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: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
problem_activity: Activity
In a previous lesson, students had a primer on histograms—how they are drawn, how they differ from dot plots, and what information they can tell us. In this activity, students practice drawing a histogram for a given data set and using it to answer statistical questions. To help students understand the lengths involved in the data set, students are asked to draw various lengths used to group the worms in the first histogram.
As students organize the data set and draw their histogram, notice any challenges or questions students come across. If a question is raised by multiple groups, consider discussing it with the whole class. Also pay attention to how students use the histogram to identify a “typical” length—some might describe it in terms of the size of the bins (e.g., “a typical length is between 20 and 40 mm”); others might choose a value within a bin or a boundary between bins. Invite them to share their reasonings later.
Launch
Arrange students in groups of 2. Provide access to centimeter rulers.
Consider giving students a brief overview of the context for the problems in the activity. Tell students that there are nearly 6,000 species of earthworms in the world.  Some earthworms help the environment, while others (generally not native to the region in which they are found) may harm the environment.  Earthworms that are native to a particular region of the world are often raised, by farmers, in terrariums (a container or bin similar to an aquarium but it contains soil and leaves). The terrarium-raised earthworms provide bait for people who fish, provide food for various wildlife, and decompose food waste into soil.  Food waste and water are added to the terrariums as food for raising and growing worms.  Soil produced by the worms as they eat the food waste is often used as fertilizer.
Explain that the lengths of the worms in the bins provide information about the ages of the worms, which can be useful for the farmer. In this activity, students will organize the lengths of the earthworms in several terrariums or bins.
Give students 8–10 minutes of quiet work time, and then 3–4 minutes to discuss their work and complete the activity with a partner.
Action and Expression: Internalize Executive Functions.
Activate or supply background knowledge by demonstrating how to use a ruler to draw a line segment for a given length.
Supports accessibility for: Memory; Conceptual processing
Student Facing
An earthworm farmer set up several containers of a certain species of earthworms so that he could learn about their lengths. The lengths of the earthworms provide information about their ages. The farmer measured the lengths of 25 earthworms in one of the containers. Each length was measured in millimeters.
[IMG]
Using a ruler, draw a line segment for each length:
20 millimeters
40 millimeters
60 millimeters
80 millimeters
100 millimeters
Here are the lengths, in millimeters, of the 25 earthworms.
6
11
18
19
20
23
23
25
25
26
27
27
28
29
32
33
41
42
48
52
54
59
60
77
93
Complete the table for the lengths of the 25 earthworms.
|                   length                    | frequency |
|---------------------------------------------|-----------|
|  0 millimeters to less than 20 millimeters  |           |
| 20 millimeters to less than 40 millimeters  |           |
| 40 millimeters to less than 60 millimeters  |           |
| 60 millimeters to less than 80 millimeters  |           |
| 80 millimeters to less than 100 millimeters |           |
Use the grid and the information in the table to draw a histogram for the worm length data. Be sure to label the axes of your histogram.
[IMG]
Based on the histogram, what is a typical length for these 25 earthworms? Explain how you know.
Write 1–2 sentences to describe the spread of the data. Do most of the worms have a length that is close to your estimate of a typical length, or are they very different in length?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Here is another histogram for the earthworm measurement data. In this histogram, the measurements are in different groupings.
[IMG]
Based on this histogram, what is your estimate of a typical length for the 25 earthworms?
Compare this histogram with the one you drew. How are the distributions of data summarized in the two histograms the same? How are they different?
Compare your estimates of a typical earthworm length for the two histograms. Did you reach different conclusions about a typical earthworm length from the two histograms?
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
When determining frequencies of data values, students might lose track of their counting. Suggest that they use tally marks to keep track of the number of occurrences for each bin.
When drawing the histogram, students might mistakenly use bar graphs as a reference and leave spaces between the bars. Ask them to look at the bars in other histograms they have seen so far and to think about what the gaps might mean considering that the bars are built on a number line.
Activity Synthesis
Ask one or two students to display their completed histograms for all to see and briefly describe the overall distribution. Then, select a few other previously identified students to share their responses and explanations for the last two questions.
Focus the discussion on how identifying center and spread using a histogram is different than doing so using a dot plot. Discuss:
“In a histogram, are we able to see clusters of values in the distribution?”
“Can we see the largest and smallest values? Can we tell the overall spread?”
“How do we identify the center of a distribution?”
From the various estimates that students give for a typical earthworm length (and from earlier exercises), students should begin to see that identifying a typical value of a distribution is not a straightforward or precise process so far. Explain that in upcoming lessons they will explore how to describe a typical value and characterize a distribution more systematically.
Representing, Conversing: MLR7 Compare and Connect.
As the selected students show and describe their histograms, invite pairs to discuss: “What is the same and what is different?” about their own histograms and reasoning. This will help students better understand there are multiple valid ways for describing center and spread.
Design Principle(s): Maximize meta-awareness; Support sense-making


correct_option_index: 1