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

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.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

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.
problem_activity: Optional activity
This task is provided for optional, additional practice for understanding variability in context.
The mathematical purpose of this activity is to compare data sets, and to interpret the measures of center and measures of variability in the context of the problem.
Launch
Keep students in the same groups. Give students 5 minutes to work on the questions and follow with a whole-class discussion.
Action and Expression: Develop Expression and Communication.
To help get students started, display sentence frames such as “In this situation, a larger/smaller value for the standard deviation means _____.” or “Both dot plots are different because….”
Supports accessibility for: Language; Organization
Student Facing
For each situation, you are given two graphs of data, a measure of center for each, and a measure of variability for each.
Interpret the measure of center in terms of the situation.
Interpret the measure of variability in terms of the situation.
Compare the two data sets.
The heights of the 40 trees in each of two forests are collected.
mean: 44.8 feet, standard deviation: 4.72 feet
[IMG]
mean: 56.03 feet, standard deviation: 7.87 feet
[IMG]
The number of minutes it takes Lin and Noah to finish their tests in German class is collected for the year.
mean: 29.48 minutes, standard deviation: 5.44 minutes
[IMG]
mean: 28.44 minutes, standard deviation: 7.40 minutes
[IMG]
The number of raisins in a cereal with a name brand and the generic version of the same cereal are collected for several boxes.
mean: 289.1 raisins, standard deviation: 19.8 raisins
[IMG]
mean: 249.17 raisins, standard deviation: 26.35 raisins
[IMG]
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?
One use of standard deviation is it gives a natural scale as to how far above or below the mean a data point is. This is incredibly useful for comparing points from two different distributions.
For example, they say you cannot compare apples and oranges, but here is a way. The average weight of a granny smith weighs 128 grams with a standard deviation of about 10 grams. The average weight of a navel orange is 140 grams with a standard deviation of about 14 grams. If we have a 148 gram granny smith apple and a 161 gram navel orange, we might wonder which is larger for its species even though they are both about 20 grams above their respective mean. We could say that the apple, which is 2 standard deviations above its mean, is larger for its species than the orange, which is only 1.5 standard deviations above its mean.
How many standard deviations above the mean height of a tree in forest A is its tallest tree?
How many standard deviations above the mean height of a tree in forest B is its tallest tree?
Which tree is taller in its forest?
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
Identify students who struggle making the connections between the summary statistics and the problem context. Ask them what information the standard deviation conveys. Tell them to focus less on the actual value of the standard deviation and more on which of the two distributions have a greater standard deviation and what that might mean.
Activity Synthesis
The purpose of this discussion is for students to explain the mean and the standard deviation in context.
For each set of graphs, select students read their answers for all three prompts. When necessary, prompt students to revise their language to include the terms shape, measure of center, and variability.
Speaking: MLR 8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language (shape, measure of center, and variability). Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking and to clarify, if necessary. Call students' attention to any words or phrases that helped clarify the original statement. This will provide more students with an opportunity to produce language as they explain the mean and standard deviation in context.
Design Principle(s): Support sense-making


correct_option_index: 3