few_shots: Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Your response:
{
  "explanation": "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. Solving this problem does not require use of the Pythagorean identity.",
  "answer": "B"
}

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Your response:
{
  "explanation": "This task shows one quadratic equation with one variable, x.",
  "answer": "B"
}

Example 3:
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 $P$.


Standard options: 
A. 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.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Your response:
{
  "explanation": "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.",
  "answer": "A"
}

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Distinguish between situations that can be modeled with linear functions and with exponential functions.

For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
The purpose of this activity is to review how to create a scatter plot using technology and come up with a function to model the data. This work is similar to the work in the associated Algebra 1 lesson, but includes more explicit instructions and steps toward analyzing data and choosing an appropriate model (MP4).
Launch
Display the table of data for all to see, and ask students whether they think a linear or exponential model would be a better fit for this data. After a minute of quiet think time, ask a few students to share their reasoning.
If needed, demonstrate using graphing technology to input the given data and create a scatter plot.
If the graphing technology platform you are using allows it, consider showing students how to create sliders. In Desmos, simply type
\(y=mx+c\)
in the expressions list, and then click “all” to add sliders for both
\(m\)
and
\(c\)
. You may need to adjust the maximum value of one or both sliders to be able to use them to create a good model for this data.
Student Facing
In the spring, an agave plant sends up a flower spike. Here are some data collected from an agave plant in a garden in Tucson, AZ, starting on April 2:
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "day"}, {"_attributes": {"scope": "col"}, "_value": "height in inches"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {"_value": "17"}]}, {"td": [{"_value": "1"}, {"_value": "23"}]}, {"td": [{"_value": "2"}, {"_value": "29"}]}, {"td": [{"_value": "3"}, {"_value": "37"}]}, {"td": [{"_value": "4"}, {"_value": "45"}]}, {"td": [{"_value": "5"}, {"_value": "52"}]}, {"td": [{"_value": "6"}, {"_value": "62"}]}, {"td": [{"_value": "7"}, {"_value": "70"}]}, {"td": [{"_value": "8"}, {"_value": "80"}]}]}]}]}
Use graphing technology to create a scatter plot, using days as the first coordinate and height as the second coordinate.
Would a linear or exponential model be a better fit for this data?
Create a function that is a good model for the data. If you chose an exponential model, start with the equation
\(y=a \boldcdot b^x\)
and select values for
\(a\)
and
\(b\)
. If you chose a linear model, start with the equation
\(y=mx+c\)
, and select values for
\(m\)
and
\(c\)
.
Graph your equation on that same coordinate plane as your scatter plot. Adjust the numbers you used in the equation to improve your model.
Explain what each number in your equation means in this situation.
Use your model to predict the height of the flower spike on day 10.
Describe any limitations on the domain of the function modeling the data.
Student Response
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Activity Synthesis
Display the scatter plot using technology. Explore some of the different equations students used as a model. Discuss which model is the better fit, and what makes a model a good fit. Invite students to share their responses to the last three questions.


correct_option_index: 0

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