As a warm-up, students are challenged to a race: be the first find the sum of the first 100 positive integers. The first one to come up with the correct sum wins a small prize (like a Jolly Rancher, "Math Buck" or chance to shoot a free-throw).
When a winner has been declared, I ask the winner to demonstrate strategy. The story of 8-year-old Gauss adding the first 100 integers in under 10 seconds is pretty famous, and it is sometimes the case that students demonstrate the legedary method without prompting. If not, though, I tell students the story, which is outlined here at NRich Maths and also attached as a file here: Gauss Story. This is short, interesting story of a clever boy finding a shortcut by looking for and making use of structure [MP7]
Students work in table groups to complete a Series Exploration, a challenging lesson in that guides students to the partial sum formula for an arithmetic sequence. Based on Gauss's method presented in the warm-up, students generalize for sums of other arithmetic sequences. This requires them to to abstract a given situation and represent it symbolically. [MP2] Through trial and error they learn the technique does not apply to terms of a geometric sequence.
As students work, I circulate around the room taking note of what they struggle with and offering hints as necessary. I try not to be "too helpful," purposefully presenting the content as an exploration rather than direct instruction so that students develop the ability to make sense of a problem independently and persevere in solving it. [MP1]
When students have had 45 or 50 minutes to work Series Exploration, we come together as a group to summarize their findings. I ask each group to share out the formula they came up with by using a Quick Poll. Because the quick poll presents the results in a bar chart format, we can quickly see if more than one group came up with the same formula. Typically, students share one of three formulas below that can effectively be used to add terms of an arithmetic sequence.
However, occasionally students will come up with other expressions - some of which are not viable. We test each suggestion by using the formula to add the first 10 even numbers, which we verify by calculating the sum directly. [MP3] If different but viable formulas for an arithmetic series are presented, we take time to perform algebraic manipulations on the non-standard expression to show that it can be written as the standard formula that we will use for the rest of the unit - (Sn=(a1+an)/2 *n).
In the final 5 minutes, students complete Exit Ticket: Intro to Series which asks students to use an efficient method of adding terms of arithmetic sequences. I hope that they will turn to the arithmetic series formula at this point If many students are still listing all the numbers out and adding them up I will start the next day's lesson with a review of using the formula.