few_shots: Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Thought: This task shows one quadratic equation with one variable, x.
Answer: B

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Standard options: 
A. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ³ 1.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
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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for free access to Student Response.
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
Teachers with a valid work email address can
click here to register or sign in
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.


correct_option_index: 0