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: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) ^( )Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
problem_activity: Narrative
The purpose of this activity is for students to transition from reasoning about division concretely or visually (using base-ten diagrams) to doing so more abstractly (by writing equations). It also reinforces the connections between multiplication and division.
Students make sense of three different strategies of dividing 78 by 3 and attend to the connections between the visual and numerical representations of the same quotient. As they do so, they practice reasoning quantitatively and abstractly (MP2).
During the synthesis, discuss how place value units play a role in all three strategies.
MLR8 Discussion Supports.
Synthesis: Display Lin, Priya, and Tyler’s strategies. As students share their observations, annotate the display to illustrate connections. For example, annotate where students see the divisor, dividend, and quotient on each diagram.
Advances: Listening, Representing
Launch
Groups of 2
“Take a couple minutes to make sense of Lin, Priya, and Tyler’s work.”
2 minutes: quiet think time
Activity
“Work with your partner to make sense of Lin, Priya, and Tyler’s work and to complete the activity.”
7–8 minutes: partner work time
Monitor for students who:
make connections between the equations and the base-ten diagram
recognize how Tyler’s equations and reasoning are different from Priya’s equations
Identify students who can explain these connections or distinctions to share during the synthesis.
Student Facing
Lin, Priya, and Tyler found the value of
\(78 \div 3\)
. Their work is shown. Make sense of each student’s work.
Lin
[IMG]
Priya
\(\begin{align} 3\times 10&= 30\\ 3\times 10&= 30\\ 3\times \phantom{0}6 &= 18 \\ \overline {\hspace{5mm}3 \times 26} &\overline {\hspace{1mm}=78 \phantom{000}} \end{align}\)
Tyler
\(\begin{align} 3\times 20&= 60\\ 3\times \phantom{0}6 &= 18 \\ \\20 + 6 &= 26 \end{align}\)
How are the three students’ work alike?
How are they different?
Student Response
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Activity Synthesis
Invite students to share how Lin, Priya, and Tyler’s work are alike and how they are different.
Select previously identified students to share additional connections that they noticed.
“Why does it make sense that Priya and Tyler wrote multiplication equations to find the value of a quotient?” (Multiplication and division can represent the same equal-groups situation. To divide 78 by 3 is to find how many are in each of 3 groups or how many groups of 3 there are. We can multiply up to 78 to find the answer.)
Consider asking: “What new ideas about dividing numbers did you learn and would like to try? Talk to a new partner about why you’d like to try them.”


aligns: yes