Patio Problem: Sequences and Functions

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Objective

SWBAT determine an arithmetic sequence and linear function that model a situation.

Big Idea

Arithmetic sequences can be modeled by linear functions that both have the same common difference.

Launch

20 minutes

Either give students with a hard copy of arith_sequence_function_launch.doc or display it on the projector.  

To begin, allow students to ask any questions up front as a class or make any observations.  Do not answer any direct questions, only clarifying questions.  Let students do most of the heavy lifting by making sense of the problem from their own perspective.  Give students about 5 minutes to think about the problem and construct a plan for solving it (MP1).

NOTE: Some items like graph paper, unit blocks, algebra tiles, calculators, etc. may be useful to solve this problem.  Have a place in the classroom where these items are kept and let students know that they can use the materials whenever needed.  This will help students develop practice MP5 by needing to choose their own tools when they feel it would be appropriate.

While students are working, try to gather some information on the different ways students are thinking about the problem.  The five questions (a-e)  below the patio daigrams will help to scaffold student exploration.  These questions set a path towards increasingly sophisticated understanding of the question.

After about five minutes, let students begin to work in pairs on the problem.  In some cases, it will make sense to have students choose their partners.  You could also pair students up based on the way they seem to be thinking about the problem.  Students could also simply work with the student they are sitting next to.

Some students may not be able to answer questions d and e while working individually or in pairs. Encourage students to keep examining the diagrams and thinking about the number of white tiles. If they are not able to come up with an explicit rule you can help guide their thinking during the classroom discussion when students share out their results.  

Continue to monitor student work and make note of various solution methods.  The way that you sequence the solution methods when students share will be important.  You want to have the solution methods grow in complexity from one to the next.  You also want to continuously draw comparisons between the methods. 

Students should have the work for their partnership shown in an organized way.  In the next portion of the lesson, students will share their work using the document camera.  If you don't have access to a document camera you may want to have students do their work on chart paper so that they can explain it to the class (MP3).

Discussion

15 minutes

Students typically solve the patio problem in one of three ways (arith_sequence_function_methods.doc). Students who are able to get to part (e) they may have come up with a second way to represent the problem.  I encourage you to sequence the student's explanations from least complicated to most complicated.  Also continue to make note of the fact that each subsequent patio is two white blocks larger than the previous (recursive sequence).  Below is a sample description of three of the methods in the order I would have students present them.

Method 1

If possible, call on a student who did not get the explicit formula and have them share their thinking with the class.  Many students will notice that there are 3 white tiles on each side of the patio that remain constant.  This leads to an idea of +6.  There is also one white tile on the top and one on the bottom for each gray tile.  If the number of gray tiles is x, this means there would be x+x or 2x tiles.  Guide students to write this formula explicitly as f(x) = 2x + 6 (f(x) is the number of white tiles). 

Method 2

Students could look at the top and bottom of the patio and notice that the top row will always have 2 more than the number of gray tiles (x + 2).  The bottom row will also have two more than the number of gray tiles.  The top and bottom together will have 2(x + 2) white tiles.  This leaves just the two tiles that are next to the gray tiles on the right and left.  This number will be constant and so the function would be f(x) = 2(x+2)+2.

Method 3

This solution does not come up as often and could be more difficult to understand for some students.  This is why this method is reserved for last.  The height of the patio will always be 3 and the length of the patio will always be x + 2.  The total area of the patio will be 3(x+2).  This area includes the gray tiles so we need to subtract that value out so the final function would be 3(x+2)-x.

I want to reiterate that you want the students to be presenting these various solution methods. If all students in the class look at the problem the same way (which is unlikely) you can ask them to think about another solution method in order to complete part (e) of the problem.

Question the students to determine how much bigger each subsequent patio is from the one before it.  Guide them to the understanding that each patio is two tiles bigger than the one before.  Then have students show where in the picture this +2 would come from.  Explain to students that this is the common difference.  Ask student to think about how we would model this understanding on a graph.  Guide them toward the understanding that for each new patio +1 we would have to add two white tiles +2.   This will get students to think about the rate of change for this function.  You can also question students about the domain and range for this function.

Closure

5 minutes

During the class, the students most likely came up with three different expressions to reprsent the function.

f(x) = 2x + 6

f(x) = 2(x+2) + 2

f(x) = 3(x+2) - x

Have students show why these three methods all represent the same function.  Then have them verify that this function is correct by plugging in the patio number x and showing that the output value is the correct number of white tiles.  Lastly, have students show where the idea of +2 is represented in their function.  In other words, what is making the number of white tiles go up by two each time.

The last part of this question will be helping students to develop the concept of slope and rate of change.  The fact that this rate of change is constant is what makes this a linear function.