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: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
problem_activity: Optional activity
This activity is optional. It gives students an opportunity to reason about and to graph the rules of piecewise functions without a context.
Students are given the equations that define two piecewise functions, along with strips of paper, each containing a part of a graph and a portion of the horizontal axis (no scale is shown). Their job is to arrange the strips, apply a scale on each axis, and add open and closed circles to the graph to accurately represent the function values at each interval of input.
Unlike the piecewise functions students saw earlier in the lesson, each function here contains intervals defined by non-constant linear expressions. Support students as needed in reasoning about those intervals. If desired, this activity could be done over two class periods. (Students could piece together the graph of the first function in one class period and do the same for the second function at another time.)
To create an accurate graph, students need to make sense of the values, expressions, and inequality symbols in each equation and persevere in discerning their connections to the graph (MP1).
Here are images of the graphs on the blackline master for reference and planning. Cut each graph along the dashed lines and the vertical axis before giving it to students.
Function
\(f\)
[IMG]
Function
\(g\)
[IMG]
Launch
Arrange students in groups of 2–3. Give each group the strips for function
\(f\)
(pre-cut from the blackline master), glue or glue stick, and a sheet of paper on which to adhere the pieces.
Tell students that they are to arrange the pieces according to the rules that define
\(f\)
, as shown in the activity statement, and to add a label and a scale to each axis after the pieces are arranged. Explain that each piece contains a part of the horizontal axis, and the gridlines are 1 unit apart. Encourage students to double check their graph before gluing the pieces on another sheet of paper.
Consider discussing students' graphs of function
\(f\)
before giving students the pieces for function
\(g\)
and asking them to do another round.
Student Facing
Your teacher will give your group strips of paper with parts of a graph of a function. Gridlines are 1 unit apart.
Arrange the strips of paper to create a graph for each of the following functions.
\(\displaystyle f(x)=\begin{cases} \begin{align} \text-5,&\qquad \text-10< x< \text-5 \\ x,&\qquad \text-5\leq x< 0 \\ 1, &\qquad 0\leq x< 3\\ x-2, &\qquad 3\leq x< 8\\6, &\qquad 8\leq x < 1 0\ \end{align} \end{cases}\)
\(\displaystyle g(x)=\begin{cases} \begin{align} 5.5,&\qquad \text-10< x\leq \text-8 \\ 4,&\qquad \text-8< x\leq \text-3 \\ \text-x, &\qquad \text-3< x\leq 2\\ \text-3.5, &\qquad 2< x\leq 5 \\ x-5, &\qquad 5< x\leq 10 \end{align} \end{cases}\)
To accurately represent each function, be sure to include a scale on each axis and add open and closed circles on the graph where appropriate.
Student Response
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click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to display their completed graphs and their reasoning. In particular, discuss how students determined the appearance of the intervals defined by non-constant linear expressions such as
\(x\)
,
\(x-4\)
, or
\(\text-x\)
.
If different groups created different graphs for the same function, ask them to analyze one another's work and try to reach an agreement.
If time permits, ask students to identify the domain and range of each function.
Speaking, Representing: MLR8 Discussion Supports.
Make sure students can explain their visual display and how they reasoned about the rules for different intervals. If needed, give them additional time to prepare their explanation. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


correct_option_index: 1