Modeling with Sequences

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SWBAT work flexibly with arithmetic and geometric sequences models and determine the best algebraic representation to work with.

Big Idea

Arithmetic and geometric sequences can be used to model real quantities and gain useful information.

Warm-Up Review

20 minutes

I begin the lesson by sending students a set of formative assessment questions on their calculators, Representing Sequences Algebraically.  The goal is to assess whether students need support in representing sequences algebraically before tomorrow's quiz.  

The assignment from the previous night will have been a challenge for students so I expect there to be several questions on these problems.  I circulate around the classroom checking homework with the homework rubric, and encourage students to check homework against the answers that are projected on the board and the answers of the students next to them.  

If they still have questions after checking in with each other, I ask that they write the number of any problems on the board that they would like to see explained.  I ask for student volunteers to explain and of the requested problems and make sure to do one or two myself as well.

Practice and Reinforce

20 minutes

Students work in groups of three to complete a sequence puzzle that reinforces what we have learned about the numeric and algebraic representations of sequences.  The goal of this activity is for students to practice matching up symbolic and numeric representations of arithmetic and geometric sequences. [MP2]  

I discuss why I use groups of three and the formative benefits of this activity in my sequence dominoes reflection video.  When this activity is used the first time, it will be necessary to print 6-10 sets of  sequence dominoes, one for each group.  Printing sets on different colors of card stock makes it easier to keep the sets of pieces together. Each group receives a set of 24 cards of the same color that have either a sequence formula on one side and a few terms of a sequence shown.  The students are told to match up the pieces, domino-style, so that they have a trail that starts at the card labeled "start" and ends at the one labeled "finish."  

When students have finished, I project an image of the completed puzzle on the board and ask groups to check their work.  We discuss briefly which of the pairings were most difficult [MP2 and MP3] and any strategies students had for making those easier.

Exploring Applications of Sequences

30 minutes

Thus far in our study of sequences we have focused primarily on mathematical applications of sequences without a real world context.  To introduce the idea that sequences can be used to model real world phenomena [MP4], students work in the same groups as the previous activity to complete the Sequence Word Problem Match Up.  This activity is designed to help students abstract a given situation and represent it symbolically [MP2] without getting distracted by substitutions or calculations.  

Like many of the other activities I use, the cards are printed out in advance on card stock so that class time is not wasted on cutting and so that I can reuse the cards from year to year.

The numbered cards in this set have modeling scenarios and the lettered cards have an explicit sequence formula.  The final product of each group is a completed chart (included at the bottom of the Sequence Word Problem Match Up) indicating the correct matching of the scenarios and equations.  I collect these small charts and look them over to inform the notes I give students next.

Guided Practice and Direct Instruction

25 minutes

After I have looked over the charts from the previous activity, I ask student to take notes on using sequences to solve applied problems. [MP4]  I remind them that scenarios in which the same quantity is added or subtracted each period is arithmetic and that scenarios in which a quantity is increasing or decreasing by a given percentage are generally geometric.  We work a few examples of each type together and then students begin work on Worksheet: Sequence Applications, which is a collection of application problems. Homework is then selected from this worksheet, which includes some review problems in addition to applications.  Students will have a quiz on sequences in the next class period.