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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. 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.

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


Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

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 $P$.


Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

Now, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: 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.
problem_activity: Activity
The purpose of this activity is to strengthen connections between zeros of a function (input(s) when the output is 0), the output when the input is 0, and the coordinates of the intercept(s) of a graph of the function.
Launch
To ensure that students understand the task before beginning, display one of the functions for all to see, and test an input value that is
not
a zero of the function. For example, evaluate
\(a(\text-5)=\text-5 - 5\)
. Is the output 0? No, the output is -10. So -5 is not an input that would give an output of 0. Draw students’ attention to the list of possible inputs, and tell them that the task is to find the input or inputs for each function that gives an output of 0.
Student Facing
Here are the definitions of some functions, followed by some possible inputs for the functions.
\(a(x)=x - 5\)
\(b(x)=x + 5\)
\(c(x)=x-3\)
\(d(x)=x+1\)
\(f(x)=3x - 6\)
\(g(x)=3x + 6\)
\(h(x)=(x+5)(x+3)\)
\(m(x)=(x+1)(x-3)\)
\(n(x)=(3x-6)(x-5)\)
Possible inputs: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5.
For each function, decide which input or inputs would give an output of 0.
Here are graphs of
\(b\)
,
\(f\)
, and
\(m\)
. Label each intercept with its coordinates, and be prepared to explain how you know.
[IMG]
Student Response
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Activity Synthesis
Display the correct responses to the first question and ask students to check their work. Invite a student to demonstrate how they know that
\(g(\text-2)=0\)
and
\(m(3)=0\)
(or choose a different function that any students struggled with). The important takeaway is not to look for shortcuts or patterns, but rather understand by evaluating the function why a given number is a zero of the function.
Invite students to share their responses and reasoning to the second question. If not mentioned, emphasize that:
The
\(y\)
-intercept of a graph corresponds to the output when the input is 0, so to find it, evaluate the function when
\(x\)
is 0. For example,
\(b(0)=5\)
, so the coordinates are
\((0,5)\)
of an intercept of the graph.
The
\(x\)
-intercepts correspond to the values found in the first question, the inputs that give an output of 0.
Any time you are dealing with an intercept, one of the coordinates must be 0. The coordinates of an
\(x\)
-intercept are
\((\text{something},0)\)
. The coordinates of a
\(y\)
-intercept are
\((0,\text{something})\)
.


aligns: no