standard_description: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
problem_activity: Activity
The pattern in this activity creates the Fibonacci sequence. One purpose of this task is for students to understand that sometimes it is straightforward to define a sequence recursively and difficult to determine a definition for the
\(n^{\text{th}}\)
term. This task also challenges students to expand their notion of recursive definitions since the current term in a Fibonacci sequence depends on the previous two terms rather than just the previous term.
This sequence was chosen to provide a contrast to the cake activity. With the cake, students had a situation whose domain does not reasonably extend above a certain value given physical constraints. Since this sequence is a mathematical construct, the domain has no upper bound restrictions on it.
Launch
Distribute 1 piece of graph paper to each student. Ask them to complete the first question and pause. Ensure that they are interpreting the description of the pattern correctly before they proceed with the rest of the activity.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
On graph paper, draw a square of side length 1. Draw another square of side length 1 that shares a side with the first square. Next, add a 2-by-2 square, with one side along the sides of
both
of the first two squares. Next, add a square with one side that goes along the sides of the previous two squares you created. Next, do it again.
Pause here for your teacher to check your work.
Write a sequence that lists the side lengths of the squares you drew.
Predict the next two terms in the sequence and draw the corresponding squares to check your predictions.
Describe how each square’s side length depends on previous side lengths.
Let
\(F(n)\)
be the side length of the
\(n\)
th
square. So
\(F(1)=1\)
and
\(F(2)=1\)
. Write a recursive definition for
\(F\)
.
Student Response
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Student Facing
Are you ready for more?
Is the Fibonacci sequence arithmetic, geometric, or neither? Explain how you know.
Look at quotients
\(\frac{F(1)}{F(0)}, \frac{F(2)}{F(1)}, \frac{F(3)}{F(2)}, \frac{F(4)}{F(3)}, \frac{F(5)}{F(4)}\)
. What do you notice about this sequence of numbers?
The 15th through 19th Fibonacci numbers are 610, 987, 1597, 2584, 4181. What do you notice about the quotients
\(\frac{F(16)}{F(15)}, \frac{F(17)}{F(16)}, \frac{F(18)}{F(17)}, \frac{F(19)}{F(18)}\)
?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may not realize that it is okay to use both
\(F(n-1)\)
and
\(F(n-2)\)
when defining a function recursively. Encourage these students to think about how the “previous side lengths” for term
\(F(n)\)
can be expressed using function notation.
Activity Synthesis
Display the sequence 1, 1, 2, 3, 5, 8, 13 and the recursive definition
\(F(1)=1, F(2)=1, \)
\(F(n)=F(n-1)+F(n-2)\)
,
\(n\ge3\)
for all to see and check their answer against. Either give students a minute to think about how to write a definition for the
\(n^{\text{th}}\)
term of the sequence, or just tell them that it is more complicated to derive than class time allows. An important modeling point to make is that the goal is to create a model that suits the needs of what you are trying to do. If we wanted to know the value of
\(F(100)\)
, then a spreadsheet and a recursive definition is likely going to be faster than working out a definition for the
\(n^{\text{th}}\)
term.
Ask students, "What is the domain of this function? Does it have any restrictions?" (The integers
\(n\)
where
\(n\ge1\)
or equivalent. We can always add another square onto the image, even if only in our imaginations once the squares get too big.) Select students to share their responses.
If time allows, tell students that we could define the Fibonacci sequence as
\(F(0)=1, F(1)=1, \)
\(F(n)=F(n-1)+F(n-2)\)
,
\(n\ge2\)
and we would still get the same list of numbers.


aligns: no