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.


Thought: 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.


Thought: 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.


Thought: 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: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
problem_activity: Activity
The mathematical purpose of this activity is to introduce students to curves representing normal distributions. Students compare normal curves with different means and standard deviations to determine how those statistics affect the curve. In particular, ensure students understand that normal curves are entirely defined by the mean and standard deviation, that the mean defines the center of the distribution, and the standard deviation affects the spread of the distribution.
Student Facing
These curves represent
normal distributions
with different means and standard deviations. What do you notice?
mean = 10. standard deviation = 1
[IMG]
mean = 10. standard deviation = 2
[IMG]
mean = 8. standard deviation = 2
[IMG]
mean = 12. standard deviation = 1
[IMG]
mean = 10. standard deviation = 0.8
[IMG]
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The equation for the curve for a normal distribution is
\(f(x) = \frac{1}{\sigma \sqrt{2\pi}} \boldcdot e^{\frac{-(x-\mu)^2}{2\sigma^2}}\)
where
\(\sigma\)
is the standard deviation,
\(\mu\)
is the mean, and
\(e\)
is a particular number close to 2.718.
Notice the part of the function that is
\((x - \mu)\)
. Using your understanding of transformations of functions, how does changing
\(\mu\)
affect the graph of the function?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The important things for students to notice are:
The curves are entirely defined by the mean and standard deviation.
The curve is symmetric around the mean.
The standard deviation affects the width and height of the curve. A larger standard deviation is wider and shorter. A smaller standard deviation is thinner and taller.
Here are some questions for discussion:
“Why do you think the normal curve for a larger standard deviation is wider and shorter than one with a smaller standard deviation?” (We would expect a larger standard deviation to mean that data is more spread out, so it makes sense that is wider. Since the area must remain 1 under the curve, it has to get shorter to get wider.)
“What is the shape of the normal curve?” (Bell-shaped and symmetric)
“Where is the line of symmetry for each curve?” (
\(x = 10\)
,
\(x = 8\)
,
\(x = 12\)
, and
\(x = 10\)
, respectively)
Tell students,
“Although it is difficult to see in these graphs, the
\(x\)
-axis is a horizontal asymptote for normal curves. As you follow the curve to the right (or left), the curve gets closer and closer to the
\(x\)
-axis without ever actually reaching it.”
“Normal curves are bell-shaped and are most reliably used to model bell-shaped or approximately bell-shaped distributions.”
“The area under the normal curve (between the curve and the
\(x\)
-axis) is always 1. When the standard deviation increases, the curve spreads out to the right and left. To maintain the same area, though, the height of the peak must decrease.”


correct_option_index: 1

2