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: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
problem_activity: Optional activity
This optional activity gives students another opportunity to use what they have learned about all operations to model and solve a problem in a baking context. Students need to make sense of the problem and persevere in solving it (MP1).
Students may approach the problem in different ways (by drawing diagrams, making computations, reasoning verbally, etc.). Students may also choose different operations to obtain the information they need. For instance, instead of dividing by a fraction, they may perform repeated subtraction. Notice the different methods students use and identify strategies or explanations that should be shared with the class.
Launch
Consider arranging students in new groups of 2. Give students 7–8 minutes of quiet work time and then 1–2 minutes to discuss their response with their partner. Ask students to be prepared to explain their reasoning.
Conversing, Writing: MLR5 Co-craft Questions.
Use this routine to increase students' awareness of the language of mathematical questions. Display the problem about baking cookies, but withhold the final section that includes the question. Give students 2–3 minutes to write down any mathematical questions they have about the situation presented, then to share their questions with a partner. Circulate and listen for questions that involve combining fractional amounts of ingredients. Reveal the actual question and ask students to compare their questions with the original.  Listen for and amplify any questions involving the relationships between the two quantities in this task (number of cups of flour and number of cups of butter).
Design Principle(s): Cultivate conversation; Support sense-making
Student Facing
Mai, Kiran, and Clare are baking cookies together. They need
\(\frac 34\)
cup of flour and
\(\frac 13\)
cup of butter to make a batch of cookies. They each brought the ingredients they had at home.
Mai brought 2 cups of flour and
\(\frac 14\)
cup of butter.
Kiran brought 1 cup of flour and
\(\frac 12\)
cup of butter.
Clare brought
\(1\frac 14\)
cups of flour and
\(\frac34\)
cup of butter.
If the students have plenty of the other ingredients they need (sugar, salt, baking soda, etc.), how many whole batches of cookies can they make? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Consider combining every 2–3 groups of 2 students and having students discuss their responses and methods in larger groups of 4–6.
If time permits, reconvene as a class to highlight a couple of strategies and reflect on the effectiveness and efficiency of students’ strategies. For example, if some students performed repeated addition instead of multiplying (or repeated subtraction instead of dividing), ask if repeated addition (or subtraction) is always as efficient as multiplication (or division), or under what circumstances one method might be preferred over the other.


aligns: no