and one bonding pair. Counting the electrons surrounding the phosphorus atom gives us a total of ten--eight valence shell electrons and two additional electrons. 

There is one other few polyatomic ion where this rule applies as well:

Phosphorus has five bonding pairs, making for a total of ten electrons involved in bonding that are attached to the phosphorus atom.

 

Sulfur

Sulfur is another element that commonly participates in expanded octets. The most relevant molecule is here:

Sulfur uses a total of twelve electrons in bonding here, in the form of six single bonds. The fluroine atoms all obey the Octet Rule, while Sulfur uses four additional electrons beyond those in its valence shell to bond with as many as six fluorine atoms. 

Of course, there are also polyatomic ions where this ability comes in handy:

Note that when an expanded octet is in an anion, one or more other atoms will have the improper number of lone pairs and bonding pairs. Two oxygen atoms in each of these ions happens to be that way. Selenium is in the same group on the periodic table as sulfur and thus follows the same pattern of expanded octets and number of bonding pairs:

The last ion here involves tellurium, another nonmetal that is directly below Selenium in the periodic table. 

 

Chlorine and the Other Halogens

There is a subset of chlorine's ions that use an expanded Octet. It is actually part of chlorine's oxyanion series:

The last ion here does not violate the Octet Rule; it is shown for the sake of helping to reveal a pattern. For each oxygen atom added after the first, chlorine uses two additional electrons to form its covalent bonds! Also note that in each case, only one oxygen atom has an improper number of lone pairs. 

Other Halogens can also form these oxyanions. As all of the halogen atoms need one more valence shell electron to obtain a full octet, their oxyanion structures look essentially the same as those of chlorine. Here are the relevant structures:

Using this Article's Contents in a Classroom

The main thing to remember here is that this content emphasizes the identification of patterns. In particular, the two emphasized are how bonding patterns occur between elements in the same group on the periodic table, and the number of electrons used by a central atom in bonding in oxyanions. I do not recommend asking students to memorize the twenty structures shown here, especially as many of them are very uncommon in high school chemistry classes beyond demonstrating expanded octets. 

That said, you can provide students an opportunity to identify patterns themselves and predict the structure of new molecules.

Example 1: Draw the perchlorate ion on the board, \(ClO_4^{-}\). Then remark that bromine and iodine are directly below fluorine on the periodic table and ask them to draw the perbromate and periodate ions' Lewis Structures, \(BrO_4^{-}\) and \(IO_4^{-}\), respectively. 

Thanks to http://www.wolframalpha.com/widgets/view.jsp?id=689aa5a01c216d8b16ed0250cebdc702 for the Lewis Structure diagrams.