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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
problem_activity: Activity
The purpose of this activity is for students to extend what they learned about square roots and
\(\frac12\)
exponents to cube roots and
\(\frac13\)
exponents.
Launch
Tell students that they will use the same reasoning as in the previous activity to make sense of numbers to the
\(\frac12\)
power or
\(\frac13\)
power. Give a few minutes of quiet work time followed by a brief, whole-class discussion.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check that students are finding the exact values and remind them of how they could use the exponent rules to check their answers if needed. Invite students to show their thinking by rewriting the fractional exponents.
Supports accessibility for: Memory; Organization
Student Facing
Use the exponent rules and your understanding of roots to find the exact value of:
\(25^{\frac12}\)
\(15^{\frac12}\)
\(8^{\frac13}\)
\(2^{\frac13}\)
Student Response
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Activity Synthesis
The key takeaway is that extending exponent rules to these fractions means that
\(b^{\frac12}=\sqrt{b}\)
and
\(b^{\frac13}=\sqrt[3]{b}\)
. Select students to share their responses for
\(15^{\frac12}\)
and
\(2^{\frac13}\)
.
Conclude the discussion by reminding students that these expressions which involve the
\(\sqrt{}\)
symbol are often called
radical
expressions because the technical name for the symbol is the
radix
(which is also where the word root comes from). Expressions like
\(\sqrt{15}\)
and
\(\sqrt[3]{2}\)
are examples of radical expressions.
Speaking: MLR8 Discussion Supports.
As students share their responses for
\(15^{\frac{1}{2}}\)
and
\(2^{\frac{1}{3}}\)
, press for details by asking how they know that
\(15^{\frac{1}{2}}\)
is equal to
\(\sqrt{15}\)
and
\(2^{\frac{1}{3}}\)
is equal to
\(\sqrt[3]{2}\)
. Also ask students which properties of exponents they used to show that the exponential and radical expressions are equivalent. Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. This will help students justify why
\(b^{\frac{1}{2}} = \sqrt{b}\)
and
\(b^{\frac{1}{3}} = \sqrt[3]{b}\)
.
Design Principle(s): Support sense-making; Optimize output (for justification)


aligns: no