few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
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).

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


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

Example 3:
Concept/skill: 
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.

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 $P$.


Answer: 
Yes. 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.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Interpret expressions that represent a quantity in terms of its context.
problem_activity: Activity
The goal of this activity is for students to reason about what horizontal intercepts a graph of a polynomial must have based on the structure of the factored form of the polynomial (MP7). Students begin by considering a polynomial equation and graph and reasoning about why the graph window is insufficient since it only shows two out of three intercepts. Students then consider the horizontal intercepts of a different polynomial through the lens of selecting an appropriate viewing window before checking with graphing technology.
Launch
Provide access to devices that can run Desmos or other graphing technology.
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
To help students consider the graph window of a polynomial, start by only displaying Mai’s graph and comment. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving the terms “parabola” or “quadratic,” along with any questions about the scale on the axes. This will help students create questions about Mai’s statement and graph as they explore an appropriate graph window.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. Ask that each discussion group member is equally prepared to report back to the whole group. As students work, encourage groups to use peer supports to ensure all group members are prepared to report out if selected.
Supports accessibility for: Language; Social-emotional skills; Attention
Student Facing
Mai graphs the function
\(p\)
given by
\(p(x)=(x+1)(x-2)(x+15)\)
and sees this graph.
[IMG]
She says, “This graph looks like a parabola, so it must be a quadratic.”
Is Mai correct? Use graphing technology to check.
Explain how you could select a viewing window before graphing an expression like
\(p(x)\)
that would show the main features of a graph.
Using your explanation, what viewing window would you choose for graphing
\(f(x)=(x+1)(x-1)(x-2)(x-28)\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Select some different windows for graphing the function
\(q(x) = 23(x-53)(x-18)(x+111)\)
. What is challenging about graphing this function?
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
If students have trouble deciding what the constant term of
\(f(x)\)
is, remind them that it can be found quickly by multiplying the constant terms of each linear factor. Students can use the distributive property to understand why this is true.
Activity Synthesis
Select students to share how they selected a viewing window for
\(f(x)=(x+1)(x-1)(x-2)(x-28)\)
. In particular, ask students to state what structure in the factored form they focused on in order to select their window size.
If students question whether we can really be sure we’re seeing all the horizontal intercepts, tell them that in future lessons, they will learn more about what functions are doing to the far left and right of a graph and how we can be sure there are no other horizontal intercepts outside our viewing window. Later in this unit, they will see that all zeros of a function correspond to factors of the function.


aligns: yes