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: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
problem_activity: Activity
Students reason abstractly and quantitatively about temperatures over time graphed on coordinate axes (MP2). The goal of this activity is for students to use inequalities to describe the location of points on a coordinate grid in one direction. This activity also introduces the idea of vertical difference on the coordinate plane using a familiar context. Students may use previous strategies such as counting squares, but are not expected to explicitly add or subtract using negative numbers.
Launch
Arrange students in groups of 2. Allow students 3–4 minutes of quiet work time and 1–2 minutes to check results with their partner. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students about the sea levels and elevation number line representations.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
The coordinate plane shows the high and low temperatures in Nome, Alaska over a period of 8 days. The axis labeled
\(T\)
represents temperatures in degrees Fahrenheit. The axis labeled
\(d\)
represents the day.
[IMG]
What was the warmest high temperature?
Write an inequality to describe the high temperatures,
\(H\)
, over the 8-day period.
What was the coldest low temperature?
Write an inequality to describe the low temperatures,
\(L\)
, over the 8-day period.
On which day(s) did the
largest
difference between the high and low temperatures occur? Write down this difference.
On which day(s) did the
smallest
difference between the high and low temperatures occur? Write down this difference.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Before doing this problem, do the problem about taxicab distance in an earlier lesson.
The point
\((0,3)\)
is 4 taxicab units away from
\((\text{-}4,3)\)
and 4 taxicab units away from
\((2,1)\)
.
Find as many other points as you can that are 4 taxicab units away from
both
\((\text{-}4,3)\)
and
\((2,1)\)
.
Are there any points that are 3 taxicab units away from both points?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Because the high temperature on day 7 was positive and the low temperature was negative, some students may not notice that the difference was 10 degrees on this day as well. Consider prompting them to use tracing paper to compare the temperature differences on days 6 and 7.
Activity Synthesis
This discussion should lead to two key takeaways. First, students express the range of values for the low and high temperatures using inequalities. Second, students share strategies for finding a difference between two values on the coordinate plane.
Ask students to share their inequalities for
\(H\)
and
\(L\)
. It is expected that students have inequalities that describe the maximum high temperature for
\(H\)
and the minimum low temperature for
\(L\)
, but the discussion should bring out that each variable has 4 statements that capture its possible values:
\(L>\text-3\)
,
\(L=\text-3\)
,
\(L < 26\)
, and
\(L=26\)
for the variable
\(L\)
and
\(H>2\)
,
\(H=2\)
,
\(H < 28\)
, and
\(H=28\)
for the variable
\(H\)
.
Ask students to share their strategies for finding the vertical distance between points. Push them to explain how they took the scale of the vertical axis into account. Invite students to explain how they used the context to make sense of their answers.
Listening, Speaking: MLR8 Discussion Supports.
To support students in producing statements about details in the inequalities, provide sentence frames for students to use when they explain their reasoning to their peers, such as: “This inequality should/should not also be equal to because _____”, “A situation uses an equal sign when _____”, “I know that the inequality _____ is correct because _____.”
Design Principle(s): Support sense-making


aligns: yes