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


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.

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

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 $P$.


Concept/skill: 
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.

Answer: 
Yes. 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.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
problem_activity: Activity
In this activity, students think a little more deeply about the data we would like to know and how that compares to the data we can collect easily and quickly (MP1). They are presented with a statistical question that does not have an obvious answer. Students are then asked to consider ways they might begin gathering data to answer the question, but are asked to realize that the data they could reasonably collect is not everyone addressed by the question. Following the activity, the discussion defines the terms
population
and
sample
.
Launch
Arrange students in groups of 2.
Set up the context by asking students, “Do people who are the only child have more pets?” then to provide a possible explanation for their answer. For example, maybe only children do have more pets because the family can better afford to take care of an animal with only 1 child. Maybe they do not because smaller families may live in smaller places and not have room for a lot of pets.
Give students 5 minutes of partner work time followed by 5 minutes of whole-class discussion.
Student Facing
Consider the question: Do people who are the only child
have more pets
?
Earlier, we used information about the people in your class to answer a question about the entire school. Would surveying only the people in your class give you enough information to answer this new question? Explain your reasoning.
If you had to have an answer to this question by the end of class today, how would you gather data to answer the question?
If you could come back tomorrow with your answer to this question, how would you gather data to answer the question?
If someone else in the class came back tomorrow with an answer that was different than yours, what would that mean? How would you determine which answer was better?
Student Response
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Activity Synthesis
The purpose of the discussion is to show the difference between the data we would like to have to answer the question and the data we have available.
Some questions for discussion:
“If we had all the time and money in the world and wanted to answer this question, who would we need to collect data from?” (Everyone in the world.)
“What would you do with the data collected from everyone to answer the questions?” (Find the mean and MAD of the data from the two sets and compare them like we did in previous lessons.)
“Why is it unreasonable to actually collect all the necessary data to answer the question?” (There are too many people to collect data from. There is not enough time to get to everyone in the world, and I cannot travel everywhere.)
“Since it may be difficult to guess an answer without doing
any
research, but we cannot get all of the data we want, what data could you get that would help estimate an answer?” (It would be good to ask a few people in different parts of the world and try to get different groups represented.)
Define population and sample. A
population
is the entire pool from which data is taken. Examples include (depending on the question) “all humans in the world,” “all 7th graders at our school,” or “oak trees in North America.” In this usage, it does not have to refer only to groups of people or animals. A
sample
is the part of the population from which data is actually collected. Examples (related to the population examples) include “5 people from each country,” “the first 30 seventh graders to arrive at our school,” or “8 oak trees from the forest near our school.”
Ask students, "What is the population for the question about only children and their pets?" (Everyone in the world.) Note that we would need data from everyone, including those who don't have pets or do have siblings.
Ask students, "What might be a sample we could use to answer the question?" (The students in our class, my neighbors, a few people from different countries.) After getting several responses, ask, "What might be the benefits and drawbacks of each of these samples?" (Some may be more convenient, but would not represent the population as well or vice-versa.)
Explain: While it is best to have data for the entire population, there are many reasons to use a sample.
More manageable. With very large populations, the amount of data can be hard to collect and work with, so a smaller subset may still be informative and easier to work with. Example: Find the average size of a grain of sand.
Necessary. Sometimes it is impossible to reach the entire population, so a sample is all that is available. Example: Find the average lifespan of tuna.
Speed. Sometimes a rough estimate is all that is needed and a sample of data is enough to estimate the population characteristic. Example: Out of curiosity, what is the median number of apps on smartphones.
Cost. Sometimes it is very costly to obtain the data needed, so a sample can reduce the cost. Example: Find the average amount of hydrogen in moon rocks.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: populations and sample.
Supports accessibility for: Conceptual processing; Language
Representing, Speaking: MLR2 Collect and Display.
To help students make sense of the terms “sample” and “population”, draw a diagram of a few circles inside a larger circle on a visual display. Label the large outer circle “population” and the small inner circles “sample.” As students respond to the question “What is the population for the question about only children and their pets?”, write the population on the visual display. As students respond to the question “What might be a sample we could use to answer the question?”, write the samples in different inner circles on the visual display. Listen for and amplify words and phrases that define these terms, such as “part of” or “entire.” This will help students visualize a sample as part of a population and understand that there are multiple samples inside a population.
Design Principle(s): Support sense-making (for representation); Maximize meta-awareness


aligns: no