few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
problem_activity: Optional activity
In this activity, students consider a doubling relationship where the exponent is a variable. Monitor for students who connect this activity to the lessons on exponents, or who recognize that the quantities in this relationship are changing with respect to each other in a different manner than previous examples they have seen. Have these students share during the discussion.
Digital
Launch
Give students 5-7 minutes of quiet work time, followed by a whole-class discussion.
Receptive/Expressive Language: Peer Tutors.
Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding.
Student Facing
A researcher who is studying mosquito populations collects the following data:
| day in                                   the study                                   \((d)\) | number of                                   mosquitoes                                   \((n)\) |
|----------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------|
|                                              1                                               |                                                2                                                 |
|                                              2                                               |                                                4                                                 |
|                                              3                                               |                                                8                                                 |
|                                              4                                               |                                                16                                                |
|                                              5                                               |                                                32                                                |
The researcher said that, for these five days, the number of mosquitoes,
\(n\)
, can be found with the equation
\(n = 2^d\)
where
\(d\)
is the day in the study. Explain why this equation matches the data.
Use the ordered pairs in the table to graph the relationship between number of mosquitoes and day in the study for these five days.
Describe the graph. Compare how the data, equation, and graph illustrate the relationship between the day in the study and the number of mosquitoes.
If the pattern continues, how many mosquitoes will there be on day 6?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Give students 5-7 minutes of quiet work time, followed by a whole-class discussion.
Student Facing
A researcher who is studying mosquito populations collects the following data:
| day in                           the study                           \((d)\) | number of                           mosquitoes                           \((n)\) |
|------------------------------------------------------------------------------|----------------------------------------------------------------------------------|
|                                      1                                       |                                        2                                         |
|                                      2                                       |                                        4                                         |
|                                      3                                       |                                        8                                         |
|                                      4                                       |                                        16                                        |
|                                      5                                       |                                        32                                        |
The researcher said that, for these five days, the number of mosquitoes,
\(n\)
, can be found with the equation
\(n = 2^d\)
where
\(d\)
is the day in the study. Explain why this equation matches the data.
Use the ordered pairs in the table to graph the relationship between number of mosquitoes and day in the study for these five days.
[IMG]
Describe the graph. Compare how the data, equation, and graph illustrate the relationship between the day in the study and the number of mosquitoes.
If the pattern continues, how many mosquitoes will there be on day 6?
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
The purpose of this discussion is to help students make connections between the table, graph, and equation that describe this situation. Ask students to share who connected this activity to the lessons on exponents, or who recognize that the quantities in this relationship are changing with respect to each other in a different manner than previous examples they have seen.
Consider asking some of these questions:
“Which of the data, equation, or graph gives you more of a feel for how the mosquito population changes with the day in the study?”
“Think back to the lesson on exponents with the dot pattern and the Genie activity. How do those situations compare with this mosquito situation? How would tables of data, graphs, and equations compare?”
“What are some similarities and differences you noticed between all the relationships you've seen in the last few lessons?”
Speaking, Listening: MLR8 Discussion Supports.
To support whole-class discussion, display sentence frames such as, “The dot pattern is similar to/ different than the mosquito situation because ________,” and “The genie situation is similar to/ different than the mosquito situation because ________.” This routine will support rich and inclusive discussion about the similarities and differences between all of the relationships students have seen in the last few lessons.
Design Principle(s): Optimize output (for comparison); Cultivate conversation


aligns: no